Time Charter Contracts in the Shipping Industry

– A Fair Valuation Perspective

Renathe Elven

MSc Finance

Supervisor: Peter Løchte Jørgensen

Department of Economics and Business

Aarhus University

Business and Social Sciences

August 2013

I

Abstract

This thesis studies a specific type of asset lease – namely time charter contracts with purchase

options. Time charter contracts are common in the shipping industry and give the charterer the

operational control of the vessel leased, whereas the option to purchase the vessel gives the

charterer the right, but not the obligation, to purchase the vessel at the options’ expiration. The

options embedded in such contracts are often complex in nature such that they are granted for free

rather than for their fair value. The intention of this thesis is to introduce fair valuation of the total

value of time charter contracts with embedded options by introducing two potential models for

valuation purposes. Both models are one-factor models that are assumed to model the main source

of risk in the shipping industry – namely the freight rate. The two adopted models ensure the freight

rate to evolve in continuous time, and one of them allows for the derivation of analytic solutions for

some simple freight rate contingent claims. The other model values the vessel underlying the

contract, as well as an embedded European option to buy the vessel by implementing Monte Carlo

simulation.

Acknowledgements

I will like to thank my supervisor, Peter Løchte Jørgensen, for his help and valuable comments during

this period.

II

Table of Contents

Abstract .................................................................................................................................................... I

Acknowledgements .................................................................................................................................. I

List of Tables ............................................................................................................................................ V

List of Figures ........................................................................................................................................... V

1 Introduction ..................................................................................................................................... 1

1.1 Motivation ............................................................................................................................... 1

1.2 Aim........................................................................................................................................... 2

1.3 Structure .................................................................................................................................. 2

2 Theoretical Framework ................................................................................................................... 3

2.1 Introduction to Important Terms in the Shipping Industry ..................................................... 3

2.1.1 Freight Rates .................................................................................................................... 3

2.1.2 Spot Freight Rates ........................................................................................................... 3

2.1.3 Time Charter Equivalent Spot Freight Rates ................................................................... 4

2.2 The Shipping Industry .............................................................................................................. 4

2.2.1 Agents in the Shipping Industry ...................................................................................... 4

2.2.2 The Different Shipping Segments .................................................................................... 5

2.2.3 Vessels in the Shipping Industry ...................................................................................... 7

2.2.4 The Shipping Market Model ............................................................................................ 7

2.3 Costs in the Shipping Industry ............................................................................................... 13

2.4 Business Risks in Shipping ..................................................................................................... 14

2.4.1 Price Risk ........................................................................................................................ 14

2.4.2 Credit Risk ...................................................................................................................... 15

2.4.3 Pure Risk ........................................................................................................................ 15

2.4.4 Summing Up - Analyzing and Managing Freight Rate Risk ............................................ 15

2.5 The Four Shipping Markets ................................................................................................... 16

2.5.1 The Freight Market ........................................................................................................ 16

2.5.2 The Sale and Purchase Market ...................................................................................... 18

2.5.3 The Newbuilding Market ............................................................................................... 18

2.5.4 The Demolition Market ................................................................................................. 18

2.6 Time Charter Contracts with Embedded Options ................................................................. 19

2.7 Background for the Models Selected .................................................................................... 21

III

2.8 The Dynamics of Freight Rates .............................................................................................. 24

2.8.1 Shipping Market Cycles ................................................................................................. 24

2.8.2 Freight Rate Dynamics ................................................................................................... 24

2.9 The Ornstein-Uhlenbeck Process .......................................................................................... 25

2.9.1 The Solution to the Ornstein-Uhlenbeck Process ......................................................... 28

2.10 The Geometric Mean Reversion Process .............................................................................. 31

2.10.1 The Solution to the Geometric Mean Reversion Process .............................................. 32

3 Analysis Section ............................................................................................................................. 35

3.1 Two Models – Different Characteristics: A Comparison ....................................................... 36

3.2 Applications of the Ornstein-Uhlenbeck Process: Introducing Valuation of Freight Rate

Contingent Claims ............................................................................................................................. 38

3.2.1 Derivation of the Fundamental Partial Differential Equation ....................................... 39

3.3 Valuation of some simple Freight Rate Contingent Claims ................................................... 44

3.3.1 Claim to Receive Spot Freight Rate Flow from Time to Time .................................. 44

3.3.2 Fixed for Floating Freight Rate Swap ............................................................................. 46

3.3.3 The Value of a Vessel ..................................................................................................... 49

3.4 European Option to Buy a Vessel .......................................................................................... 52

3.5 Applications of the Geometric Mean Reversion Process: Vessel and European Option

Valuation ........................................................................................................................................... 55

3.5.1 Monte Carlo Simulation ................................................................................................ 56

3.5.2 The Value of a Vessel ..................................................................................................... 56

3.5.3 European Option to Buy a Vessel .................................................................................. 59

3.6 The Valuation Results: Comparisons ..................................................................................... 61

4 Limitations ..................................................................................................................................... 62

4.1 Limitations Caused by the Models Selected.......................................................................... 62

4.1.1 The Parametric Property of the Models ........................................................................ 62

4.1.2 One-Factor ..................................................................................................................... 63

4.2 The Assumptions ................................................................................................................... 64

4.2.1 Constant Market Price of Freight Rate Risk ................................................................... 64

4.2.2 Constant Risk-Free Interest Rate ................................................................................... 65

5 Summary and Conclusions ............................................................................................................ 66

6 List of References .......................................................................................................................... 68

IV

7 Appendix ........................................................................................................................................ 70

7.1 The Ornstein-Uhlenbeck Process – Detailed Solution ........................................................... 70

7.2 The Ornstein-Uhlenbeck process - Derivation of the Mean and the Variance ..................... 72

7.2.1 The Time Conditional Mean ........................................................................................ 72

7.2.2 The Time Conditional Variance ................................................................................... 73

7.3 The Geometric Mean Reversion Process – Detailed Solution ............................................... 74

7.4 The Ornstein-Uhlenbeck Process – Claim to Receive Spot Freight Rate Flow from Time to

Time 77

7.5 The Ornstein-Uhlenbeck Process - European Option to Buy the Vessel ............................... 80

7.6 The VBA Codes ...................................................................................................................... 84

V

List of Tables

Table 1: The variables in the shipping market model ..............................................................................8

Table 2: Base case parameter values .................................................................................................... 35

Table 3: The dependence of fair time charter rates.............................................................................. 47

Table 4: The value of a 5-year time charter contract ............................................................................ 49

Table 5: The dependence of vessel values ............................................................................................ 51

Table 6: Value of European option to buy a vessel ............................................................................... 54

Table 7: The value of a 5-year time charter contract with European purchase option ........................ 55

Table 8: The dependence of vessel values ............................................................................................ 58

Table 9: Value of European option to buy a vessel ............................................................................... 60

List of Figures

Figure 1: Simulated spot freight rate from the Ornstein-Uhlenbeck process ....................................... 36

Figure 2: Simulated spot freight rate from the Geometric Mean Reversion process ........................... 37

Figure 3: Simulated and expected value of a vessel.............................................................................. 52

1

1 Introduction

Through the last decade, the shipping industry has been subject to extreme volatility in freight rates.

Over the period from 2003 to mid-2008, freight rates increased by almost 300 per cent to

exceptional levels. This large increase in freight rates was followed by a corresponding drop of 95 per

cent over the last quarter of 2008. Such high volatility in the market also implies a shipping industry

that is extremely risky. Therefore, the last decade’s market fluctuations have changed the way the

shipping industry views and manages its risks, and accordingly the derivatives market for freight have

accelerated and a commoditization of the freight market is present. Today, agents that may not be

involved in the underlying physical market, such that investment banks, hedge funds and other

traders, can be seen participating in the shipping industry.

1.1 Motivation

Thus, the accelerated derivatives market for freight has opened up possibilities for hedging and

managing risk stemming from large volatility in freight rates. This leads to the topic of this thesis,

namely fair valuation of time charter contracts with embedded options. Time charter contracts are

common contractual agreements in the shipping industry which will be described in detail later.

Options embedded in time charter contracts are wildly used and can be considered as a tool used to

hedge against freight rate risk, this will also be described in detail later. More specific, the options

which will be examined and valued here are call options which enable the charterer to purchase the

vessel either during the contract period, or at the end of the contract period. The options serve as an

insurance against undesirable movements in freight rates.

Time charter contracts with purchase options are interesting from both academic and practical

business management perspectives as they can be very complex and of significant economic

importance. Jørgensen and Giovanni (2010) mention that they are aware of several shipping

companies having a total net asset value where more than half of it stems from an estimated value of

their portfolio of time charter contracts with purchase options. Thus, properly valuation of these

contracts is extremely important in order to both support the stock markets’ valuation of shipping

companies, and in order to assist managers of such companies in the general process of operation

and risk management of their companies. High volatility in freight rates may create either a

significant decrease in a shipping company’s total reported net asset value or a significant increase in

its total net asset value.

2

The complex nature of time charter contracts with purchase options makes fair valuation a difficult

task. The need for development and analysis of good valuation models are therefore increasingly

important. According to Alizadeh and Nomikos (2009), embedded options in the shipping industry

are very often granted for free or for a nominal fee without being properly valued.

1.2 Aim

Therefore, this thesis aims to shed light on the importance of fair valuation of time charter contracts

with embedded options, as well as valuations of such contracts and a comparison between models.

Two models will be introduced for valuation purposes and their valuation abilities will be compared.

The model that will be used to obtain total values of a time charter contracts with embedded

purchase options is adopted from Jørgensen and Giovanni (2010). The other model is adopted from

Tvedt (1997) and will be introduced as an alternative model for valuations to the one from Jørgensen

and Giovanni (2010).

1.3 Structure

This thesis is divided into three main parts; a theoretical framework, an analysis section and a

discussion section where limitations are elucidated.

First, a theoretical framework of the shipping industry will be established. A fundamental

introduction of the comprehensiveness of the shipping industry will be given, where the shipping

market model will be emphasized. This simplified model describes the mechanisms that make freight

rates evolve in accordance with the market cycles. Further, the time charter contracts with

embedded options in the shipping industry will be described, as well as the background for the

model selection. Finally, the dynamics of freight rates will be described forming the basis for the two

models selected before the theoretical framework ends by a detailed description of these models,

and for that purpose also a derivation of the two processes’ solution will be done.

As an introduction to the analysis section the two models’ characteristics will be compared and

discussed. Evidences of one model being more appropriate in freight rate modeling will be

presented. Further, derivations will be done in order to value time charter contracts with purchase

options. To a greater or lesser extent, both models will be applied for valuations.

At the end, some of the limitations regarding the two models will be presented before the thesis will

be summarized and concluded.

3

2 Theoretical Framework

This section will first give a basic understanding of the shipping industry before it continues with a

description of time charter contracts with embedded options. Further, the background that clarifies

the reasons for the two models chosen will be presented, as well as a description of the freight rate

dynamics. Finally, the two models adopted for the valuations will be presented in detail in which the

solutions to both of them also will be derived.

2.1 Introduction to Important Terms in the Shipping Industry

To prevent confusions, some relevant terms in relation to the shipping industry will be described in

this first section of the theoretical framework. This will serve as a soft introduction to the

comprehensive shipping industry.

2.1.1 Freight Rates

Freight rates represent the cost of providing the service of seaborne transportation (Alizadeh and

Nomikos 2009). Hence, freight rates are not tangible assets and can therefore not be stored.

Kavussanos and Visvikis (2006) do also stress this special feature of freight services where they

describe the demand for freight services as a derived demand. This is due to the fact that the freight

service provided by the vessel is “gone” if it is not utilized at the time it is available. Again, the freight

service is non-storable and it cannot be carried forward in time.

Freight rates evolve through time according to market cycles that are prevalent in the shipping

industry. These will be described later.

2.1.2 Spot Freight Rates

The spot freight rate represents the cost of providing seaborne transportation today. It reflects the

continuous balance between supply and demand for shipping services (Alizadeh and Nomikos 2009).

Factors determining supply and demand will be described in Section 2.2.4 where the freight rate

mechanism will be examined.

4

2.1.3 Time Charter Equivalent Spot Freight Rates

The time charter equivalent spot freight rate represents the spot freight rate less the voyage costs

(Tvedt 1997). During this thesis, it is actually the time charter equivalent spot freight rate that is

applied in the derivations. For reasons explained later, applying the time charter equivalent spot

freight rate is in fact beneficial when valuing time charter contracts with purchase options.

What is already important to note is that market-quoted spot freight rates embed varying degree of

costs, and thus the time charter equivalent spot freight rates will not be directly comparable with the

market-quoted spot freight rates. This is important to have in mind when comparing with market

data.

2.2 The Shipping Industry

This section will give an introduction to the shipping industry that plays a central role in the global

economy and has also been at the forefront of global development through times. The main assets in

the shipping industry are the vessels that can transport cargo from one part of the world to another.

Already, it is worth mentioning that competition is an important key word in relation to this

intriguing industry, and as for the bulk shipping segment that this thesis aim to address, the condition

of perfect competition is present. When nothing else is stated, the whole section will be based on

Stopford (2009).

2.2.1 Agents in the Shipping Industry

In the following, a description of the most important agents participating in the shipping industry will

be given. Hence, some agents will be left outside this explanation since they are less important

according to the aim of this thesis. The groups that will be explained here are the shipowners, the

charterers and the shipbrokers.

The shipowner has vessels for hire and enters the market with a vessel available – free of cargo. The

vessel has particular characterizations and the vessel’s availability will be described in the contractual

agreements. The shipowner may be looking for a short charter for the vessel or a long charter, in

which will be dependent on the shipowner’s strategy.

The charterer can be either an individual or an organization; common to both of them is that they

have a volume of cargo they need to transport from one location to another. The vessel type needed

will be determined by the physical characteristics of the cargo.

5

The shipbroker operates as an intermediary between shipowners and charterers. According to who

have a need, the shipbroker will be contacted and his task is to discover what cargoes or vessels are

available, what expectations the shipowners/charterers have about what they will be paid or pay,

and what is reasonable given the state of the market. The deal for their client is negotiated, and

often in tense competition with other brokers.

2.2.2 The Different Shipping Segments

Three different segments constitute the shipping industry; these are the liner shipping segment, the

bulk shipping segment and the specialized shipping segment. This clearly division of the industry is

necessary in order to meet specific needs of different customers; anything from grain to cars can be

transported by sea. Shipping companies’ characteristics differ depending on which segment they

operate in, this is due to differences in both the transported cargo, and in the dynamics of how the

agreement between the shipowner and the charterer is conducted. The characteristics of each

shipping segment will be described below. Despite the differences in shipping companies’

characteristics across these three segments, one shipping company can often operate in more than

one segment and thus contribute to intense competition for the same cargo. Therefore, it is not

convenient to treat the shipping industry as a series of isolated segments, but rather as a single

market. Investors in the shipping industry move their investments from one segment to another, and

if there is a supply-demand imbalance in one of the segments, this will also move on to the other

segments.

As this thesis aims to address the bulk shipping segment, liner shipping and specialized shipping will

only shortly be described, whereas the bulk shipping segment will be examined in more detail.

The liner shipping segment offers transport for cargoes that are too small to fill a single vessel and

thus need to be grouped with others for transportation. Such cargoes are often highly valued and can

be delicate in nature; the shipper thus often requires a special shipping service with a fixed tariff

rather than a fluctuating market rate. Cargoes transported in the liner shipping segment are called

general cargo and include loose cargo, containers and pallets. This creates complex administrative

tasks and makes this segment very different from the bulk shipping segment. The specialized shipping

segment is recognized by properties lying in between the liner shipping segment and the bulk

shipping segment. This leads to a somewhat indefinite distinction between the specialized shipping

segment and the two other segments. Cargoes that are considered special include cars, forest

products, chemicals and refrigerated products.

6

The bulk shipping segment supplies transport for cargoes that need to be transported in large

homogeneous shiploads. Generally a whole vessel is hired for transportation of one type of cargo,

but it is also possible to carry different bulk cargoes in a single vessel. If so, each cargo occupies a

separate hold or possibly even part of a hold. Commodities that often need to be transported in

bulks are raw materials and bulky semi-manufactures. The bulk shipping segment is divided into dry

bulk and liquid bulk. Dry bulk cargo transported in shiploads is mainly raw materials such as iron ore,

coal and grain. Liquid bulk cargo includes crude oil, oil products, and liquid chemicals. Cost

minimization of providing safe transport, as well as effective management of vessel investments is

the main focus within the bulk shipping business. Costs can be minimized due to the characteristics

of the bulk segment in which few transactions are handled. Typically, a vessel completes about six

voyages with a single cargo each year.

As already mentioned, the bulk shipping segment is subject to conditions of perfect competition. This

creates rather volatile freight rates and prices (Kavussanos and Visvikis 2006). Perfect competition

arises due to the many buyers and sellers of freight services. They negotiate on a relatively

homogenous product, the freight service, and have no barriers to entry or exit the market. In

addition, the freight markets are well organized markets. The product can be assumed to be almost

perfectly homogenous since, in a particular route-trade, there are no significant differences in the

quality of the freight service offered. The fact that the product is almost perfectly homogenous

relates to some minor differences in vessel characteristics and customer relations that may exist

between shipping companies.

Another contribution to the condition of perfect competition is the availability of information in the

freight markets. The Baltic Exchange, among others, brings together participants wishing to buy or

sell the freight service. Relevant information on fixtures1, prices, and cargoes/vessels available are

collected and disseminated to the market. Such information is used by shipowners and charterers to

make informed decisions in the freight markets (Kavussanos and Visvikis 2006). The feature of

perfect competition arising in the bulk shipping segment, resulting in volatile freight rates, makes it

especially interesting to address this particular segment; investors and charterers are to a great

extent exposed to risk, and proper risk management becomes extremely important.

In summary, these three shipping segments face different tasks depending on the value and volume

of cargo, the number of transactions handled, and the commercial systems employed.

1 Stopford (2009) explains a fixture as an agreement where a vessel is chartered or a freight rate is agreed on.

Further, he explains that the arrangement happens in much the same way as any major international hiring or subcontracting operation.

7

2.2.3 Vessels in the Shipping Industry

In order to give a comprehensive picture of the shipping industry a description of different vessel

types is, at this point, appropriate. Due to the differences in cargoes transported by sea, both within

each segment and also across the segments, vessels are built and adjusted to fit the cargoes they

transport. The result is several different vessel types and sizes in the world fleet2. However, this

section will only give a presentation of the different vessel types in the bulk shipping segment.

There exist four different types of vessels in the dry bulk sector where each of them are classified by

their size measured in dead weight tons (dwt). Handy bulk carriers are the smallest ones, those

vessels are of 10 000 to 40 000 dwt. Handymax bulk carriers are of 40 000 to 60 000 dwt, Panamax

are of 60 000 to 100 000 dwt, and Capesize are the largest vessels of over 100 000 dwt. As any other

physical asset vessels do also have a limited lifetime. The shipping industry is in continuous

technological progress and the vessels operating in the industry therefore suffer from obsoleteness

rather fast. Logically, as a vessel grows old or gets obsolete its value will also decrease, this reduction

in value continues until the vessel’s age lies between 20 and 30 years which is when it is normally

scrapped.

According to Kavussanos and Visvikis (2006), a useful tool to understand why different vessel sizes

are necessary is the Parcel Size Distribution (PSD) of each commodity. A parcel is an individual

consignment of cargo for shipment, each commodity can be transported in different parcel sizes, and

these different parcel sizes constitute the Parcel Size Distribution. Based on the observation that

some commodities are typically moved in larger sizes than others, the PSD function describes the

range of parcel sizes in which that commodity is transported. In addition to the parcel size

distribution, port and seaway restrictions have created differences in types and sizes of vessels.

2.2.4 The Shipping Market Model

To understand the freight market and how the freight market cycles are generated the mechanisms

determining spot freight rates will be explained next. These are the economic mechanisms that the

shipping industry uses to regulate supply and demand.

The ten most important economic variables determining supply and demand are collected in a

simplified model called “The Shipping Market Model”. Ten variables which have an influence on the

shipping market are listed and the purpose is to leave out less relevant details in order to create a

2 All existing vessels.

8

picture of how the spot freight rates are determined. Five of the variables influence demand in the

shipping market, and the other five influences supply. The variables are listed in Table 1 below.

Demand Supply

1. The World Economy 1. The World Fleet

2. Seaborne Commodity Trades 2. Fleet Productivity

3. Average Haul 3. Shipbuilding Production

4. Random Shocks 4. Scrapping and Losses

5. Transport Costs 5. Freight Revenue

Table 1. Ten variables in the shipping market model. Source: Stopford (2009), page 136.

First, the demand for sea transport will be examined. According to Kavussanos and Visvikis (2006),

demand for freight services is a derived demand; the charterer’s demand is not for the vessel, but for

the service the vessel provides.

2.2.4.1 The World Economy

The world economy is considered as the most important single influence on vessel demand. This is

due to the world economy generating most of the demand for sea transport. Events in the world

economy that generate demand for sea transport is import of raw materials for the manufacturing

industry, and the trade in manufactured products. The relationship between the world industry and

the demand for sea transport is complex and consists of two aspects of the world economy; the

business cycle and the trade development cycle. These two aspects may create changes in the

demand for sea transport.

The business cycle has an important influence on the demand for sea transport in the short-term.

Fluctuations in the world economy are directly transferred to the shipping market. In this way, the

foundation for the cyclical behavior of freight rates is set.

The trade development cycle is related to the long-term relationship between sea transport and the

world economy. It says something about the speed of industrial growth relative to the speed of sea

trade growth. The sea trade growth of individual regions will change as time goes by due to both the

9

change in a country’s economic structure, and due to the ability of local resources of food and raw

materials to meet local demand.

2.2.4.2 Seaborne Commodity Trades

This variable explains the relationship between sea trade and the industrial economy both in the

short-term and in the long-term.

Short-term volatility arises due to seasonality in some trades. An example of such a trade is grain,

which is subject to seasonal variations caused by harvests. Due to the difficulties in planning

transportation of seasonal agricultural commodities, shippers rely heavily on the spot charter market

when demand for sea transport arises. Thus, fluctuations in the grain market have larger impact on

the spot charter market than other trades, like for example iron ore. Tonnage requirements in the

transportation of iron ore are almost always met through long-term contracts.

Demand affected by long-term trends in commodity trade is best identified by studying the economic

characteristics of the industries that produce and consume the traded commodities. Overall, there

are four types of changes that affect the demand for seaborne transport in the long-run; changes in

the demand for that particular commodity, changes in the source from which supplies of the

commodity are obtained, changes due to a relocation of processing plant changing the trade pattern,

and changes in the shipper’s transport policy.

Changes in demand for that particular commodity may have an effect on the tonnage requirements if

this particular commodity is imported. If, for example, the country decides to replace the imported

commodity by a domestic commodity, this will naturally affect the demand for seaborne transport.

Changes in the source from which supplies of the commodity are obtained happen when new sources

are discovered. These new sources may happen to be located near countries that earlier imported

the same commodity, this may result in imports of this commodity being redundant. Thus, the

demand for sea transport is changed.

Changes due to relocation applies to industrial raw materials, and may affect both the volume of

cargo transported by sea, and the type of vessel used to transport this cargo. Raw materials are often

“transformed” several times before the final product is made. If the “transformation” is done before

it is shipped rather than after, the volume and characteristics of the vessel may be changed.

Changes in the shipper’s transport policy relates to, for example, switching between using long-term

contracts and using the spot charter market. This will again affect the demand for sea transport.

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2.2.4.3 Average Haul

Cargo shipped over larger distances generates more demand for sea transport than cargo shipped

over shorter distances. Therefore, demand is affected by the length of where the cargo is shipped.

The demand of sea transport is measured in ton miles making sure that the distance effect is taken

account for. Ton miles are defined by the tonnage of cargo shipped, multiplied by the average

distance over which it is transported.

2.2.4.4 Random Shocks

Random shocks can have major impact on the economic system, and in turn affect the cyclical

process. Random shocks of more or less severity include weather changes, wars, new resources and

commodity price changes. Economic shocks do often have the most important influence on the

shipping market, the reason why is that the timing is usually unpredictable and they bring about a

sudden and unexpected change in vessel demand.

Political events often have an indirect effect on vessel demand. Examples of such events are a

localized war, a revolution or strikes.

2.2.4.5 Transport Costs

Raw materials will only be transported from other destinations around the world if the

transportation costs are at a relatively low level, or if the quality of a product can be increased to a

level that gives major benefits.

Improved efficiency, bigger vessels and more effective organization of the shipping operation result

in reduced transport costs and higher quality of service. Thus, also the amount of seaborne transport

increases.

Even though these five factors influencing demand for sea transport are a simplified picture of

reality, they give an indication of the complex nature of seaborne demand. On the other hand, the

supply for sea transport is quite different in nature as also will be seen in the following.

The supply for seaborne transport is characterized as being slow in its adaptation to changes in

demand. This is due to the time-lag created in response to an increase in demand; several years are

needed for the completion of a new vessel. Responding to a decrease in demand is also a slow

11

process, once a vessel is built it is estimated to have a lifetime of 15-30 years (Stopford 2009).

Therefore, if a large surplus is to be removed, that process can take several years resulting in a time-

lag between the decreased demand and the surplus reduction.

2.2.4.6 The World Fleet

This variable measures the total amount of vessels existing all over the world. Scrapping old vessels

and deliveries of new vessels determine the rate of fleet growth, which can be positive or negative.

The world fleet contains all the different vessel types and sizes. Since a new vessel is estimated to

have a lifetime of 25 years on average only a few vessels are scrapped each year. The pace of

adjustments to changes in the market is therefore measured in years.

When demand for seaborne transport does not turn out as expected, supply is adjusted. This is the

key feature of the shipping market model (Stopford 2009).

2.2.4.7 Fleet Productivity

The productivity of the vessels that constitute the world fleet can vary. This creates a flexibility

element since a vessel often has several days where it does not transport cargo. Such “ineffective”

activities include ballast time, cargo handling, incidents, repair, lay-up, waiting, short-term storage

and long-term storage.

The fleet productivity is measured in ton miles per dead weight ton and depends upon four main

factors; speed, port time, deadweight utilization and loaded days at sea.

Speed is measured by the time a vessel uses on a voyage. New vessels are often designed to go

faster, but this reduces the transport capacity of the vessel. Also, older vessels are often subject to

hull fouling which will reduce the maximum operating speed. Port time relates to the time a vessel is

at port. Factors that determine the efficiency at port include the physical performance of the vessels

and terminals, and the organization of the transport operation. The deadweight utilization measures

how much of the total cargo capacity that is lost due to bunkers, stores, etc. Loaded days at sea are

the time where the vessel actually transports cargo at sea. It is desirable to increase loaded days at

sea to improve the efficiency of the world fleet.

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2.2.4.8 Shipbuilding Production

Shipbuilding is an important adjustment factor to the world fleet. In times of increased demand,

shipbuilding can increase the world fleet to meet the demand required. But building new vessels is a

lengthy process, therefore, when deciding to build a new vessel it is important that the need for this

vessel in the future is identified. It can be difficult to predict future demand, and if the prediction

turns out to be wrong, this can lead to excess of vessels in relation to required demand.

2.2.4.9 Scrapping and Losses

The balance between delivery of new vessels and scrapping of old ones (or losses) determines the

growth rate of the world fleet. A new vessel is estimated to have a lifetime between 15-30 years,

indicating the difficulty in estimating exactly when the vessel is to be scrapped. The reason why is

that scrapping depends on the balance of a number of factors; age, technical obsolescence, scrap

prices, current earnings and market expectations. These factors create some flexibility to the

shipowner in deciding when a vessel is to be scrapped.

2.2.4.10 Freight Revenue

The freight rate is the most important regulator of the supply of sea transport. Freight rates are used

by the market to motivate decision-makers to adjust capacity in the short-term, and to find ways of

reducing their costs in the long-term.

The shipping industry consists of two main pricing regimes; the freight market and the liner market.

As explained above, the liner market can be thought of as a retail shipping business; transport of

cargo in small quantities is offered to many customers. The freight market (bulk shipping) is totally

different, this can be thought of as a wholesale operation; transport of cargo in shiploads is offered

to few customers at individually negotiated prices.

In the short-term, supply is adjusted in response to prices by changing the vessels operation speed

and move to and from lay up. In the long-term, investment decisions such as ordering new vessels

and scrapping old ones, are heavily influenced by the freight rate. The supply and demand

adjustment mechanism will be explained in more detail in Section 2.8 where freight rate dynamics

are examined.

13

2.2.4.11 Summing Up

According to Tvedt (1997) it is usual to assume that demand is quite inelastic to freight rates. The

shipping industry is characterized by large scale operations, and the cost of transportation at sea is a

minor share of the total oil price. Therefore, only to a very small extent, demand is supposed to

depend on freight rates. Further, he points out that the supply can be quite inelastic to freight rates

in the short run when there are no vessels available. The reason is that speed and efficiency in

loading and discharging only to a limited degree can be increased. On the other hand, when freight

rates have been low for a while many vessels may have been laid up. This makes it possible to

increase short run supply by re-entering vessels that are laid up.

2.3 Costs in the Shipping Industry

In this section, costs in the shipping industry will shortly be described. They are divided into four

categories; capital costs, operation costs, voyage costs, and cargo-handling costs. The whole section

will be based on Alizadeh and Nomikos (2009).

Capital costs are related to interest payments and capital repayments. These costs depend on how

the shipowner or the shipping company has financed their vessel purchases, and on the interest rate

level. Fleet financing can take several forms, some of which include full equity, bank loans, bonds,

public offerings and private placements. Shipping companies with high operational and financial

capabilities may enjoy better financial agreements than shipping companies with relatively lower

levels of credit and collateral. Operating costs are costs arising from the day-to-day running of the

vessel. These costs are generally in the responsibility of the shipowner, and incur whether the vessel

is active or idle. Operating costs include, among others, crew wages, stores and provisions,

maintenance, and insurance. Such costs do not vary over time, but they grow at a constant rate

normally in line with inflation. Voyage costs are costs related to a specific voyage. These costs include

fuel costs, port charges, pilotage and canal dues. The specific voyage undertaken, and the type and

size of the vessel are factors that decide the level of costs. Cargo-handling costs arise from the

loading, stowage, lightering and discharging of the cargo.

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2.4 Business Risks in Shipping

The shipping industry is considered as one of the most volatile industries where participants in the

markets are exposed to substantial financial and business risks. Fluctuations in freight rates, bunker

prices, vessel prices, and even from fluctuations in the level of interest rates and exchange rates are

all reasons why this is an extremely risky industry (Alizadeh and Nomikos 2009). All these factors

have an impact on the cash flows of shipping investment and operations, thus they also influence the

profitability of shipping companies as well as their business viability. Alizadeh and Nomikos (2009)

divide business risk in shipping into three categories; price risk, credit risk and pure risk. In the

following, these will be described and risk stemming from fluctuations in freight rates will be relied

most weight.

2.4.1 Price Risk

Price risk refer to a shipping company’s costs and earnings which is uncertain and outside of direct

control of the shipping company. The first source of price risk is freight rate risk which refers to the

variability in earnings of a shipping company due to changes in freight rates. Alizadeh and Nomikos

(2009) argue that this may be the most important source of risk for a shipping company due to the

direct impact volatility in freight rates have on the profitability of the company. The management of

risk arising from freight rates will be described after the other types of business risk in shipping have

been presented.

The second source of price risk is the operating-costs risk which refer to volatility in a shipping

company’s’ costs. Sharp and unanticipated changes in, for example, bunker prices will have a major

impact on the operating profitability of shipping companies and vessel operators. The third source of

price risk is the risk arising from exposure to changes in interest-rates. Most vessels in the shipping

industry are financed through term loans priced on a floating rate basis, thus unanticipated changes

in interest rates may create cash flow and liquidity problems for companies which may no longer be

able to service their debt obligations. The fourth and last source of price risk is asset-price risk arising

from fluctuations in the price of the assets of the companies. In the shipping industry, vessels are the

most important assets, they are often used as collateral in vessel-finance transactions and a

reduction in vessel value may therefore affect the creditworthiness of a shipowner and its ability to

service debt obligations. Volatility in vessel values will also affect a shipping company’s balance

sheet.

15

2.4.2 Credit Risk

Credit risk refers to counter-parties to transactions and their ability to perform their financial

obligations in full and on time. Therefore, credit risk is also known as counter-party risk. In the

shipping industry most of the deals, trades and contracts are negotiated directly between the

counterparties, their trust and commitment to honor the agreement therefore becomes extremely

important.

2.4.3 Pure Risk

Pure risk relates to a decrease in the value of the shipping company’s assets due to physical damage,

accidents and losses. Also, risk of loss due to physical risks, technical failure and human error in the

operation of the assets of a company are covered. In addition, the risk of legal liability for damages as

a result of actions of the company is covered.

2.4.4 Summing Up - Analyzing and Managing Freight Rate Risk

For the shipowner to be able to analyze the freight rate risks which he is exposed to, it is convenient

to consider the vessels as investments – as assets in portfolios (Kavussanos and Visvikis 2006).

Through the freight services that vessels offer to charterers a stream of income is generated and the

level of this income is dependent on the freight rate level at each point in time. Also, capital

gains/losses created by selling the vessels at a price higher/lower than what they were bought at is

part of the shipowner’s investment strategy. To manage the risks arising from freight rates

Kavussanos and Visvikis (2006) point out the use of financial derivatives. They explain that financial

derivatives have been used in the shipping industry since 1985, but also that the popularity of these

derivatives are far less popular than those available in other sectors of the economy. The time

charter contracts with purchase options, which this thesis aims to analyze and valuate, is an

instrument used to protect shipowners and charterers from risk arising from fluctuations in freight

rates.

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2.5 The Four Shipping Markets

Within the shipping industry, markets play an extremely important role in the operation of the

international sea transport. Stopford (2009) mentions the nineteenth-century economist, Jevons,

who provided a definition of a market where the basic principles is still very suitable to the shipping

industry. The definition is quoted below.

Originally a market was a public place in a town where provisions and other objects were

exposed for sale; but the world has been generalized, so as to mean any body of persons who

are in intimate business relations and carry on extensive transactions in any commodity. A

great city may contain as many markets as there are important branches of trade, and these

markets may or may not be localized. The central point of a market is the central exchange,

mart or auction rooms where traders agree to meet and transact business … But this

distinction of locality is not necessary. The traders may be spread over a whole town, or

region or country and yet make a market if they are … in close communication with each

other. (Jevons, 1871, Ch. IV)

Within the shipping industry there exist four different markets trading in different commodities. The

freight market trades in sea transport, the sale and purchase market trades second-hand vessels, the

newbuilding market trades new vessels, and the demolition market deals in vessels for scrapping.

Since this thesis aims to model the valuation of contracts on sea transport, it is the freight market

that is considered, and thus the freight market will be relied most weight in the following. However,

for completeness the main characteristics of the three other markets will also be described. The

whole section is based on Stopford (2009).

2.5.1 The Freight Market

The freight market is the marketplace where sea transport is bought and sold. Within the freight

market different sectors are developed in order to support the different vessel types. The freight

rates within each sector often behave quite differently from each other in the short term, but since it

is the same broad group of agents participating in the shipping industry, what happens in one sector

eventually ripples through into the others. There exist two different types of transactions in the

freight market; the freight contract and the time charter. The freight contract is a fixed type of

contract where the shipper buys transport from the shipowner at a fixed price per ton of cargo, and

is used by shippers who prefer to pay an agreed sum and leave the management of the transport to

the shipowner. On the other hand, the time charter contract is based on the spot freight market, and

17

the vessel is hired by the day. Experienced vessel operators who prefer to manage the transport

themselves are the users of time charter contracts.

Four different contractual agreements are used in the freight market; the voyage charter, the

contract of affreightment, the time charter, and the bare boat charter. A voyage charter agrees on a

fixed cargo price, measured in price per ton. In a contract of affreightment, the shipowner agrees to

transport a series of cargo parcels for a fixed price per ton. These series of cargo parcels are agreed

to be transported within a fixed time interval, for example within two months. The details of each

voyage are in the concern of the shipowner. The vessels are then used in an efficient manner by,

among others, switching cargo between vessels and arrange backhaul cargoes.

The time charter contract takes a step further and gives the charterer the operational control of the

vessels that carry the cargo. When the charterer has the operational control he instructs the master

where to go and what cargo to load and discharge. Responsibilities left for the shipowner are

ownership and management of the vessel. The length of the charter can vary from the time taken to

complete a single voyage, to a period of months or years. When a vessel is chartered, the shipowner

continues to pay the operating costs of the vessel. Operating costs include crew, maintenance and

repair. Commercial operations, voyage expenses3 and cargo handling costs are left to the charterer.

Since time charters hand over the voyage costs to the charterer it is convenient to apply the time

charter equivalent spot freight rate when agreeing upon a time charter contract. By subtracting the

voyage costs from the spot freight rate time charter rates will reflect the net freight earnings through

shipping operations (Alizadeh and Nomikos 2009). Therefore, when valuing time charter contracts

with purchase options later on, the time charter equivalent spot freight rate will be applied.

A bare boat charter can be arranged if the charterer wishes to have full operational control of the

vessel without owning it. The owner of the vessel does not need to be a professional shipowner, it

can also be an investor buying a vessel, and then entering into a bare boat charter. The charter

period usually spans from ten to twenty years. Management of the vessel and operating and voyage

costs is in the charterer’s responsibility.

3 Voyage expenses include bunkers, port charges and canal dues (Stopford 2009).

18

2.5.2 The Sale and Purchase Market

In the sale and purchase market second-hand vessels are traded with high intensity. Generally, the

sale and purchase transactions are carried out through shipbrokers who have been instructed by the

shipowner to find a buyer for the vessel. Most commonly, competition is created by offering the

vessel through several broking companies. The sale and purchase market generates price volatility,

and “asset play”4 can result in high profits being an important source of income for shipping

investors.

2.5.3 The Newbuilding Market

As its name indicates, the newbuilding market trades vessels that are not yet built. For the building

process, specifications of the vessel must be decided. The shipyard generally has their own standard

designs, and it is therefore desirable that the buyer choose one of those. This will ease the

negotiation process, the pressure on design and estimating resources will be reduced, and the

shipyards standard designs are normally cheaper to build than a customized design.

2.5.4 The Demolition Market

The demolition market has similarities to the second-hand market, but now it is the scrap yards that

are the customers instead of the shipowners. When a shipowner is not able to sell his vessel in the

second-hand market, he offers it on the demolition market. Also here, a broker generally handles the

sale. Scrap values are determined by negotiation and depend on the availability of vessels for scrap

and the demand for scrap metal.

With the four shipping markets described, and with the different contractual agreements in the

freight market in hand, it is time to move on to the primary theme of this thesis; namely time charter

contracts and their embedded options.

4 Described by Stopford (2009) as well-timed buying and selling the vessels.

19

2.6 Time Charter Contracts with Embedded Options

This section will give an understanding of the various options that often are embedded in time

charter contracts in the shipping industry. Some of them are quite complex in nature, and for

valuation purposes advanced numerical methods are necessary. Therefore, this section serves as an

introduction to both simple options and more complex options, whereas simple European options

will be valued later on.

Time charter contracts with embedded options are common in the shipping industry. When agreeing

upon a time charter, options to extend the lease period and options to buy the vessel are often

embedded in the lease contract. The option to extend the lease period makes it possible for the

charterer to lengthen the life of the contract period (Hull 2012). The option to buy the vessel is a so

called purchase option (call-option) and gives the charterer the opportunity to buy the vessel at a

predetermined price. These options serve as an insurance for the charterer against undesirable

movements in the freight rate level over the contract period as he can terminate the contract by

purchasing the vessel.

Options embedded in contracts in the shipping industry are real options. This is options on physical

assets (Hull 2012), such as the vessels in the shipping industry. According to Alizadeh and Nomikos

(2009) the underlying assets of real options are cash flows affected by managerial decisions. In the

shipping industry, owning a vessel results in cash flows when operating it in the freight market; by

offering seaborne transport the shipowner captures the freight earnings.

Jørgensen and Giovanni (2010) point out that these embedded options can have more or less

complex properties, in addition to being of significant economic value. This makes such contracts

interesting from both academic and practical business management perspectives. Due to the

economic significance of such contracts, the need for development and analysis of good valuation

models is increasing. Good developed valuation models will support the stock market’s valuation of

shipping companies and assist managers in the general process of operation and risk management of

their companies.

Embedded options can have different styles, the most common types of options in the shipping

industry is European options, American options and Bermudan options.

European options can only be exercised at a predetermined date (the expiration date) in the future

(Hull 2012). If the embedded options in a time charter contract are of European style, the lease

period can only be extended at one specific predetermined date. If the option to extend is exercised,

the option to purchase the vessel normally also is extended until the end of the lease period. When

20

the option to purchase the vessel is of European style, the vessel can only be bought at one

predetermined date in the future. If the option is not exercised at that point in time, the purchase

option ceases to exist. Normally, the expiration date of the purchase option is at the end of the

contract period.

American options can be exercised at any point in time until the end of the contract period (Hull

2012). If the embedded options in the time charter contract are of American style, the charterer can

extend the lease period or buy the vessel at any point in time in the contract period. Also here, if the

option to extend is exercised, the purchase option is normally also extended. If the purchase option

is exercised, the time charter contract ceases to exist.

Bermudan options are American options with non-standard features. The option holder have the

possibility to exercise the option at several predetermined dates in the contract period (Hull 2012). If

the embedded options in the time charter contract are of Bermudan style, the charterer can choose

to extend the lease period or purchase the vessel at several dates in the lease period. These dates

are often set to once a year. As before, if the option to extend is used, the option to purchase the

vessel is normally also extended.

Since options are financial derivatives, the availability of reliable price information on the underlying

asset is a necessary condition. Alizadeh and Nomikos (2009) emphasizes that available price

information on the underlying freight market is necessary in order to trade derivatives on freight.

Further, they explain that the price information on the underlying freight market needs to be

continuous, measurable and fully transparent.

The Baltic Exchange is the leading provider of freight market information (Alizadeh and Nomikos

2009). When derivative transactions are priced and settled in the freight market, they generally rely

on freight indices provided by the Baltic Exchange. Several different freight indices exist due to both

the differences between segments in the shipping industry, but also due to different vessel types and

sizes. Since this thesis focus on the dry bulk segment, the Baltic Dry Index (BDI) is the index of

interest. The BDI is an index calculated as the equally weighted average of the indices related to the

different vessel sizes in the dry bulk segment (Alizadeh and Nomikos 2009). This index is used as a

general market indicator reflecting the movements in the dry bulk segment.

In addition to being of European, American or Bermudan style, options embedded in time charter

contracts are written on the underlying average spot freight rate over the defined contract period;

21

they are path dependent5 and therefore of Asian style (Koekebakker, Adland et al. 2007). As

mentioned before, freight rates cannot be delivered in their physical form, they represent a cost of a

freight service that cannot be stored or carried forward in time (Kavussanos and Visvikis 2006). Their

non-storable feature is one reason why it is convenient to write freight rate claims on an average of

spot freight rates over a defined period of time. Further, Koekebakker, Adland et al. (2007) explain

that a charterer operating in the spot freight market is exposed to freight rate fluctuations during

some period of time. In order to capture freight rate fluctuations over a defined period of time, it is

convenient to treat freight rate contingent claims as path-dependent derivatives.

Another aspect worth mentioning is the freight revenue process when operating a vessel in the spot

freight market. The duration of a voyage can differ from a few weeks to several months; the

expected freight revenue from each voyage is given by an estimated average of the forecasted

fluctuations of the freight rates over the voyage’s duration. Therefore, the spot freight rate is itself

implicitly average based since it refers to fluctuations over a specific time period (Koekebakker,

Adland et al. 2007).

2.7 Background for the Models Selected

This section will give a presentation of previous empirical findings that support the two models

adopted for valuations. Different arguments are presented that together form an empirical

foundation for adopting both the Ornstein-Uhlenbeck process and the Geometric Mean Reversion

process for valuation purposes. Due to the similarities between the two models, the following

arguments will apply to both processes. But first, clarifying arguments for the two models adopted

will be presented.

The main intention of this thesis is to introduce the concept of fair valuation of time charter

contracts with purchase options. For this purpose, the Ornstein-Uhlenbeck process is adopted and

analytic solutions for the valuations will be derived. The Ornstein-Uhlenbeck process is a one-factor

model with mean reversion properties which, as will be argumented for in the following, serves as

well suited when modeling freight rates.

However, two models will be introduced for valuation purposes and the second one is the Geometric

Mean Reversion process. As its name indicates, mean reversion is also a property of this process and

one factor is present also here. The Geometric Mean Reversion process is considered as more

5 According to Hull (2012) path-dependent options are options where the payoff depends on the path followed

by the price of the underlying asset, not just its final value.

22

realistic in freight rate modeling compared to the Ornstein-Uhlenbeck process, arguments pointing in

that direction will be examined later. Nevertheless, the Ornstein-Uhlenbeck process will be applied

for the valuation of time charter contracts with purchase options due to the nice analytic solutions

from Jørgensen and Giovanni (2010).

Due to the lack of analytic solutions from the Geometric Mean Reversion process, valuation of time

charter contracts with embedded options requires numerical routines such as the Finite Difference

method6. Such routines are complex and time intensive to implement, but Monte Carlo simulation7

can be used to value vessels and European options from the Geometric Mean Reversion process. This

will be done later on as an introduction to further studies of the use of this model.

Now that the model choices have been clarified it is time to move on to empirical arguments for why

these models are specially suited for freight rate modeling. First, both models are driven by one

factor which is assumed to be the spot freight rate. Stopford (2009) points out four factors that are

influential on the vessel value; freight rates, age, inflation and shipowners’ expectations for the

future. Further he says that freight rates are the one factor that primarily influences vessel prices. As

the freight market goes up and down, so will this continue to the sale and purchase market.

Also, Jørgensen and Giovanni (2010) argue that the spot freight rate is the major source of

uncertainty in the shipping industry, which results in the spot freight rate representing the main

source of business risk in shipping. By adopting the Ornstein-Uhlenbeck process and choose the spot

freight rate as the stochastic factor evolving through time, simple and more complex freight rate

contingent claims in the shipping industry can be fairly valued, in addition to valuation of the vessels

involved in these contingent claims. The Ornstein-Uhlenbeck process ensures the spot freight rate to

evolve through time according to a mean reverting stochastic process. Empirical arguments

according to the mean reversion property will be presented below, whereas an economic reasoning

for mean reversion in freight rate movements will be presented in the next section where freight rate

dynamics are examined.

Fair valuation of time charter contracts with purchase options include valuation of the vessels

underlying the contract. Therefore, another important justification for the models adopted and the

spot freight rate as the uncertain factor in the models is the evidence of high correlation between

spot freight rates and vessel prices. Adland and Koekebakker (2007) introduce their research by

mentioning that freight rates are relatively highly correlated with vessel values since peaks and

6 Finite Difference methods value a derivative by solving the differential equation that the derivative satisfies

(Hull 2012). 7 A procedure for randomly sampling changes in market variables in order to value a derivative (Hull 2012). This

procedure will be described in detail in Section 3.5.1.

23

troughs in the freight market have a tendency to quickly work their way into the sales and purchase

market. They propose that the most important factor determining the price of a vessel is the vessel’s

age; this is due to the depreciation of the vessel’s value during its lifetime. Further, they expect the

one-year time charter rate to be the second important factor to the price of a vessel. In their

research, they find strong evidence of correlation between freight rates and vessel prices. Also, they

find that the vessel value is an increasing function of the freight rate level (Adland and Koekebakker

2007).

Another argument for adopting both the Ornstein-Uhlenbeck process and the Geometric Mean

Reversion process for valuation purposes is the evidence of mean reversion in spot freight rates. In

his work, Adland (2000) finds only very slight mean reversion for low and medium values of the spot

freight rate. But when the freight rate increases beyond 35 000 per day, the mean reversion is

stronger. This is the same as saying that the drift decreases since stronger mean reversion imply

stronger “attraction” towards the long-term mean reversion level. Further, he explains that the

decline in drift at high freight rates prevents the freight rate from exploding towards infinity.

The evidence of mean reversion in freight rates is also acknowledged by Koekebakker, Adland et al.

(2006) who state that the freight rate is expected to be mean reverting in one sense or another. They

apply basic maritime theory to show why the spot freight rate process must be mean reverting

(Koekebakker, Adland et al. 2006). The same theory is also to be found in Stopford (2009). In Section

2.8 below, where the freight rate dynamics are described, the economic reasoning will be presented.

To sum up, the presented arguments point in the direction of a model that takes account of one

variable evolving in a stochastic manner through time (the spot freight rate), and which also have

mean reversion properties. Both the Ornstein-Uhlenbeck process and the Geometric Mean Reversion

process have these properties, and it is therefore reasonable to assume these models to be

appropriate models for the valuations.

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2.8 The Dynamics of Freight Rates

The following section will describe the dynamics of spot freight rates and how they evolve in a

stochastic manner through time. To understand the evolvements of spot freight rates a short

description of the predominant market cycles in the shipping industry is essential. Thereafter, the

freight rate dynamics will be described by first introducing the elasticity of freight rates in different

market conditions, and then the behavior of freight rates will be described.

2.8.1 Shipping Market Cycles

The shipping industry is pervaded by market cycles. Stopford (2009) identifies three components of a

typical cyclical time series. The first is the long-term cycles driven by technical, economic or regional

change. A long-term cycle moving upwards is good for business, whereas a long-term cycle moving

downwards is bad for business. The second is the short-term cycles characterized by short-lived

movements and are important drivers of the shipping market cycle. Short-term cycles are often

referred to as “business cycles”, they fluctuate up and down and a complete cycle can last anything

from 3 to 12 years from peak to peak. The third and last component is the seasonal cycles which are

regular fluctuations within the year. As for the dry bulk shipping segment, weak markets appear

often during July and August due to relatively little grain being shipped. Thus, seasonal cycles occur in

response to seasonal patterns of demand for sea transport.

Now that the shipping market cycles are shortly described it is time to move on to the dynamics of

freight rates, which in fact is the factor that fluctuates according to the market cycles.

2.8.2 Freight Rate Dynamics

According to Alizadeh and Nomikos (2009), the balance between supply and demand for shipping

services is at any point in time reflected by the spot freight rates. The crucial factors for demand and

supply are the ones listed in Table 1. When the shipping market is in recession and spot freight rates

are at very low levels, an overcapacity is developed and many vessels cannot find employment, are

laid up, slow steam or even carry part cargo. When the market is in such a recession, changes in

demand due to external factors8 can be absorbed by the extra available capacity. Thus, the impact on

spot freight rates would be relatively small and they are considered as being inelastic to changes in

demand. Gradually, market conditions will improve and freight rates will increase. Available vessels

8 Nomikos and Alizadeh (2009) propose seasonal changes in trade and random shocks (events) as external

factors.

25

will be employed again until the world fleet is fully utilized and any increase in supply is only possible

by increasing productivity through increasing speed and shortening port stays and ballast legs. When

market conditions are good and spot freight rates are at high levels, any changes in demand due to

external factors would create a relatively large movement in spot freight rates; spot freight rates are

considered elastic in relation to changes in demand.

Good market conditions imply high spot freight rates. Tvedt (1997) explains that very high spot

freight rates may appear in shorter periods of time due to the demand for seaborne transport being

inelastic to changes in spot freight rates, and that there is a short-term upper limit to supply. Further,

he explains that good market conditions with high levels of spot freight rates will not be a persistent

situation over time. The potential for supply adjustment, both newbuilding and demolition, will

guarantee that very high or very low spot freight rates will not be a persistent situation over time

(Adland and Cullinane 2006). High spot freight rates will tempt shipowners to order new vessels in

order to capture high freight earnings. Due to the time lag from ordering a new vessel until delivery,

a gambling position is created for the shipowner. Often, the market clears at a rate that is not high

enough to cover the investment costs of a new vessel. If the shipowner manages to order a new

vessel in time such that the rates are still high when the vessel is delivered, he may catch a high

reward. But ordering a vessel when the spot freight rates are high is often too late because the rates

will probably be back to normal low levels when the vessel is ready to be delivered.

Summing up, the shipping market cycles and the continuous stochastic evolution of spot freight rates

go hand in hand; spot freight rates will evolve in accordance with the current state of the shipping

market and its cycles. This serves as an economic reasoning for the mean reverting behavior of spot

freight rates which reinforces the empirical findings described in the Section 2.7 above.

2.9 The Ornstein-Uhlenbeck Process

In the following, the model adopted to value time charter contracts with purchase options, namely

the Ornstein-Uhlenbeck process, will be described in detail, along with the derivation of the solution

to the process. The Ornstein-Uhlenbeck process has been applied to the shipping industry by, among

others, Bjerksund and Ekern (1995), Tvedt (1997) and Jørgensen and Giovanni (2010).

The basic underlying assumption is that the stochastic component of the instantaneous cash flow

from owning a vessel, the spot freight rate, is characterized by an Ornstein-Uhlenbeck process

(Bjerksund and Ekern 1995). When the spot freight rate is said to be stochastic, its value will change

over time in an uncertain way (Hull 2012). Spot freight rates evolve in a stochastic manner and their

26

values can change at any point in time, they are therefore defined to be continuous-time stochastic

variables. When spot freight rates are assumed to follow an Ornstein-Uhlenbeck process, they are

ensured to behave like continuous-time stochastic variables due to the special feature of the model

being an Itô process. An Itô process consist of two terms; a drift term which is a function of the value

of the underlying variable9 and time, and a variance term which also is a function of the value of the

underlying variable and time. The variance term contains a standard Wiener process which is a

particular type of a stochastic process with mean of zero and variance of one per year, it is denoted

(Hull 2012). Both the drift term and the variance term are liable to change over time.

Bjerksund and Ekern (1995) assume that the instantaneous cash flow generated by an operating

vessel, , may be described as follows:

(1)

where represents the size of the cargo, represents the operating cost-flow rate, and represent

the uncertain spot freight rate (annualized) at time per unit of cargo. For ease of notification, it is

assumed that the spot freight rate is prevailing for the vessel as a whole and net of all costs. Thus,

and (Jørgensen and Giovanni 2010), and is left as the instantaneous cash flow from

operating a vessel. As described in Section 2.1.3 where time charter equivalent spot freight rates are

introduced, this assumption causes lack of the ability for direct market comparisons as market-

quoted freight rates embed varying degree of costs.

9 The freight rate in this case.

27

As argumented for in Section 2.7 ,where the background for the model selected is examined, and in

Section 2.8, where the dynamics of freight rates are examined, in addition to following Bjerksund and

Ekern (1995), it is assumed that the spot freight rate can be modeled by the following stochastic

differential equation:

(2)

where is the speed of mean reversion, is the constant long-term mean, is the instantaneous

volatility of spot freight rates, and is a standard Wiener process defined on some probability

space (Jørgensen and Giovanni 2010). The model is an Itô process where the first term is

considered as the drift term, whereas the last term is considered as the variability term. This process

is identical to the one Vasicek (1977) proposed for modeling interest rate dynamics.

In line with Bjerksund and Ekern (1995), technical descriptions of the model, the drift term, ensures

that the process always are pushed back to its long-term mean. When , the drift term is

negative and the process will be pushed up to its long-term mean. On the other hand, when ,

the drift term is positive and the process will be pushed down to its long-term mean. Further, they

explain that higher values of will create a stronger tendency of the stochastic process to move back

towards its long-term mean. Thus, the higher the , the higher the degree of mean reversion.

Further, they explain that the second term characterizes the volatility of the process with being

an increment of a standard Wiener process with characteristics as explained above.

Important to remember is that the model measures time in years, whereas actual spot freight rates

are normally quoted on a daily basis. This is important to have in mind when comparing with market

data where have to be considered instead of . is one day measured in years, where one

year is assumed to contain 360 days (Jørgensen and Giovanni 2010).

Due to the properties of the Wiener process being standard normal distributed, Equation (2) implies

that future spot freight rates are normally distributed (Jørgensen and Giovanni 2010). The density

function10 will therefore have the classic “bell shaped” curve. In order to derive the mean and the

variance of the distribution of future spot rates the solution to Equation (2) have to be derived, this

will be done in the following section.

10

Plots the shape of the distribution curve of the random variable that is considered (Skovmand 2012), which in this case is the spot freight rate.

28

2.9.1 The Solution to the Ornstein-Uhlenbeck Process

The solution to the process can be derived from the stochastic differential equation in Equation (2).

This is desirable both in order to calculate the freight rate at time , and in order to derive the

mean and the variance of the distribution of future spot freight rates. From the mean of the

Ornstein-Uhlenbeck process the expected return from operating a vessel in the spot freight market

over a defined period, can be calculated. The variance of the Ornstein-Uhlenbeck process calculates

the volatility in spot freight rates over the same period. Both measures are useful for agents in the

shipping industry as they give an indication of how the market will develop in the future.

First, consider how to obtain the freight rate at time which is given by today’s freight rate plus

the sum of the dynamics of the spot freight rate evolving from time to time :

(3)

Equation (3) has to be solved explicitly in order to ensure the availability of analytic solutions. In

order to do this, a temporary variable have to be introduced (Hammer, Hafsaas et al. 2011):

(4)

It is desirable to obtain the dynamics of this temporary variable since the dynamics of the spot

freight rate are considered in Equation (2). The dynamics of Equation (4) are given by Ito’s lemma11:

(5)

(6)

11

Describe the behavior of functions of stochastic variables. Such a function can be the price of a derivative which is dependent on the underlying stochastic variable and time (Hull 2012).

29

Now, the next step is to insert Equation (2) for in the equation above:

(7)

Manipulations12 of Equation (7) give:

(8)

Continuing by integrating from time zero to time :

(9)

Calculating

:

Continuing with Equation (9):

(10)

12

A detailed derivation is attached in the Appendix.

30

(11)

(12)

Calculating

:

Finally, the solution to Equation (2) becomes:

(13)

From Equation (13) the time conditional mean and variance of the normal-distributed future spot

freight rate can be stated as follows:

(14)

(15)

where the detailed derivations of Equations (14) and (15) are attached to the Appendix.

31

2.10 The Geometric Mean Reversion Process

The Geometric Mean Reversion process will now be presented and the solution will be derived. As

mentioned earlier, this process is more realistic in modeling freight rates, but to the author’s

knowledge analytic solutions do not exist. Therefore, when calculating vessel values and European

option values numerical procedures need to be implemented. Those will be explained in more detail

below. Following Tvedt (1997), the increment of the process is given by the following stochastic

differential equation:

(16)

where is the speed of mean reversion, is the long-term mean reversion level, is the

(annualized) spot freight rate at time , is the instantaneous volatility of spot freight rates, and

is the increment of a standard Wiener process.

As Tvedt (1997) describes, the Geometric Mean Reversion process is mean reverting and is also

downwards restricted since it is not possible to take the natural logarithm of spot freight rates that

equals zero or are negative - zero is an absorbing level.

Another important feature of the Geometric Mean Reversion process is that it ensures the volatility

to be progressively increasing in the freight rate level. This occurs in the last term of Equation (16)

where the volatility parameter is multiplied by the time spot freight rate level. In his research,

Adland (2000) finds evidence that the volatility is increasing in the freight rate level. More specific, he

finds that the diffusion function, , is close to linear for low and medium freight rates, while it is

increasing progressively for very high freight rates (Adland 2000). Assuming that the spot freight rate

follows the Geometric Mean Reversion process is thus appropriate in relation to his findings.

In the following, the derivation of the solution to the Geometric Mean Reversion process in Equation

(16) will be done, whereas the time conditional mean and variance only will be presented. The

same applications for the solution, the mean and the variance as for the Ornstein-Uhlenbeck process

are applicable also here.

32

2.10.1 The Solution to the Geometric Mean Reversion Process

The solution to Equation (16) is derived following Tvedt (1997). In order to simplify later calculations

he suggests starting by dividing the whole process by and multiplying it with the integrating factor

:

(17)

Further, he continues by defining a function . Again, the increments of

this temporary function are desirable. They are given by Ito’s lemma:

(18)

(19)

Rearranging:

(20)

In order to proceed towards the solution two things have to be done. First, it is desirable to

substitute the Geometric Mean Reversion process for . In order to ease the notification,

and are defined such that . This gives:

33

where the first and the second term tend to zero, and where the last term tends to such that

.

Then, by rearranging Equation (17) to

and substituting the

process for Equation (20) becomes:

(21)

The next step is to rearrange this equation and integrate it from time zero to time :

(22)

(23)

Finally, the solution is obtained by rearranging Equation (23) such that is alone, the

rearrangement step by step is attached to the Appendix. The freight rate level at time , , can be

expressed as:

(24)

with mean and variance:

(25)

34

(26)

where:

35

3 Analysis Section

This section starts by an examination of the differences between the two considered models; the

Ornstein-Uhlenbeck process and the Geometric Mean Reversion process. Evidences of the Geometric

Mean Reversion process being more appropriate in freight rate modeling will be presented and

discussed. Further, the Ornstein-Uhlenbeck process and the Geometric Mean Reversion process will

be examined separately. When investigating the Ornstein-Uhlenbeck process, valuation of simple

freight rate contingent claims will be introduced and for that purpose the partial differential equation

will be established. Analytic solutions will be derived before they are used to the valuation of simple

freight rate contingent claims. Finally, European option values to buy the vessel will be calculated,

which in turn is used to obtain the total value of time charter contracts with purchase options. Next,

the Geometric Mean Reversion process will be investigated and the purpose is to obtain vessel

values and European option values to buy the vessel by Monte Carlo simulation. Finally, the section

ends by a comparison of vessel values and option values obtained by these two models.

As for the calculations throughout this section base case parameter values will be used. They are

adopted from Tvedt (1997) where he has estimated parameter values in relation to both processes.

The base case parameter values are presented in Table 2 below. However, when tables are

presented for different volatilities and different spot freight rates, those specific varying parameter

values will be adopted from Jørgensen and Giovanni (2010).

Parameter Ornstein-Uhlenbeck Process Geometric Mean Reversion

Process

14 371 10,45

1 094 0,1184

0,00247 0,0033

1,5% 1,5%

Table 2. Base case parameter values for both processes.

36

3.1 Two Models – Different Characteristics: A Comparison

This section will give an examination of the differences between the Ornstein-Uhlenbeck process and

the Geometric Mean Reversion process. At a first glance they look quite similar, and they do in fact

have some properties in common. But due to some important dissimilarities they behave quite

differently from each other.

In real life, negative values of the spot freight rate are impossible. This is due to the shipowner’s

option to lay up his vessel if the operational costs are not covered. In fact, due to the option to lay

up, the spot freight rate will generally never fall below the given lay up level (Tvedt 1997). The

Ornstein-Uhlenbeck process fails to take account of the impossibility of spot freight rates becoming

negative since the process is not downwards restricted. Large volatility will often give negative spot

freight rates because the process is normally distributed around a given mean. This is demonstrated

in Figure 1 below where the process is simulated over a period of almost six years with time steps

equal to one observation of the spot freight rate each day. The base case parameter values in Table 2

are applied and presented in the figure description. However, the volatility is adopted from

Jørgensen and Giovanni (2010) in order to capture the high volatility effect. From Figure 1 it is clearly

that the spot freight rates over the horizon can take on both negative and positive values. This is

somewhat unrealistic compared to a real life situation.

Figure 1. Simulated spot freight rate from the Ornstein-Uhlenbeck process. =0,00247, =14 371,

=14 500, =1/360 and =7 000.

-10 000,00

-5 000,00

-

5 000,00

10 000,00

15 000,00

20 000,00

0 1 2 3 4 5 6

Fre

igh

t R

ate

Val

ue

Time

Simulated Freigt Rate Following an Ornstein-Uhlenbeck Process

Freight Rate

37

In relation to negative freight rates, what actually can happen in reality during short intervals is that

the spot freight rates may become so low that the estimated time charter equivalent spot rate will

become negative. This happens when the voyage income is less than the total of fuel consumption

and harbor and channel costs, but in these cases the shipowner will probably rather lay up his vessel

than keeping it in operation. Therefore, the market almost always clears at a positive time charter

equivalent spot freight rate and the Geometric Mean Reversion process, which is downwards

restricted, may give a more appropriate and realistic description of the spot freight rate.

As Figure 2 below clearly indicate, the simulated Geometric Mean Reversion process only give

positive values of the spot freight rate and is therefore also more realistic in relation to a real life

situation. Again, the base case parameter values from Table 2 are applied and given in the figure

description. However, in order to capture the high volatility effect the volatility is set equal to one13.

Figure 2. Simulated spot freight rate from the Geometric Mean Reversion Process. =0,0033, =10,45,

=14 500, =1/360 and =1,0000.

13

Inspired by Tvedt (1997) where he at one point has set the volatility parameter equal to 0,93.

-

5 000

10 000

15 000

20 000

25 000

30 000

35 000

0 1 2 3 4 5 6 7 8

Fre

igh

t R

ate

Val

ue

Time

Simulated Freight Rate Following a Geometric Mean Reversion Process

Freight Rate

38

3.2 Applications of the Ornstein-Uhlenbeck Process: Introducing Valuation

of Freight Rate Contingent Claims

In the following, some simple freight rate contingent claims whose value, , depend only on

the spot freight rate factor process, , and time will be valued. Before this can be done, the

fundamental partial differential equation that must satisfy in order to give the correct value

of such contingent claims will be established. According to Hull (2012), a key property of the

fundamental partial differential equation is that no variables affected by the risk preferences of

investors are involved. This opens for risk-neutral valuation of claims dependent on only the spot

freight rate factor process and time; the risk-neutral valuation result14 can be applied. When no risk

preferences enter the partial differential equation a simple assumption that all investors are risk-

neutral can be made. Thus, the risk-free rate of interest, , is assumed to be the expected rate of

return on all investments. Also, over the contract period the risk-free rate of interest is assumed to

be constant. The assumption of a risk-neutral world does therefore simplify the computations of the

claims which are dependent on only the spot freight rate factor process and time.

The spot freight rate essential in the partial differential equation is representing the price of a

service, and is therefore not a physical asset. In order to price freight rate contingent claims by the

partial differential equation, an assumption of the existence of a physical asset whose price is

perfectly correlated with the spot freight rate need to be done (Jørgensen and Giovanni 2010). They

explain that this is necessary to the application of standard no arbitrage arguments (Hull 2012).

Further they explain that the vessels underlying the option contracts could be used, but hedging

strategies based on trading vessels are very unpractical. Instead, they rely on the assumption of the

existence of a liquid market for the relevant futures. Such markets do in fact exist in the real world;

the Oslo-based International Maritime Exchange (IMAREX)15 is an example.

When both the assumption of a risk-neutral world and the assumption of the existence of a liquid

market for the relevant futures are done, Ito’s lemma and a risk-neutralizing hedge argument16 can

be used to derive the fundamental partial differential equation that must satisfy in order to

give the correct value of the option contract.

14

The present value of any cash flow in a risk-neutral world can be obtained by discounting its expected value at the risk-free rate (Hull 2012). The mathematical expression will be presented later. 15

Writes freight rate derivatives upon the freight indices obtained from the Baltic Exchange and from Platts. 16

Transformation of the “true” probability measure to the risk-neutral probability measure which will be derived later.

39

3.2.1 Derivation of the Fundamental Partial Differential Equation

From Equation (2), the single factor spot freight rate process is given as:

where is the increment of a standard Wiener process under the “true” probability measure, . It

is argumented that derivatives prices, , influenced by the factor process, , and time must

have dynamics described by (Vasicek 1977).

(27)

where:

and where is the market price of freight rate risk. In order to discount by the risk-free interest

rate, , and thus apply the risk-neutral valuation result, the “true” probability measure have to be

changed to the risk-free probability measure, . In short, the risk-neutral valuation principle states

that a derivative can be valued by “(a) calculating the expected payoff on the assumption that the

expected return from the underlying asset equals the risk-free interest rate and (b) discounting the

expected payoff at the risk-free rate” (Hull, 2012, page 630).The transformation from the “true”

probability measure to the risk-free probability measure can be done by applying Girsanov’s

Theorem. This procedure will be done in the following. Define:

(28)

40

where

is now the increment of a standard Wiener process under the risk-free probability

measure, . is again the market price of freight rate risk. Then:

(29)

(30)

(31)

In the following, an assumption of a constant market price of freight rate risk is done such that

reduces to . Therefore, the factor process under is given as:

(32)

(33)

(34)

(35)

41

Define

. Then:

(36)

This is the dynamics of under the risk neutral probability measure, . The mean and the variance

are given as in Equations (14) and (15). The only difference is that is replaced by . Finally, Ito’s

lemma can be used to characterize the dynamics of the -process under :

(37)

The next step is to substitute the Itô process for and in Equation (37). The Itô process

under the risk-neutral probability measure is given in Equation (36):

To ease the calculations, and are defined such that .

Calculating :

42

In the limit, tends to zero, tends to , and tends to (Campbell, Lo et al. 1997).

Hence, the first and the second term tend to zero, and the last term tends to . This reduces

equation to . Inserting both and into Equation (37):

(38)

This is equal to:

(39)

(40)

(41)

43

According to Jørgensen and Giovanni (2010), absence of arbitrage requires that:

(42)

(43)

where is the dividend rate received by the claim. By introducing as the

absolute cash flow from the claim the following partial differential equation is the one must

satisfy:

(44)

Or, equivalently:

(45)

This is the partial differential equation that all claims depending on the spot freight rate process, ,

and time need to satisfy in order to be priced arbitrage free. Solving for will give the price of a

particular freight rate contingent claim. Another way of solving Equation (45) is to manipulate the

probabilistic Feynman-Kac representation of the solution to the partial differential equation. A partial

differential equation can then be solved by an expectation of the discounted payoff of the derivative,

, modified replacing the “true” probability measure, , by a risk-free probability measure,

(Duffie 2001). By doing this, the risk-free interest rate, , can be used as the expected rate of return

when discounting backwards in time. The probabilistic Feynman-Kac representation of the solution

to the partial differential equation takes the following form (Jørgensen and Giovanni 2010):

44

(46)

where would typically be the expiration date at which the value of the claim would be given as a

known function of . Equation (46) is quite intuitive, it indicates that the time value of the claim is

the expected value of the continuous flow of net dividends plus the value of the payoff at the

maturity date, both discounted back to time .

Finally, when the partial differential equation is established, with its solution, the valuation of some

simple freight rate contingent claims can be done.

3.3 Valuation of some simple Freight Rate Contingent Claims

The following section will present derivations of analytic solutions for evaluating simple freight rate

contingent claims, as well as valuation results for each simple contingent claim.

3.3.1 Claim to Receive Spot Freight Rate Flow from Time to Time

The first claim that will be valued is the claim to receive the spot freight rate on a continuous basis

from time , when the current spot freight rate is , to time . To do this, the probabilistic Feynman-

Kac representation from Equation (46) is used. In the following, the main steps of how Equation (46)

is used to derive an analytic solution of the value of such a claim are shown, whereas a detailed step

by step representation is attached to the Appendix. Defining the current value of receiving the spot

freight rate on a continuous basis as:

(47)

45

The freight rate process, :

(48)

Inserting in Equation (47) gives:

( ) (47)

Since the expectation of the increment of a standard Wiener process equals zero (Hull 2012), the last

term becomes equal to zero and only the two first terms are left. Further:

(49)

(50)

(51)

(52)

where

is an annuity factor of the present value using the discount rate of

receiving a unitized continuous cash flow for years (Jørgensen and Giovanni 2010).

46

3.3.2 Fixed for Floating Freight Rate Swap

When a shipowner and a charterer agrees on a time charter contract, the shipowner receives a fixed

daily freight rate over the lifetime of the contract, and leaves the operation of the vessel in the

charterers responsibility. If the shipowner had not chartered his vessel, he could have operated it in

the spot freight market himself and received the floating spot freight rate over the same period of

time. Therefore, when a shipowner enters into a time charter contract, he agrees to receive a fixed

daily freight rate in exchange for a floating one. This is the same mechanism as an interest rate swap;

an exchange of a fixed rate of interest on a certain notional principal for a floating rate of interest on

the same notional principal (Hull 2012).

It is in the shipowner’s interest to determine the constant freight rate, , fixed at time that will

be equivalent to receiving the variable spot freight rate over the period from time to time . In this

way, the value of the freight rate swap will be equal to zero at initiation. This is also a property equal

to an interest rate swap agreement (Hull 2012).

To determine the fixed freight rate that will make the shipowner indifferent in his choice of whether

to operate the vessel on his own, or charter it, Jørgensen and Giovanni (2010) apply Equation (52)

and solve for :

(53)

Equation (53) indicate that the present value of the fixed freight rate have to be equal to the present

value of receiving the spot freight rate on a continuous basis from time to . When ensuring this

equality, the shipowner will be indifferent from receiving the fixed freight rate over the horizon, or

receiving the floating spot freight rate over the same horizon. Solving for :

(54)

where will be the fair continuously paid time charter rate (fixed freight rate) during the life of

the contract.

47

In the following, some fair valued time charter rates for different values of the speed of mean

reversion and current spot freight rates will be presented. The calculation of these fair valued time

charter rates is done by applying Equation (54) above. The base case parameter values from Table 2

are applied and given in the table description. As for this example, both the spot freight rates and the

fair time charter rates are presented in daily values. It is worth mentioning that when the analytic

solutions are used for valuations the parameter values have to be scaled up into yearly values so that

the time units are proportionate to each other. This is the case for all remaining calculations during

the next sections.

Spot Freight Rate,

5 000 10 000 15 000 20 000 25 000 30 000 35 000 40 000

0,10 6 974 10 921 14 868 18 814 22 761 26 708 30 655 34 602

0,25 8 981 11 857 14 733 17 609 20 484 23 360 26 236 29 112

0,50 10 881 12 743 14 605 16 467 18 329 20 191 22 053 23 915

1,00 12 466 13 483 14 499 15 515 16 531 17 548 18 564 19 580

2,00 13 406 13 921 14 436 14 951 15 466 15 981 16 496 17 011

5,00 13 983 14 190 14 397 14 604 14 811 15 018 15 225 15 432

10,00 14 177 14 280 14 384 14 488 14 591 14 695 14 799 14 902

Table 3. The dependence of fair time charter rates (daily) on speed of mean reversion, and current

spot freight rate, . =1,5% and =14 371 (daily).

The results in Table 3 indicate that when the spot freight rate is 15 000 per day, which is close to the

long-term mean of 14 371 per day, the fair time charter rate will also lie close to the long-term mean

for all values of the speed of mean-reversion parameter. This is in line with intuition; when the spot

freight rate is close to the long-term mean it is not expected to increase or decrease wildly the next

five years, thus also the fair time charter rates should lay close to the spot freight rate.

The story is different when the current spot freight rate differs from the long-term mean. For current

spot freight rates that are below the long-term mean, the fair time charter rates are far from the

long-term mean when the speed of mean reversion-parameter takes on low values, but are closer to

the long-term mean when the speed of mean reversion-parameter takes on high values. This makes

sense since when the speed of mean reversion is high, it is expected that the spot freight rate will

48

quickly return to the long-term mean. Therefore, the fair time charter rate should be close to the

long-term mean. Opposite, when the speed of mean reversion is low, it is expected that the spot

freight rate will return slowly to its long-term mean. The fair time charter rate should in that case be

set closer to the current spot freight rate to secure as low difference between the two rates as

possible during the contract period.

For current spot freight rates lying above the long-term mean the opposite is present; when the

speed of mean reversion is high the fair time charter rate is closer to the long-term mean than is the

case when the speed of mean-reversion is low. Also this makes perfectly sense; low speed of mean

reversion creates expectations of the spot rate to slowly return to the long-term mean. And again,

high speed of mean reversion creates expectations of the spot freight rate to quickly return to the

long-term mean, implying a fair time charter rate closer to the long term mean to reduce the

possibility of large differences between these two rates.

At the time a time charter contract is agreed on and the fair time charter rate is fixed, the value of

the swap is, as mentioned before, equal to zero. However, Equation (47) clearly indicates that the

value of receiving the current spot freight rate on a continuous basis during the contract period is

based on an expectation. This implies uncertainty that the spot freight rate actually will evolve as

expected. If the spot freight rate moves in other directions than expected, the fair time charter rate

will also be different than the one that is fixed in the contract. This difference between the “new” fair

time charter rate and the one that was set at initiation gives the swap contract a value - positive or

negative. It is possible to calculate the value of a time charter contract entered into at an earlier

point in time. Jørgensen and Giovanni (2010) present this valuation formula for a contract that

receives the floating spot rate and pays a fixed time charter rate:

(55)

where time , is the prevailing fair time charter rate and is the contracted fixed time

charter rate. This equation indicate that if the prevailing fair rate is higher than the contracted fixed

time charter rate the swap has a positive value, and otherwise if the contracted fixed time charter

rate is higher than the prevailing fair time charter rate. The annuity factor, , ensures that

the value of the swap equals the discounted value of the continuous flow of the spread difference.

49

Valuation of a swap for different spot freight rates, fixed contracted time charter rates, and fair time

charter rates will be done next. The base case parameter values from Table 2 are applied and

presented in the table description. Table 4 below show the value of a 5-year time charter contract

which is dependent on the current fair time charter rate, and the previously contracted time charter

rate. In the table, spot freight rates, current fair time charter rates and previously contracted rates

are presented in daily values, whereas the values for the 5-year contract are yearly17 values.

The Value of a 5-Year Time Charter Contract

Spot and Current Time

Charter Rate

, Previously Contracted Rate

5 000 10 000 20 000 30 000

5 000 5 057 98 690 -8 572 091 -25 913 655 -43 255 218

10 000 10 027 8 716 815 46 033 -17 295 530 -34 637 094

20 000 19 966 25 953 063 17 282 282 -59 282 -17 400 845

30 000 29 905 43 189 312 34 518 531 17 176 967 -164 596

Table 4. The value of a 5-year time charter contract. r = 1,5%, = 14 371 (daily), = 0,00247.

As expected, when the previously contracted time charter rate differ from the current fair time

charter rate, the swap contract does either have a positive value or a negative value. A current fair

time charter rate higher than the previously contracted time charter rate gives the swap contract a

positive value. Otherwise, a current fair time charter rate lower than the previously contracted time

charter rate results in a negative value of the swap contract. Also, when the difference between the

two rates is large, the value of the swap is also large - positive or negative.

3.3.3 The Value of a Vessel

According to Jørgensen and Giovanni (2010), a vessel is a physical asset that earns rents to its owner

through the flow of net spot freight rates during the vessel’s lifetime. Therefore, they recommend

using Equation (47) to value a vessel. But since the vessel will have a scrap value18 when it reaches

the end of its useful economic life, an extension of this formula has to be applied. Assuming that the

17

One year is assumed to contain 360 days. 18

When the vessel is scrapped the remaining steel can be sold to the steel industry (Stopford 2009).

50

final service date, , of a vessel as well as its scrap value, , are known with certainty, the vessel

value can be calculated using this extension of Equation (52):

(56)

Equation (56) signify that the valuation formula for a vessel follows a Gaussian process with

deterministically time-varying drift and diffusion coefficients (Jørgensen and Giovanni 2010).

Therefore, future vessel values at any time are normally distributed under both the “true”

probability measure and the risk-neutral probability measure. The time conditional mean and

variance under both probability measures are given below (Jørgensen and Giovanni 2010):

(57)

(58)

(59)

In Table 5 below, vessel values as a function of different spot freight rates and varying remaining

vessel lifetimes is calculated using Equation (56). The parameter values are again the ones from Table

2 and are given in the table description. The spot freight rates are given in daily values whereas the

vessel values are given in yearly values.

51

Remaining Vessel Life ( )

5 10 15 20 30

5 000 13 408 189 21 392 759 28 968 933 36 152 125 42 957 810

10 000 22 026 314 37 908 156 52 721 046 66 535 648 79 418 068

15 000 30 644 438 54 423 553 76 473 159 96 919 171 115 878 325

20 000 36 262 562 70 938 950 100 225 272 127 302 694 152 338 583

25 000 47 880 687 87 454 347 123 977 385 157 686 217 188 798 841

30 000 56 498 811 103 969 744 147 729 498 188 069 740 225 259 098

35 000 65 116 936 120 485 141 171 481 610 218 453 263 261 719 356

40 000 73 735 060 137 000 539 195 233 723 248 836 786 298 179 613

Table 5. The dependence of vessel values on remaining vessel value ( ) and the spot freight rate

( ). =0,00247, =14 371 per day, =1,5% and =5 000 000.

In line with intuition, Table 5 clearly indicates that vessel values are an increasing function of both

the spot freight rate and the remaining vessel life. Higher spot freight rate gives higher cash inflow to

the shipowner and therefore also the vessel value is increased. Longer remaining lifetime of a vessel

also imply higher vessel value compared to a vessel with shorter remaining lifetime.

In Figure 3 below, Equation (56) is used to calculate the evolution of a vessel’s value over a lifetime of

25 years. This is the jagged line where the simulated vessel value is calculated over its lifetime. The

expected vessel value is also calculated under the risk-neutral probability measure using Equation

(57) above. For this purpose, all parameter values are adopted from Jørgensen and Giovanni (2010)

and are presented in the figure description. In line with intuition, we can see that the value of a

vessel is expected to decrease as it ages.

52

Figure 3. Simulated and expected value of a vessel. =0,25, =0,05, =25, =5 000 000, =20 000,

σ=5 000 and Δt=1/360.

3.4 European Option to Buy a Vessel

As described earlier, a European option to buy a vessel gives the charterer the right, but not the

obligation, to buy the vessel at expiration. If the option is not exercised, the contract with the

purchase option will expire and the vessel is handed over to the shipowner. Purchase options in

shipping contracts are common practice, but simple European options are generally not used. As

mentioned earlier, complex options happen to be used more often in shipping contracts. Although

European purchase options are somewhat unrealistic in relation to real life, such options will be

valued in order to demonstrate the interpretations of the nice analytic solutions derived through the

sections above.

Following Jørgensen and Giovanni (2010), the current date is denoted by and the expiration date of

a European option to buy a vessel is denoted by . Also, the vessel must be scrapped at date for a

value of . Further, will denote the exercise price19 of the purchase option. The payoff function at

expiry is then given as:

(60)

19

The price at which the vessel may be bought at in the time charter contract (Hull 2012).

0

20000000

40000000

60000000

80000000

100000000

120000000

0 5 10 15 20 25 30

Val

ue

Time

Simulated and Expected Vessel Value

Simulated Vessel Value

Expected Vessel Value

53

Due to the analytic solution for the vessel value in Equation (56), an analytic solution for the time

value of this European call option can be derived (Jørgensen and Giovanni 2010). In the following,

only the results will be presented, whereas a detailed step by step derivation will be attached to the

Appendix. Thus, from the Appendix the time value of the European call option in Equation (60) is

given as:

(61)

where:

(62)

(63)

(64)

(65)

and where and denote the standard normal cumulative probability and density functions,

respectively.

Now, when the framework for valuing a European option to buy a vessel is established, purchase

options for varying spot freight rates and freight rate volatilities will be valued using Equation (61).

The parameter values are again taken from Table 2 and are given in the table description. Also, the

spot freight rates and the freight rate volatilities are daily values, whereas the option values are

annualized values.

54

Freight Rate Volatility,

1 000 3 000 5 000 7 000 9 000

Spot Freight

Rate,

5 000 44 1 421 456 7 120 274 14 938 207 23 654 772

10 000 126 219 5 886 695 14 715 677 24 191 691 33 895 366

15 000 6 998 523 16 859 312 26 828 385 36 813 079 46 802 986

20 000 31 459 010 35 670 469 43 790 420 52 928 505 62 438 019

25 000 59 276 730 60 198 581 65 052 295 72 326 786 80 698 860

30 000 87 118 860 87 249 455 89 463 264 94 518 454 101 339 629

Table 6. Value of European option to buy a vessel. =1,5%, =0,00247, =14 371 (daily),

=5 000 000, =0, =5, =25 and =93 000 000.

The option values in Table 6 confirm well-known option theory; as the freight rate volatility increase,

so does the option value. Higher volatility imply higher probability of upside gains, thus the option

value also increases. In addition, increased spot freight rate will also increase the option value since

the cash flow from owning the vessel is increased.

Since the option is European, the total value of the time charter contract with a purchase option can

safely be decomposed into its leasing contract component and its option contract component

(Jørgensen and Giovanni 2010). The total value of the 5-year time charter contract with an

embedded European purchase option can be found by adding the value of the 5-year contract to the

value of the embedded purchase option. The value of the 5-year contract is calculated by using

Equation (55). Table 7 below show total fair contract value for varying spot freight rates and fixed

time charter rates. Again, the base case parameter values from Table 2 are applied and given in the

table description.

55

Fixed Time Charter Rate

5 000 10 000 15 000 20 000 25 000 30 000

Spot Freight

Rate,

5 000 7 218 964 -1 451 817 -10 122 599 -18 793 381 -27 464 162 -36 134 944

10 000 23 432 491 14 761 710 6 090 928 -2 579 854 -11 250 635 -19 921 417

15 000 44 163 324 35 492 542 26 821 760 18 150 979 9 480 197 809 416

20 000 69 743 483 61 072 702 52 401 920 43 731 138 35 060 357 26 389 575

25 000 99 623 483 90 952 701 82 281 920 73 611 138 64 940 356 56 269 575

30 000 132 652 577 123 981 795 115 311 013 106 640 232 97 969 450 89 298 668

Table 7. The value of a 5-year time charter contract with European purchase option. r=1,5%,

=0,00247, =14 371 (daily), =5 000 (daily), =5 000 000, =0, =5, =25, =93 000 000.

Table 7 indicates that the 5-year time charter contract with an embedded European option to buy

the vessel can take on a negative value. Generally, this is the case when the spot freight rate is

sufficiently low compared to the fixed time charter rate. This is a logical result since the spot freight

rate represents the cash inflow to the charterer, and the fixed time charter rate has to be paid; cash

inflow that is lower than the cash outflow creates a loss to the charterer, and the contract value

therefore becomes negative and unfavorable. This is also the reason why the contract value

decreases as the fixed time charter rate increases; higher rate which have to be paid will reduce the

contract value.

3.5 Applications of the Geometric Mean Reversion Process: Vessel and

European Option Valuation

As described in Section 2.6 where time charter contracts with embedded options were examined, the

options embedded in time charter contracts are path-dependent which imply that the payoff

depends on the path followed by the price of the underlying asset, not just its final value. In this case,

the payoff will depend on the path in which the freight rate follows over the contract period; the

freight rate represents the price of operating a vessel in the spot freight market. To the author’s

knowledge there exist no analytic solutions to the Geometric Mean Reversion process. Thus, in order

to value both the vessel and the option to buy the vessel, numerical procedures must be applied. For

this purpose Monte Carlo simulation is implemented.

56

3.5.1 Monte Carlo Simulation

Monte Carlo simulation is a numerical procedure which is applied when analytic results do not exist

(Hull 2012). Especially, when the derivatives are path-dependent, as is the case for the options

embedded in the time charter contracts, Monte Carlo simulation is a highly popular tool. The idea

underlying Monte Carlo simulation is the feature of random sampling. When Monte Carlo simulation

is applied to the valuation of an option the risk-neutral valuation result is used; several paths are

sampled to obtain the expected payoff in a risk-neutral world and then the average payoff is

discounted at the risk-free rate in order to achieve the option value. An assumption of a constant

risk-free interest rate is done when applying Monte Carlo simulation, which fit perfectly in this case

since the risk-free interest rate was assumed to be constant in Section 3.2 where valuation of simple

freight rate contingent claims was introduced. The main drawback with Monte Carlo simulation is

that it is extremely time consuming in the achievement of the required level of accuracy. This will be

exemplified later.

3.5.2 The Value of a Vessel

This section will introduce the use of Monte Carlo simulation when valuing a vessel applying the

Geometric Mean Reversion process. The model adopted to the valuation is obtained from Tvedt

(1997). However, some simplifying assumptions will be done for consistency compared to the ones

done when investigating the Ornstein-Uhlenbeck process. The simplifying assumptions done will be

explained when proceeding. According to Tvedt (1997) the instantaneous cash flow from operation

and lay up until the vessel is scrapped, is given by:

(66)

where is the time charter equivalent spot freight rate, is the operation costs except for voyage

related costs20, is the costs of keeping the vessel mothballed and is an indicator function of the

event , where . Keeping the vessel in operation is the optimal policy for the

shipowner when the spot freight rate plus lay up costs are above the operation costs. When this is

the case, is equal to one and . Otherwise, is equal to zero and . Again, for

ease of notation it is assumed that the spot freight rate is quoted for the entire vessel and net of all 20

Those are subtracted from the spot freight rate in order to obtain the time charter equivalent spot freight rate.

57

costs. This results in the spot freight rate being the instantaneous net profits from an operating

vessel:

(67)

Further, Tvedt (1997) explains that when the vessel reaches its maximum age at time , its value

must be equal to the value of the vessel as scrap, . But if the value of the vessel as a going concern

is less than the demolition value, the vessel may be scrapped before the estimated end of its lifetime.

Formally, the termination date is equal to the stopping time given by:

(68)

where is as defined below. Equation (68) indicates that the termination date is reached the first

time that the value of the vessel as a going concern is equal to or below the scrap value.

Finally, the value of a vessel at time can be presented. It is represented by the market value of the

cash flow generated from time to , and is given by:

(69)

where is the risk-neutral probability measure. Equation (69) indicates that the vessel value equal

the discounted sum of the spot freight rates from time to , plus the discounted scrap value.

Again, to the author’s knowledge, no analytic solutions exist to Equation (69). To calculate the vessel

value Monte Carlo simulation is therefore implemented in a Visual Basic for Applications (VBA)

function, the codes will be attached in the Appendix. In the following, the procedure will shortly be

explained.

58

First, vessel values for several points in time have to be calculated. Random numbers are therefore

generated from the standard normal distribution for the increment of the standard Wiener process,

, in the Geometric Mean Reversion process. The Geometric Mean Reversion process is then

applied to obtain thousand different values for the freight rate evolving from time to time by

Monte Carlo simulation. Each simulated freight rate is in turn used in Equation (68) to calculate

several different vessel values in time . Finally, to obtain the vessel value in time , the vessel values

are averaged over the thousand simulated paths. The VBA code will be attached to the Appendix.

Vessel values as a function of different spot freight rates and varying remaining vessel life are

presented in the table below. Also here, the base case parameter values given in Table 2 are applied

and presented in the table description below. The number of yearly subdivisions21 is set to 200, and

the number of simulated paths is set to 1000.

Remaining Vessel Life (T-t)

5 10 15 20 25

X(t)

5 000 12 957 020 20 024 764 26 148 386 31 171 237 35 707 173

10 000 21 412 215 35 540 770 47 441 184 57 675 263 66 562 843

15 000 29 623 056 50 647 868 68 102 516 84 102 145 96 709 234

20 000 37 703 051 65 869 093 89 329 727 110 266 254 127 514 082

25 000 46 113 328 81 182 557 110 679 791 134 738 187 156 181 838

30 000 54 495 751 95 208 781 130 570 383 160 705 809 185 371 697

35 000 62 634 579 110 674 635 152 576 992 185 889 464 213 720 792

40 000 71 269 729 125 249 842 171 712 908 211 957 959 244 816 516

Table 8. The dependence of vessel values for different freight rates and different remaining vessel

lifetimes. =0,0033, =10,45, =1,5%, =0,1184 and =5 000 000.

Table 8 clearly indicates that vessel values are an increasing function of both the spot freight rate and

the remaining vessel life. As is in line with intuition, the longer the remaining vessel lifetime, the

higher the vessel value. Also, higher spot freight rates indicate higher vessel value which is also in line

21

One year is divided into 200 equally time steps.

59

with intuition; higher spot freight rates generate higher cash inflow from owning a vessel, which in

turn will increase the value of owning this vessel.

3.5.3 European Option to Buy a Vessel

When an analytic solution does not exist, a concept called Monte Carlo on Monte Carlo simulation

can be used to calculate the value of a European option to buy a vessel. This is done by a new

simulation of freight rates using the Geometric Mean Reversion process. For each simulated spot

freight rate from time to time vessel values are calculated using Equation (69) which is done by

the same procedure as the calculation of vessel values above. In time , the payoff function is given

by:

(70)

where is the exercise price of the purchase option. For each vessel value in time the payoff

function is calculated. To find the option value in time , the discounted average of the payoff

functions obtained in time is calculated using the risk-neutral valuation result:

(71)

where is the risk-free interest rate which is assumed to be constant over the defined period. Also

this is done in VBA and the codes will be attached in the Appendix. Option values for different spot

freight rates and different volatilities are presented in Table 9 below. The number of yearly

subdivisions is again sat to 200, and the number of simulated paths has been sat to 500. The

parameter values are again obtained from Table 2 and are presented in the table description. The

exercise price is obtained from Jørgensen and Giovanni (2010).

60

Freight Rate Volatility, σ

0,1 0,3 0,5 0,7 0,9

Spot Freight

Rate, X(0)

5 000 0 183 775 702 012 1 097 792 260 950

10 000 0 3 028 103 5 536 125 8 553 248 5 832 592

15 000 6 379 6 169 871 11 481 198 1 921 091 855 705

20 000 216 678 8 900 324 33 333 346 14 663 615 11 260 378

25 000 1 415 484 15 739 370 38 285 950 17 309 911 26 778 091

30 000 2 970 948 20 728 850 16 371 935 86 278 677 27 623 172

Table 9. Value of European purchase option. Dependence on spot freight rate and freight rate

volatility. =0,0033, =10,45, =1,5%, Maturity Option=15 years, Maturity Vessel=25 years,

=93 000 000, Scrap Value=5 000 000.

The European option value is expected to be an increasing function of both the spot freight rate and

the freight rate volatility. As the spot freight rate increases, the vessel will become more valuable due

to higher cash inflow, and thus the option to buy such a valuable vessel will also increase. When the

freight rate volatility increases the probability of large upside gains will increase, thus, the option

value will also increase. Most of the values in Table 9 are acting in line with what is expected, but

some values are somewhat strange. There can be several reasons for these strange results, but an

obvious reason is that extremely few paths are Monte Carlo simulated. A modest number of 500

simulations are done due to the limited abilities of Microsoft Office Excel22.

The efficiency of Monte Carlo simulation can be increased by implementing variance-reduction

techniques. The antithetic variates method is an example of such a technique; correlation is created

across simulated paths in order to reduce the variance of the sum of random variables (Campbell, Lo

et al. 1997). This technique ensures a doubled amount of simulated paths by utilizing the symmetry

present in the normal distribution. Each simulated path can be reflected through its mean to produce

a mirror-image with the same statistical properties (Campbell, Lo et al. 1997); a mirrored random

variable is created with exactly the same statistical properties, but with the opposite sign.

22

A simulation of 500 paths ran a whole night before it was done.

61

3.6 The Valuation Results: Comparisons

This section will shortly comment the vessel values and the European option values obtained from

both the Ornstein-Uhlenbeck process and the Geometric Mean Reversion process.

Important to note is that estimation procedures lie outside the scope of this thesis. This creates

difficulties when comparing the valuation results. For correct comparison the parameter values to

both models should have been estimated from the exact same dataset in order to ensure that they

matches relatively to each other. This may be the reason why the vessel valuations give ambiguous

results; it is expected that the Geometric Mean Reversion process will give higher vessel values as

negative values are impossible. Also comparisons of the option values becomes difficult.

By comparing the vessel values from the Ornstein-Uhlenbeck process and the Geometric Mean

Reversion process in Table 5 and Table 8 respectively, they clearly indicate quite similar results.

Generally, the vessel values obtained from the Ornstein-Uhlenbeck process lie above the values

obtained from the Geometric Mean Reversion process, which is ambiguous as the Geometric Mean

Reversion process is expected to give higher values. Also, as both the spot freight rate increases and

the remaining lifetime increases, so does the difference between the values also increase.

As for the European option values, comparison becomes difficult due to both the estimation issue,

and due to the reduced accuracy when obtaining values from the Geometric Mean Reversion

process. A modest amount of 500 paths are Monte Carlo simulated, which in fact is too few when

accurate results are desirable. However, the valuations serve as an introduction to further studies

where Finite Difference methods could have been implemented for more accurate results.

62

4 Limitations

When choosing a mathematical model for valuation purposes there will always be a trade-off

between analytical tractability and goodness of fit to observations. This is important to have in mind

when considering the valuation results during this thesis. This section will examine some of the

simplifications done when modeling freight rates which create a gap between the valuation results

and reality.

4.1 Limitations Caused by the Models Selected

By choosing the Ornstein-Uhlenbeck process and the Geometric Mean Reversion process for

valuation purposes, important considerations going beyond the properties of the model will

automatically not be taken into account. Some of the considerations left behind when choosing

these two models will be discussed in this section.

4.1.1 The Parametric Property of the Models

Both the Ornstein-Uhlenbeck process and the Geometric Mean Reversion process are parametric in

nature; their distributions are assumed to belong to a parametric23 family (Jorion 2007), which in this

case is the normal distribution. In his article, Adland (2000) proposes a non-parametric model to

model the time charter equivalent spot freight rate by using monthly data over a period of ten years.

By doing this, misspecification of the density function’s shape of the time charter equivalent spot

freight rates is avoided, and estimations may appear more accurate compared to estimations based

on a misspecified model. By estimating the marginal density of the time charter equivalent spot

freight rate he finds that the distribution is slightly skewed to the right. This result does not

correspond to the assumption of the time charter equivalent spot freight rates being normally

distributed in both the Ornstein-Uhlenbeck process and in the Geometric Mean Reversion process.

Adland (2000)’s findings imply misspecification when assuming normally distributed time charter

equivalent spot freight rates. Subsequent to this, the results appearing in this thesis will need to be

considered with this possible misspecification in mind. But also Adland (2000)’s results have

shortcomings; a non-parametric model opens for greater estimation error than parametric models

do, in addition he stresses that too few observations are obtained in order to get a statistically

confident estimate.

23

To obtain parameter values in parametric models estimation procedures are necessary (Jorion 2007).

63

4.1.2 One-Factor

The model adopted to the valuation of time charter contracts with purchase options is a one-factor

model with an assumption of the spot freight rate being the factor influencing the value of a vessel.

This assumption will disregard all information that is not embedded in the current spot freight rate

level and the dynamics of the spot freight rate process observed in the past. Important fundamental

market information that is expected to influence future freight rate dynamics is thus not taken into

account at all (Adland and Strandenes 2007). A gap is then created between theory and reality due to

the existence of several factors influencing vessel values.

According to Alizadeh and Nomikos (2009), factors influencing vessel values can be classified into two

groups; vessel-specific factors and market-specific factors. Vessel-specific factors generate a more

intuitive understanding of why vessel values vary as those are related to the particulars and

condition of each vessel. Examples of such factors are size, type, age and general condition. Market-

specific factors relates to the general state of the freight market. Those factors are more complex

than the vessel-specific factors. Current and expected freight rate levels and market conditions are

among the most important market-specific factors.

Stopford (2009) argues that the vessel’s age is the second most important factor in relation to vessel

values. As also the vessel valuation results from both models indicate, a newer vessel is worth more

than an older vessel. As a vessel gains age, it may lose performance and also higher maintenance

costs may appear. Thus, this will lead to a lower vessel value in total, and when its market value falls

below the scrap value the vessel is likely to be sold for scrapping. Due to the importance of the

vessel’s age, a multifactor model, where both the freight rate and the vessel’s age are taken into

account, could have been considered when valuing vessels. This would have included more realism

to the model.

64

4.2 The Assumptions

This section will address some of the simplifying assumptions done through this thesis, and describe

how more reality would have been included if they were not taken.

4.2.1 Constant Market Price of Freight Rate Risk

Since the risk-free interest rate have been applied as the discount factor the true probability

measure, , had to be transformed to the risk-neutral probability measure, . To do this

transformation Girsanov’s Theorem was used, and the market price of freight rate risk, , was

introduced and integrated in the Ornstein-Uhlenbeck process. Further, the market price of interest

rate risk was assumed to be constant.

The market price of freight rate risk can be thought of as the difference between implied forward

freight rate24 and the expected spot freight rate (Adland 2003). Therefore, the value of the market

price of risk is closely related to the expectations of the term structure of freight rates in the future.

The assumption of this being constant has been disproved by Adland (2003). He presents qualitative

arguments and suggests that the market price of risk in the freight market in bulk shipping must be

time varying and depend on the state of the spot freight market and the duration of the time charter

in a systematic fashion. He argues for the existence of other factors than volatility in freight rates

that support the hypothesis of a non-constant market price of risk. Finally, he finds it reasonable to

consider the market price of risk as an increasing function of the spot freight rate level.

An earlier research also done by Adland (2000) supports his conclusions discussed above. This is a

quantitative based research which contribute with two separate results; that shipowners are not

compensated for the risk associated with trading in the spot freight market when freight rates are

low, and that the market price of risk is an increasing function of the freight rate level (Adland 2000).

The first result may appear due to the lack of considering other risk factors than the volatility of

freight rates when determining the market price of risk; when freight rates are low, the volatility of

freight rates is also low, and the risk associated with trading in the spot market may be considered as

small. The second result support his qualitative arguments discussed above that the market price of

risk is increasing in the freight rate level. This indicate that shipowners are compensated for bearing

freight rate risk when freight rates are at high levels; operating in the spot freight rate market is

more risky in this case since high freight rates implies high volatilities.

24

The freight rate evolvements in the future.

65

4.2.2 Constant Risk-Free Interest Rate

Throughout this thesis the risk-free interest rate is assumed to be constant. The risk-free interest rate

do in fact vary through time, thus this is another assumption made to simplify the calculations and

for the ability to apply the risk-neutral valuation result. To accommodate the shortcoming of

constant risk-free interest rate, one approach could be to model the time structure of the risk-free

interest rate outside the models. This could be done by using the Vasicek (1977) model which is

identical to the Ornstein-Uhlenbeck process. The estimated time structure could then be

implemented in the model as a simple variable.

66

5 Summary and Conclusions

The aim of this thesis was to shed light on the importance of fair valuation of embedded options in

time charter contracts, with options to purchase the vessel underlying the contract as the primary

area of research. As these options can be very complex in nature, they are often granted for free

rather than for their fair value. This creates misleading information of a shipping company’s total net

asset value as the embedded options are of highly economic significance to the company. Also, large

volatility in freight rates can lead to large decreases or increases in the option values, which in turn

can affect the shipping company’s viability. Thus, important to stress is also the proper management

of the risks a shipping company faces. Also, this thesis aimed to introduce two possible models for

the valuations where each of them was examined and their properties were compared. This gave an

indication of one model being more appropriate in freight rate modeling than the other.

First, the comprehensive shipping industry was introduced. The shipping market model that

describes the mechanisms controlling the shipping market cycles shortly described in Section 2.8.1,

was examined. Further, various costs and risks occurring in this industry was described, as well as a

description of the four existing markets. With the theories of the shipping industry in hand,

embedded options in time charter contracts were examined before a comprehensive description of

the two models could be established.

The first model introduced for valuation purposes was the Ornstein-Uhlenbeck process which has

been applied for valuation of time charter contracts with embedded purchase options. The solution

to this process was derived before the derivation of the partial differential equation that all freight

rate dependent claims must satisfy in order to be priced arbitrage free. Further, the partial

differential equation enabled for derivations of analytic solutions to simple freight rate contingent

claims - equal to the approach seen in Jørgensen and Giovanni (2010). The analytic solutions were

then applied to valuations of simple freight rate contingent claims, as well as vessel valuation,

European options to buy the vessel and finally, the total fair value of the time charter contract with

purchase options.

The second model introduced for valuation purposes was the Geometric Mean Reversion process. To

the author’s knowledge no analytic solutions exist and numerical routines were therefore needed for

valuations. Vessel values and European options to purchase a vessel were valued by the application

of Monte Carlo simulation. The Geometric Mean Reversion process was introduced as an alternative

and more realistic valuation model. However, for valuation of time charter contracts with purchase

options, implementation of Finite Difference methods would have been necessary. This may be an

interesting area of further research.

67

The Ornstein-Uhlenbeck process gave analytic solutions such that time charter contracts with

purchase options for different fixed time charter rates and spot freight rates could be valued. The

results imply that the value of the time charter contract increase in the spot freight rate level, but

decrease in the fixed time charter rate level. This was also expected as higher spot freight rates imply

higher cash inflow and thus an increase in the value of the option to purchase the vessel. Whereas

higher fixed time charter rates creates a larger gap from the long-term mean of 14 371, and thus also

increasing the gap between the fixed time charter rate that would have been fair from the agreed

fixed time charter rate.

The Geometric Mean Reversion process facilitated for the use of Monte Carlo simulation. Vessel

values and European option values for purchasing the vessel were obtained. However, accuracy in

the results was reduced at the expense of an illustration of how Monte Carlo on Monte Carlo

simulation can be applied to obtain option values when analytic solutions do not exist. A larger

amount25 of simulated paths would have been beneficial in relation to more accurate results, but this

would have been very time consuming due to the shortcomings of Microsoft Office Excel.

The intention behind the introduction of both models was the opportunity for comparisons – both

the model characteristics and the valuation results. The comparison of the valuation results

highlighted the weakness when lack of estimation is present. In order to obtain correct comparisons

between the valuation results obtained from both models, the parameter values need to match

relatively to each other. This could have been ensured by using the exact same dataset when

conducting the estimation procedure to both models, and at least the ambiguous vessel values

obtained could have been avoided.

The introduction of two models opened for an examination of model characteristics and a thoroughly

review of which model that would have been best fitted to freight rate modeling. The discussion

points in the direction of the Geometric Mean Reversion process being most appropriate, which is

due to the fact that the process is downwards restricted with zero as an absorbing level. With the

impossibility of freight rates being negative, this property becomes valuable compared to the

Ornstein-Uhlenbeck process which is not downwards restricted. Thus, this thesis ends by concluding

that the Geometric Mean Reversion process is more realistic in freight rate modeling compared to

the Ornstein-Uhlenbeck process.

25

A suggestion of 20 000 to 30 000 simulated paths would have been beneficial.

68

6 List of References

Adland, R. and K. Cullinane (2006). "The Non-Linear Dynamics of Spot Freight Rates in Tanker Markets." Transportation Research: Part E 42(3): 211-224.

Adland, R. and S. Koekebakker (2007). "Ship Valuation Using Cross-Sectional Sales Data: A Multivariate Non-Parametric Approach." Palgrave Macmillan Journals: 105-118.

Adland, R. and S. P. Strandenes (2007). "A Discrete-Time Stochastic Partial Equilibrium Model of the Spot Freight Market." Journal of Transport Economics & Policy 41(2): 189-218.

Adland, R. O. (2000). "A Non-Parametric Model of the Timecharter-Equivalent Spot Freight Rate in the Very Large Crude Oil Carrier Market." Foundation for Research in Economics and Business Administration.

Adland, R. O. (2003). The Stochastic Behavior of Spot Freight Rates and the Risk Premium in Bulk Shipping. The Department of Ocean Engineering, Massachusetts Institute of Technology. Ph.D in Ocean Systems Management

Alizadeh, A. H. and N. K. Nomikos (2009). Shipping Derivatives and Risk Management. Faculty of Finance, Cass Business School, City University, London, Pargrave Macmillan.

Bjerksund, P. and S. Ekern (1995). Contingent Claims Evaluation of Mean-Reverting Cash Flows in Shipping. Real Options in Capital Investment: Models, Strategies, and Applications. L. Trigeorgis. London, Preager: 207-219.

Campbell, J. Y., et al. (1997). The Econometrics of Financial Markets. Princeton, New Jersey 08540, Princeton University Press.

Duffie, D. (2001). Dynamic Asset Pricing Theory. Princeton and Oxford, Princeton University Press.

Hammer, H., et al. (2011). Valuing Time Charter Contracts with Purchase and Extension Options. Bergen, Norges Handelshøyskole.

Hull, J. C. (2012). Options, Futures, And Other Derivatives, Pearson Education Limited.

Jorion, P. (2007). Value at Risk - The New Benchmark for Managing Financial Risk, The McGraw-Hill Companies.

Jørgensen, P. L. and D. D. Giovanni (2010). "Time Charters with Purchase Options in Shipping: Valuation and Risk Management." Applied Mathematical Finance 17(5): 399-430.

69

Kavussanos, M. G. and I. D. Visvikis (2006). Derivatives and Risk Management in Shipping, Witherby Shipping Business.

Koekebakker, S., et al. (2006). "Are Spot Freight Rates Stationary?" Journal of Transport Economics & Policy 40(3): 449-472.

Koekebakker, S., et al. (2007). "Pricing freight rate options." Transportation Research: Part E 43(5): 535-548.

Skovmand, D. (2012). Supplementary Notes on: Linear Algebra, Probability and Statistics for Empirical Finance.

Stopford, M. (2009). Maritime Economics, Routledge, Taylor & Francis Group, London and New York.

Tvedt, J. (1997). "Valuation of VLCCs Under Income Uncertainty." Maritime Policy & Management: 159-174.

Vasicek, O. A. (1977). "An Equilibrium Characterization of the Term Structure." Journal of Financial & Quantitative Analysis 12(4): 627-627.

70

7 Appendix

The Excel spreadsheets where the tables and figures have been obtained are saved in a USB stick

enclosed to this paper.

7.1 The Ornstein-Uhlenbeck Process – Detailed Solution

The freight rate at time is given by today’s freight rate plus the sum of the dynamics of the

spot freight rate evolving from time to time :

A temporary variable is introduced in order to solve the equation above explicitly:

The dynamics of this temporary variable are given by Ito’s lemma:

The Ornstein-Uhlenbeck process is substituted for above:

71

Manipulations of the equation above give:

Continuing by integrating from time zero to time :

Calculating

:

Using this result in the continuation:

72

Calculating

:

Finally, the solution to the Ornstein-Uhlenbeck process becomes:

7.2 The Ornstein-Uhlenbeck process - Derivation of the Mean and the

Variance

7.2.1 The Time Conditional Mean

Taking the expectation of each term in the solution to the Ornstein-Uhlenbeck process:

Since the expectation of a Wiener process equals zero the mean of the process becomes:

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7.2.2 The Time Conditional Variance

Taking the variance of each term in the solution to the Ornstein-Uhlenbeck process:

Since the variance of the constants in the two first terms equals zero we are left with the following

expression:

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7.3 The Geometric Mean Reversion Process – Detailed Solution

To simplify later calculations:

A temporary variable is introduced in order to solve the equation above explicitly:

The increments of this temporary variable are given by Ito’s lemma:

Rearranging:

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Now, it is desirable to substitute the process for . In order to ease the notifications

and are defined such that . This gives:

where the first and the second term tend to zero, and where the last term tends to such that:

Then, by rearranging the manipulated Geometric Mean Reversion process to

and substituting the process for

:

The next step is to rearrange this equation and integrating it from time zero to time :

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What is left now is to rearrange this equation such that is alone:

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Finally, the freight rate level at time , , can be expressed as:

7.4 The Ornstein-Uhlenbeck Process – Claim to Receive Spot Freight Rate

Flow from Time to Time

Defining the current value of receiving the spot freight rate on a continuous basis as:

The freight rate process, :

Inserting in the expression for :

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Since the expectation of a standard Wiener process equals zero (Hull 2012), the last term equals zero

and only the two first terms are left. Further:

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where

is an annuity factor.

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7.5 The Ornstein-Uhlenbeck Process - European Option to Buy the Vessel

Define

and the value of a vessel is then given by:

The next step in the derivation of an analytic formula for the option value is to evaluate

+. Again, by following Jørgensen and Giovanni (2010) consider:

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where is the density function that describes freight rates (Skovmand 2012). The density

function plots the shape of the distribution curve of the random variable that is considered, which is

the freight rate in this case. Since the freight rates are assumed to be normally distributed by their

density function plots the classic “Bell Curve” which takes the form (Skovmand 2012):

which in this case will look like this (Jørgensen and Giovanni 2010):

Continuing by inserting the expression for the density function gives:

Further, define

. Then:

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Finally, the time t value of the European call option is given by premultiplying the above result with

times the discount factor:

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where:

and where and denote the standard normal cumulative probability and density functions,

respectively.

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7.6 The VBA Codes

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