Exotic Options – Products and Applications

MA598 Project 1

Harri Donaie, Samuel Hughes-Narborough, Eric Kelie, Trizer Nankunda, George

Thomas 2015/16

Supervisor: Pradip Tapadar

ekdk2@kent.ac.uk

gt210@kent.ac.uk

hmd8@kent.ac.uk

srh38@kent.ac.uk

tn88@kent.ac.uk

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CONTENTSChapter 1 - Introduction.........................................................................................................................2

A Brief History of Derivative Markets..............................................................................................2

General Features................................................................................................................................2

Why Use Derivatives..........................................................................................................................2

Chapter 2 – Forward Contracts and Vanilla Options .........................................................4

Forward Contracts.............................................................................................................................4

Call Options.........................................................................................................................................5

Put Options.........................................................................................................................................8

Chapter 3 – Exotic Options ........................................................................................................11

Lookback Option..............................................................................................................................11

Compound Option............................................................................................................................14

Binary Options.................................................................................................................................15

Asset-Or-Nothing.........................................................................................................................16

Cash-Or-Nothing..........................................................................................................................16

Touch option................................................................................................................................17

Barrier Options................................................................................................................................18

Mountain Range Options.................................................................................................................19

Atlas..............................................................................................................................................20

Himalayan.....................................................................................................................................21

Summary......................................................................................................................................23

Bibliography ......................................................................................................................................25

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CHAPTER 1 - INTRODUCTION

A BRIEF HISTORY OF DERIVATIVE MARKETSIn 1851 the Chicago Board of Trade (CBOT) offered the first recorded forward contract (a contract to buy or sell an asset in the future at a set price) and followed this in 1865 by developing standardised agreements, known as ‘futures’. At this time the CBOT also began the first ‘clearing’1 operation by requiring a ‘margin’ to be posted by buyers and sellers in the grain markets and in 1926 formed the Board of Trade Clearing Corporation to guarantee its trades.. Throughout the rest of the 1900’s the CBOT and its newer rivals expanded their operations, launching more contract options in a variety of commodities, and by 1999 you could obtain from the exchanges derivative contracts in silver, foreign currency, stock indexes, agricultural goods and even the weather. [1] Derivatives are now a large part of trading in markets across the world and investors are continually developing new types of derivative contract.

GENERAL FEATURESDerivatives are a category of security where the price of the security is related to an underlying asset or assets. They act as a contract between two (or more) investors and the value of the derivative will depend on price movements in the underlying asset. [2] For each derivative there will be a long party and a short party, the long party is the buyer of the derivative (and the ‘holder’) and the short party is the seller of the derivative (or ‘writer’). Derivatives usually have an expiration date, a time in the future at which the underlying asset is bought and sold; a strike price (K), the price at which the derivative will be traded for at the expiration date; and premium, the price paid upfront by the long party to the short party. There exist derivatives without expiration date or strike price and in some instances, such as with forward contracts, it is possible for the premium to be 0. Derivatives will have contract specifications and there will be a subsection in the contract defining specifically what is being traded for within the contract, so as not to cause any ambiguity upon expiry. It also specifies the strike price, or the price to be paid to the recipient upon maturity. This can vary depending upon the grade or quality of the commodity being traded in certain cases - Hull [3] gives the example of corn bushels. Additionally the contract will specify whether it is of an American or European type, with the former able to be exercised at any point until the end date of the contract, and the latter only exercisable upon the final maturity point.

In the following chapters we will look at a variety of derivatives; explaining how they work and looking at the benefits and drawbacks to an investor.

WHY USE DERIVATIVESInvestors use derivatives for three main reasons, hedging, speculating and arbitraging. The use of hedging can be thought of as the primary use of derivatives, and the main benefit of hedging is reducing future uncertainty by protecting against losses. If an investor knows that at time T they will be required to buy or sell an asset they can purchase a derivative contract to take the opposite side of what they would like to happen. Imagine a derivative with desired expiration date T and strike price K. The investor can pay a premium p to the party taking the short side of the derivative. Now the investor holds the long side of the derivative and is protected against any adverse price movements between now and the expiration date. Hedging gives the benefit of knowing cash flows will be K in the future but at the expense of paying p right now.

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When an investor has an opinion of future price movements they can speculate on their opinions to profit if they are correct. While hedging is for the risk averse by removing uncertainty and protecting against adverse price movements, speculation is considered risk taking and aims to profit from price movements. An investor who is looking to speculate would search for assets that they believe are incorrectly priced or that will change in price before the derivative expiry. The speculator takes the appropriate position on a derivative so that they may profit if the asset changes in price. An investor in a short position will profit if the asset price moves by less than p against their short position, while the long party profits if the market moves by more than p for the long position. Taking the short side of a derivative carries a larger risk, with the possibility of unlimited losses.

The third main use of derivatives is arbitraging. Arbitrageurs are a type of investor who attempts to profit from price inefficiencies in the market by making simultaneous trades that offset each other, capturing risk-free profits. [4] They exploit inefficiencies in markets that create discrepancies in pricing. If it is possible to both sell a derivative for p and buy it for p + a, the arbitrageur takes both positions and profits p + a – p = a. The trades are offset and the arbitrageur has gained risk free profit. Arbitrageurs ensure that the law of one price (which states that a good must sell for the same price in all locations) holds as their trading makes prices quickly conform to the law.

The long derivative positions have a maximum guaranteed outgoing of p but a maximum potential income of the difference in K and the asset price at t=T. The short derivative position is the inverse of this.

In the following chapters a variety of vanilla and exotic options will be covered. For ease of understanding examples will use Company XYZ who will have shares trading at 100p/share on 01/01/2016, assuming a risk free interest rate r.

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CHAPTER 2 – FORWARD CONTRACTS AND VANILLA OPTIONS

FORWARD CONTRACTSForward contracts are one of many financial derivative instruments to exist on the modern market . They are an agreement between two parties to buy/sell an asset at a specified date in the future at a specified price.

Estimating demand for crops is usually difficult and more often than not supply exceeds demand. Markets like the CBOT were created in the 19 th century, giving farmers the opportunity to bring their commodities and sell them at either spot (current) price or at a forward price (an agreed price for future delivery). “[F]orward contracts […] were the forerunners to today's futures contracts”. The introduction of forward contracts reduced the likelihood that farmers would lose crops and profits, helping to “stabili[s]e supply and prices in the off-season”. [5]

Forward contracts are classed as over-the-counter instruments (OTCs), as they are non- standardised and traded in decentralised markets. They are private agreements between counterparties which can be traded anywhere. Agreements can be made in informal environments, and the terms and conditions in are flexible and subject to negotiation.

The underlying assets in forwards are often commodities, including, but not limited to oil, precious metals, grain or cotton. Over the years the uses of forward contracts have evolved to include financial instruments such as foreign currency or shares.

As forward contracts are OTC, the details of contracts are private; and due to a lack of transparency the size of the forward market is unknown. In any market, information deficiencies can cause issues for market participants. [6] One possible issue is default risk arising from the absence of a formal clearing house. [5] Taking into account the "too big to fail" 1

ideology, the likelihood of systematic default has become more of an issue. Another drawback to having a lack of transparency is an inability to gauge demand in the market for assets, which could result in inefficient pricing of contracts.

Forward contracts can be deliverable or non-deliverable. In a deliverable contract, the underlying asset moves from the short to the long party, in exchange for the specified payment. Non-deliverable contracts do not see their underlying assets move however, only money is exchanged. They have two main elements, the fixing2 and settlement dates3.

The principal use of forward contracts is to minimise risk and hedge against unfavourable price movements for both the buyer and seller. It allows both long and short parties to ‘lock in’ a price that they would be happy to trade for the underlying asset, protecting the seller from potential price drops, whilst protecting the buyer from increases in price. As entering into a forward

1The idea that a business has become so large and ingrained in the economy that a government will provide assistance to prevent its

failure. "Too big to fail" describes the belief that if an enormous company fails, it will have a disastrous ripple effect throughout the economy. [41]2

Fixing date is the day and time whereby the comparison between the NDF rate and the prevailing spot rate is made. [42]3 Settlement date (or delivery date) is the day when the difference is paid or received. It is usually one or two business days after the fixing date

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contract locks in a price, losses for the long party are limited however for the short party, the loss can be unlimited as here the payoff is ◻ − ◻◻ and ◻◻ can be large. [7]

Forward contracts are also used by speculators to make profits. If a speculator believes an asset is mispriced, and notices an extra positive or negative return that they believe can be exploited they will enter into a forward contract. [8]

Another use of forward contract is covered interest arbitrage, a trading strategy allowing investors to capitalise on interest rate differentials4 between two countries by using a forward contract to eliminate exposure to exchange rate risk [9]5.

Using our example scenario, a specification for a forward contract might look similar to the following:

In this situation, the investor already holds a portfolio of XYZ stocks and intends to sell them on July 1. They would, however like to protect from a fall in the price of XYZ over the period. To ensure this, they write a forward contract to sell the share in 6 months, at the current price ◻ = 100p. At the end of the period, if the price has fallen below 100p, the investor will have realised a net gain, and the inverse if the price has risen.

CALL OPTIONSA call option is a contract where the holder of the option has the right, but not the obligation, to buy a specified asset from the writer at a specified strike price ◻, at or before a specified future date ◻, depending on whether it is of European or American style.

The writer therefore has an obligation to sell the shares to the holder of the option depending on whether the holder chooses to exercise the option. For this commitment, the holder of the call option is required to make an up-front payment known as the option price, denoted by ◻ (◻ in the case of American options), to the writer. However, the writer of the option also has to pay a margin requirement to the exchange as collateral for their obligation to sell the underlying asset.

Call options are said to be in-the-money when ◻ < ◻◻, while if ◻ > ◻◻, then the option is not profitable and is out-of-the-money. The intrinsic value of a call option is max(◻◻ − ◻, 0), the difference between the stock price and the strike price, where this is positive, and 0 otherwise.If buying the stock at the strike price at time T is not profitable, then the holder has the right not to exercise the option and it will expire unused. However, this is the maximum loss the option holder can face (option premium) and theoretically, the gain from the option is unlimited.

4 The differential that measures the gap in interest rates between two similar interest-bearing assets5 This scenario does not happen often as investors will jump at any opportunity to make risk-free profit. As a result of this unbalanced demand, market forces will restore natural equilibrium. [42]

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If we consider a European call option, then the payoff for an investor holding the long position is max(◻◻ − ◻, 0). This represents the fact that the option is exercised because the price of the asset is greater than the strike price (◻◻ > ◻). Conversely, the payoff for the writer, who holdsthe short position of the call option is [− max(◻◻ − ◻, 0)]. This can be seen graphically in Figure1.

Figure 1

Figure 2

It can be seen that, in terms of payoff, the writer of the contract must take the unfavourable position of seeing either no payoff or a loss. As a result, the writer receives an option premium upon inception of the contract in order to compensate for the potential loss. This premium is paid by the holder of the contract, and taking this into account, we see in the profit diagram in

Figure 2. We see that the profit for the holder of the call is max(◻◻ − ◻, 0) − ◻, and for the writer

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the profit is − max(◻◻ − ◻, 0) + ◻. We formalise the payoff and profit equations for the holder by max(◻◻ − ◻, 0) and max(◻◻ − ◻, 0) − ◻, respectively, while the same equations for the writer are calculated by taking their respective negative of the holder’s equations, as they arecounterparties in a closed system.

Black-Scholes pricing equations have been a standard pricing model for European options on an extensive variety of underlying assets and we use the concept of risk neutrality to derive these equations. There are several assumptions made during the analysis of Black Scholes which are omitted here for brevity, but can be found in Hull chapter 15. [3].

Call options offer unlimited profit and limited risk opportunities to investors, hence they are very attractive financial instruments. Large profits are made when the price of the stock rises above the strike price, at which the holder has the right to buy. The holder can then sell this stock at the market price and make a profit. Conversely, if the stock price falls below the market price, then the maximum loss incurred by the holder is capped at the option premium paid for the option. Investors also use call options to hedge trading positions. A covered call is an example of a hedge position where an option provides some compensation for the risk that the price of the underlying asset will decline. [11] With this strategy, an investor writes a call option against shares of stock they already own and in exchange receive an option premium. Investors write options that are far out-of-the money as their aim is to generate income from the option premium as well as reduce the chances of their stock getting called away. However, this strategy only provides a limited hedge which covers the investors only down to the breakeven price [stock price – call price].

Speculators find call options very appealing too as they can take advantage of how they think the stock is going to move. If they predict a rise in the stock price, they can buy call options and hence large profits are made and if they predict a fall in the stock price, then they can apply the covered call strategy and still gain income from the option premium. In the case of arbitrageurs, an arbitrage opportunity exists if there is divergence between the value of calls and puts with the same strike price and expiration date hence they would step in and make risk free trades.

Returning to our example, Company XYZ, an investor who believes that XYZ’s share price,◻◻ = 100p, is going to rise sharply due to the company’s improved performance pays 3p to purchase a call option contract, of length 3 months, and with a strike price K=100p.

If the price of the shares of XYZ rises to 110p per share on the maturity date as anticipated by the shareholder, then ◻◻ > ◻, and the option is in-the-money. The investor will then exercise the option and invoke their right to buy 1 share of XYZ stock at 100p and sell it immediately on the open market for 110p. The investor then makes a profit of 10p (1 share × (110p current price – 100p strike price) and a net profit of 7p (10p profit − 3p option premium).

However, if the stock price fell below the strike price, say to 85p, then ◻◻ < ◻and the option is out-of-the-money. The investor would rather let the call option expire worthless and lose 3p, while the writer will still gain the option premium of 3p.

Given the above, the contract specification for the call option on Company XYZ’s shares would likely include the following:

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PUT OPTIONSA put option is a contract that gives the holder, or the long party, the right (but not the obligation) to sell a specific amount of an underlying asset at some point in the future, for a upfront price. The asset is sold to the writer, or short party, who would then be obligated to purchase the asset. [12]

As the holder will only exercise his option should it be beneficial for them, the writer is paid an upfront fee to compensate for the risk of loss they are taking, known as an option premium (denoted by ◻ for European options, and ◻ for American). The holder will exercise their option where ◻ − ◻◻ > 0. In this situation, the put option is called in-the-money, as the holder can selltheir holding at a price higher than the current market price, and realise a profit of ◻ − ◻◻ − ◻.Alternatively, if ◻ − ◻◻ < 0 the option is called out-of-the-money. Additionally, if ◻ − ◻◻ = 0, theoption is called at-the-money, and has no intrinsic value. The option expires without being exercised if it is out-of-the-money, as it would result in a loss for the long party, while an at-the- money or on-the-money option would result in a loss for the holder of premium fee ◻.

Suppose an investor wanted to sell their single share in Company XYZ in 3 months, but is concerned that the value of the stock may fall significantly in the period. They take the long position in a put contract for premium fee ◻. This allows the investor to sell the stock at the strike price K to the counterparty even if the share price falls below ◻◻ . Should this occur i.e.◻ > ◻◻ and the investor will exercise their put option, and sell their holding for K, effectivelyreceiving a payoff of ◻ − ◻◻. After the sale, the investor has realised a return of ◻ − ◻◻ − ◻, while conversely, the short party makes a return of ◻ + ◻◻ − ◻.

We formalise these situations by defining the payoff and profit equations for the holder as max(◻ − ◻◻, 0) and max(◻ − ◻◻, 0) − ◻, respectively. As the writer is the counterparty in this closed system, their respective equations are simply the negatives of those for the holder. Wedisplay these positions graphically in Figures 3 and 4.

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Figure 3 – Payoff diagram without premium fee p

Figure 4 – Profit diagram with premium fee p

For the short party, purchasing the asset when the option is in-the-money results in a net loss of◻ + ◻◻ − ◻. If ◻ is over or under-priced there exists an arbitrage opportunity. For example consider if ◻ is under-priced, this can be exploited by the investor by purchasing a put option and the stock simultaneously, financed by borrowing ◻ + ◻◻ at interest rate ◻. At the expiration date, the arbitrageur must pay back the loan. If ◻◻ < ◻ the investor sells the stock through their put option. However if ◻◻ > ◻ they sell their stock directly in the market at ◻◻, both creating arisk-free profit. A similar process can be applied to call options, where the holder will buy the call option then short sell the stock. The ‘principle of no arbitrage’ states simply that such opportunities do not exist, as such a mispricing would be taken advantage of immediately, and thus market forces would lead to the price returning to a fair level.

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The most obvious use of a put is as a type of insurance. An example of this is the ‘protective put strategy’, a risk-management strategy that investors can use to guard against the loss of unrealised gains. The put option acts like an insurance policy, it costs money, which reduces the investor's potential gains from owning the security, but it also reduces their risk of losing money if the security declines in value. For example, if an investor purchased a stock in Company XYZ for £1 that is now worth £2 but they have not yet sold it, they have unrealised gains of £1. If they do not want to sell the stock yet (perhaps because they think it will appreciate further) but they want to make sure they do not lose the £1 in unrealised gains, they can purchase a put option for strike price ◻ = £2 for that same stock that will protect them for as long as the option contract is in force. If the stock continues to increase in price, say, going up to £3, the investor can benefit from the increase. If the stock declines from £2 to £1.50 or even£0.01, the investor is able to limit their losses because of the protective put.

Another benefit of purchasing a put (or call) contract is that once it is purchased the buyer then has the right, but not the obligation to sell the contract onto another investor, note this is selling the contract and not the stock. These contracts can be traded on what is called an ‘options exchange’.

From our example, the contract specification for a put option on Company XYZ’s shares would likely include the following:

Strike Price £2Premium £0.10Inception Date January 1, 2016Expiration Date April 1, 2016Option Style EuropeanUnderlying Asset

Ordinary shares in Company XYZ

Contract Multiplier

1 share

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CHAPTER 3 – EXOTIC OPTIONS

LOOKBACK OPTIONA lookback option is a European option, usually traded on stocks, for which the strike price depends on the price of the asset over the lifetime of the option. There exist two types of lookbacks, floating and fixed, named as such for the calculation of their respective strike prices. [13]

A floating strike price lookback option differs from a vanilla option in that it has no predefined strike price; instead it is decided at maturity to be equal to the optimal value of the asset since the start date of the option. In the case of a floating lookback call, the strike price is the minimum price, while it is the maximum price achieved over the period for floating lookback put options.

The payoff of the floating lookback options is therefore max(0, ◻◻ − ◻◻◻◻) for call, and max(0, ◻◻◻◻ − ◻◻) for put options, where ◻◻ is the price of the asset at maturity, ◻◻◻◻ is the lowest price achieved over the period [0, ◻], and ◻◻◻◻ is the maximum price over the same period [0, ◻]. Note that if ◻ = 0, ◻◻◻◻ = ◻◻◻◻ = ◻◻.

Thus, the profit functions are max(0, ◻◻◻◻ − ◻◻) − ◻◻◻ for floating puts and max(0, ◻◻ − ◻◻◻◻) −◻◻◻ for floating calls, where ◻◻◻ and ◻◻◻ denote the price for a floating put and call lookback option, respectively, at time ç = 0.

It should be noted here, that in the absolute worst case for an investor holding a floating lookback option, upon maturity the asset price ◻◻ would be the lowest (highest) price historically for a call (put). In both of these cases, the payoff would be zero, and the option would be at-the-money. We therefore see that floating lookbacks are only ever at or in-the- money. As a result, they will always be exercised by the holder, and so is an option by name only, not by nature.

While the payoff and profit functions are relatively straightforward, the values of these options are considerably more complex to calculate. Hull section 26.11 gives the formulae used in pricing these options [14].

The other type of lookback option, the fixed strike price lookback, differs from the floating option in that it has a fixed strike price, and the value of the asset at time ◻ becomes the variable that “looks back” at the historic prices.

As opposed to being set upon maturity, the strike price, ◻, is specified at the inception of the contract. In this case, the option acts more closely to a standard vanilla option, with the exception that upon maturity, the current asset price is replaced by the optimum price of the asset over the time period (the maximum for calls, and minimum for puts). As a result, the payoff functions for a fixed strike price lookback call and put option are max(◻◻◻◻ − ◻, 0) andmax(◻ − ◻◻◻◻, 0), respectively. The pricing formulae are again available in Hull section 26.11[14]. We denote the price of a European style fixed strike price lookback put option at timeç = 0 by ◻◻◻◻ , and ◻◻◻◻ for call options.

Share price of Company XYZ125

12011511010510095908580

Smax

K

0

Smin

T

Time, t

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As a result, the profit functions for fixed strike lookback options are max(◻ − ◻◻◻◻, 0) − ◻◻◻◻ for puts and max(◻◻◻◻ − ◻, 0) − ◻◻◻◻ for calls.

Figure 5

Comparison between how the payoffs are calculated can be seen in Figure 5, with the green brackets denoting payoffs for floating puts (top) and calls (bottom), and blue showing fixed strike calls (top) and puts (bottom).

In a manner similar to how Bermudan options exist as a compromise of sorts between American and European options6, there exist variants of the lookback option, that exhibit features of both fixed and floating lookbacks, for example the limited period floating strike lookback option, wherein the lookback period is only active for the early part of the life of the option. These types of options typically cost less than the “full” lookback period options. There also exists a Russian option, which is, in essence, an American-style lookback option, with no pre-set expiry date. The holder will receive the maximum value of the asset since the option’s inception upon exercise, the date of which is at the holder’s discretion. Note that the flexibility and practically guaranteed returns available cause this option to have a very high premium. [15]

As these options allow the holder to exercise at the optimum price retrospectively at the end of the contract, they are extremely attractive to investors. Due to this, and the obvious potential downside to the short party, the price is very high compared to a vanilla put or call. The price of the options also depends greatly upon the frequency at which the price of the underlying asset is measured, with more frequent observations giving a better price for the long party, in terms of optimality, and as a result a higher price. The formulae used by Hull [3] assume continuously

6 Bermudan options may only be exercised on certain dates, usually at regular intervals, during the life of the contract. [52]

Sm

ax - K -

Sm

ax - S

T - Share

Price,

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observed asset prices, and can therefore be seen as an upper bound for discretely observed prices.

One way that floating strike lookback options are useful to investors is that they are able to speculate on whether they think an asset’s value will reach a great high/low during the life of the contract, and guaranteeing the optimal strike price to capitalise on said speculation. They can do this by making a choice between a put or call option, depending on which direction they expect the asset price to move. This logic could be easily applied to hedging also, in that the best possible price is obtained over the period. Fixed strike lookback options are popular to investors as they can again speculate that an asset’s price will move far beyond their chosen strike price, while protecting them from any loss at the same time.

Due to the high price of lookback options, they are likely to be less favourable to an investor looking to implement a hedging strategy. That is not to say, however, that lookback options could not be utilised in order to gain the best price for an asset, merely that the option premium may offset any relatively small gains, due to its cost. There have been academic papers published claiming to have designed a hedging strategy for these options, however the assumptions required for these to hold are, generally speaking, unrealistic for practical use.

The holder of a fixed strike lookback option does not have to decide upon the best time to exercise their option, as they would with an American vanilla option, in order to maximise profits. They do, however, need to determine a strike price at the inception of the contract. A floating option holder is solely relying on whether the final price of the underlying asset is vastly different from the optimal price achieved to determine their payoff in the European case. Nothing further needs to be decided in this case other than the length of the option. Regardless, they have paid the option premium and ‘locked in’ the best price available until maturity.

On the other hand, a writer of a lookback option will realise a profit if and only if the price of the underlying at maturity has moved by less than the premium of the contract – which, considering the high price of these options, is a possibility. In this case, the writer will see a gain, as theirpremium P at time T is worth ◻◻◻◻ (> ◻ as ◻◻◻ > 1 ∀◻, ◻ > 0), while they are obligated tobuy/sell an asset that has moved in value by some amount lower than P. This does come coupled with the chance of large losses in the event that the underlying asset moves unfavourably to them by more than P. [16]

Returning again to our example of Company XYZ, we take an investor who, looking to maximise profits over the period from t=0 to T by purchasing shares in XYZ, purchases a 3-month floating lookback call option for ◻◻◻ = £0.05. From Figure 5 above, showing XYZ’s share price over the period, we see that at time ç = ◻, ◻◻ = £1.05, ◻◻◻◻ = £0.84 and ◻◻◻◻ = £1.20. As ◻◻ > ◻◻◻◻, theoption is exercised, and the shares are purchased, resulting in a payoff of £0.21 (£1.05 finalprice – £0.84 minimum price). The investor thus yields a profit of £0.16 (£0.21 payoff – £0.05 option premium).

Assume alternately that the investor, with a holding of shares that they are looking to sell for the best price, purchases a fixed strike lookback put option for ◻◻◻◻ = £0.16, with strike price◻ = £1.10. Upon maturity, ◻◻ is replaced by ◻◻◻◻, which produces the optimal payoff for the investor of £0.26 (£1.10 strike price – 84p minimum price). The profit for the holder of the option therefore is £0.10 (£0.26 payoff – £0.16 premium).

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COMPOUND OPTIONA compound option is a vanilla option for which the underlying asset is another vanilla option. There are four types of compound options:

Call on a call (CoC) option – gives the right but not the obligation to buy a call option Call on a put (CoP) option – gives the right but not the obligation to buy a put option Put on a call (PoC) option – gives the right but not the obligation to sell a call option Put on a put (PoP) option – gives the right but not the obligation to sell a put option

Compound options have two strike prices (K*, K) and two expiration dates (t*, T). K*, t* are for the compound option Co and K, T for the underlying vanilla option Uo. The expiration date for Co

will be at a point before the expiration of Uo (T > t*). The premium paid for Co will be based on

K* and the volatilities of the Uo and the underlying asset. Exercising a compound call option requires making premium payment of K* and upon exercising the compound option the investor receives the Uo, taking a long position on the Uo. Exercising a put compound option results in the investor receiving K* in exchange for the Uo, taking a short position on the Uo. The payoffs for the long positions on compound options are:

Call on a call option: max{Cstd – K*, 0}

Call on a put option: max{Pstd – K*, 0}

Put on a call option: max{K* - Pstd, 0}

Put on a put option: max{K* - Cstd, 0}

Where Cstd and Pstd represent the payoff for the Uo at t*. The short positions payoffs aremax{◻◻◻◻◻ 0 , ◻◻◻◻◻◻◻◻◻ݑ }.

For more details on how a compound option is priced refer to Economics and Mathematics of Financial Markets - Jakša Cvitanić and Fernando Zapatero p. 245.

A key advantage to the long position for compound options when compared to vanilla options is that it allows for much more increased leverage; the practice of using borrowed funds to make investments. Increasing leverage increases potential gains and losses. As more options can be bought if the gain exceeds the cost of borrowing per option investors can make greater gains from leverage. If, however, the gain from the options is less than the borrowing cost, or a loss is made, losses will be greater. Compound options also have the appeal of a less expensive premium than that of vanilla options.

Long positions on compound options are generally used when the outcome of a future event is uncertain. In such instances it may be better to purchase a compound option (and thus risk less up front) that can be exercised or left to expire once the outcome is known. In this way the investor hedges by reducing potential losses and by waiting until there is greater certainty before making larger investments.

The drawback to the short position on a compound option is that the investor can potentially make losses of the difference of K* and Uo premium less Co premium, which is theoretically

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unlimited. Long position holders face making two premium payments that might cost more in total than buying the Uo at t=0 would have.

It is apparent that the reason for taking a short position on compound options is speculative. An investor believes that the price of options will go in one direction and by making a bet against it they profit the premium if the belief is correct. The fact that price fluctuations of a compound option are dependent on the volatility of an asset and a vanilla option of the asset suggests that speculators are speculating on the volatility of volatility.

Similarly to how vanilla options are themselves used for risk management, compound options can also be used to reduce risk. Taking our example of Company XYZ, if the share price has been especially volatile there will be a high level of uncertainty in the market. In such market conditions investors will be reluctant to pay the high market premiums associated with vanilla options, and may alternatively look to the cheaper compound options. A volatile and uncertain market could change suddenly and for investors it is better to lose only the smaller compound option premium.

A real world circumstance that could lead to this situation is if Company XYZ were bidding for a contract the outcome of which will be known at time t. If Company XYZ is successful investors expect the price to rise between t and T, and would prefer to buy calls or sell puts with expiration date of T to profit from the increase in XYZ share price. If Company XYZ fails in their bid the share price is expected to fall, and investors would want to buy puts and sell calls. As the outcome of the bid is unknown investors could buy CoC or PoP options with expiration dates t and T, strike price K* and K, if they think the bid will be successful (and CoP and PoC if they think it will be unsuccessful). Then at time t when the outcome is known exercise the compound option if profitable for a vanilla option that expires at T when the price change from the outcome is expected to have impacted the share price. In this way investors do not have to stake as much in up-front costs as they would have had they invested in vanilla options before the outcome was known, thus reducing their potential losses and risk.

BINARY OPTIONSBinary options are a relatively new way to trade price fluctuations within the financial markets. They offer an easy and efficient way for investors to gain exposure to assets and manage the associated risks. Binary options have two potential payoffs, a fixed amount of profit if the option expires in-the-money, or nothing if the option expires out-of-the-money. [17]

Traits found in both binary and traditional options are strike price (K), premium (P) and expiration date, however there are differences. One contrasting feature is that the premium (P) for a binary option is chosen by the investor and not determined by the market. Another difference is the expiration dates: binary options offer shorter timeframes than traditional vanilla options, which can last from weeks to years, whereas binaries usually have expiration intervals as short as a single minute and up to 24 hours. Other brokers may allow investors to choose the expiries that suit the investor’s trading strategy. [18]

When investors choose to trade binaries they attempt to predict the movement of the underlying asset within a given time frame. Ownership of assets does not change hands, instead cash transactions are made. An investor wishing to gain exposure to Company XYZ could

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purchase a binary option. In exchange for an upfront premium, the contract has a potential payoff which is known to the investor at inception.

To receive the payoffs associated with binary options the investor’s prediction on whether the price of the underlying will increase or decrease must be correct. Another distinguishing feature here is that the conditions that must be met to receive the payoff are more specific. Depending on what type of binary option a certain price must be met, exceeded or lie in a range. If the investor’s prediction is correct and all conditions are met the payoff will be received, the amout being a percentage of the initial premium7. If the investor makes an incorrect prediction they will receive nothing at all, which is similar to the payoff of vanilla option as this also depends on the price of the underlying asset relative to its strike price. [19]

Binary options have several variations to their structure, each of which has its own specific conditions and can be used by speculators or hedgers in different market conditions. Irrespective of what type of binary option a premium is always paid by the investor at the outset.

ASSET-OR-NOTHING - A derivative security where the investor doesn't receive any payoff unless the investor predicts the correct market movement for the underlying. The price of the underlying must exceed the strike price for an asset-or-nothing call option, or remain below the strike price for a put. The payoff is equal to the asset’s price (◻◻) at the time of expiry, meaning there is potential for high returns. For example an investor buys an asset or nothing call optionpriced at 100p at t=0. At maturity the asset price is 140p. Because the payoff from this option is equal to the asset’s price at maturity the investor receives 140p. If the investor used the 100p to buy Company XYZ common stock the return would have been 40p. [20]

CASH-OR-NOTHING - In this binary option the price of the underlying asset must reach

or exceed the strike price for a cash-or-nothing call, or fall below for a put. Providing the price of the underlying asset is above/below the strike price of the option contract at maturity for call and put options respectively the investor will receive a payoff. The payoff associated with this type of binary option is a predetermined amount, and will be a percentage of the initial premium.

Figure 6 shows a cash-or-nothing call option on XYZ stock, at time 0 (3pm) the price is 100p. The lifespan of the option is 1 hour, to receive the payoff the price of the underlying must be above 100p by maturity (4pm). At maturity XYZ's share price is 120p, hence the investor receives a payoff. The difference from vanilla products is that the payoff received from is a percentage of the premium rather than ◻◻ − ◻◻ (the payoff from common stock). As binaries

paya percentage of the premium there is potential for higher returns. Rather than receive 20p frombuying 100p worth of shares with a digital call option the investor can receive up to 100% profit.

7 The percentage of the initial premium received is dependent on the type of binary option and the

broker. In the case of an asset-or-nothing binary, the payoff is not a fixed amount but one unit of the asset.

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TOUCH OPTION - Touch options have two variations: no touch and one touch.No touch is used when an investor anticipates the price of the underlying will remain within a certain level without touching the upper and lower bounds. A one touch option is used when an investor believes the price of the underlying will touch or surpass the bounds. At time 0 levels are created above and below the current market price. If the price of the underlying does not touch either level before the contract expires the investor receives a % of the premium.

A one touch option with an duration of 1 hour on XYZ stock is shown in figure 7. At time 0 (11:45) the price of the underlying is 100p, the lower and upper bounds are 80p and 100p respectively. Profits are made if the market reaches the boundary level at any point from the beginning of the contract (t = 0) till expiry. We see that during the contract the price touches the upper bound, and the investor receives a payoff (up to 100%) of the initial premium. [21] [22]

The lack of regulation in the binary options market increases the risk to investors, leaving binary option trading prone to fraud. In the UK binary options are treated as a gambling product rather than a financial product, so binary option brokers are not regulated by the FCA8. [23]

Similarly to traditional vanilla options, elements considered when pricing binary options include the market price, strike price, term to maturity, current interest rate and yield. [24]

8 FCA - Financial Conduct Authority.

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BARRIER OPTIONSA barrier option is an exotic option whose payoff depends on whether the price of the underlying asset reaches a certain level known as the barrier, H, during a certain time period. The value of this option depends on the path of the price of the underlying asset during the options contract term. Due to this, it is considered a path dependent option. Barrier options are traded in OTC markets and can be specified as being monitored continuously or on a daily basis.

Barrier options take one of two forms: knock-out or knock-in. The feature that differentiates between these two forms is their ‘existence’ during the life of the contract. Knock-out options begin ‘active’, but become worthless if the asset price ◻◻ reaches the barrier H, while knock-in

options start worthless, but are ‘activated’ if ◻◻ reaches H. [3] Since barrier options are pathdependent, the name of the option depends on the path of the price of the underlying asset. An upward movement in the price of the underlying leads to an up-and-out and an up-and-in barrier option while a downward movement leads to a down-and-out and down-and-in barrier option.9

There are eight standard categories of barrier options which are summarised in Table 1.

Table 110

Up-and-out options cease to exist11 when the underlying asset price ◻◻ reaches ◻ > ◻◻ at time ç ∈ (0, ◻], while down-and-in options come into existence when the underlying asset price falls below the knock in price ◻ < ◻◻. For options that ‘exist’, the payoff is that of a vanilla European option, with expiry T and strike price K. Since the value of a regular call option is equal to thevalue of an up-and-out call ◻◻◻ plus an up-and-in call ◻◻◻, we have ◻◻◻ = ◻ − ◻◻◻ 12. For a down- and-in put option, the payoff is zero unless the asset crosses a pre-defined barrier ◻ < ◻◻ at some time ç. If the barrier is crossed then the payoff is that of a vanilla European put option. Thepricing formulae can be found in Hull pp. 581 [3].

Returning to our example, Company XYZ, we take an investor looking to purchase a down-and- in put option. The option has a strike price of 90p and a barrier of H=70p. If the price of the

9 An option with the suffix “-and-out” is a knock-out option, and “-and-in” a knock-in.10 “Below spot” refers to H being below the price of the underlying asset at time t=0, while “Above spot” implies ◻ > ◻◻11 An option that ceases to exist has a payoff of 0 with probability 1.12 This is true of other combinations of barrier options where the payoffs sum to that of a vanilla option

◻

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underlying falls below 70p before maturity, the put option is activated and the holder has the right to sell the stock at the strike price of 90p, for a return of 90p − ◻◻. On the other hand, if the stock doesn’t fall below H, then the put option isn’t activated and expires worthless.

Barrier options appeal to investors as they are less expensive than vanilla options due to the reduced payoff opportunities. Barrier options can also be useful to investors who want to hedge against the risk of prices rising. Take for example, a down-and-out call, with an investor who needs to buy a large quantity of, say, steel in three months’ time. They are comfortable with the current price of steel however they want to hedge against the price of steel rising. They do not expect the price to fall at any time in the near future. They do not want to hedge the purchase with a forward contract because, if they were wrong and the price dropped, they are still obliged to buy at the forward price. A vanilla call option would be a good alternative; however it is more expensive than barrier options. The investor would therefore prefer a down-and-out call because it is less expensive than a standard vanilla call option and yet it retains the upside protection. Hence barrier options provide a hedging opportunity at a lower premium than a standard option.

Barrier options are also attractive in that they provide additional flexibility, as investors are able to express their view on the price movement of an asset in the option’s contract. For speculators, barrier options are very profitable if future price movements of the underlying asset are correctly predicted. A speculator who anticipates significant price movements in the underlying asset is better off purchasing a knock in option and if he predicts a small price movement, he is better off purchasing a knock out option compared to its equivalent vanilla counterpart. If the speculator is correct, a barrier option will have a lower premium, yet carry the same payoff upon maturity.

Despite the fact that barrier options are attractive to some investors, they carry more risk compared to vanilla options. This is because before the option expires in-the-money, there are a number of conditions13 to be satisfied and hence the probability of loss is higher.

Due to the fact that barrier options are path dependent, valuation of these options is tricky compared to standard vanilla options. Barrier options are also prone to market risk as an option can become inactive due to a significant, yet unexpected, price movement. Also barrier options may not be attractive to investors due to the fact that they are traded only in OTC markets and some investors may not be keen to buy and sell contracts that are not traded on public exchanges due to default risk.

MOUNTAIN RANGE OPTIONSMountain range options are a family of exotic options based on multiple underlying assets. They were first created in 1998 by Société Générale, a French securities firm. This group of options combines the characteristics of both range and basket options, where a basket option’s underlying asset consists of several assets, rather than a single asset. The payoff of a range option, if the investor’s prediction is correct, is their initial investment plus the predetermined percentage of that amount. The return on a basket option depends on a linear combination of several underlying asset prices, each with a predetermined weighting. The payoffs of mountain range options are based upon the performances of the stocks at predetermined time periods.

13 The conditions include the path taken by ◻ and the barrier level ◻.

◻

◻ ◻ ◻

◻ ◻ ◻

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Because of this, they have sampling or recalculation dates, where the best or worst performing stocks from the basket are removed. Therefore, holders must constantly re-evaluate the parameters that affect the current value.

The price of a mountain range option is often difficult to quantify as it is based on multiple variables, in particular it depends on the correlation between the individual securities in the basket. The complexity of determining a fair market value for these options makes standard formulas, such as the Black-Scholes [25] method for vanilla options, nearly impossible to apply. Instead, they are usually priced using Monte Carlo simulation methods [26].

Being struck on several underlying assets, mountain range options are particularly relevant for investors who want to cover several positions with one derivative. Instead of monitoring multiple options written on individual assets, a mountain range option can be structured to achieve the same coverage. The advantage of this feature is that the combined volatility will be lower than the volatility of the individual assets. A lower volatility will result in a cheaper option price, which can significantly decrease the costs incurred by hedging. They are traded over-the- counter, typically by private banks and institutional investors such as hedge funds.

Mountain range options can be further subdivided into 5 types, depending on the contract specification terms. This section will discuss Atlas and Himalayan options, though Altiplano, Annapurna and Everest options are also common.

ATLASNamed after the Greek mythological titan Atlas, this is a call option based on the mean or average of a basket of stocks. At maturity date T, some of the best and worst performing stocks are removed from the group of underlying assets, at which point the payoff is calculated using the remainder of the stocks.

Such an option is typically only used by institutional investors, as it requires a high level of resources and tolerance to risk being a holder of what can be a very complex security. However, as time gets closer to the maturity date, the price and value of the Atlas option becomes more apparent as the best and worst stocks begin to separate themselves from the other stocks.

Let the basket contain ◻ stocks, and ◻◻ denote the value of stock ◻. Then we can calculate the ◻th

smallest return denoted by ◻◻ using

◻ ◻ ◻◻ ◻ ◻ ◻◻◻ = ◻◻◻ ◻◻◻ , ◻◻ , … , ◻◻◻

◻ ◻ ◻◻ ◻ ◻

◻ ◻ ◻ ◻◻◻ = ◻◻÷ ◻◻◻ , ◻◻ , … , ◻◻◻

◻ ◻ ◻◻ ◻ ◻ ◻

where ◻◻ ≤ ◻◻ ≤ ⋯ ≤ ◻◻ ≤ ⋯ ≤ ◻◻

The Atlas option will then remove a fixed number of stocks ◻◻ from the lowest performing assets in the basket and ◻◻ number of stocks from the best performing assets. Note that◻◻ + ◻◻ < ◻ in order to make sure there is at least one remaining stock in the basket to compute the payoff. Assuming strike price ◻ and maturity time ◻, the payoff is calculated by

◻

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1 ◻◻◻◻ ◻ − ◻, 0◻◻◻÷ ◻◻ − (◻ + ◻◻

◻ ◻◻◻◻◻◻◻◻

Atlas options can be advantageous to a holder as it can remove a stock which is doing extremely poorly and would otherwise reduce the payoff if using a simple vanilla option. However, a flaw of this option is that it also removes the best performing asset to provide the holder with a payoff of the average of the stock returns.

HIMALAYANHimalayan options have predetermined periodic measurement dates in the contract specification. On each date a payoff is made based on the best performing stock in the basket of assets. Immediately after this payoff, the security is removed from the basket and this process is repeated until there is one security remaining. The total payoff received by the holder will be the sum of all the periodic payments. Similarly to Atlas options, Himalayan options are very difficult to value as the content and volatility of the basket changes over time as securities are periodically removed at the dates specified in the contract. Because of this, they are usually only traded by institutional investors.

An investor may believe that a particular asset will perform well in the first time period, but then begin to decline as time reaches the maturity date. They may then use a Himalayan option to their advantage as they will sell the best performer at the end of the first period. For example, an investor believes Company XYZ will increase its share price from 100p to 110p within the first month due to high reported earnings. However, they are also aware that a directly competing organisation Company ABC are in the process of creating an innovative product that will outperform Company XYZ when it is released in two months’ time, thus reducing Company XYZ’s share price. Here the investor may then purchase a Himalayan option with expiry date in two months’ time and monthly periodic eliminations of the best performing asset at the end of each month. At the end of the first month, the holder’s predictions were correct and Company XYZ outperformed Company ABC by reaching a share price of 110p compared to Company ABC’s 90p. The holder now sells their share of Company XYZ and the option will now remove Company XYZ from the basket. At the end of the following month, Company ABC’s new product has dominated the market and increased ABC’s share price to 150p, whereas XYZ’s has fallen to 80p. However, since the share in XYZ has already been sold and removed from the basket, its performance in the final month is irrelevant. The option will now sell the share of ABC for 150p and the maturity date has been reached, thus ending the option.

In more general terms, a Himalayan option with notional amount ◻ (amount invested in each asset), and maturity date ◻ starts with a basket of ◻ assets. The contract specification will include the ◻ payoff times such that

ç◻ = 0 < ç◻ < ç◻ … < ç◻ = ◻

At payoff time ç◻ the rate of return of all assets currently in the basket are calculated, and the asset with the largest return is denoted by ◻◻◻ where 1 ≤ ◻◻ ≤ ◻. The payoff of this derivative is then computed using

◻◻ç◻◻◻◻◻ = ◻ ∗ Max ◻

◻◻◻,◻◻ −

◻◻◻,◻◻◻◻◻,◻◻

, 0◻

)

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Then ◻◻◻ is immediately removed from the basket. The procedure is repeated until maturity date◻, at which time the last payoff occurs and the basket is empty, thus completing the option.

Himalayan options can be complex to utilise as they involve the prediction of high volatility at specific time periods in order to become optimal. However, they are particularly useful since they constantly take the best performing asset, thus giving all stocks numerous occasions to outperform the others.

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SUMMARYForward contracts offer investors a level of security that cannot be obtained investing in real time. Investors can lock in prices, enabling the investor to either secure profits or hedge against unfavourable price movements and limit losses. Despite high levels of default risk and a lack of transparency in the forward market, forwards are a powerful tool for hedging portfolio risk or obtaining attractive returns from investment portfolios.

Call options are very appealing financial instruments to investors because they offer unlimited profits, as there is no limit to how high the stock price could rise above the strike price, and limited risk opportunities since the risk is limited to the option price paid at the outset. Call options can also be bought to insure against potential losses, say a rise in the price of the security for a short position holder in the underlying. Speculators also find call options very attractive because they can take advantage of how they think the price will move by buying call options in case they predict a significant rise in the stock price. As for arbitrageurs, risk free profits can be made when there is divergence between the value of the calls and puts at the same strike price and expiration date.

Put options are particular beneficial to investors as it limits their exposure, since the maximum the holder can lose is simply the premium fee they pay for the contract. However, the profit is also limited to the value of the underlying asset going to 0, unlike call options where the potential profit is theoretically unlimited. They can also be used as an insurance policy to speculators who wish to protect their unrealised gains.

Lookback options offer an investor the ability to maximise their return on a vanilla option over a certain period, as if they were able to alter the strike price, or inception and maturity dates. Speculators can therefore make use of these options if they think the underlying asset price will move greatly in a particular direction. As a result, these options are best suited to volatile stocks, or stocks in a market where sudden extreme movements can be expected due to recent information. In order to realise a profit, the asset price must move by a considerable degree, since the premium price is often very high due to the attractiveness of the option.

Compound options offer investors an opportunity to hedge against the outcome of future events. Because premiums are cheaper for compound options than vanilla options investors can risk less capital when choosing to invest in a compound option and thus stand to have reduced losses if events make it no longer viable to exercise for the vanilla option. Compound options also offer the opportunity to take advantage of leverage to make larger profits for speculators. The drawback of compound options is the potential of higher premium payments to acquire the underlying vanilla option – a trade-off for the reduction of risk.

Binary options are a modern financial instrument with many different structural variations which allow them to be easily traded. The duration of contracts are far shorter than for forward contracts and vanilla options and for each variation specific conditions must be met to receive payoffs. Binaries offer returns of up to 100% of the initial premium paid by investors. Although fairly risky as there is no way to recuperate any losses that have materialised, binaries can be traded by retail investors so are highly accessible, allowing speculators and hedgers alike to make returns.

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Barrier options are very popular financial instruments mainly because they are cheaper than their equivalent vanilla counterpart due to the limited payoff opportunities they offer. During an upward movement in the market, investors benefit from call barrier options and during a downward movement, hedgers can use put barrier options to mitigate risk. Despite the fact that they are very useful when investors want to buy an asset at a favourable price, barrier options are difficult to hedge due to the volatility of the market thus these options need to be monitored regularly.

Mountain Range options are complicated options involving several underlying assets at one time. They can be particularly advantageous to hedge funds as the combined volatility will be lower than the volatility on each individual asset, which in turn creates lower costs for large institutional firms. A flaw of these options is that they are too complex to price using standard pricing methods, hence do not appeal to lone investors.

These only make for a tiny proportion of the exotic derivative option contracts available on the market. With academia and businesses always exploring innovative ways to generate large payoffs, the number of options available are ever increasing. Hull [3] gives information on a wider range of options than those discussed here, however there exists no comprehensive list of options available.

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BIBLIOGRAPHY[1] CME Group. Accessed (2016, March) Timeline of CME Achievements. [Online].

http://www.cmegroup.com/company/history/timeline-of-achievements.html

[2] Investopedia. Accessed (2016, March) Derivative - What is a 'Derivative'. [Online]. http://www.investopedia.com/terms/d/derivative.asp

[3] J Hull, "Options, Futures, and other Derivatives," in Options, Futures, and other Derivatives.

New Jersey: Pearson, 2015, pp. 218-223.

[4] Investopedia. Accessed (2016, March) Arbitrageur - DEFINITION of Arbitrageur. [Online]. http://www.investopedia.com/terms/a/arbitrageur.asp

[5] Investing Answers. Accessed (n.d.) Forward Contract. [Online]. http://www.investinganswers.com/financial-dictionary/options-derivatives/forward- contract-4892

[6] Investopedia. Accessed (n.d.) Futures Fundamentals: A Brief History. [Online]. http://www.investopedia.com/university/futures/futures1.asp

[7] Chron. Accessed (n.d.) How does a Forward Contract work? [Online]. http://smallbusiness.chron.com/forward-contract-work-18907.html

[8] P Serou. International Finance: Theory into Practice. [Online]. http://press.princeton.edu/releases/Sercu/Lecture_slides/05CHForwards2_slide.pdf

[9] Investopedia. Accessed (2016, March) Interest Rate Differential Defiintion. [Online]. http://www.investopedia.com/terms/i/interest-rate-differential.asp

[10] Richard D Bateson, Financial Derivative Instruments. London: Imperial College Press, 2010.

[11] David A Dubofsky, Thomas W Miller, and JR, Derivatives valuation and risk management.Oxford: Oxford University Press, 2003.

[12] Investopedia. Accessed (2016, March) Put Option Definition. [Online]. http://www.investopedia.com/terms/p/putoption.asp

[13] Investopedia. Accessed (2016, March) Lookback Option Definition. [Online]. http://www.investopedia.com/terms/l/lookbackoption.asp

[14] J Hull, "Options, Futures, and other Derivatives," in Options, Futures, and other Derivatives.

New Jersey: Pearson, 2015.

[15] Yue Kuen Kwok. (n.d.) Advanced Topics in Derivative Pricing Models. [Online]. https:// www.math.ust.hk/~maykwok/courses/MATH6380/Topic2.pdf

Page 26

[16] Optiontradingpedia. Accessed (2016, March) LookBack Options. [Online]. http://www.optiontradingpedia.com/lookback_options.htm

[17] Investopedia. InvestopediA. [Online]. http://www.investopedia.com/terms/b/binary- option.asp

[18] Options Advice. Accessed (2016, March) Expiry Times. [Online]. http://www.optionsadvice.co.uk/education/expiry-times/

[19] Binary Options U. Accessed (n.d.) Binary Options 101. [Online]. http://www.binaryoptionsu.com/binary-options-101/

[20] Investopedia. Asset-Or-Nothing Call Option. [Online]. o http://www.investopedia.com/terms/a/aonc.asp

[21] Alpari. Accessed (n.d.) Binary option types. [Online]. https://alpari.com/en/binary_options/types/

[22] Profitf. Accessed (n.d.) Types of Binary options. [Online]. http://www.profitf.com/articles/binary-options-education/types-binary-options/

[23] FCA. Accessed (2016, March) Potential Changes in Regulation. [Online]. https:// www.fca.org.uk/news/statements/binary-options--potential-changes-in- regulation-

[24] EZTrader. Binary Option Pricing. [Online]. http://www.eztrader.com/en/binaryoptions- guide/binaryoption-pricing.aspx

[25] Investopedia. Accessed (n.d.) Options Pricing: Black-Scholes Model. [Online]. http://www.investopedia.com/university/options-pricing/black-scholes-model.asp

[26] Palisade. Accessed (n.d.) Monte Carlo Simulation. [Online]. http://www.palisade.com/risk/monte_carlo_simulation.asp

[27] Tradeking. Accessed (n.d.) Tradeking. [Online]. https:// www.tradeking.com/education/options/put-options-explained

[28] Richard D Bateson, Financial Derivative Instruments. London: Imperial College Press, 2011.

[29] Risk Encyclopedia. Accessed (n.d.) Asian Option (Average Option). [Online]. http://www.riskencyclopedia.com/articles/asian-option-average-option/

[30] Investopedia. Accessed (n.d.) What is the difference between notional value and market value? [Online]. http://www.investopedia.com/ask/answers/050615/what-difference- between-notional-value-and-market-value.asp

[31] P Jorion and S Khoury, Financial Risk Management. Oxford: Blackwells Publishers, 1996.

Page 27

[32] The Options Guide. Accessed (n.d.) The Options Guide. [Online]. http://www.theoptionsguide.com/call-option.aspx

[33] London Stock Exchange. (n.d.) London Stock Exchange Group. [Online]. http://www.lseg.com/sites/default/files/content/documents/contract_specification_FTS E_100_options_0.pdf

[34] London Metal Exchange. Accessed (n.d.) London Metal Exchange. [Online]. https:// www.lme.com/en-gb/metals/non-ferrous/copper/contract- specifications/options/

[35] Eurex Exchange. (2015, October) Eurex Exchange. [Online]. http://www.eurexchange.com/blob/238416/8bf96a3ace04cfde0b287b371d45146d/dat a/contract_specifications_en_ab-2015_10_26.pdf

[36] V Jhawar. Accessed (2015, May) Investopedia. [Online]. http://www.investopedia.com/articles/active-trading/050815/top-strategies-trading- exotic-options.asp

[37] FIA. (2014) FIA. [Online].https://fimag.fia.org/sites/default/files/content_attachments/2014 FIA Annual Volume Survey %E2%80%93 Charts and Tables.pdf

[38] Investopedia. Accessed (2016, January) Why Forward Contracts Are The Foundation Of All Derivatives. [Online]. http://www.investopedia.com/articles/active- trading/102313/why-forward-contracts-are-foundation-all-derivatives.asp

[39] Investopedia. Accessed (n.d.) Non-deliverable forward - NDF. [Online]. http://www.investopedia.com/terms/n/ndf.asp

[40] EZTrader. Accessed (2016, March) Binary Option Pricing. [Online]. http://www.eztrader.com/en/binaryoptions-guide/binaryoption-pricing.aspx

[41] Alpari. Accessed (2016, March) Binary option types. [Online]. https://alpari.com/en/binary_options/types/

[42] Binary Options U. Accessed (2016, March) Binary Options 101. [Online]. http://www.binaryoptionsu.com/binary-options-101/

[43] Investopedia. Accessed (2016, March) Covered Interest Arbitrage. [Online]. http://www.investopedia.com/terms/c/covered-interest-arbitrage.asp

[44] Investing Answers. Accessed (2016, March) Forward Contract. [Online]. http://www.investinganswers.com/financial-dictionary/options-derivatives/forward- contract-4892

[45] Investopedia. Accessed (2016, March) Futures Fundamentals: A Brief History. [Online].

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http://www.investopedia.com/university/futures/futures1.asp

[46] Chron. Accessed (2016, March) How does a Forward Contract work? [Online]. http://smallbusiness.chron.com/forward-contract-work-18907.html

[47] Profitf. Accessed (2016, March) Types of Binary options. [Online]. http://www.profitf.com/articles/binary-options-education/types-binary-options/

[48] Investopedia. Accessed (2016, March) Asset-Or-Nothing Call Option. [Online]. o http://www.investopedia.com/terms/a/aonc.asp

[49] Investopedia. Accessed (2016, March) Binary Option. [Online]. http://www.investopedia.com/terms/b/binary-option.asp

[50] Investopedia. Accessed (n.d.) Covered Interest Arbitrage. [Online]. http://www.investopedia.com/terms/c/covered-interest-arbitrage.asp

[51] Investopedia. Accessed (2016, March) Too Big To Fail Definition. [Online]. http://www.investopedia.com/terms/t/too-big-to- fail.asp? layout=infini&v=4B&adtest=4B

[52] ICAP. (2016, March) NDF Project Description. [Online]. http://www.icap.com/what-we- do/global-broking/~/media/Files/I/Icap-Corp/pdfs/icap-ndfs.pdf

[53] Investopedia. Accessed (2016, March) Bermuda Option Definition. [Online]. http://www.investopedia.com/terms/b/bermuda.asp