mms derivatives lec 2

Financial Futures

By Vaibhav Kabra M.F.S.M, F.R.M.

Determination of Forward and Futures Prices

by Vaibhav 3

Short Sales• A short seller does not own securities that he sells

• In order to execute a short sale the following has to be done by the short seller

• Borrow and sell securities simultaneously through the broker

• The short seller has to return the securities when demanded by the lender/broker or when the short sale is closed out

• A deposit is kept with the broker

by Vaibhav 4

Short Squeeze• Short Squeeze

• Whenever the lender/broker runs out of securities to lend, the short seller is forced to close his position, this situation is short squeeze

• Short selling rules

• Pay all the dividends to the lender

• Deposit collateral with lender to guarantee the eventual repurchase of the security

by Vaibhav 5

Forwards Valuation

Pricing Model: Assumptions

1. Borrowing and lending at risk free rate

2. Same tax rates on all net profits

3. No transaction costs

4. No short sales restriction

5. Arbitrage opportunities arise and are exploited

by Vaibhav 6

Forwards Valuation• The equation for forward price of an asset paying no dividend

is given by

F0 = S0 * erT

An arbitrage opportunity will arise if the above equation does not hold true

For e.g.If F0 > S0 * erT , then one would profit by selling the forward and

buying the asset with the borrowed funds

If F0 < S0 * erT , then one would profit by selling the asset, lending out the proceeds and buying the forward

by Vaibhav 7

Forwards ValuationImpact of Interim Cash Flows on Forward Prices

(Benefit to the Owner of Stock/Commodity)

• If the underlying pays a known amount of cash over the life of the forward contract, the equation for the forward price becomes

F0 = (S0 – I)* erT

• Since the owner of the forward contract does not receive any of the cash flows from the underlying asset between contract origination and delivery, the present value of these cash flows must be deducted from the Spot Price

• Note that “I” is the present value of the cash flows over T years

by Vaibhav 8

Forwards ValuationImpact of Dividends on Forward Prices


• When the underlying asset for a forward contract pays a dividend, the equation for Forward Price is

F0 = S0 * e(r-q)T

“q” is the continuously compounded dividend yield

by Vaibhav 9

Forwards ValuationImpact of Lease Rates on Forward Prices


• Lease Rate is the amount of interest a commodity lender requires• The lease rate signifies the amount of return that an investor

requires after buying and then lending the commodity• A commodity lender can earn the lease rate by buying a commodity

and immediately selling it forward.

• Now the forward price when the lease rate comes in the picture is

F0 = S0 e ( r – δ )T

where δ = lease rate • The lease rate is the income earned if and only if the commodity is

loaned out

by Vaibhav 10

Forwards ValuationImpact of Storage Costs on Forward Prices

(Cost to the Owner of Stock/Commodity)

• Holding a commodity requires storage costs. Hence the forward price must be greater than the spot price to compensate for the physical storage costs.

• The owner will only hold the commodity if the forward price is greater than or equal to the expected spot price plus storage costs.

F0 = S0 * e(r+λ)T

“λ” is the continuous annual storage cost

F0 >= S0 * erT + λ(0,T)

λ(0,T) is the FV of storage costs for one unit of commodity from time 0 to T

by Vaibhav 11

Forwards ValuationImpact of Convenience Yield on Forward Prices

(Benefit to the Owner of Stock/Commodity)• If the owners of the commodity need the commodity for their

business, holding a physical inventory of the commodity creates value.

• For example, assume a manufacturer requires a specific commodity as a raw material.

• To reduce the risk of running out of inventory and slowing down production, excess inventory is held by the manufacturer.

• This reduces the risk of idle machines and workers. In the event that the excess inventory is not needed, it can always be sold.

• Holding an excess amount of a commodity for a non monetary return is referred to as convenience yield

F0 = S0 * e(r-c)T

“c” is the continuous compounded continuous yield

by Vaibhav 12

Forwards ValuationComparing Lease Rates, Storage Costs and Convenience Yield.

The basic equation relating forward and spot prices

F0 = S0 * erT

• This expression says that if there are no cost or benefits associated with buying and holding the commodity, the forward price is just the spot price compounded at the risk free rate over the holding period .

• If there are benefits (e.g.: lease rates , convenience yield ) to buying the commodity today , the holder is willing to accept a lower forward price . The forward price is reduced by the benefit

F0 = S0 e ( r – δ )T

where δ = lease rate

F0 = S0 * e(r-c)T

or c = the convenience yield

by Vaibhav 13

Forwards ValuationComparing Lease Rates, Storage Costs and Convenience Yield.

• If there are costs such as storage costs associated with the purchasing of commodity today the forward price is increased by the cost

F0 = S0 * e(r+λ)T

Where λ = storage cost

• If costs and benefits are combined the equation for forward price becomes

F0 = S0 * e(r-c- δ +λ)T

by Vaibhav 14

Concept Checker

• Calculate the 9 month forward rate for a bushel of Corn that has a spot rate of $10 and an annual lease rate of 6%. The appropriate continuously compounding annual risk free rate for the commodity is equivalent to 9%.

A. 9.75B. 10.22C. 11.5D. 12

by Vaibhav 15

Concept Checker

• Suppose the spot price today for a bushel of corn is $4 , the continuously compounded interest rate is 8% , and the monthly storage costs are 3.5% .Calculate the 9 month forward price.

A. 4.26B. 5.5C. 3.5D. 2

by Vaibhav 16

Concept Checker

• Calculate the 3 month forward price for a bushel of soybeans if the current spot price is $3 per bushel , the effective monthly interest rate is 1% , and the monthly storage costs are $ 0.04 per bushel.

by Vaibhav 17

Backwardation and Contango

• Backwardation is the situation where the future contract price is lower than the spot price for a commodity.

• As the futures contract approaches maturity, it will trade at higher and higher prices to finally meet the future spot price.

• Backwardation = Futures Price < Spot Price.

• Backwardation is evident when there is a significant benefit to hold the asset.

• For e.g. Lease Rates, Dividends, Convenience yields

by Vaibhav 18


• Contango is the situation where the future contract price is higher than the spot price for a commodity.

• As the futures contract approaches maturity, it will trade at lower and lower prices to finally meet the future spot price.

• Contango = Futures Price > Spot Price.

• Contango is evident when there is a NO significant benefit to hold the asset.

by Vaibhav 19


by Vaibhav 20

Factors impacting Gold , Corn , Natural Gas & Oil

Gold• Gold has low storage costs.

• The forward price tends to be a gradually increasing function of maturity which implies a presence of lease rate.

– Direct exposure to gold : own gold

– indirect exposure to gold :long position in gold futures.

• You forgo a “lease rate” if you own gold and you also bear storage costs

• In a synthetic long position in gold: no storage costs, but credit risk exposure.

by Vaibhav 21


Natural Gas• Price is impacted largely due to Seasonal Demand and Storage

Cost

• Natural Gas : Characteristics

• Expensive to transport overseas due to its physical properties

• Costly to store

• Exposed to seasonal demand with a characteristic peak in the winter

• Natural gas is constantly produced and seasonally demanded

by Vaibhav 22


Corn • Corn is seasonal commodity . • In theory corn price rises between harvests due to

storage costs• But In reality the price varies year to year.• Corn is seasonally produced and constantly

demanded

by Vaibhav 23


Oil

• Easier to transport than natural gas • Thus easy to compare price world wide• Long run forward price is less volatile than the

short term forward price

by Vaibhav 24

Commodity Spread

• A commodity spread occurs when a commodity is an input in the production of other commodity

• Example: Crush Spread• Soybean used in production of soybean oil• Hold long (short) position in soybean and short (long) position in

soybean oil • This is called crush Spread

• Example : Crack Spread• Crude oil , heating oil & gasoline• Crack spread : 7-4-3 spread : 7 gallons Crude oil , 4 gallons heating oil

& 3 gallons gasoline• Thus an oil company can lock in crude oil input and output (finished

goods) price

by Vaibhav 25

Commodity Spread

• Suppose we plan on buying crude oil in 1 month to produce gasoline and hitting oil for sale in 2 months . The one month future price for crude oil is currently $42.5 per barrel. The 2 month future price for gasoline and heating oil are $45 per barrel and $43.5 per barrel respectively . What is the 7-5-2 crack (commodity) spread.

A. 2.07/barrelB. 6/barrelC. 14.5/barrelD. 22.09/barrel

Hedging Strategies using Futures

by Vaibhav 27

Short Hedge

• This occurs when the hedger shorts the future contract

• This strategy is employed when the hedger is currently in a long position and wants to hedge against price decrease

• A short hedge is appropriate when the hedger already owns an asset and expects to sell it at some time in the future.

• Example : A wheat producer sells wheat futures to lock in the price today and hedge against price decrease

by Vaibhav 28

Long Hedge

• This occurs when the hedger buys the future contract

• This strategy is employed when the hedger is currently in a short position and wants to hedge against price increase

• A long hedge is also appropriate when the hedger expects to buy it at some time in the future.

• Example : A wheat whole seller buys wheat futures to lock in the price today and hedge against price increase

by Vaibhav 29

Basis Risk• A perfect hedge occurs when the

characteristics of the hedged asset and the futures contract (the hedging instrument ) perfectly match.

• When this is the case the gains on the futures position and the losses on the hedged asset cancel each other. However this is not always the case.

• When the characteristics of the hedged asset and the hedging instrument do not match due to different underlying assets used or different maturities of the assets, it may lead to Basis Risk.

Basis

Prices

Present MaturityTime

Futures

Cash

by Vaibhav 30

Basis Risk• Basis = Spot Price (Hedged Asset) - Futures Price (Hedging

Instrument)

• The change in the basis is unavoidable. The change in basis over the hedge horizon is termed basis risk, and it can work either for or against the hedger

• The basis will be zero at maturity if the Hedged asset and the asset underlying the hedging instrument are the same

• When the spot price increase at a faster rate than the futures price, the basis increases and it is referred to as “Strengthening of Basis”

• When the Futures price increase at a faster rate than the Spot price, the basis decrease and it is referred to as “Weakening of Basis”

by Vaibhav 31

Sources of Basis Risk

• Interruption in the convergence of the futures and spot prices

• Changes in the Cost of Carry

• Maturity or Duration Mismatch

• Liquidity Mismatch

• Credit Risk Mismatch

by Vaibhav 32

Optimal Hedge Ratio

• Optimal Hedge Ratio accounts for an imperfect relationship between the spot and futures positions.

• Optimal Hedge Ratio leverages the degree of correlation between the rates.

• Hedge Ratio = ρS,F * (σS / σF),

• ρS,F = the correlation between the spot prices and the futures prices• σS = the standard deviation of the spot price• σF = the standard deviation of the futures price

F

S

by Vaibhav 33

Optimal Hedge Ratio

• Also, ρS,F = CovS,F / (σS * σF)

• Hedge Ratio = CovS,F / (σS * σF) * (σS / σF)

• Hedge Ratio = CovS,F / σF2 = βS,F,

by Vaibhav 34

Hedging withStock Index Futures

• This is an application where the equity portfolio is hedged using the stock index futures

• Here the β of hedged portfolio is the hedge ratio

• Thus the number of contracts (N) required to completely hedge the equity position is given by

• Number of contracts

= βPortfolio*(Portfolio Value/Value of Futures Contract)

= βPortfolio*[(Portfolio Value / (Futures Price*Contract Multiplier)]

by Vaibhav 35

Adjusting the Portfolio Beta

• When the portfolio manager wants to shift the portfolio beta (β) to a target beta (β*) the number of contracts required are

• Number of Contracts = (β* - β) (P/A)

• The equation can result in negative or positive values

• Negative value means selling futures or reducing systematic risk

• Positive values means buying futures or increasing systematic risk

by Vaibhav 36

Concept Checkers• A fund manager has a $100 million portfolio with a beta of 0.75 .

The manager has bullish expectations for the next couple of months and plans to use futures contract on the S&P 500 to increase the portfolio’s beta to 1.8 . Given the following information, which strategy should the fund manager follow :

• The current level of the S&P index is 1250 • Each S&P futures contract delivers $250 times the index • The risk free interest rate is 6% p.a.

A. Enter into a long position of 323 S&P futures contract B. Enter into a long position of 336 S&P futures contract C. Enter into a long position of 480 S&P futures contract D. Enter into a short position of 240 S&P futures contract

by Vaibhav 37

Concept Checkers

• A company expects to buy 1million barrels of West Texas intermediate crude oil in 1 year . The annualized volatility of the price of a barrel of WTI is calculated at 12 % . The company chooses to hedge by buying a futures contract on Brent Crude . The annualized volatility of the Brent Futures is 17% and the correlation co efficient is 0.68 .

Calculate the variance minimizing hedge ratioA. 0.62B. 0.53C. 0.48D. 0.42

by Vaibhav 38

Concept Checkers

• Consider an equity portfolio with market value of $100 million and a beta of 1.5 with respect to the S&P 500 index. The current S&P 500 index level is 1000 and each futures contract is for delivery of $250 times the index level . Which of the following strategy will reduce the beta of the equity portfolio to 0.8

A. Long 600 S&P 500 futures contract B. Short 600 S&P 500 futures contractC. Long 280 S&P 500 futures contractD. Short 280 S&P 500 futures contract

Thank You !

mms derivatives lec 2

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