# Lecture Note 4 _Forwards and Futures_

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```					Financial Risk Management
Fall 2007

Week 5: Forward and Futures
Introduction
Contract Types

•   Forward Contract : a contract between two parties for one party to buy something
from the other at a later date at a price agreed upon today
–   Exclusively over-the-counter

•   Futures: a contract between two parties for one party to buy something from the
other at a later date at a price agreed upon today
–   Subject to a daily settlement of gains and losses and guaranteed against the risk that
either party might default
–   Exclusively traded on a futures exchange

•   Option on Future : a contract between two parties giving one party the right to buy
or sell a futures contract from the other at a later date at a price agreed upon
today
–   Also known as commodity options or futures options
–   Exclusively traded on a futures exchange
Price and Value of a Contract

•   For futures or forward, price is the contracted rate of future purchase, but value is
something different.
–   The price of a forward contract does not change over the life of the contract.
–   The price of a futures contract does not change over the life of the contract, but new
contracts with new prices are created and introduced to the exchange.
–   At the beginning of a contract, value = 0 for both futures and forwards.
–   Value of a contract changes over time as the price of the underlying asset changes.

•   Notation
–   Vt(0,T) is the value of a forward contract created at time 0 and expiring at time T.
–   F(t,T) is the price at time t of a forward contract expiring at time T.
–   vt(T) is the value at time t of a futures contract expiring at time T
–   ft(T) is the price of a futures contract created at time t and expiring at time T
Value of a Forward Contract

•   Forward price at expiration:
F(T,T) = ST .
–   That is, the price of an expiring forward contract is the spot price.

•   Value of forward contract at expiration:
V T (0,T) = ST - F(0,T).
–   An expiring forward contract allows you to buy the asset, worth ST , at the forward price
F(0,T).
–   The value to the short party is -1 times this.
Value of a Forward Contract (continued)

•   The Value of a Forward Contract Prior to Expiration
–   A: Go long forward contract at price F(0,T) at time 0.
–   B: At t go long the asset and take out a loan promising to pay F(0,T) at T
–   At time T, A and B are worth the same, ST – F(0,T).
–   Thus, they must both be worth the same prior to T.
–   So V t (0,T) = S t – F(0,T)(1+r)-( T-t)
•   Example
–   Go long 45 day contract at F(0,T) = \$100.
–   Risk-free rate = 0.10.
–   20 days later, the spot price is \$102.
–   The value of the forward contract is 102 - 100(1.10)-25/365 = 2.65.
Value of a Forward Contract (continued)

•   Example
–   Go long 45 day contract at F(0,T) = \$100.
–   Risk-free rate = 0.10.
–   20 days later, the spot price is \$102.
–   The value of the forward contract is 102 - 100(1.10)-25/365 = 2.65.

•   Margin : deposit required to maintain an open position
–   initial margin : the amount that must be deposited on the day the transaction is opened.
–   maintenance margin : the amount that must be maintained everyday after the
transaction is opened.

•   Daily Settlement :
–   settlement price :
•   a committee composed of clearinghouse officials establish the settlement price at the end of
each day.
•   It is usually an average of the prices of the last few trades of the day.
–   marking-to-market :
•   If the settlement price exceeds the previous day’s settlement price, a dollar amount equal to the
difference is credited to the accounts of those holding long positions.
•   If the difference is negative – i.e., the settlement price is below the previous day’s settlement
price, a dollar amount equal to the difference is credited to the accounts of those holding short
positions.
•   On any given day when the balance is greater than the maintenance margin, the excess over the
initial margin can be withdrawn.
•   Same amount as credited to the accounts of certain positions are charged to the accounts of
those holding opposite positions.
•   If the balance of an account falls below the maintenance margin, the account holder receives a
margin call and must deposit enough funds to bring the balance back up to the initial margin
requirement. The additional funds deposited are called the variation margin.
=-(\$97,406.25-\$97,843.75)

=-(\$97,781.25-\$97,406.25)       =\$2,937.50
-\$375.00

Margin call

Variable Margin

Futres price increased by 2,656.25   -2,656.25     2,656.25
Value of a Futures Contract

•   Suppose that you purchase a futures contract at a price ft(T)* during day t.
–   If the settlement price at the end of the day is determined as ft(T), then an amount
v t(T) = ft(T) - ft(T)*
should be credited to your account after marking-to-market.

•   Suppose that ft-1 (T) is the settlement price at day t-1.
–   If the settlement price at the end of day t is determined as ft(T), then an amount
v t(T) = ft(T) – ft-1(T)
should be credited to your account after marking-to-market.

•   Any profit or loss is accumulated in the margin account. Hence the value of the
margin account at the end of trading day t after marking to market is

where IM is the initial margin and f-1(T) = f0(T)* .
Closing a Futures Position

•   Most futures traders close out their positions prior to expiration.
–   They simply take a new position that is the opposite of the existing positions.
–   The process is called offsetting.

•   Suppose you close a long futures position on date t, by selling identical futures
contract at a price ft(T)*.
–   Value of the existing (long) position (net of IM) at the end of the expiration date is

–   Value of the new (short) position (net of IM) at the end of the expiration date is

–   After netting, net profit or loss becomes
Delivery and Cash Settlement of Futures

•   None-Cash-Settled Futures : three-day delivery process
–   Position day : The holder of a short position who intends to make delivery notifies the
clearinghouse of its desire to deliver.
–   Notice of intention day : The exchange selects the holder of the oldest long position to
–   Delivery day : Short make delivery and the long pays the short.

•   Cash-Settled Futures
–   The settlement price on the last trading day is fixed at the closing spot price of the
underlying instrument.
–   All contracts are marked to market on that day. All positions are deemed closed.
Types of Futures Contracts
•   Agricultural Commodities
–   Grains : wheat, corn, oats, soybeans, rice
–   Livestock : cattle, hogs
–   Food products : coffee, cocoa, orange juice, sugar
–   Others: cotton, wool, wood
•   Natural Resources
–   Metals : gold, silver, copper, platinum, palladium
–   Energy products
•   Miscellaneous Commodities
–   fertilizer, shrimp, electricity, rubber, glass, cement, potatoes, peanuts, sunflower seeds,
inflation, peas, flax, etc
•   Foreign Currencies
•   Treasury Bills and Eurodollars
•   Treasury Notes and Bonds
•   Equities
–   Index futures
–   Futures on individual stocks
Pricing Forwards and Futures
Forward Price

•   The Value of a Forward Contract Prior to Expiration
V t (0,T) = S t – F(0,T)(1+r)-( T-t)

•   At time 0, the forward price F(0,T) is determined so that neither of the seller and
the buyer finds an arbitrage opportunity.
–   Set V0(0,T) = 0 and get
F(0,T) = S 0 (1+r)T
Futures Price

•   Futures price at expiration:
fT (T) = ST .

•   At time 0, the futures price should equal the forward price if we ignore effects of marking-
to-market.
f0(T)* = F(0,T) = S0 (1+r)T

•   At time t, the price of a newly introduced futures contract should equal the forward price
introduced on the same day.
ft(T)* = F(t,T) = St (1+r)T -t

•   Forward and futures prices will be equal
– One day prior to expiration
– More than one day prior to expiration if
•   Interest rates are certain
•   Futures prices and interest rates are uncorrelated
–   Futures prices will exceed forward prices if futures prices are positively correlated with
interest rates.
•   Default risk can also affect the difference between futures and forward prices.
Carry Arbitrage: Equities

•   Forward and Futures Pricing When the Underlying Generates Cash Flows
–   For example, dividends on a stock or index
–   Assume one dividend DT paid at expiration.

•   Buy stock, sell futures guarantees at expiration that you will have DT + f0(T).
–   Present value of this must equal S0, using risk-free rate. Thus,
f0 (T) = S 0 (1+r)T - DT .
–   If D0 represents the present value of the dividends, the model becomes
f0(T) = (S0 – D0 )(1+r)T.

•   For multiple dividends, let DT be compound future value of dividends.
–   See Figure 9.1, p. 293 for two dividends.
–   Dividends reduce the cost of carry.
Carry Arbitrage: Equities (continued)

•   For dividends paid at a continuously compounded rate of dc,

•   Example: S 0 = 50, rc = 0.08, dc = 0.06, expiration in 60 days (T = 60/365 = 0.164).
– f0(T) = 50e(0.08   - 0.06)(0.164)   = 50.16.
Carry Arbitrage: Equity Forward Contracts

•   When there are dividends, to determine the value of a forward contract during its
life

V t(0,T) = S t – Dt,T – F(0,T)(1 + r)-( T-t)

where Dt,T is the value at time t of the future dividends to time T

•   If dividends are continuous,
Carry Arbitrage: Currencies
•   Interest Rate Parity
–    The relationship between spot and forward or futures prices of a currency.
–    Same as carry arbitrage model in other forward and futures markets.
–    Proves that one cannot convert a currency to another currency, sell a futures, earn the
foreign risk-free rate, and convert back without risk, earning a rate higher than the
domestic rate.

•   Let
–    S0 = spot rate in domestic currency per foreign currency.
–    r =domestic interest rate
–    r = foreign interest rate
–    T = holding period

•   Strategy
–    Take (borrow) S0(1+ r)-T units of domestic currency.
–    Convert the domestic currency to the foreign current at the spot rate S0.
–    Sell forward contract to deliver one unit of foreign currency at T at price F(0,T).
–    Buy (1+ r)-T units of foreign currency. Hold foreign currency and earn rate r.
Carry Arbitrage: Currencies (continued)

•   Liability of S0(1+ r)-T units of domestic currency grows to S0(1+ r)-T(1 + r)T
units of domestic currency.

•   At time T you will have one unit of the foreign currency.
–   Deliver foreign currency and receive F(0,T) units of domestic currency.

•   In the absence of arbitrage, F(0,T) should equal to the liability in domestic
currency.
–   At time 0, net cost of the combined transactions was 0.

•   Therefore
F(0,T) = S 0 (1+ r)-T (1 + r)T
–   Sometimes written as
F(0,T) = S0(1 + r)T/(1 + r)T
Carry Arbitrage: Currencies (continued)

•   Example (from a European perspective):
–   S0 = €1.0304.
–   U. S. rate is 5.84%.
–   Euro rate is 3.59%.
–   Time to expiration is 90/365 = 0.2466.

F(0,T) = €1.0304(1.0584)-0.2466(1.0359)0.2466 = €1.025

•   If forward rate is actually €1.03, then it is overpriced.
–   Buy US bond of (1.0584)-0.2466 = \$0.9861 for 0.9861(€1.0304) = €1.0161.
–   Sell one forward contract at €1.03.
–   Earn 5.84% on \$0.9861. This grows to \$1.
–   At expiration, deliver \$1 and receive €1.03.
–   Return is (1.03/1.0161)365/ 90 - 1 = 0.0566 (> 0.0359)
–   This transaction is called covered interest arbitrage.

•   Let
–    s = the cost of storing an asset
–    i = the interest rate for the period of time the asset is owned (this is the cost when one
borrows money for purchase of asset)

What you pay between today and time T
•   First assume no uncertainty of future price.
S 0 = S T - s - iS 0      Time value of what you pay today

What you pay today              What you get at time T
•   If we now allow uncertainty but assume people are risk neutral, we have
S 0 = E(S T ) - s - iS 0

•   If we now allow people to be risk averse, they require a risk premium of E().
Now
S 0 = E(S T ) - s - iS 0 - E()
Pricing Models and Risk Premiums (continued)

•   Cost of carry : q =s + iS 0
–   s + iS0 is the cost that one should pay if, instead of purchasing futures or forwards, one
decides to purchase the asset today and carry it till expiration date.
–   Define iS0 as the net interest, which is the interest foregone minus any cash received.

•   Theoretical Fair Price
•   Buy asset in spot market, paying S0; store and incur costs
•   sell futures contract at price f0(T);.

•   At expiration, make delivery. Profit:
P = f0 (T) - S 0 - q
– This must be zero to avoid arbitrage; thus,
f0 (T) = S 0 + q
– See Figure 9.2, p. 300.
– Note how arbitrage and quasi-arbitrage make this hold.
Futures Contracts of Different Expirations

•   Expirations of T2 and T1 where T2 > T1.

•   Then
f0 (T 1 ) = S 0 + q 1
and
f0 (T 2 ) = S 0 + q 2

•   Spread between two futures prices will be

f0 (T 2 ) - f0 (T 1 ) = q 2 - q 1 .
Pricing Models and Risk Premiums (continued)

•   Contango is a market situation when
f0 (T) > S 0 .
–   See Table 9.2, p. 303.
–   Contango represents positive cost of carry.

•   Convenience yield, denoted by c, is an additional return from holding asset.
–   Convenience yield is positive when the asset is in short supply.
–   Convenience yield may also be a non-pecuniary return.
•
•   When f0 (T) < S 0 , convenience yield is c is larger than the cost of carry.
–   Market is said to be at less than full carry and in backwardation or inverted.
–   See Table 9.3, p. 304.

•   Market can be both backwardation and contango.
–   See Table 9.4, p. 304.
Pricing Models and Risk Premiums (continued)
f0 (T) = E(ST )
–   Market consists of only speculators.
–   See Figure 9.4, p. 306.

Note that E(fT (T )) = E(ST )
E(fT (T)) > f0 (T).
price to increase.
–   When hedgers go short futures, they transfer risk premium to speculators who go long
futures.
E(S T ) = f0 (T) + E(). futures expect to earn a risk
–   See Figure 9.5, p. 308.                                    premium.
–   Normal contango: E(ST ) < f0(T)                           Sellers are willing to pay the
–   Normal backwardation: f0(T) < E(ST )                      risk premium if they want to
hedge.
Put-Call-Forward/Futures Parity

•   Put-call-forward/futures parity says:

Pe (S 0 ,T,X) = Ce (S 0 ,T,X) + (X - f0 (T))(1+r)-T

•   Numerical example using S&P 500.
•   On May 14,
•   S&P 500 at 1337.80
•   June futures at 1339.30
•   June 1340 call at 40
•   June 1340 put at 39.
•   Expiration of June 18 so T = 35/365 = 0.0959.
•   Risk-free rate at 4.56%.

Identical payoffs
at expiration
Put-Call-Forward/Futures Parity (continued)

•   Numerical example using S&P 500.
•   On May 14,
–   S&P 500 at 1337.80
–   June futures at 1339.30
–   June 1340 call at 40
–   June 1340 put at 39.
–   Expiration of June 18 so T = 35/365 = 0.0959.
–   Risk-free rate at 4.56%.

•   Arbitrage Strategy : Buy put and futures for 39, sell call and bond for 40.70
and net 1.70 profit at no risk.
–   Pe(S0,T,X) = 39
–   Ce(S0,T,X) + (X - f0(T))(1+r)-T = 40 + (1340 - 1339.30)(1.0456)-0.0959 = 40.70.
–   Transaction costs would have to be considered.
Options on Futures
Intrinsic Value of an American Option on Futures

•   Minimum value of American call on futures

Ca (f0 (T),T,X) Max(0, f0 (T) - X)

•   Minimum value of American put on futures

Pa (f0 (T),T,X) Max(0,X - f0 (T))

•   Difference between option price and intrinsic value is time value.
Lower Bound of a European Option on Futures

•   For calls,

•   Portfolio A dominates Portfolio B so
Ce (f0 (T),T,X) Max[0,(f0 (T) - X)(1+r)-T ]

•   Note that lower bound can be less than intrinsic value even for calls.
Lower Bound of a European Option (continued)

•   For puts,

•   Portfolio A dominates Portfolio B so
Pe (f0 (T),T,X) Max[0,(X - f0 (T))(1+r)-T ]
Put-Call Parity of Options on Futures

•   Construct two portfolios, A and B.

Pe (f0 (T),T,X) = Ce (f0 (T),T,X) + (X - f0 (T))(1+r)-T
•   Compare to put-call parity for options on spot:
Pe(S0,T,X) = Ce(S0,T,X) - S0 + X(1+r)-T
–   If options on spot and options on futures expire at same time, their values are equal,
implying f0(T) = S0(1+r)T .
Early Exercise of Call and Put Options on Futures
•   Deep in-the-money call may be exercised early because :
1) A deep in-the-money option behaves almost identically to futures.
• It has little time value.
2) Exercise frees up funds tied up in option
• But it requires no funds to establish futures because margin requirement could be
met by putting interest-earning bonds.

•   Similar arguments hold for puts.
When deep in-the-money, the value
of the option is almost identical to
the futures
The value of a European option is
less than that of an American option
because the holder should wait till
maturity to realize the gain.
Black Model for European options on Futures

•   A European call option on futures can be priced using

where

•   For puts
Homework Assignments No. 4

•   Selected from end of chapter problems:

Chapter 8: #3, #8, #9
Chapter 9: #1, #3, #6, #23, #24

•   Due date: This homework problems will not be collected.
Answers to these problems will be posted soon.