chapter 9: principles of pricing forwards, futures, and options on futures

D. M. Chance An Introduction to Derivatives and Risk Management, 6th ed.

Ch. 9: 1

Chapter 9: Principles of Pricing Forwards, Futures, and Options on Futures

To know value is to know the meaning of the market.To know value is to know the meaning of the market.

Charles DowCharles Dow

Money Talks Money Talks (by Rosalie Maggio), 1998, p. 23(by Rosalie Maggio), 1998, p. 23


Ch. 9: 2

Important Concepts in Chapter 9

Price and value of forward and futures contractsPrice and value of forward and futures contracts Relationship between forward and futures pricesRelationship between forward and futures prices Determination of the spot price of an assetDetermination of the spot price of an asset Cost of carry model for theoretical fair priceCost of carry model for theoretical fair price Contango, backwardation, and convenience yieldContango, backwardation, and convenience yield Futures prices and risk premiumsFutures prices and risk premiums Futures spread pricingFutures spread pricing Pricing options on futuresPricing options on futures


Ch. 9: 3

Some Properties of Forward and Futures Prices The Concept of Price Versus ValueThe Concept of Price Versus Value

Normally in an efficient market, price = value.Normally in an efficient market, price = value. For futures or forward, price is the contracted rate of For futures or forward, price is the contracted rate of

future purchase. Value is something different.future purchase. Value is something different. At the beginning of a contract, value = 0 for both At the beginning of a contract, value = 0 for both

futures and forwards. futures and forwards. NotationNotation

VVtt(0,T), F(0,T), v(0,T), F(0,T), vtt(T), f(T), ftt(T) are values and prices of (T) are values and prices of

forward and futures contracts created at time 0 and forward and futures contracts created at time 0 and expiring at time T.expiring at time T.


Ch. 9: 4

Some Properties of Forward and Futures Prices (continued) The Value of a Forward ContractThe Value of a Forward Contract

Forward price at expiration: Forward price at expiration: F(T,T) = SF(T,T) = STT. . That is, the price of an expiring forward contract is That is, the price of an expiring forward contract is

the spot price.the spot price. Value of forward contract at expiration:Value of forward contract at expiration:

VVTT(0,T) = S(0,T) = STT - F(0,T). - F(0,T). An expiring forward contract allows you to buy the An expiring forward contract allows you to buy the

asset, worth Sasset, worth STT, at the forward price F(0,T). The , at the forward price F(0,T). The value to the short party is -1 times this.value to the short party is -1 times this.


Ch. 9: 5

Some Properties of Forward and Futures Prices (continued) The Value of a Forward Contract (continued)The Value of a Forward Contract (continued)

The Value of a Forward Contract Prior to ExpirationThe Value of a Forward Contract Prior to Expiration A: Go long forward contract at price F(0,T) at time 0.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 B: At t go long the asset and take out a loan promising to pay

F(0,T) at TF(0,T) at T

• At time T, A and B are worth the same, SAt time T, A and B are worth the same, STT – F(0,T). – F(0,T).

Thus, they must both be worth the same prior to t. Thus, they must both be worth the same prior to t.

• So VSo Vtt(0,T) = S(0,T) = Stt – F(0,T) – F(0,T)-(T-t)-(T-t)

• SeeSee Table 9.1, p. 306Table 9.1, p. 306.. Example: Go long 45 day contract at F(0,T) = $100. Risk-Example: Go long 45 day contract at F(0,T) = $100. Risk-

free rate = .10. 20 days later, the spot price is $102. The value free rate = .10. 20 days later, the spot price is $102. The value of the forward contract is 102 - 100(1.10)of the forward contract is 102 - 100(1.10)-25/365-25/365 = 2.65. = 2.65.


Ch. 9: 6

Some Properties of Forward and Futures Prices (continued)

The Value of a Futures ContractThe Value of a Futures Contract Futures price at expiration: Futures price at expiration:

ffTT(T) = S(T) = STT..

Value during the trading day but before being marked Value during the trading day but before being marked to market:to market: vvtt(T) = f(T) = ftt(T) - f(T) - ft-1t-1(T).(T).

Value immediately after being marked to market: Value immediately after being marked to market: vvtt(T) = 0.(T) = 0.


Ch. 9: 7

Some Properties of Forward and Futures Prices (continued)

Forward Versus Futures PricesForward Versus Futures Prices Forward and futures prices will be equalForward and futures prices will be equal

One day prior to expirationOne day prior to expiration More than one day prior to expiration ifMore than one day prior to expiration if

• Interest rates are certainInterest rates are certain• Futures prices and interest rates are uncorrelatedFutures prices and interest rates are uncorrelated

Futures prices will exceed forward prices if futures Futures prices will exceed forward prices if futures prices are positively correlated with interest rates.prices are positively correlated with interest rates.

Default risk can also affect the difference between Default risk can also affect the difference between futures and forward prices.futures and forward prices.


Ch. 9: 8

A Forward and Futures Pricing Model

Spot Prices, Risk Premiums, and the Cost of Carry for Spot Prices, Risk Premiums, and the Cost of Carry for Generic AssetsGeneric Assets First assume no uncertainty of future price. Let s be the First assume no uncertainty of future price. Let s be the

cost of storing an asset and i be the interest rate for the cost of storing an asset and i be the interest rate for the period of time the asset is owned. Thenperiod of time the asset is owned. Then SS00 = S = STT - s - iS - s - iS00

If we now allow uncertainty but assume people are risk If we now allow uncertainty but assume people are risk neutral, we haveneutral, we have SS00 = E(S = E(STT) - s - iS) - s - iS00

If we now allow people to be risk averse, they require a If we now allow people to be risk averse, they require a risk premium of E(risk premium of E(). Now). Now SS00 = E(S = E(STT) - s - iS) - s - iS00 - E( - E())


Ch. 9: 9


Spot Prices, Risk Premiums, and the Cost of Carry for Spot Prices, Risk Premiums, and the Cost of Carry for Generic Assets (continued)Generic Assets (continued) Let us define iSLet us define iS00 as the net interest, which is the interest as the net interest, which is the interest

foregone minus any cash received.foregone minus any cash received. Define s + iSDefine s + iS00 as the cost of carry. as the cost of carry.

Denote cost of carry as Denote cost of carry as .. Note how cost of carry is a meaningful concept only for Note how cost of carry is a meaningful concept only for

storable assetsstorable assets


Ch. 9: 10


The Theoretical Fair PriceThe Theoretical Fair Price Do the followingDo the following

Buy asset in spot market, paying SBuy asset in spot market, paying S00; sell futures ; sell futures contract at price fcontract at price f00(T); store and incur costs.(T); store and incur costs.

At expiration, make delivery. Profit:At expiration, make delivery. Profit:• = f= f00(T) - S(T) - S00 - -

This must be zero to avoid arbitrage; thus,This must be zero to avoid arbitrage; thus,• ff00(T) = S(T) = S00 + +

See See Figure 9.1, p. 313Figure 9.1, p. 313.. Note how arbitrage and quasi-arbitrage make this hold.Note how arbitrage and quasi-arbitrage make this hold.


Ch. 9: 11

A Forward and Futures Pricing Model (continued)

The Theoretical Fair Price (continued)The Theoretical Fair Price (continued) See See Figure 9.2, p. 314Figure 9.2, p. 314 for an illustration of the for an illustration of the

determination of futures prices.determination of futures prices. Contango is fContango is f00(T) > S(T) > S00. See . See Table 9.2, p. 315Table 9.2, p. 315..

When fWhen f00(T) < S(T) < S00, convenience yield is , convenience yield is , an additional , an additional

return from holding asset when in short supply or a return from holding asset when in short supply or a non-pecuniary return. Market is said to be at less than non-pecuniary return. Market is said to be at less than full carry and in backwardation or inverted. See full carry and in backwardation or inverted. See Table 9.3, p. 316Table 9.3, p. 316. Market can be both backwardation . Market can be both backwardation and contango. See and contango. See Table 9.4, p. 317Table 9.4, p. 317..


Ch. 9: 12

A Forward and Futures Pricing Model (continued) Futures Prices and Risk PremiaFutures Prices and Risk Premia

The no risk-premium hypothesisThe no risk-premium hypothesis Market consists of only speculators.Market consists of only speculators. ff00(T) = E(S(T) = E(STT). See ). See Figure 9.3, p. 319Figure 9.3, p. 319..

The risk-premium hypothesisThe risk-premium hypothesis E(fE(fTT(T)) > f(T)) > f00(T).(T). When hedgers go short futures, they transfer risk When hedgers go short futures, they transfer risk

premium to speculators who go long futures.premium to speculators who go long futures. E(SE(STT) = f) = f00(T) + E((T) + E(). See ). See Figure 9.4, p. 321Figure 9.4, p. 321..

Normal contango: E(SNormal contango: E(STT) < f) < f00(T) (T) Normal backwardation: fNormal backwardation: f00(T) < E(S(T) < E(STT))


Ch. 9: 13


Forward and Futures Pricing When the Underlying Generates Cash FlowsForward and Futures Pricing When the Underlying Generates Cash Flows For example, dividends on a stock or indexFor example, dividends on a stock or index

Assume one dividend DAssume one dividend DTT paid at expiration. paid at expiration. Buy stock, sell futures guarantees at expiration that you will have Buy stock, sell futures guarantees at expiration that you will have

DDTT + f + f00(T). Present value of this must equal S(T). Present value of this must equal S00, using risk-free , using risk-free rate. Thus,rate. Thus,

• ff00(T) = S(T) = S00(1+r)(1+r)TT - D - DTT.. For multiple dividends, let DFor multiple dividends, let DTT be compound future value of dividends. be compound future value of dividends.

See See Figure 9.5, p. 324Figure 9.5, p. 324 for two dividends. for two dividends. Dividends reduce the cost of carry.Dividends reduce the cost of carry. If DIf D00 represents the present value of the dividends, the model becomes represents the present value of the dividends, the model becomes

• ff00(T) = (S(T) = (S00 – D – D00)(1+r))(1+r)TT..


Ch. 9: 14


Forward and Futures Pricing When the Underlying Forward and Futures Pricing When the Underlying Generates Cash Flows (continued)Generates Cash Flows (continued) For dividends paid at a continuously compounded rate For dividends paid at a continuously compounded rate

of of c, ,

Example: SExample: S00 = 50, r = 50, rcc = .08, = .08, c= .06, expiration in 60 = .06, expiration in 60

days (T = 60/365 = .164).days (T = 60/365 = .164). ff00(T) = 50e(T) = 50e(.08 - .06)(.164)(.08 - .06)(.164) = 50.16. = 50.16.

)T(r0

ceST)f(0, c−=


Ch. 9: 15


Another Look at Valuation of Forward ContractsAnother Look at Valuation of Forward Contracts When there are dividends, to determine the value of a When there are dividends, to determine the value of a

forward contract during its life forward contract during its life VVtt(0,T) = S(0,T) = Stt – D – Dt,Tt,T – F(0,T)(1 + r) – F(0,T)(1 + r)-(T-t)-(T-t)

where Dwhere Dt,Tt,T is the value at t of the future dividends to is the value at t of the future dividends to TT

Or if dividends are continuous,Or if dividends are continuous,

Or for currency forwards,Or for currency forwards,

)()(t ),0(),0(V tTrtT

tcc eTFeST −−−− −=

)()(t )1)(,0()1(),0(V tTtT

t rTFST −−−− +−+= ρ


Ch. 9: 16


Pricing Foreign Currency Forward and Futures Contracts: Pricing Foreign Currency Forward and Futures Contracts: Interest Rate ParityInterest Rate Parity The relationship between spot and forward or futures The relationship between spot and forward or futures

prices of a currency. Same as cost of carry model in prices of a currency. Same as cost of carry model in other forward and futures markets.other forward and futures markets.

Proves that one cannot convert a currency to another Proves that one cannot convert a currency to another currency, sell a futures, earn the foreign risk-free rate, currency, sell a futures, earn the foreign risk-free rate, and convert back without risk, earning a rate higher and convert back without risk, earning a rate higher than the domestic rate.than the domestic rate.


Ch. 9: 17

A Forward and Futures Pricing Model (continued) Pricing Foreign Currency Forward and Futures Contracts: Interest Pricing Foreign Currency Forward and Futures Contracts: Interest

Rate Parity (continued)Rate Parity (continued) SS00 = spot rate in domestic currency per foreign currency. Foreign = spot rate in domestic currency per foreign currency. Foreign

rate is rate is ρ. Holding period is T. Domestic rate is r. Take STake S00(1+ (1+ ρ))-T-T units of domestic currency and buy (1+ units of domestic currency and buy (1+ ρ))-T-T

units of foreign currency.units of foreign currency. Sell forward contract to deliver one unit of foreign currency at Sell forward contract to deliver one unit of foreign currency at

T at price F(0,T).T at price F(0,T). Hold foreign currency and earn rate Hold foreign currency and earn rate ρ. At T you will have one

unit of the foreign currency. Deliver foreign currency and receive F(0,T) units of domestic

currency.


Ch. 9: 18


Pricing Foreign Currency Forward and Futures Contracts: Pricing Foreign Currency Forward and Futures Contracts: Interest Rate Parity (continued)Interest Rate Parity (continued)

So an investment of SSo an investment of S00(1+ (1+ ρρ))-T-T units of domestic units of domestic

currency grows to Fcurrency grows to F (0,T) units of domestic currency (0,T) units of domestic currency

with no risk. Return should be r. Thereforewith no risk. Return should be r. Therefore

• F(0,T) = SF(0,T) = S00(1+ (1+ ρρ))-T-T(1 + r)(1 + r)TT

This is called interest rate parity.This is called interest rate parity. Sometimes written asSometimes written as

• F(0,T) = SF(0,T) = S00(1 + r)(1 + r)TT/(1 + /(1 + ρρ))TT


Ch. 9: 19

A Forward and Futures Pricing Model (continued) Pricing Foreign Currency Forward and Futures Contracts: Interest Pricing Foreign Currency Forward and Futures Contracts: Interest

Rate Parity (continued)Rate Parity (continued) Example (from a European perspective): SExample (from a European perspective): S00 = €1.0304. U. S. = €1.0304. U. S.

rate is 5.84%. Euro rate is 3.59%. Time to expiration is 90/365 rate is 5.84%. Euro rate is 3.59%. Time to expiration is 90/365 = .2466.= .2466. F(0,T) = €1.0304(1.0584)F(0,T) = €1.0304(1.0584)-0.2466-0.2466(1.0359)(1.0359)0.24660.2466 = €1.025 = €1.025

If forward rate is actually €1.03, then it is overpriced.If forward rate is actually €1.03, then it is overpriced. Buy (1.0584)Buy (1.0584)-0.2466-0.2466 = $0.9861 for 0.9861(€1.0304) = €1.0161. = $0.9861 for 0.9861(€1.0304) = €1.0161.

Sell one forward contract at €1.03.Sell one forward contract at €1.03. Earn 5.84% on $0.9861. This grows to $1.Earn 5.84% on $0.9861. This grows to $1. At expiration, deliver $1 and receive €1.03.At expiration, deliver $1 and receive €1.03. Return is (1.03/1.0161)Return is (1.03/1.0161)365/90365/90 - 1 = .0566 (> .0359) - 1 = .0566 (> .0359) This transaction is called covered interest arbitrage.This transaction is called covered interest arbitrage.


Ch. 9: 20


Pricing Foreign Currency Forward and Futures Contracts: Pricing Foreign Currency Forward and Futures Contracts: Interest Rate Parity (continued)Interest Rate Parity (continued) It is also sometimes written asIt is also sometimes written as

F(0,T) = SF(0,T) = S00(1 + (1 + ρρ))TT(1 + r)(1 + r)-T-T

Here the spot rate is being quoted in units of the Here the spot rate is being quoted in units of the foreign currency.foreign currency.

Note that the forward discount/premium has nothing to Note that the forward discount/premium has nothing to do with expectations of future exchange rates.do with expectations of future exchange rates.

Difference between domestic and foreign rate is Difference between domestic and foreign rate is analogous to difference between risk-free rate and analogous to difference between risk-free rate and dividend yield on stock index futures.dividend yield on stock index futures.


Ch. 9: 21

A Forward and Futures Pricing Model (continued) Prices of Futures Contracts of Different ExpirationsPrices of Futures Contracts of Different Expirations

Expirations of TExpirations of T22 and T and T11 where T where T22 > T > T11..

Then fThen f00(1) = S(1) = S00 + + 1 and ff00(2) = S(2) = S00 + +

Spread will be f0(2) - f0(1) = 2 - 1.


Ch. 9: 22

Put-Call-Forward/Futures Parity

Can construct synthetic futures with options.Can construct synthetic futures with options. See See Table 9.5, p. 330Table 9.5, p. 330.. Put-call-forward/futures parityPut-call-forward/futures parity

PPee(S(S00,T,X) = C,T,X) = Cee(S(S00,T,X) + (X - f,T,X) + (X - f00(T))(1+r)(T))(1+r)-T-T

Numerical example using S&P 500. On May 14, S&P 500 Numerical example using S&P 500. On May 14, S&P 500 at 1337.80 and June futures at 1339.30. June 1340 call at at 1337.80 and June futures at 1339.30. June 1340 call at 40 and put at 39. Expiration of June 18 so T = 35/365 40 and put at 39. Expiration of June 18 so T = 35/365 = .0959. Risk-free rate at 4.56.= .0959. Risk-free rate at 4.56.


Ch. 9: 23

Put-Call-Forward/Futures Parity (continued)

So PSo Pee(S(S00,T,X) = 39,T,X) = 39

CCee(S(S00,T,X) + (X - f,T,X) + (X - f00(T))(1+r)(T))(1+r)-T-T

= 40 + (1340 - 1339.30)(1.0456)= 40 + (1340 - 1339.30)(1.0456)-0.0959-0.0959 = 40.70. = 40.70. Buy put and futures for 39, sell call and bond for 40.70 Buy put and futures for 39, sell call and bond for 40.70

and net 1.70 profit at no risk. Transaction costs would and net 1.70 profit at no risk. Transaction costs would have to be considered.have to be considered.


Ch. 9: 24

Pricing Options on Futures

The Intrinsic Value of an American Option on FuturesThe Intrinsic Value of an American Option on Futures Minimum value of American call on futuresMinimum value of American call on futures

CCaa(f(f00(T),T,X) (T),T,X) Max(0, ff00(T)(T) - X)

Minimum value of American put on futuresMinimum value of American put on futures PPaa(f(f00(T),T,X) (T),T,X) Max(0,X - ff00(T)(T))

Difference between option price and intrinsic value is Difference between option price and intrinsic value is time value.time value.


Ch. 9: 25

Pricing Options on Futures (continued)

The Lower Bound of a European Option on FuturesThe Lower Bound of a European Option on Futures For calls, construct two portfolios. See For calls, construct two portfolios. See

Table 9.6, p. 332Table 9.6, p. 332.. Portfolio A dominates Portfolio B soPortfolio A dominates Portfolio B so

CCee(f(f00(T),T,X) (T),T,X) Max[0,(ff00(T)(T) - X)(1+r)-T] Note that lower bound can be less than intrinsic value Note that lower bound can be less than intrinsic value

even for calls.even for calls. For puts, see For puts, see Table 9.7, p. 333Table 9.7, p. 333.. Portfolio A dominates Portfolio B soPortfolio A dominates Portfolio B so

PPee(f(f00(T),T,X) (T),T,X) Max[0,(X - ff00(T)(T))(1+r)-T]


Ch. 9: 26


Put-Call Parity of Options on FuturesPut-Call Parity of Options on Futures Construct two portfolios, A and B.Construct two portfolios, A and B. See See Table 9.8, p. 335Table 9.8, p. 335.. The portfolios produce equivalent results. Therefore The portfolios produce equivalent results. Therefore

they must have equivalent current values. Thus,they must have equivalent current values. Thus, PPee(f(f00(T),T,X) = C(T),T,X) = Cee(f(f00(T),T,X) + (X - f(T),T,X) + (X - f00(T))(1+r)(T))(1+r)-T-T..

Compare to put-call parity for options on spot:Compare to put-call parity for options on spot: PPee(S(S00,T,X) = C,T,X) = Cee(S(S00,T,X) - S,T,X) - S00 + X(1+r) + X(1+r)-T-T.. If options on spot and options on futures expire at If options on spot and options on futures expire at

same time, their values are equal, implying fsame time, their values are equal, implying f00(T) = (T) = SS00(1+r)(1+r)TT, which we obtained earlier (no cash flows)., which we obtained earlier (no cash flows).


Ch. 9: 27


Early Exercise of Call and Put Options on FuturesEarly Exercise of Call and Put Options on Futures Deep in-the-money call may be exercised early becauseDeep in-the-money call may be exercised early because

behaves almost identically to futuresbehaves almost identically to futures exercise frees up funds tied up in option but requires exercise frees up funds tied up in option but requires

no funds to establish futuresno funds to establish futures minimum value of European futures call is less than minimum value of European futures call is less than

value if it could be exercisedvalue if it could be exercised See See Figure 9.6, p. 337Figure 9.6, p. 337.. Similar arguments hold for putsSimilar arguments hold for puts Compare to the arguments for early exercise of call and Compare to the arguments for early exercise of call and

put options on spot.put options on spot.


Ch. 9: 28


Options on Futures Pricing ModelsOptions on Futures Pricing Models Black model for pricing European options on futuresBlack model for pricing European options on futures

( )

Tdd

T

T/2(T)/X)ln(fd

where

)]XN(d)(T)N(d[feC

12

20

1

210Trc

σ

σ

σ

−=

+=

−= −


Ch. 9: 29


Options on Futures Pricing Models (continued)Options on Futures Pricing Models (continued) Note that with the same expiration for options on spot Note that with the same expiration for options on spot

as options on futures, this formula gives the same price.as options on futures, this formula gives the same price. ExampleExample

See See Table 9.9, p. 339Table 9.9, p. 339.. Software for Black-Scholes can be used by inserting Software for Black-Scholes can be used by inserting

futures price instead of spot price and risk-free rate for futures price instead of spot price and risk-free rate for dividend yield. Note why this works.dividend yield. Note why this works.

For putsFor puts

)]N(d[1(T)ef)]N(d[1XeP 1Tr

02Tr cc −−−= −−


Ch. 9: 30

See Figure 9.7, p. 341 for linkage between forwards/futures, underlying asset and risk-free bond.

Summary


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(Return to text slide)


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chapter 9: principles of pricing forwards, futures, and options on futures

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