D. M. ChanceAn 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. Charles Dow Money Talks (by Rosalie Maggio), 1998, p. 23

D. M. ChanceAn Introduction to Derivatives and Risk Management, 6th ed.Ch. 9: 2 Important Concepts in Chapter 9 n Price and value of forward and futures contracts n Relationship between forward and futures prices n Determination of the spot price of an asset n Cost of carry model for theoretical fair price n Contango, backwardation, and convenience yield n Futures prices and risk premiums n Futures spread pricing n Pricing options on futures

D. M. ChanceAn Introduction to Derivatives and Risk Management, 6th ed.Ch. 9: 3 Some Properties of Forward and Futures Prices n The Concept of Price Versus Value u Normally in an efficient market, price = value. u For futures or forward, price is the contracted rate of future purchase. Value is something different. u At the beginning of a contract, value = 0 for both futures and forwards. n Notation u V t (0,T), F(0,T), v t (T), f t (T) are values and prices of forward and futures contracts created at time 0 and expiring at time T.

D. M. ChanceAn Introduction to Derivatives and Risk Management, 6th ed.Ch. 9: 4 Some Properties of Forward and Futures Prices (continued) n The Value of a Forward Contract u Forward price at expiration: F F(T,T) = S T. F That is, the price of an expiring forward contract is the spot price. u Value of forward contract at expiration: F V T (0,T) = S T - F(0,T). F An expiring forward contract allows you to buy the asset, worth S T, at the forward price F(0,T). The value to the short party is -1 times this.

D. M. ChanceAn Introduction to Derivatives and Risk Management, 6th ed.Ch. 9: 5 Some Properties of Forward and Futures Prices (continued) n The Value of a Forward Contract (continued) u The Value of a Forward Contract Prior to Expiration F A: Go long forward contract at price F(0,T) at time 0. F 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, S T – F(0,T). Thus, they must both be worth the same prior to t.At time T, A and B are worth the same, S T – F(0,T). Thus, they must both be worth the same prior to t. So V t (0,T) = S t – F(0,T) -(T-t)So V t (0,T) = S t – F(0,T) -(T-t) See Table 9.1, p. 306.See Table 9.1, p. 306.Table 9.1, p. 306Table 9.1, p. 306 F Example: Go long 45 day contract at F(0,T) = $100. Risk-free rate = days later, the spot price is $102. The value of the forward contract is (1.10) -25/365 = 2.65.

D. M. ChanceAn Introduction to Derivatives and Risk Management, 6th ed.Ch. 9: 6 Some Properties of Forward and Futures Prices (continued) n The Value of a Futures Contract u Futures price at expiration: F f T (T) = S T. u Value during the trading day but before being marked to market: F v t (T) = f t (T) - f t-1 (T). u Value immediately after being marked to market: F v t (T) = 0.

D. M. ChanceAn Introduction to Derivatives and Risk Management, 6th ed.Ch. 9: 7 Some Properties of Forward and Futures Prices (continued) n Forward Versus Futures Prices u Forward and futures prices will be equal F One day prior to expiration F More 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 u Futures prices will exceed forward prices if futures prices are positively correlated with interest rates. u Default risk can also affect the difference between futures and forward prices.

D. M. ChanceAn Introduction to Derivatives and Risk Management, 6th ed.Ch. 9: 8 A Forward and Futures Pricing Model n Spot Prices, Risk Premiums, and the Cost of Carry for Generic Assets u First assume no uncertainty of future price. Let s be the cost of storing an asset and i be the interest rate for the period of time the asset is owned. Then F S 0 = S T - s - iS 0 u If we now allow uncertainty but assume people are risk neutral, we have F S 0 = E(S T ) - s - iS 0 u If we now allow people to be risk averse, they require a risk premium of E(  ). Now F S 0 = E(S T ) - s - iS 0 - E(  )

D. M. ChanceAn Introduction to Derivatives and Risk Management, 6th ed.Ch. 9: 9 A Forward and Futures Pricing Model n Spot Prices, Risk Premiums, and the Cost of Carry for Generic Assets (continued) u Let us define iS 0 as the net interest, which is the interest foregone minus any cash received. u Define s + iS 0 as the cost of carry. u Denote cost of carry as . u Note how cost of carry is a meaningful concept only for storable assets

D. M. ChanceAn Introduction to Derivatives and Risk Management, 6th ed.Ch. 9: 10 A Forward and Futures Pricing Model n The Theoretical Fair Price u Do the following F Buy asset in spot market, paying S 0 ; sell futures contract at price f 0 (T); store and incur costs. F At expiration, make delivery. Profit: = f 0 (T) - S 0 -   = f 0 (T) - S 0 -  u This must be zero to avoid arbitrage; thus, f 0 (T) = S 0 + f 0 (T) = S 0 +  u See Figure 9.1, p Figure 9.1, p. 313Figure 9.1, p. 313 u Note how arbitrage and quasi-arbitrage make this hold.

D. M. ChanceAn Introduction to Derivatives and Risk Management, 6th ed.Ch. 9: 11 A Forward and Futures Pricing Model (continued) n The Theoretical Fair Price (continued) u See Figure 9.2, p. 314 for an illustration of the determination of futures prices. Figure 9.2, p. 314Figure 9.2, p. 314 u Contango is f 0 (T) > S 0. See Table 9.2, p Table 9.2, p. 315Table 9.2, p. 315  When f 0 (T) < S 0, convenience yield is, an additional return from holding asset when in short supply or a non-pecuniary return. Market is said to be at less than full carry and in backwardation or inverted. See Table 9.3, p Market can be both backwardation and contango. See Table 9.4, p  When f 0 (T) < S 0, convenience yield is , an additional return from holding asset when in short supply or a non-pecuniary return. Market is said to be at less than full carry and in backwardation or inverted. See Table 9.3, p Market can be both backwardation and contango. See Table 9.4, p. 317.Table 9.3, p. 316Table 9.4, p. 317Table 9.3, p. 316Table 9.4, p. 317

D. M. ChanceAn Introduction to Derivatives and Risk Management, 6th ed.Ch. 9: 12 A Forward and Futures Pricing Model (continued) n Futures Prices and Risk Premia u The no risk-premium hypothesis F Market consists of only speculators. F f 0 (T) = E(S T ). See Figure 9.3, p Figure 9.3, p. 319Figure 9.3, p. 319 u The risk-premium hypothesis F E(f T (T)) > f 0 (T). F When hedgers go short futures, they transfer risk premium to speculators who go long futures.  E(S T ) = f 0 (T) + E(). See Figure 9.4, p  E(S T ) = f 0 (T) + E(  ). See Figure 9.4, p. 321.Figure 9.4, p. 321Figure 9.4, p. 321 u Normal contango: E(S T ) < f 0 (T) u Normal backwardation: f 0 (T) < E(S T )

D. M. ChanceAn Introduction to Derivatives and Risk Management, 6th ed.Ch. 9: 13 A Forward and Futures Pricing Model (continued) n Forward and Futures Pricing When the Underlying Generates Cash Flows u For example, dividends on a stock or index F Assume one dividend D T paid at expiration. F Buy stock, sell futures guarantees at expiration that you will have D T + f 0 (T). Present value of this must equal S 0, using risk-free rate. Thus, f 0 (T) = S 0 (1+r) T - D T.f 0 (T) = S 0 (1+r) T - D T. u For multiple dividends, let D T be compound future value of dividends. See Figure 9.5, p. 324 for two dividends. Figure 9.5, p. 324Figure 9.5, p. 324 u Dividends reduce the cost of carry. u If D 0 represents the present value of the dividends, the model becomes f 0 (T) = (S 0 – D 0 )(1+r) T.f 0 (T) = (S 0 – D 0 )(1+r) T.

D. M. ChanceAn Introduction to Derivatives and Risk Management, 6th ed.Ch. 9: 14 A Forward and Futures Pricing Model (continued) n Forward and Futures Pricing When the Underlying Generates Cash Flows (continued)  For dividends paid at a continuously compounded rate of,  For dividends paid at a continuously compounded rate of  c,  Example: S 0 = 50, r c =.08, =.06, expiration in 60 days (T = 60/365 =.164).  Example: S 0 = 50, r c =.08,  c  =.06, expiration in 60 days (T = 60/365 =.164). F f 0 (T) = 50e ( )(.164) =

D. M. ChanceAn Introduction to Derivatives and Risk Management, 6th ed.Ch. 9: 15 A Forward and Futures Pricing Model (continued) n Another Look at Valuation of Forward Contracts u When there are dividends, to determine the value of a forward contract during its life F V t (0,T) = S t – D t,T – F(0,T)(1 + r) -(T-t) F where D t,T is the value at t of the future dividends to T u Or if dividends are continuous, u Or for currency forwards,

D. M. ChanceAn Introduction to Derivatives and Risk Management, 6th ed.Ch. 9: 16 A Forward and Futures Pricing Model (continued) n Pricing Foreign Currency Forward and Futures Contracts: Interest Rate Parity u The relationship between spot and forward or futures prices of a currency. Same as cost of carry model in other forward and futures markets. u 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.

D. M. ChanceAn Introduction to Derivatives and Risk Management, 6th ed.Ch. 9: 17 A Forward and Futures Pricing Model (continued) n Pricing Foreign Currency Forward and Futures Contracts: Interest Rate Parity (continued)  S 0 = spot rate in domestic currency per foreign currency. Foreign rate is  S 0 = spot rate in domestic currency per foreign currency. Foreign rate is . Holding period is T. Domestic rate is r.  Take S 0 (1+ ) -T units of domestic currency and buy (1+ ) -T units of foreign currency.  Take S 0 (1+  ) -T units of domestic currency and buy (1+  ) -T units of foreign currency. F Sell forward contract to deliver one unit of foreign currency at T at price F(0,T).  Hold foreign currency and earn rate  Hold foreign currency and earn rate . At T you will have one unit of the foreign currency. F F Deliver foreign currency and receive F(0,T) units of domestic currency.

D. M. ChanceAn Introduction to Derivatives and Risk Management, 6th ed.Ch. 9: 18 A Forward and Futures Pricing Model (continued) n Pricing Foreign Currency Forward and Futures Contracts: Interest Rate Parity (continued)  So an investment of S 0 (1+  ) -T units of domestic currency grows to F (0,T) units of domestic currency with no risk. Return should be r. Therefore F(0,T) = S 0 (1+  ) -T (1 + r) TF(0,T) = S 0 (1+  ) -T (1 + r) T F This is called interest rate parity. F Sometimes written as F(0,T) = S 0 (1 + r) T /(1 +  ) TF(0,T) = S 0 (1 + r) T /(1 +  ) T

D. M. ChanceAn Introduction to Derivatives and Risk Management, 6th ed.Ch. 9: 19 A Forward and Futures Pricing Model (continued) n Pricing Foreign Currency Forward and Futures Contracts: Interest Rate Parity (continued) u Example (from a European perspective): S 0 = € U. S. rate is 5.84%. Euro rate is 3.59%. Time to expiration is 90/365 = F F(0,T) = €1.0304(1.0584) (1.0359) = €1.025 u If forward rate is actually €1.03, then it is overpriced. F Buy (1.0584) = $ for (€1.0304) = € Sell one forward contract at €1.03. F Earn 5.84% on $ This grows to $1. F At expiration, deliver $1 and receive €1.03. F Return is (1.03/1.0161) 365/ =.0566 (>.0359) F This transaction is called covered interest arbitrage.

D. M. ChanceAn Introduction to Derivatives and Risk Management, 6th ed.Ch. 9: 20 A Forward and Futures Pricing Model (continued) n Pricing Foreign Currency Forward and Futures Contracts: Interest Rate Parity (continued) u It is also sometimes written as F F(0,T) = S 0 (1 +  ) T (1 + r) -T F Here the spot rate is being quoted in units of the foreign currency. u Note that the forward discount/premium has nothing to do with expectations of future exchange rates. u Difference between domestic and foreign rate is analogous to difference between risk-free rate and dividend yield on stock index futures.

D. M. ChanceAn Introduction to Derivatives and Risk Management, 6th ed.Ch. 9: 21 A Forward and Futures Pricing Model (continued) n Prices of Futures Contracts of Different Expirations u Expirations of T 2 and T 1 where T 2 > T 1.  Then f 0 (1) = S 0 + f 0 (2) = S 0 +  Then f 0 (1) = S 0 +  1 and f 0 (2) = S 0 +   u u Spread will be   f 0 (2) - f 0 (1) =  2 -  1.

D. M. ChanceAn Introduction to Derivatives and Risk Management, 6th ed.Ch. 9: 22 Put-Call-Forward/Futures Parity n Can construct synthetic futures with options. n See Table 9.5, p Table 9.5, p. 330Table 9.5, p. 330 n Put-call-forward/futures parity u P e (S 0,T,X) = C e (S 0,T,X) + (X - f 0 (T))(1+r) -T n Numerical example using S&P 500. On May 14, S&P 500 at and June futures at June 1340 call at 40 and put at 39. Expiration of June 18 so T = 35/365 = Risk-free rate at 4.56.

D. M. ChanceAn Introduction to Derivatives and Risk Management, 6th ed.Ch. 9: 23 Put-Call-Forward/Futures Parity (continued) u So P e (S 0,T,X) = 39 u C e (S 0,T,X) + (X - f 0 (T))(1+r) -T u = 40 + ( )(1.0456) = u Buy put and futures for 39, sell call and bond for and net 1.70 profit at no risk. Transaction costs would have to be considered.

D. M. ChanceAn Introduction to Derivatives and Risk Management, 6th ed.Ch. 9: 24 Pricing Options on Futures n The Intrinsic Value of an American Option on Futures u Minimum value of American call on futures  C a (f 0 (T),T,X) f 0 (T)  C a (f 0 (T),T,X)  Max(0, f 0 (T) - X) u Minimum value of American put on futures  P a (f 0 (T),T,X) f 0 (T)  P a (f 0 (T),T,X)  Max(0,X - f 0 (T)) u Difference between option price and intrinsic value is time value.

D. M. ChanceAn Introduction to Derivatives and Risk Management, 6th ed.Ch. 9: 25 Pricing Options on Futures (continued) n The Lower Bound of a European Option on Futures u For calls, construct two portfolios. See Table 9.6, p Table 9.6, p. 332Table 9.6, p. 332 u Portfolio A dominates Portfolio B so  C e (f 0 (T),T,X) f 0 (T)  C e (f 0 (T),T,X)  Max[0,(f 0 (T) - X)(1+r) -T ] u Note that lower bound can be less than intrinsic value even for calls. u For puts, see Table 9.7, p Table 9.7, p. 333Table 9.7, p. 333 u Portfolio A dominates Portfolio B so  P e (f 0 (T),T,X) f 0 (T)  P e (f 0 (T),T,X)  Max[0,(X - f 0 (T))(1+r) -T ]

D. M. ChanceAn Introduction to Derivatives and Risk Management, 6th ed.Ch. 9: 26 Pricing Options on Futures (continued) n Put-Call Parity of Options on Futures u Construct two portfolios, A and B. u See Table 9.8, p Table 9.8, p. 335Table 9.8, p. 335 u The portfolios produce equivalent results. Therefore they must have equivalent current values. Thus, F P e (f 0 (T),T,X) = C e (f 0 (T),T,X) + (X - f 0 (T))(1+r) -T. u Compare to put-call parity for options on spot: F P e (S 0,T,X) = C e (S 0,T,X) - S 0 + X(1+r) -T. F If options on spot and options on futures expire at same time, their values are equal, implying f 0 (T) = S 0 (1+r) T, which we obtained earlier (no cash flows).

D. M. ChanceAn Introduction to Derivatives and Risk Management, 6th ed.Ch. 9: 27 Pricing Options on Futures (continued) n Early Exercise of Call and Put Options on Futures u Deep in-the-money call may be exercised early because F behaves almost identically to futures F exercise frees up funds tied up in option but requires no funds to establish futures F minimum value of European futures call is less than value if it could be exercised u See Figure 9.6, p Figure 9.6, p. 337Figure 9.6, p. 337 u Similar arguments hold for puts u Compare to the arguments for early exercise of call and put options on spot.

D. M. ChanceAn Introduction to Derivatives and Risk Management, 6th ed.Ch. 9: 28 Pricing Options on Futures (continued) n Options on Futures Pricing Models u Black model for pricing European options on futures

D. M. ChanceAn Introduction to Derivatives and Risk Management, 6th ed.Ch. 9: 29 Pricing Options on Futures (continued) n Options on Futures Pricing Models (continued) u Note that with the same expiration for options on spot as options on futures, this formula gives the same price. u Example F See Table 9.9, p Table 9.9, p. 339Table 9.9, p. 339 u Software for Black-Scholes can be used by inserting futures price instead of spot price and risk-free rate for dividend yield. Note why this works. u For puts

D. M. ChanceAn Introduction to Derivatives and Risk Management, 6th ed.Ch. 9: 30 See Figure 9.7, p. 341 for linkage between forwards/futures, underlying asset and risk-free bond.Figure 9.7, p. 341 Summary

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