Copyright © 2001 by Harcourt, Inc. All rights reserved.1 Chapter 9: Principles of Forward and Futures Pricing A good part of the pricing is about sticking a wet finger in the air, and hoping you get it right.. Eric Bakker AsiaRisk, February 1999

Copyright © 2001 by Harcourt, Inc. All rights reserved.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

Copyright © 2001 by Harcourt, Inc. All rights reserved.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.

Copyright © 2001 by Harcourt, Inc. All rights reserved.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 short is -1 times this.

Copyright © 2001 by Harcourt, Inc. All rights reserved.5 Some Properties of Forward and Futures Prices (continued) n The Value of a Forward Contract (continued) u Go long a contract at price F(0,T). u At time t (prior to expiration), go short a new contract expiring at T but with price F(t,T). u Value of first contract will be F V t (0,T) = (F(t,T) - F(0, T))(1+r) -(T-t). F See Table 9.1, p u Example: Go long 45 day contract at F(0,T) = $100. Risk-free rate = days later, contracts maturing in 25 days have price of $104. Value of first contract = F ( )(1.10) -25/365 =

Copyright © 2001 by Harcourt, Inc. All rights reserved.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.

Copyright © 2001 by Harcourt, Inc. All rights reserved.7 Some Properties of Forward and Futures Prices (continued) n Forward Versus Futures Prices u One day prior to expiration: F f t (T) = F(t,T). See Table 9.2, p u Two days or more prior to expiration, assuming known interest rate: F f t-1 (T) = F(t-1,T). See Table 9.3, p. 365.

Copyright © 2001 by Harcourt, Inc. All rights reserved.8 Some Properties of Forward and Futures Prices (continued) n Forward Versus Futures Prices (continued) u Why forward and futures prices might not be equal F Uncertain interest rates. See Table 9.4, p Futures > forward when futures prices move together with rates.Futures > forward when futures prices move together with rates. Futures < forward when futures prices move together with interest rates.Futures < forward when futures prices move together with interest rates. Futures = forward when futures prices are unrelated to interest rates.Futures = forward when futures prices are unrelated to interest rates. F Default risk can also affect forward-futures differential.

Copyright © 2001 by Harcourt, Inc. All rights reserved.9 Some Properties of Forward and Futures Prices (continued) n Forward Versus Futures Prices (continued) u Comparing forward and futures prices of Treasury bills u See Figure 9.1, p n The Term Structure Constructed with Futures u Term structure of interest rates: relationship between the interest rate and the maturity of a loan or bond u Spot rate: rate on a transaction to start now u Forward rate: rate agreed on now on a transaction to start later

Copyright © 2001 by Harcourt, Inc. All rights reserved.10 Some Properties of Forward and Futures Prices (continued) n The Term Structure Constructed with Futures (continued) u Derivation of forward rate: Buy 2 year bond at 8% or 1 year bond at 6% plus forward contract to buy 1 year bond in 1year at r%. To avoid arbitrage, we must have (1.08) 2 = (1.06)(1 + r). So r = u Let us construct the term structure, which is the set of spot rates, from the futures prices, which serve as forward rates. We use T-bill futures on November 15. u T-bill futures quoted using IMM Index method. For example, Index = This implies the discount is = 3.16.

Copyright © 2001 by Harcourt, Inc. All rights reserved.11 Some Properties of Forward and Futures Prices (continued) n The Term Structure Constructed with Futures (continued) u Futures prices is (91/360) = (futures actually uses 90 days but we want a 91 day rate so we ignore this slight difference). This contract expires in 32 days on December 17. u The return is (100/ ) 365/ = This is the rate r(32,123). Using the spot rate on a 32 day T-bill with the same procedure gives r(0,32) = u Now the spot rate for 123 days, r(0,123), is F [(1.0305) 32/365 (1.0327) 91/365 ] 365/ =.0321

Copyright © 2001 by Harcourt, Inc. All rights reserved.12 Some Properties of Forward and Futures Prices (continued) n The Term Structure Constructed with Futures (continued) u We continue to do this as far out as we can. u See Figure 9.2, p. 375 showing how the shorter-term spot rates and forward rates link together to construct longer-term spot rates. u Remember that first we get the discount rate, then convert it to a price and then convert the price to a yield. Then we link the spot and futures yields together to obtain the longer term spot yield.

Copyright © 2001 by Harcourt, Inc. All rights reserved.13 A Forward and Futures Pricing Model n Spot Prices, Risk Premiums, and the Cost of Carry 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(  )

Copyright © 2001 by Harcourt, Inc. All rights reserved.14 A Forward and Futures Pricing Model n Spot Prices, Risk Premiums, and the Cost of Carry (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

Copyright © 2001 by Harcourt, Inc. All rights reserved.15 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 9.3, p. 379 u Note how arbitrage and quasi-arbitrage make this hold.

Copyright © 2001 by Harcourt, Inc. All rights reserved.16 A Forward and Futures Pricing Model (continued) n The Theoretical Fair Price (continued) u See Figure 9.4, p. 381 for an illustration of the determination of futures prices. u Contango is f 0 (T) > S 0. See Table 9.5, 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 9.6, 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 9.6, p. 383.

Copyright © 2001 by Harcourt, Inc. All rights reserved.17 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 9.5, p 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 9.6, p  E(S T ) = f 0 (T) + E(  ). See Figure 9.6, p u Normal contango: E(S T ) < f 0 (T) u Normal backwardation: f 0 (T) < E(S T )

Copyright © 2001 by Harcourt, Inc. All rights reserved.18 A Forward and Futures Pricing Model (continued) n The Effect of Intermediate 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 9.7, p. 391 for two dividends. u Dividends reduce the cost of carry.

Copyright © 2001 by Harcourt, Inc. All rights reserved.19 A Forward and Futures Pricing Model (continued) n The Effect of Intermediate Cash Flows (continued)  For dividends paid at a continuously compounded rate of,  For dividends paid at a continuously compounded rate of ,  Example: S 0 = 50, r c =.08, =.06, expiration in 60 days (T = 60/365 =.164).  Example: S 0 = 50, r c =.08,  =.06, expiration in 60 days (T = 60/365 =.164). F f 0 (T) = 50e ( )(.164) =

Copyright © 2001 by Harcourt, Inc. All rights reserved.20 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. u u See Figures 9.8, p. 394 and 9.9, p. 395 for example with Eurodollars.

Copyright © 2001 by Harcourt, Inc. All rights reserved.21 See Figure 9.10, p. 396 for linkage between forwards/futures, stock and risk-free bond. Summary