Lecture_12 - 12 Forwards and Futures Under stochastic...

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1 12 Forwards and Futures Under stochastic interest rates, we will show that forward and futures contracts are distinct securities and that forward and futures prices are (usually) different quantities. This distinction is not uniformly understood. There are still textbooks available which confuse these two contacts. One contract cannot be used (in a simple fashion) to arbitrage the other, contrary to occasional practice. The futures contracts analyzed here are simplified versions of the exchange-traded futures contracts (see Chapter 1).
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2 A Forwards Consider a forward contract issued on a T 2 -maturity zero-coupon bond with delivery date T 1 where 2 1 T T . The time t forward price on this contract is denoted F(t, T 1 :T 2 ) . The purpose of this section is to use the contingent claim valuation methodology to give an alternative characterization of the forward price. Given there are no arbitrage opportunities and markets are complete, the subsequent analysis proceeds independently of the particular economy (with one, two, or N 3 factors) studied. Let ) ( t v denote the value of this forward contract at time t, when it is initiated.
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3 We value this forward contract using the risk- neutral valuation procedure. The time 1 T payoff is ) 2 : 1 , ( ) 2 , 1 ( T T t F T T P . Applying the risk-neutral valuation formula gives ) ( ) 1 ( ) 2 : 1 , ( ) 2 , 1 ( ~ ) ( t B T B T T t F T T P t E t v = . (12.1)
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4 By market convention, the forward price ) 2 : 1 , ( T T t F is determined such that at initiation, the forward contract has zero value. The forward price ) 2 : 1 , ( T T t F is determined such that . 0 ) ( ) 1 ( ) 2 : 1 , ( ) 2 , 1 ( ~ = t B T B T T t F T T P t E We now solve this equation. . 0 ) ( ) 1 ( ) 2 : 1 , ( ~ ) ( ) 1 ( ) 2 , 1 ( ~ = t B T B T T t F t E t B T B T T P t E (12.2)
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5 But, ) ( ) 1 ( ) 2 , 1 ( ~ ) 2 , ( t B T B T T P t E T t P = . Similarly, ) ( ) 1 ( 1 ~ ) 1 , ( t B T B t E T t P = . We also know that ) 2 : 1 , ( T T t F is a constant at time t.
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6 Combined, these facts imply that expression (12.2) can be rewritten as: . 0 ) 1 , ( ) 2 : 1 , ( ) 2 , ( = T t P T T t F T t P (12.3) Rearranging terms gives the desired result: ) 1 , ( ) 2 , ( ) 2 : 1 , ( T t P T t P T T t F = . (12.4) This expression is easy to understand if one recognizes the ratio 1 /P(t, T 1 ) as the future value of a dollar received at time T 1 . Then expression (12.4) can be interpreted as the future value at time T 1 of the T 2 -maturity zero-coupon bond – hence, a forward price.
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7 The self-financing trading strategy that synthetically replicates a forward contract is easily determined. The synthetic forward contract is constructed with what is called a "cash and carry" strategy.
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8 Table 12.1: Cash Flows to a Cash and Carry Trading Strategy Time T 1 Cash Flows Forward Contract P(T 1 , T 2 ) F(t, T 1 : T 2 ) Synthetic Forward Buy T 2 -maturity bond 1 2 ) Sell F(t, T 1 2 )– F ( t , T 1 2 ) •1 T 1 -maturity bonds SUM 1 2 1 2 )
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9 After the initiation date, the forward contract can have nonzero value.
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This note was uploaded on 10/31/2010 for the course NBA 5550 taught by Professor Jarrow,robert during the Fall '08 term at Cornell University (Engineering School).

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Lecture_12 - 12 Forwards and Futures Under stochastic...

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