Lecture_08 - 1 8 Bond Trading Strategies-Theory A The...

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Unformatted text preview: 1 8 Bond Trading Strategies-Theory A The One-Factor Economy We assume that there are no arbitrage opportunities with respect to the smallest class of trading strategies 1 Φ . In this class of trading strategies, we choose a particular 1 τ-maturity zero-coupon bond and all trades take place with respect to it and the money market account. Taken as given (exogenous) are the stochastic process for the 1 τ-maturity zero-coupon bond, P( , 1 τ ) and (u(t, 1 τ ;s t ), d(t, 1 τ ;s t )) , and the stochastic process for the spot rate, r( ) , and (u(t,t+ 1 ;s t )) . The purpose is to price all the remaining zero-coupon bonds in the market. 2 1 Complete Markets First, we want to show that the market is complete with respect to the trading strategies 1 Φ . To do this, consider any arbitrary simple contingent claim with maturity date 1 τ- 1, denoted x( 1 τ- 1 ; 1 1 − τ s ) . 3 In this case, we need to choose the number of units of the money market account n ( 1 τ- 2 ; 2 1 − τ s ) and the number of 1 τ-maturity zero-coupon bonds purchased ) 2 1 ; 2 1 ( 1 − − τ τ τ s n at time 1 τ- 2 such that ) 2 1 ; 1 1 ( ) 2 1 ; 1 , 2 1 ( ) 2 1 ; 1 , 2 1 ( ) 2 1 ; 2 1 ( 1 ) 2 1 ; 2 1 ( ) 3 1 ; 2 1 ( ) 2 1 ; 2 1 ( u s x s u s P s n s r s B s n − − = − − − − − − + − − − − − − τ τ τ τ τ τ τ τ τ τ τ τ τ τ τ τ τ (8.1a) and ) 2 1 ; 1 1 ( ) 2 1 ; 1 , 2 1 ( ) 2 1 ; 1 , 2 1 ( ) 2 1 ; 2 1 ( 1 ) 2 1 ; 2 1 ( ) 3 1 ; 2 1 ( ) 2 1 ; 2 1 ( d s x s d s P s n s r s B s n − − = − − − − − − + − − − − − − τ τ τ τ τ τ τ τ τ τ τ τ τ τ τ τ τ . (8.1b) 4 The solution is given by ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ − − − − − − − − − − − − − − − − − − = − − 2 1 ; 1 , 2 1 2 1 ; 1 , 2 1 2 1 ; 2 1 3 1 ; 2 1 2 1 ; 1 , 2 1 2 1 ; 1 1 2 1 ; 1 , 2 1 2 1 ; 1 1 2 1 ; 2 1 τ τ τ τ τ τ τ τ τ τ τ τ τ τ τ τ τ τ τ τ τ τ s d s u s r s B s d u s x s u d s x s n (8.2a) and . 2 1 ; 1 , 2 1 2 1 ; 1 , 2 1 2 1 ; 1 , 2 1 2 1 ; 1 1 2 1 ; 1 1 2 1 ; 2 1 1 ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ − − − − − − − − − − − − = − − τ τ τ τ τ τ τ τ τ τ τ τ τ τ τ τ s d s u s P d s x u s x s n (8.2b) 5 As expression (8.2) makes clear, this time 1 τ- 2 portfolio choice only utilizes information available at time...
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This note was uploaded on 12/09/2008 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_08 - 1 8 Bond Trading Strategies-Theory A The...

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