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10-applications4(f)

# 10-applications4(f) - More Applications of Linear Pricing...

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Primbs, MS&E 345 1 More Applications of Linear Pricing

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Primbs, MS&E 345 2 Exchange one asset for another A generalization of Black-Scholes Black’s model with stochastic interest rates Bond options Caplets, etc. Futures, forwards, forward rates, and swap rates Swaptions Interest rate derivatives
Primbs, MS&E 345 3 Exchange one asset for another This is equivalent to the payoff at time T . + - )) ( ) ( ( 1 2 T S T S - = + ) ( )) ( ) ( ( ) 0 ( 1 1 2 1 0 1 T S T S T S E S c S Q Then our pricing formula is where ) ( 1 1 2 2 1 2 1 2 dz dz S S S S d σ σ - = from previous calculations To value this, we will let S 1 be the numeraire. 1 1 1 1 1 1 dz S dt S dS σ μ + = 2 2 2 2 2 2 dz S dt S dS σ μ + = Consider two assets: and the option to exchange asset 2 for asset 1 at time T . dt dz dz E ρ = ] [ 2 1 - = + 1 ) ( ) ( 1 2 1 T S T S E S Q

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Primbs, MS&E 345 4 - = + 1 ) ( ) ( ) 0 ( 1 2 1 0 1 T S T S E S c S Q where ) ( 1 1 2 2 1 2 1 2 dz dz S S S S d σ σ - = Note that: ) 2 , 0 ( ~ ) ( 2 1 2 2 2 1 1 1 2 2 σ ρσ σ σ σ σ - + - N dz dz 2 1 2 1 2 2 2 2 ˆ σ ρσ σ σ σ - + = So letting: z d S S S S d ˆ ˆ 1 2 1 2 σ = ( 29 [ ] + - 1 T f E Hence, letting we just need to evaluate: 1 2 S S f = z fd df ˆ ˆ σ = where Exchange one asset for another
Primbs, MS&E 345 5 ( 29 [ ] + - 1 T f E z fd df ˆ ˆ σ = where ( 29 [ ] ) ( ) ( 1 2 1 0 d N d N f f E T - = - + Evaluating gives T T f d σ σ ˆ 2 / ˆ ) ln( 2 0 1 + = T d d σ ˆ 1 2 - = where: Substituting in terms of S 1 and S 2 gives the final answer: ) ( ) 0 ( ) ( ) 0 ( 2 1 1 2 0 d N S d N S c - = T T S S d σ σ ˆ 2 / ˆ )) 0 ( / ) 0 ( ln( 2 1 2 1 + = T d d σ ˆ 1 2 - = 2 1 2 1 2 2 2 2 ˆ σ ρσ σ σ σ - + = where Exchange one asset for another

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Primbs, MS&E 345 6 Exchange one asset for another A generalization of Black-Scholes Black’s model with stochastic interest rates Bond options Caplets, etc. Futures, forwards, forward rates, and swap rates Swaptions Interest rate derivatives
Primbs, MS&E 345 7 Futures contracts and the risk neutral measure: time 0 time T time/position mark to market (at time t+dt) time T value T t T s df ds r ) exp( 0 = t ) exp( 0 + dt t s ds r T t dt t s df ds r ) exp( 0 + ) exp( ) exp( 0 + + T dt t s T t dt t s ds r df ds r t (start) t+dt

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Primbs, MS&E 345 8 Futures contracts and the risk neutral measure: time 0 time T time/position mark to market (at time t+dt) time T value t (start) t ) exp( 0 + dt t s ds r T t T s df ds r ) exp( 0 T t dt t s df ds r ) exp( 0 + T-dt ) exp( 0 T s ds r T dt T T s df ds r - ) exp( 0 T dt T T s df ds r - ) exp( 0 Total Cost: = 0 ) )( exp( 0 T t T T T s f f ds r - = Total Payoff: = T t T s T s df ds r ) exp( 0 ) )( exp( 0 T t T T s f S ds r - = Since f T T =S T t+dt
Primbs, MS&E 345 9 Plug into our risk neutral pricing formula: The futures price is the expected price of the stock at time T in a risk neutral world.

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10-applications4(f) - More Applications of Linear Pricing...

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