soinc789

soinc789 - the risk-free interest rate is zero What is the...

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E3106. Prof. Kou, Fall 2005. Solutions to In-Class Homework Problems 7, 8, 9. 7. Let Y ( t )= σ B ( t )+ μt . Show that the process A ( t ) is a martingale, where A ( t )=exp { Y ( t ) μt 1 2 σ 2 t } . Solution: E [ A ( t ) | F s ]= E [exp { Y ( t ) μt 1 2 σ 2 t }| F s ] = E [exp { Y ( t ) Y ( s ) μt 1 2 σ 2 t } e Y ( s ) | F s ] = e Y ( s ) e μt 1 2 σ 2 t E [exp { Y ( t ) Y ( s ) }| F s ] = e Y ( s ) e μt 1 2 σ 2 t E [exp { Y ( t ) Y ( s ) } ] = e Y ( s ) e μt 1 2 σ 2 t e μ ( t s )+ σ 2 ( t s ) / 2 = e Y ( s ) e μs s σ 2 / 2 = A ( s ) . Thus, A ( t ) is a martingale. 8. Consider a European call option, which is written on a stock whose current value is $8 at time 0, with the strike price $9 and expiration date in one month. Assume for simplicity that one month later, when the option can be exercised, the stock price can either appreciate to $10 or depreciate to $6. Assume that the risk-free interest rate is zero. What is the price of the call option at time 0?
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Unformatted text preview: the risk-free interest rate is zero. What is the price of the call option at time 0? Solution: We know that the option price is give by C (0) = 1 R μ C u R − d u − d + C d u − R u − d ¶ . In our case, C u = (10 − 9) + = 1 , C d = (6 − 9) + = 0 , R = 1 , u = 10 8 , d = 6 8 . Thus C (0) = 1 1 μ 1 1 − 6 8 10 8 − 6 8 + 0 10 8 − 1 10 8 − 6 8 ¶ = 0 . 5 . 9. Is it true that N ( t ) ≥ n + 1 if and only if S n < t ? Solution: No. As a counterexample, suppose S n = t and S n +1 = t . Then N ( t ) ≥ n + 1 , but S n = t. 1...
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This document was uploaded on 10/18/2011.

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