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sol2 - ACTSC/STAT 446/846 Assignment 2 Solutions 1 Binomial...

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ACTSC/STAT 446/846 – Assignment 2 – Solutions 1. Binomial Model [10pts] Using a binomial tree with n = 8 time steps, S (0) = 100, expiration of 1 year (and therefore h = 1 / 8), r = 5%, σ = 0 . 3 and δ = 3 . 5%. (a) Compute European and American put option prices for K = 90 , 110. (b) Compute European and American call option prices for K = 90 , 110. See the calculations in CalculA2.xls. We first build the tree of stock prices using u = exp(( r - δ ) h + σ h ) = 1 . 114 and d = exp(( r - δ ) h - σ h ) = 0 . 9011. We then compute the option value at time 0 recursively, using H ( i, j ) = e - rh ( qV ( i + 1 , j + 1) + (1 - q ) V ( i + 1 , j )) , where q = ( e ( r - δ ) h - d ) / ( u - d ) = 0 . 4735 and in the case of the American option, V ( i, j ) = max( E ( i, j ) , H ( i, j ), where E ( i, j ) is the exercise value in node ( i, j ), while in the European case, V ( i, j ) = H ( i, j ). In both cases, the recursion is initilaized using V ( N, j ) = E ( N, j ). call put K Euro Amer Euro Amer 90 17.48 17.52 6.53 6.65 110 8.68 8.69 16.76 17.22 (c) Repeat (a) and (b) but assuming that δ = 0. Comment on what you observe. call put K Euro Amer Euro Amer 90 19.87 19.87 5.48 5.74 110 10.13 10.13 14.77 15.76 As we know, when δ = 0 the American and European calls have the same value. With no dividends, the stock prices are higher, and this makes the call option more valuable while the put options become less valuable. (d) Repeat (a) and (b) but with r = 0 and δ = 3 . 5%. Comment on what you observe. call put K Euro Amer Euro Amer 90 14.95 15.55 8.39 8.39 110 7.03 7.23 20.47 20.47 Now it is the American and European put that have the same value. When r is 0, there is no advantage in exercising earlier based on the time-value of money for the put option, and since δ > 0, it is best to hold on to the stock as long as possible.

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sol2 - ACTSC/STAT 446/846 Assignment 2 Solutions 1 Binomial...

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