Question

1. Consider a single-period binomial model with r = 1/3, B0 = 1, S0 = 2, d = 5/4, u = 3/2, and p = 1/2. (You can take the sample space ? to be {?1,?2}, with ?1 corresponding to the stock price going "up", and ?2 corresponding to the stock price going "down".) (a) Compute B1. (b) Compute S1(?1) and S1(?2), and the probability of each outcome. (c) For the trading strategy ? = (2, 4), compute V0(?), V1(?)(?1), and V1(?)(?2). (d) Let X be a European call option with strike price $2.50 and expiration time T = 1. (i) Find X(?1) and X(?2). (ii) Find the replicating strategy ? = (?1, ?1) for X (iii) Find the manufacturing cost for that strategy. That is, compute V0(?). (e) Give an example of arbitrage opportunity if the claim X can be purchased for C0 = 1/16 (dollars) at time 0. 2. Repeat the steps of Exercise 1, with the following data: r = 1/4, B0 = 1, S0 = 3, d = 1, u = 2, and p = 3/4. Use the trading strategy ? = (3, ?2) in part (c) . The contingent claim X is now a European put option with strike price K = $4. Use C0 = 1 in doing part (e).

          1. Consider a single-period binomial model with r = 1/3, B0 = 1, S0 = 2, d = 5/4, u = 3/2, and p = 1/2. (You can take the sample space ? to be {?1,?2}, with ?1 corresponding to the stock price going "up", and ?2 corresponding to the stock price going "down".)
(a) Compute B1.
(b) Compute S1(?1) and S1(?2), and the probability of each outcome.
(c) For the trading strategy ? = (2, 4), compute V0(?), V1(?)(?1), and V1(?)(?2).
(d) Let X be a European call option with strike price $2.50 and expiration time T = 1.
(i) Find X(?1) and X(?2).
(ii) Find the replicating strategy ? = (?1, ?1) for X
(iii) Find the manufacturing cost for that strategy. That is, compute V0(?).
(e) Give an example of arbitrage opportunity if the claim X can be purchased for C0 = 1/16 (dollars) at time 0.
2. Repeat the steps of Exercise 1, with the following data: r = 1/4, B0 = 1, S0 = 3, d = 1, u = 2, and p = 3/4. Use the trading strategy ? = (3, ?2) in part (c) . The contingent claim X is now a European put option with strike price K = $4. Use C0 = 1 in doing part (e).

1. Consider a single-period binomial model with r = 1/3, B0 = 1, S0 = 2, d = 5/4, u = 3/2, and p = 1/2. (You can take the sample space ? to be ?1,?2, with ?1 corresponding to the stock price going "up", and ?2 corresponding to the stock price going "down".)
(a) Compute B1.
(b) Compute S1(?1) and S1(?2), and the probability of each outcome.
(c) For the trading strategy ? = (2, 4), compute V0(?), V1(?)(?1), and V1(?)(?2).
(d) Let X be a European call option with strike price 2.50 and expiration time T = 1.
(i) Find X(?1) and X(?2).
(ii) Find the replicating strategy ? = (?1, ?1) for X
(iii) Find the manufacturing cost for that strategy. That is, compute V0(?).
(e) Give an example of arbitrage opportunity if the claim X can be purchased for C0 = 1/16 (dollars) at time 0.
2. Repeat the steps of Exercise 1, with the following data: r = 1/4, B0 = 1, S0 = 3, d = 1, u = 2, and p = 3/4. Use the trading strategy ? = (3, ?2) in part (c) . The contingent claim X is now a European put option with strike price K =4. Use C0 = 1 in doing part (e).

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