MATH5965 - Discrete Time Financial Modelling Practice Sheet 1
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MATH5965 - Discrete Time Financial Modelling
exercise
1. Consider the following one-period binomial model. Assume the initial value of the stock is S0 . The stock price will go to either S0u or S0d at time t = 1. Assume that the one-period simple interest rate is r such that a bond B will become B0(1 + r) at time t = 1. Assume B0 = 1.
a) Suppose that d < 1 + r < u. Show that the fair price of a European call option which matures at time t = 1 with strike price at K is
┌ 1 u(+)r一d(一) d (S0u 一 K)+ + u u(一)r一d(一) 1 (S0d 一 K)+┐
by constructing a replicating portfolio.
b) Construct a replicating portfolio for European put option written on the stock S , with the strike price K and the expiration date t = 1. Compute the fair price of the European put option through a replicating portfolio.
c) Let c and p denote the prices of above call and put respectively. Check that the put-call parity relationship holds, i.e. show that
c 一 p = S0 一
d) Let P denote a risk-neutral probability in the state of S0u. Find the unique risk-neutral probability P for the discounted stock price S* = S/B, and compute the fair prices of both options using the risk-neutral valuation formula.
e) What happen if we drop the assumption that d < 1 + r < u?
2023-04-22