Consider the one-step binomial model over the period [0, T]. Let ω(u) denote the ‘up’ scenario and ω(d) the ‘down’ scenario with respective probabilities p(u) and p(d) = 1 – p(u). The underlying asset S is worth St at any point in time t and does not pay any dividend. Let Dt be the value of a derivative on S at time t, r the annual risk-free rate and denote the compound interest rate over the period [0, T].
Assume that the final price of the underlying is:
• in the ‘up’ scenario;
• in the ‘down’ scenario,
where u and d are parameters satisfying:
(a) In this question Calculate the value of a European call struck at 100. Does your result depend on the probabilities and ?
(b) In general, show that the value of the derivative at time t = 0 can be written:
where p is a function of r[T] , u and d.
(c) Verify that
(i) Verify that D0 is equal to the expected present value of the payoff DT.
(ii) Find the expected gross rate of return on S over [0, T]. Why do you think p is called the ‘risk-neutral probability’ of S going up?
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