Risk-neutral probability

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 

(d) Let 

(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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