Why do you think p is called the ‘risk-neutral probability’ of S going up?

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 D_T.

(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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