Consider the derivative with payoff function g(S(T)). Let V(S,t) denote its price.

The basic pricing principle is that (under GBM) we can write:

V(S,t) = e^-r(T-t) E[g(X(T)]

where the expectation is taken with respect to the probability distribution p(u) for the random variable X(T) that arises from the evolution over [t, T] of the stochastic process X(t):

dX = r X dt + s X dW X(t) = S(t)

(This is identical to the physical process for S(t) except that the real drift a has been replaced by r.)

As we know, X(T) is distributed lognormally.

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[a] Show that under this pricing distribution:

E[X(T)] = the forward price of the stock to date T

Hint: Suppose the derivative were a forward contract with delivery price equal to todays forward price, and you use the expectation formula to determine its value. What does the left-hand side equal:

V(S,t) = e^-r(T-t) E[g(S(T)]

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[b] Suppose that for a fixed expiry T we had put prices for all possible strikes K. Lets denote this price profile by P(K).

We reverse our pricing principle and argue that theres some density function f(u) (called the implied distribution) for S(T) so that:

P(K) = e^-r(T-t) E[Put Payoff] =

e^-r(T-t) E[MAX(K - S(T), 0)] =

e^-r(T-t) S (K u) f(u) du

What are the limits on the integral: S (K u) f(u) du ?

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[c] How can we extract f(u) from P(K)?

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[d] Whats E[S(T)] under the implied distribution?

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[e] In terms of the function f(u) write down an inequality for:

P(K+1) - P(K)