Question: Consider an option which expires at time T. The current value of the option is V(S, t). It is possible to synthesize the option using

Consider an option which expires at time T. The current value of the option is V(S, t). It is possible to synthesize the option using vanilla European calls, all with expiry at time T. We assume that calls with all exercise prices are available and buy f(E) of the call with exercise price E, which has value C(S, t; E). The value of the synthesized option is then

V(S, t) = | f(E')C(S, t; E')dE'.

Find the density of call options, f(E), that we must use to synthesize the option.
Synthesize the option payoff to find f(E).

V(S, t) = | f(E')C(S, t; E')dE'.

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