When should we have C(S,tV)?

[MattT]

Hi Mark,

In deriving a PDE for the price of an option in the case of a local vol model (V = V(S,t) ) it is assumed the option price is a function of S and t only; one then applies Ito’s lemma to C(S,t).

I had a slightly different way of thinking about this problem that appears to give a very different (seemingly wrong) answer: the price of an option should depend on the stock price, time and volatility, so we have C(S, t, V). Now, in a local vol. model, we have V = V(S,t) and are interested in C(S, t, V(S,t)) (so I see why we can think of C as a function of S and t only). However, I argued that we should equally well be able to apply Ito lemma to C(t, S_t, V(S_t, t) ) to get

dC_t = dC/dt dt + dC/dS dS_t + dC/dV dV_t + 0.5 d^2C/dS^2 (dS_t)^2 + 0.5 d^2C/dV^2 (dV_t)^2 + 0.5 d^2C/dVdS dV_tdS_t

Continuing to argue in this way results in a PDE (to be satisfied only on the surface { S, t, V(S,t) } ) which is very different to the BS PDE (replaced to have V(S,t) rather than just constant V).

1) I suspect I get the wrong result because I am assuming there is a well defined C^2 function C(S, t, V) on [0, \infty]^3 whereas my PDE would only define C on a surface in this domain. So I can’t actually define C on all of [0, \infty]^3 and moreover all the replication stuff that formally proved BS price is correct is going to fail?

2) Curiously, in the case where V is constant, or only time dependent, you could define D(S, t) = C(S, t, V(t)) and then (using the PDE you get for C by proceeding as I described above), show that D has to satisfy the expected BS PDE. I suspect this is just good fortune as opposed to being a correct way of thinking about the derivation?

This all makes me slightly concerned about more general models of the form:

dS _t = \mu S_t dt+ V^{0.5} dW^(1)_t

dV_t = f(V_t, S_t, t) dt + sigma dW^(2)_t

(where as usual W^(1) and W^(2) are (possibly correlated) BMs).

3) Presumably one ought to be careful if sigma = 0 and f = f(V_t, t) because then we just have an ODE volatility and it’s just a function of time, so we should apply Ito to C(S,t) rather than C(S, t, V).

4) In the case where sigma = 0 and f depends on S_t I am really unsure, eg. If dV_t = S_t dt say. It seems that if we assume C = C(S,t,V) we will again run into problems, because we don’t have free choice of V. How would one proceed in this situation?

Sorry for so many questions! Any help at all is really greatly appreciated. Thanks.

[mj]

well you are essentially asking how stochastic volatility models work and I'm afraid that the answer to that is to read about them. I give a brief intro in Chapter 16.

See also Gatheral's book or a lot of other texts.

I've never analyzed the transition from stochastic vol to local vol.