Deriving Theta (The Greeks)

[Delvesy888]

I have, I think, a fundamentally simple question...

I have been looking at the solution for the Theta for a call/put option based on the Garman-Kohlhagen model.

I can arrive at the solution if I differentiate f with respect to t, assuming that S_t does not depend on t.

However, of course, S_t does depend on t, and we have

partial dSt/dt = (r+q-0.5(sigma)^2)*S_t

Am I missing something fundamental about partial derivatives? Perhaps if I was to do the 'full' differentiation by treating S_t as a function of t, everything will magically cancel and I will be left with the same expression?

Any help would be greatly appreciated.

Thanks

 

[Mike Lewry]

Theta

Theta is the sensitivity of the option price to the passage of time, keeping all other parameters fixed.

This underlined bit is crucial and is there because we're considering a partial derivative.

The value of S(t) is part of "all other parameters" and so we're considering a change in time without a change in S. If we were to allow S to change as well, we'd be including a delta effect.

Does this clear things up?

 

[Delvesy888]

Ah yes, of course, under Garman-Koholhagen, f is a function of 5 variables (Greeks) and in taking a partial derivative, we keep the remaining 4 fixed, this includes S_t when talking about Theta.

Thanks very much for clearing that up.

 

[Whippet1]

In St, the "t" subscript is there to remind us that the share price varies through time. It is not an explicit variable in its own right. So, we ignore it when partially differentiating the explict time t variable to find theta.

Equally, we ignore the t subscript when differentiating partially with respect to the share price to find delta and gamma.