# Question on Local Volatility

In Gatheral's book The Volatility Surface (Wiley, 2006), a local volatility model is defined as... $$dS_t =S_t \mu_tdt + S_t \sigma(S_t, t)dZ$$ The famous Dupire Equation is given by... $$\sigma^2(K, T, S) = \frac{\partial C/\partial T}{\frac{1}{2} K^2\partial^2C/\partial K^2}$$

I have two questions...

(a) according to the definition of local vol, $$\sigma$$ is a function of t and S. How come it also depends on K in the second equation?

(b) In practice, what kind of functions do people use for $$\sigma(t, S_t)$$? Would a quadratic function or cubic function suffice?

• It is confusing I'll admit. The $S_t$ in the first definition corresponds to $K$ in the second, $t$ to $T$, and the $S$ in the second is really just there because its used to give the price of $C$ which I imagine maybe gets expanded into the BS equation to write the local vol as a function of the BS surface? That would be my guess. – will May 7 at 22:05
• For point b., I have seen pade approximants used effectively. – will May 7 at 22:20
• @will It still looks like in (2), holding all other variables constant, we get different values of local vol for different values of strikes whereas in (1), K does not affect strike at all. – user1559897 May 8 at 13:39
• you are confusing things. Think of (1) as the same function of K, T, and S, where K is St, T is t, and S is omitted, but would be S0. – will May 8 at 21:18
• @will On a high level, local vol should only depend on current stock price and time, but the dupire equation is implying that a local vol is a different number for each strike, time and stock price. I am really confused about the mismatch. – user1559897 May 9 at 13:08