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?

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    $\begingroup$ 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. $\endgroup$ – will May 7 '19 at 22:05
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    $\begingroup$ For point b., I have seen pade approximants used effectively. $\endgroup$ – will May 7 '19 at 22:20
  • $\begingroup$ @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. $\endgroup$ – user1559897 May 8 '19 at 13:39
  • $\begingroup$ 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. $\endgroup$ – will May 8 '19 at 21:18
  • $\begingroup$ @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. $\endgroup$ – user1559897 May 9 '19 at 13:08

To answer your second question, per "Pricing, Hedging and Trading Financial Instruments" by Carol Alexander, the following approaches have been proposed in literature:

  • cubic polynomials (Dumas et al., 1998)
  • piecewise quadratic functions (Beaglehole and Chebanier, 2002)
  • cubic splines (Coleman et al., 1999)
  • hyperbolic trigonometric functions (Brown and Randall, 1999)
  • Hermite Polynomials (McIntyre, 2001)

Hopefully this list can give you a place to start. A quadratic or cubic function would likely suffice however be sure that the parameterization you choose preserves the static smile property of the local volatility(i.e. the underlying price of the security should not change the volatility at any point on the surface)

  • $\begingroup$ Let's say we are using a cubic polynomial. Are we assuming that local volatility is a cubic function of S and t? Or are we saying it is a function of K, S and t? $\endgroup$ – user1559897 May 8 '19 at 13:37
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    $\begingroup$ I think it will be a function of S and t. 𝜎(S,t) is the volatility at time t if the underlying price takes price S at that time and the local volatility surface is the surface created by these volatilities. Dupire's equation has K as a parameter because it is generating a surface of volatilities using the different market prices of the European options of the underlying which each have a different strike K. $\endgroup$ – Karl L May 9 '19 at 0:16

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