# How to extract volatility smile implied by a mixture model?

If one had to extract the implied volatility smile from a local volatility model, one can simply use the relationship:

$$\sigma^2_{imp}(t, x)T = \int_t^T \sigma^2_{loc}(s, x)ds$$

with $$\sigma_{loc}$$ the dupire formula for local volatility for a given time $$t$$ and moneyness $$x$$.

With the same formula, one can extract the model forcast for the forward smile by replacing $$t$$ with a future date $$S$$, $$t.

Now suppose we have a mixture model that consists in a weighted sum of two local volatilities and the price is given by:

$$\text{Price}_{\text{mixture}} = p \cdot \text{Price}_{\text{LocVol1}} + (1-p)\cdot \text{Price}_{\text{LocVol2}}$$

How can I extract the smile from the mixture model ?

• Wouldn't you just simply have to apply the local vol estimation method on the mixture pricing as a function of $x$ and $T$? Aug 3, 2021 at 8:51
• Could you write it with the same notation? I am not sure I get the idea Aug 3, 2021 at 9:21

## 1 Answer

This would be my ansatz; there are probably people on here who might have a better solution:

I am following Gatheral's teaching notes on local volatility (eq. 5)

$$\sigma_{loc}^2(K,T,S_0)\equiv \frac{\frac{\partial C}{\partial T}}{\frac{1}{2}K^2\frac{\partial^2C}{\partial K^2}}$$

or $$\frac{\partial C}{\partial T}=\sigma^2_{loc}(K,T,S_0)\frac{1}{2}K^2\frac{\partial^2C}{\partial K^2}$$

For brevety, I'll introduce $$\sigma^2_{loc}$$, $$C_{KK}$$ and $$C_T$$ as obvious, and I'll superindex with $$(i)$$ for option or surface $$i$$. Hence, for your mixer \begin{align} \sigma^2_{loc,mix}&=\frac{wC^{(1)}_T+(1-w)C^{(2)}_T}{\frac{1}{2}K^2\left(wC^{(1)}_{KK}+(1-w)C^{(2)}_{KK}\right)}\\ &=\frac{\frac{1}{2}K^2\left(w\sigma^2_{loc,(1)}C^{(1)}_{KK} +(1-w)\sigma^2_{loc,(2)}C^{(2)}_{KK} \right)}{\frac{1}{2}K^2\left(wC^{(1)}_{KK}+(1-w)C^{(2)}_{KK}\right)}\\ &=\frac{wC^{(1)}_{KK}}{wC^{(1)}_{KK}+(1-w)C^{(2)}_{KK}}\sigma^2_{loc,(1)}+\frac{(1-w)C^{(2)}_{KK}}{wC^{(1)}_{KK}+(1-w)C^{(2)}_{KK}}\sigma^2_{loc,(2)} \end{align}

Of course, if the two surfaces imply roughly identical second derivatives, you could say $$\sigma^2_{loc,mix}=w\sigma^2_{loc,(1)}+(1-w)\sigma^2_{loc,(2)}$$, but that will most probably defeat the original idea of mixing surfaces, no?

• Thank you for the answer, that makes perfect sense. However in my case I already have a model that produces smooth implied volatility in order to avoid calculating $C_{KK}$ which numerically can be produces unsatisfactory results. Do you think I can get $C_{KK}$ using an implied volatility surface instead? Aug 11, 2021 at 10:42
• I think, practically, you do not have to calculate $C_{KK}$ numerically: Simply calculate all necessary first and second (cross) derivatives of the Black-Scholes-Merton formula (or Black formula, or whichever you use); that should result in an algebraic expression for $C_{KK}$. To be specific: $d^2C(K,\sigma(K))/dK^2 = C_{KK} + 2C_{K\sigma}\sigma_K + C_{\sigma\sigma}\sigma_K^2 + C_{\sigma}\sigma_{KK}$, where $f_{ij}$ denote partial derivatives of $f$ with respect to arguments $i,j$. Aug 11, 2021 at 11:28
• I see, I am trying to avoid to take finite difference derivatives and use the already available implied volatility Aug 12, 2021 at 12:40
• Agreed! There’s No need to do anything with FD: simply use the Black Scholes derivatives. Aug 12, 2021 at 13:04