How to extract volatility smile implied by a mixture model?

Question

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 $S$, $t<S<T$.

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 ?

Answer 1

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?