Deriving Local Volatility for a Mixture of Option Surfaces
Summary
The document asks how to obtain a volatility smile from a model whose option prices are a weighted mixture of prices generated by two local volatility surfaces. It contrasts this with a stated relationship for extracting implied volatility from a single local volatility model, then uses the Dupire equation to derive the local variance associated with the mixed call price surface.
Because option price and its strike curvature combine linearly under the mixture, the resulting local variance is a weighted average of the component local variances, with weights adjusted by each component’s contribution to the strike second derivative. It reduces to the original mixture weights only when the component surfaces have roughly equal second derivatives. The response presents this as a proposed approach, not a definitive treatment, and offers no numerical example or validation. The key practical point is that mixing volatilities directly generally does not reproduce the local volatility implied by mixing option prices.
Key ideas
- Apply the Dupire relation to the mixed option price surface to obtain its implied local variance.
- The mixture local variance weights each component by its share of the total strike curvature.
- The effective weights generally differ from the original price mixture weights.
- A simple weighted average of component variances is justified only when their strike curvatures are similar.
- The proposed derivation is not accompanied by empirical validation or a worked example.
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# How to extract volatility smile implied by a mixture model?
# 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 $S$, $t<S<T$.
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 by Kermittfrog (score 4)
https://quant.stackexchange.com/a/66279
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?Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.