Estimating Local-Volatility Vol-Vol Breakeven with Monte Carlo

Summary

The document defines vol-vol breakeven as the instantaneous covariation of log implied volatilities for two options. For a local volatility model, when both options are the same contract, it gives a formula relating this quantity to the spot sensitivity of log implied volatility, scaled by local volatility and spot. The proposed Monte Carlo procedure reprices a call from simulated paths, backs out implied volatility, and estimates the spot derivative by repeating the calculation after shifting the initial spot.

The question is how to extend the estimate beyond local volatility, such as to stochastic-volatility or local-stochastic-volatility dynamics, when the special formula no longer applies. The excerpt provides no answer or validation results. Its finite-difference setup also leaves practical estimation choices unspecified, including bump size, sampling error, and how to estimate the instantaneous quadratic variation from simulated implied-volatility paths. It is therefore a methodological question with a model-specific starting point, rather than a general Monte Carlo solution.

Key ideas

• Vol-vol breakeven is defined as covariation between log implied volatilities over time.
• For a single option in a local volatility model, the document relates breakeven to spot sensitivity.
• A Monte Carlo price can be converted to implied volatility for a baseline spot and a bumped spot.
• A finite difference of those implied volatilities estimates the required spot derivative.
• The proposed approach does not explain how to estimate generic breakeven under stochastic volatility.

Tags

# Vol-Vol Breakeven (MC Estimation)


# Vol-Vol Breakeven (MC Estimation)












I am currently reading the paper Computation of Break-Even for LV and LSV Models. This paper defines the vol-vol breakeven \begin{align*}\tag{1} B_t(T,K,T',K') &\ :=\ d\langle \ln \sigma^{T,K}_., \ln \sigma^{T',K'}_. \rangle_t \end{align*} where $\sigma^{T,K}_t$ is the implied volatility for expiry $T$ and strike $K$ at time $t$.

Now if we assume a Local Vol model like $dS_t = \sigma(t,S_t)S_t dW_t$ and use $K=K'$ and $T=T'$, then \begin{align}\tag{2} B_t(T,K,T,K) &\ =\ \left(\frac{\partial \ln \sigma^{T,K}(t, S_t)}{\partial S_t}\cdot \sigma(t,S_t) \cdot S_t\right)^2 \end{align}

Using Monte-Carlo we can generate $N$ paths using initial spot $S_t$ and compute the MC price of a call option with strike $K$ & expiry $T$, and then back out a MC-implied $\hat{\sigma}^{T,K}_{baseline}$ from that price. Then I generate another $N$ paths, but using initial spot $S_t+h$, from which I then compute $\hat{\sigma}^{T,K}_{h}$. This gives $$ \frac{\partial \ln \sigma^{T,K}(t, S_t)}{\partial S_t} \ \approx\ \frac{\ln \hat{\sigma}^{T,K}_h - \ln \hat{\sigma}^{T,K}_{baseline}}{h} $$ We know $\sigma^{T,K}(t, S_t)$, which is the local vol, and we know spot $S_t$, and hence we can compute $B_t(T,K,T,K)$ via $(2)$.

However, what if the dynamics of $dS_t$ are different (e.g. based on a Stochastic Vol model, or a Local-Stochastic Vol model, or something else), then we cannot use formula $(2)$. Assuming I can generate many paths for $S$ via MC, is there a way to approximate $B_t(T,K,T,K)$, essentially using the generic formula $(1)$?

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.