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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