# 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 $$(1)$$?