Gyöngy’s Theorem and the Heston LSV Leverage Function

# LSV leverage function calibration


# LSV leverage function calibration












Introduction

I try to understand how to calibrate an Heston-LSV model, and I have trouble with how to use Gyongy theorem. Here is the model (1): \begin{align} dS_t &= rS_t dt + \sigma_{\text{LSV}}(t, S_t)\sqrt{V_t}S_t dW^S_t\\ dV_t &= \kappa(\theta - V_t)dt + \gamma \xi \sqrt{V_t}dW^V_t\\ d\langle W^S, W^V\rangle_t &= \rho dt \end{align} where $\sigma_{\text{LSV}}$ is the leverage function,$\gamma$ is the mixing parameter, and the other parameters are those from a standard Heston model.

Let's assume I have already calibrated the Heston parameters $(\kappa, \theta, \xi, \rho)$. I struggle at understanding how to apply Gyongy theorem to link $\sigma_{\text{LSV}}$ to the local volatility $\sigma_{\text{LV}}$.

Gyongy theorem

Gyongy theorem states that if $(X_t)$ is a $n$-dimensional stochastic process such that $dX_t = \alpha_t dt + \beta_t dW^X_t$ (for $(W^X_t)$ a $m$-dimensional uncorrelated brownian motion, then there exists another $n$-dimensional process $(Y_t)$ whose one-dimensional laws are the same as those of $(X_t)$, and $(Y_t)$ follows $dY_t = a(t, Y_t)dt + b(t, Y_t)dW^X_t$ with $a(t, y) = \mathbb{E}[\alpha_t \mid X_t = y]$ and $b(t, y)b(t, y)^T = \mathbb{E}[\beta_t\beta_t^T\mid X_t = y]$.

What I tried so far

We let $dW^V_t = \rho dW^1_t + \sqrt{1-\rho^2}dW^2_t$ (and $W^S = W^1$), so that (1) rewrites \begin{align} d\underbrace{\begin{pmatrix}S_t\\V_t\end{pmatrix}}_{X_t} = \underbrace{\begin{pmatrix}rS_t\\\kappa (\theta - V_t)\end{pmatrix}}_{\alpha_t}dt + \underbrace{\begin{pmatrix}\sigma_{\text{LSV}}(t, S_t)\sqrt{V_t}S_t & 0\\ 0 & \gamma \xi \sqrt{V_t}\end{pmatrix} \begin{pmatrix}1 & 0\\\rho & \sqrt{1-\rho^2}\end{pmatrix}}_{=\begin{pmatrix}\sigma_{\text{LSV}}(t, S_t)\sqrt{V_t}S_t & 0\\ \rho \gamma \xi \sqrt{V_t} & \sqrt{1-\rho^2} \gamma \xi \sqrt{V_t}\end{pmatrix} =\beta_t} d\underbrace{\begin{pmatrix}W^1_t\\W^2_t \end{pmatrix}}_{W^X_t} \end{align}

Applying Gyongy theorem, we get $a(t, (s, v)) = \mathbb{E}\left[\begin{pmatrix}rS_t\\\kappa (\theta - V_t)\end{pmatrix} \mid S_t = s, V_t=v\right] = \begin{pmatrix}rs\\\kappa (\theta - v)\end{pmatrix}$ and \begin{align} b(t, (s, v))b(t, (s,v))^T &= \mathbb{E}\left[\beta_t\beta_t^T\mid S_t = s, V_t=v\right]\\ &= \mathbb{E}\left[\begin{pmatrix}\sigma_{\text{LSV}}^2(t, S_t)V_tS_t^2 & \rho \gamma\xi \sigma_{\text{LSV}}(t, S_t)S_tV_t\\ \rho \gamma\xi \sigma_{\text{LSV}}(t, S_t)S_tV_t & \gamma^2\xi^2V_t\end{pmatrix}\mid S_t = s, V_t=v\right]\\ &= \begin{pmatrix}\sigma_{\text{LSV}}^2(t, s)vs^2 & \rho \gamma\xi \sigma_{\text{LSV}}(t, s)sv\\ \rho \gamma\xi \sigma_{\text{LSV}}(t, s)sv & \gamma^2\xi^2v\end{pmatrix} \end{align}

And now I am stuck. From there, how do I conclude on the link between $\sigma_{\text{LSV}}$ and $\sigma_{\text{LV}}$?