Calibration Heston Local Stochastic Volatility (LSV) Model

Question

The Heston Local Stochastic Volatility (LSV) model has the following dynamics: $$dS_{t}=r S_{t} d t+L\left(S_{t}, t\right) \sqrt{V_{t}} S_{t} d W_{t},$$ $$d V_{t}=\kappa\left(\theta-V_{t}\right) d t+\eta \sqrt{V_{t}} d Z_{t},$$ $$d W_{t} d Z_{t}=\rho d t.$$ The leverage function which $L\left(S_{t}, t\right)$, which ensures that the LSV model reproduce the vanilla option quotes of a Local Volatility model satisfies the equation $$L\left(s, t\right)=\frac{\sigma_{L V}(s, t)}{\sqrt{\mathbb{E}\left[V_{t} \mid S_{t}=s\right]}}.$$ Assuming we already have a well-behave local volatility surface, to calibrate the Heston LSV model, one should (1) calibrate the Heston model and (2) calibrate the leverage function $L\left(S_{t}, t\right)$. Several approaches have been proposed for the calibration of $L\left(S_{t}, t\right)$, e.g. solving a Kolmogorov forward PDE or Markovian projection methods.

As shown in Gatheral (p. 12) for instance, we know that local variance can be seen as a conditional expectation of instantaneous variance $$\sigma^{2}_{L V}(s, t)=\mathbb{E}\left[V_{t} \mid S_{t}=s\right].$$ Therefore, is it correct to say that another method to calibrate the leverage function would be to take the ratio of local volatilities form the LV model and local volatilities generated by the pure Heston Model? Namely $$L\left(s, t\right)=\frac{\sigma_{L V}(s, t)}{\sigma_{Heston \, LV}(s, t)}.$$ Thanks in advance for your answers!

Answer 1

Under Heston LSV (HLSV) dynamics, Gatheral's equality is:

$$ \sigma_{LV}^{HLSV}(S_t,t) = \sqrt{E^{HSLV}\left[V_tL(S_t,t)^2 | S_t \right]} = L(S_t,t)\sqrt{E^{HSLV}\left[V_t | S_t \right]}, $$

as $L(S_t,t)$ is $\sigma(S_t)$-measurable, where superscript $HSLV$ is meant to remind us what is our dynamics we started with (in particular the joint probability density function for $(S_t,V_t)$ needed to compute conditional expectation $E\left[V_t | S_t \right]$).

Under (pure) Heston SV (HSV) dynamics ($L$ set to constant $1$ in HSLV), Gatheral's equality is:

$$ \sigma_{LV}^{HSV}(S_t,t) = \sqrt{E^{HSV}\left[V_t| S_t \right]}. $$

If both $\sigma_{LV}^{HLSV}$ and $\sigma_{LV}^{HSV}$ perfectly hit the market local volatility, $\sigma_{LV}^{mkt}$, calculated from the market continuum of call prices via Dupire formula, then we have:

$$ L(S_t,t)=\frac{\sqrt{E^{HSV}\left[V_t | S_t \right]}}{\sqrt{E^{HLSV}\left[V_t | S_t \right]}} = \frac{\sigma_{LV}^{HSV}(S_t, t)}{\sqrt{E^{HLSV}\left[V_t | S_t \right]}} \left(= \frac{\sigma_{LV}^{mkt}(S_t, t)}{\sqrt{E^{HLSV}\left[V_t | S_t \right]}}\right).$$

(We note, of course, that the two dynamics have very different levels of parameterization richness and that the calibrated parameters $\kappa, \theta, \eta, \rho$ will not be the same in the two models, as it is exactly the presence of $L$ that distorts them when calibrating to the same market targets.)

If

$$ \sigma_{LV}^{HSV} \not= \sigma_{LV}^{mkt} = \sigma_{LV}^{HLSV} ,$$

the above relationship fails, but we still have:

$$ L(S_t,t)=\frac{\sigma_{LV}^{mkt}(S_t, t)}{\sqrt{E^{HLSV}\left[V_t | S_t \right]}}. $$