Calibrating a Heston Local Stochastic Volatility Leverage Function
Summary
The document discusses calibration of a Heston local stochastic volatility model, which combines a stochastic variance process with a state- and time-dependent leverage function. The leverage function is chosen so the model reproduces a target local volatility surface. The answer clarifies that the conditional variance used in this relationship depends on the joint distribution generated by the model being considered.
A ratio of the local volatility surface from pure Heston to the market surface can represent the leverage function only when the pure Heston and Heston LSV models both exactly reproduce the same market local volatility. In that case the ratio follows from their respective conditional variance relationships. If pure Heston does not fit the market surface, that shortcut fails; the leverage must instead be computed using the market local volatility and the conditional variance under the Heston LSV dynamics. The discussion assumes a suitable market surface and does not provide a numerical calibration procedure.
Key ideas
- The leverage function adjusts stochastic variance so an Heston LSV model can match a local volatility surface.
- Conditional variance depends on the joint asset and variance dynamics of the model in use.
- A ratio of pure Heston and market local volatilities is valid only when both models fit the same target surface.
- Otherwise, leverage must use market local volatility and conditional variance under the LSV dynamics.
- The document outlines relationships but does not specify a full numerical calibration algorithm.
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# Calibration Heston Local Stochastic Volatility (LSV) Model
# Calibration Heston Local Stochastic Volatility (LSV) Model
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 by ir7 (score 5, accepted)
https://quant.stackexchange.com/a/63149
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:
$$ L(S_t,t)=\frac{\sigma_{LV}^{mkt}(S_t, t)}{\sqrt{E^{HLSV}\left[V_t | S_t \right]}}. $$Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.