Calibrating LSV Models for Equity Quanto Options
Summary
The document asks how to price equity quanto options with a local-stochastic volatility model, building on a proposed Heston approach that adjusts equity and variance dynamics for quanto effects. The supplied answer recommends modeling both the equity and foreign exchange markets: calibrate one LSV model to equity vanilla options and relevant exotics, and another to FX vanillas and potentially exotics. Quanto pricing then requires working under the appropriate foreign measure and accounting for dependence between the two spot processes.
The answer says correlation may be estimated from historical data or calibrated from reliable quanto vanilla prices, and describes Monte Carlo as one possible use of that dependence estimate. This is a calibration outline, not a worked derivation or implementation. It emphasizes that the task is demanding because it involves two volatility models, two leverage functions, and a correlation model; the quality of resulting prices depends on those inputs and their calibration.
Key ideas
- Quanto pricing requires accounting for the equity and FX processes under the relevant measure.
- The proposed approach calibrates separate LSV models to equity and FX vanilla markets.
- Relevant exotic options may also be included in each model’s calibration.
- Dependence between equity and FX spot processes must be estimated or calibrated.
- The answer presents a demanding calibration outline rather than a complete implementation.
Tags
Full text
# Pricing Quantos with Local-Stochastic Volatility model
# Pricing Quantos with Local-Stochastic Volatility model
I would like to price equity quanto options with the Heston Local-Stochastic Volatility model (LSV) but I am having hard time understanding how to apply quanto adjustment in such complex setup.
When it comes to the pure Heston model, after the reading of the following paper: https://www.risk.net/derivatives/2171456/quanto-adjustments-presence-stochastic-volatility
I have an idea of how to apply the quanto drift adjustment:
- calibrate the Heston Heston parameters to Vanillas
- adjust the drift in the price process $S_t$ with the product of Equity/FX correlation, FX volatility and EQ volatility $(\rho_{S,FX}\sigma_S\sigma_{FX})$
- adjust the drift in the volatility process $\nu_t$ by adjusting the long term mean $\theta$
However, in the LSV model there is local volatility entering the price process, which makes things even more complex and complicated. Unfortunately, I cannot find any resource about adjusting the LSV model for quuanto exotic options.
How one would approach the calibration of the LSV model for quanto options?
## Answer by fwd_T (score 1)
https://quant.stackexchange.com/a/75062
You will need two LSV models. The first LSV model is calibrated to vanilla and potential exotics equity derivatives. With this you will be able to price any kind of vanilla and exotic equity derivatives under the domestic measure. But for quanto vanilla or exotic options you need the ability to price in a so-called foreign measure. This is why you will need a second LSV model modelling the FX rate. This second LSV model also has to be calibrated to the vanilla and also potentially to various exotic FX options. In addition to this, there will be a (potentially stochastic) correlation between the two spot processes. This is usually calibrated based on historical data, but if you have reliable data for various quanto vanillas, you can try to calibrate it and use it for example in Monte Carlo.
In conclusion, I would say this is not an easy exercise. It involves a lot of market calibration (two stochastic volatility processes and two leverage functions) and either the historical estimation or the calibration of the fifth stochastic process: a correlation between the two. Only after this will you be able to produce reliable equity quanto prices.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.