# How to use a stochastic volatility model to price a quanto option

I want to price a quanto option using a Stochastic Volatility model (like Heston model, 1993).

Normally, what we do is:

1. Calibrate the stochastic volatility model,
2. draw a binomial tree consistent with the model, with numerous steps and
3. derive the option price from the binomial tree.

This process is given as an example in this book on page 284, where an European vanilla option is priced under a stochastic volatility model.

But what about for a quanto option? How can I draw a binomial tree to price this option? That is, a binomial tree that takes into consideration the correlation among the underlying asset and the exchange rate?

Is there some way to apply a stochastic volatility model to price a quanto option? Could you please help me with the detailed steps?

The quanto option pricing formula is given in this paper.

[EDIT - being more specific] Suppose I already have:

• The correlation among the underlying asset ($S$) and the exchange rate. I have it. I know it is, say, 0.7.
• I have a well calibrated Heston model for the volatilities of both the underlying asset ($S$) and the exchange rate.

I want to know:

• How to get, from my Heston model, the volatilities that I need to put into the quanto option pricing formula? In the formula of this paper, the vols are supposed constant. How can I treat stochastic vols using this formula? I do not know what vols I shoul put into the quanto option pricing formula. I want to know how to get the vols that I need from the Heston model.
• I don't see where on p. 284 in Alexander (2008) it says anything about using a binomial tree to price European options under a stochastic volatility model. Since there are two state variables (spot and vol.) I would expect to see something like a 2-dimensional tree if anything. But more commonly, you'd use either Monte Carlo or finite differences. Could you please clarify? – LocalVolatility Dec 19 '16 at 23:40
• Peter jaeckel has two papers on quanto pricing in stochastic local vol, here is the latter (you can find the other on his website). The conclusion is that it is not a simple thing to do. – will Dec 20 '16 at 0:28
• @LocalVolatility I was more specific in my post. Could you please read the edit? In fact, in Alexander book what we have on page 284, example II4.8 is a Monte Carlo simulation method. – John Dec 20 '16 at 11:29
• @John this is really a poor quality edit... please do something about it. I think you are confused. You need something along the lines of the model used in "Dimitroff G., Szimayer A., Wagner A., Quanto Option Pricing in the Parsimonious Heston Model, Fraunhofer-Institut fur Techno- und Wirtschaftsmathematik, 2009". – Quantuple Dec 20 '16 at 11:48
• You cannot use the paper you mention because it assumes constant vol (hence not stohastic). Assuming stochastic vol (à la Heston or whatever), you won't have any closed form formula... also, the correlation between $S$ and $FX$ is not enough anymore, because you have cross-effects (like correlation between $FX$ and the (now stochastic) equity volatility etc.) – Quantuple Dec 20 '16 at 11:51

## 2 Answers

If you consider Heston as your working modelling assumption to price quantos (caveat: you also need additional considerations regarding the fx dynamics and equity-fx dependence), then you should use the usual numerical methods associated to Heston (well, not Fourier transform-based methods since there are no semi-closed form formulae for quanto vanillas under Heston - I think the Stein&Stein stochastic vol model leads to something more tractable though), so Finite Differences or Monte Carlo.

What changes compared to standard Heston is that the dynamics of the (foreign) risky asset since it now needs to be expressed in the (domestic) risk-neutral mesure to be able to price quanto options as discounted payoff expectations.

Contrary to what happens for quantos priced under correlated equity-fx GBMs (see this question), both the drift of the equity and that of its instantaneous variance get "quanto-adjusted" (consequence of Girsanov theorem with stochastic vol, assuming equity and vol driving Brownian motions are correlated).

All that needs to be done - and it is not as easy as it seems in practice due to the absence of a liquid market, hence "implied" parameters, see the question which I have referred to earlier - is to specify parameters for the dependence structure between the various driving Brownian motions if your model (equity, variance, fx).

You can get an idea of what awaits you by reading the following paper:

[Dimitroff G., Szimayer A., Wagner A.], Quanto Option Pricing in the Parsimonious Heston Model, Fraunhofer-Institut fur Techno- und Wirtschaftsmathematik, 2009

which is available here for download.

• I edited my question in order to be more specific. I do not know how to get the vols required by the quanto option ricing formula from my Heston model. Could you please provide a step bu step? The question that you cited was psoted by me. I think that I really advanced in my research and this is the last thing that I need to tackle quano options. Tks! – John Dec 20 '16 at 11:38

this paper may also be of interest:

"Quanto Implied Volatility Smile"

Cesarini et al

https://ssrn.com/abstract=2388093