# Calibration of the GARCH Diffusion Model

The GARCH diffusion model  is one of the running examples of bivariate stochastic volatility models in my first book. (Others include the well-known Heston ’93 model and the so-called 3/2-model). As with most finance models, it comes in two versions: a real-world (aka P-measure) version and a risk-neutral (aka Q-measure) version. The latter is used to value options.

For equity applications, the stochastic process pair is $\{S_t,v_t\}$. Here $S_t \ge 0$ is a stock price and $v_t = \sigma_t^2 > 0$ is the associated stochastic variance rate. Then, the risk-neutral version, as an SDE (stochastic differential equation) system, reads

$\displaystyle \left \{ \begin{array}{l}dS_t = (r-q) S_t dt + \sigma_t S_t dB_t, \\d v_t = \kappa (\bar{v} - v_t) + \xi \, v_t \, dW_t, \quad dB_t \, dW_t = \rho \, dt. \end{array}\right. \quad\quad\quad\quad\quad (1)$

It’s a very simple and natural model which, unfortunately, lacksÂ  exact solutions for option values or transition densities. To determine the unknown parameters in (1), one needs to calculate option prices (numerically) and fit the model to option chains, a procedure generally called “calibration”. To do this efficiently and accurately for this model (and many others) requires a PDE approach. (There have been some earlier calibrations of this model via Monte Carlo).

The lack of an efficient — or, apparently, any — PDE calibration for this model prompted Yiannis Papadopoulos and me to perform one. Our methods and first results were recently posted on the arXiv:

A First Option Calibration of the GARCH Diffusion Model by a PDE Method

Yiannis’ own announcement may be found in his blog here. You will also find at that link a free downloadable GARCH diffusion calibrator demo that Yiannis has developed. (Windows executable). You can run it on a sample option chain that he supplies, and see a calibration in well under a minute (11 seconds on my desktop). Or you can run it on your own data, simply by imitating the provided data file format.

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