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 . Here is a stock price and is the associated stochastic variance rate. Then, the risk-neutral version, as an SDE (stochastic differential equation) system, reads
It’s a very simple and natural model which, unfortunately, lacks 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:
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.