I'm trying to price European options with a GARCH volatility model. What I have is a program that calibrates the GARCH volatility process for a stock which I intend to use to value a derivative on the underlying. My initial thought would be to simple discretize the geometric brownian motion and simulate with a two factor model for a number of steps, at each time step updating the volatility according to the GARCH process and then using that volatility to simulate 1 step ahead. Is this the correct approach?

In Risk Management and Financial Institutions [HULL] it gives this equation for the expected future volatility on day $n+t$ . From this equation I gather that the expected average volatility between today and day $n+t$ would simply be average of all the volatility forecasts for the $t$ days. Now since in option pricing we are only interested in the average volatility of the time period would it be correct to simply take this volatility forecast and simulate the stock process exactly using the solution to GBM? Would this give the same result as the approach above?

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If you want to do it formally, the most basic texts on the subject are Duan (1995) and Heston and Nandi (2000). In the later case, they actually propose a quasi-analytical formula for pricing European options under their GARCH model. The model has this form under the physical measure: \begin{align} ln S_{t+1} - ln S_t &= r_{ft+1} + \lambda h_{t+1} - \frac{1}{2} h_{t+1} + \sqrt{h_{t+1}} z_{t+1}, \; z_{t+1} \sim N(0,1) \\ h_{t+1} &= \sigma^2 + \pi( h_t - \sigma^2 ) + \alpha h_t(z_t^2 - 1 - 2 \gamma z_t ) \end{align} I write it differently than Heston and Nandi (2000) because I want to highlight the convexity correction. Specifically, given the normality of disturbances, this is the choice you make to ensure that gross returns have the form you'd expect, i.e. the risk-free rate plus some compensation for the volatility risk: \begin{align} E\left( \frac{S_{t+1}}{S_t} \big| F_t \right) = \exp \left( r_{ft+1} + \lambda h_{t+1} \right). \end{align}

This is by no means the best model you can find, but it captures some things. In particular, if you compute the correlation between returns and variance, it's controlled by $\gamma$ -- it introduces an asymmetry in the response of variance which in turn allows you capture the "leverage effect" (Black, 1976) whereby returns appear to be negatively correlated with measures of volatility and variance (at least for stock market indexes, it seems true).

Now, to risk-neutralize this model, you have to choose a pricing kernel explicitly or implicitly. If you use an exponentially affine pricing kernel, you basically have to define a distorted shock $\eta_{t+1}$ such that the return on your stock is the risk-free rate. Then, you can exploit the fact that this model implies the conditional characteristic function of $S_t$ is exponentially affine in state variables ($S_t$ and $h_t$) to derive a closed-form pricing formula. That formula is given in many texts. You could look up Heston and Nandi (2000) and use the formula they give, bearing in mind $\lambda^{HN200} = \lambda^{Mine} - 1/2$ to map it to the way I wrote the model above. More generally, Christoffersen, Dorion, Jacobs and Wang (2010) detailed and compared many GARCH option pricing models. Christoffersen, Heston and Jacobs (2013) proposed a way to improve pricing through a quadratic pricing kernel. Bobaoglu, Christoffersen, Heston and Jacobs (2018) also proposed to combine interesting features like a component structure (you have a long term trend-like value for variance and a short term value that fluctuates around it) with non-gaussian disturbances (so you get fat tails).

Now, what must you do? Well, as long as your GARCH model remains in the (exponentially) affine class (meaning, the characteristic function is exponentially linear in state variables), there exists a quasi-closed form solution for pricing. Otherwise, you're forced to do Monte Carlo simulation. Either way, calibrating the model to price or hedge options requires option data (ideally, both stock and option data). You could use a method of moment estimator for $\lambda$ by forcing it to match the empirical value of the risk premium for some estimate of it and using some value for the long-run variance of the underlying.


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