Other models

An extremely popular family of models in discrete time are the ARCH or GARCH type specifications, where the volatility is time varying. The attempt to capture a number of stylized facts, more of which will be discussed later, such as the volatility variations of asset returns. For example a GARCH-M $\left( 1,1\right)$ model would be of the form

$$S_{k} = S_{k-1}\exp \left\{ \sigma Y_{k}+\left( r+\lambda \sigma -\frac{1}{2}\sigma _{k}^{2}\right) \right\}$$

$$\sigma _{k}^{2} = \alpha +\beta \sigma _{k-1}^{2}+\gamma \sigma _{k-1}^{2}Y_{k-1}^{2}$$

In our framework, the fact that $\sigma _{k+1}$ is $\mathscr{F}_{k}$-measurable is important when one wants to develop a derivative pricing model. Such a model is discussed in Duan (1995, Mathematical Finance). By using exactly the same arguments of the previous discussion [either the Girsanov theorem approach or the equilibrium approach] one can derive a risk neutral measure $Q$ under which the stock price evolves, namely

$$S_{k} = S_{k-1}\exp \left\{ \sigma Y_{k}^{Q}+\left( r-\frac{1}{2}\sigma _{k}^{2}\right) \right\}$$

$$\sigma _{k}^{2} = \alpha +\beta \sigma _{k-1}^{2}+\gamma \sigma _{k-1}^{2}\left( Y_{k-1}-\lambda \right) ^{2}.$$

Figure 6.2 demonstrates the time variation of filtered volatilities, estimated using around 1500 daily returns.

Figure 6.2: The volatility across time of a GARCH model fitted into daily returns. The maximum likelihood estimated parameters [in percentage terms] are: $\omega=1.19$, $\alpha=0.75$, $\beta=0.19$ and $\mu=0.23$

Unfortunately, although a risk neutral measure is perfectly defined, the expectation under this measure is not feasible. The most efficient procedure is constructed in Duan and Simonato (1999, Journal of Economic Dynamics and Control), where the state spaces are discretized. A variant of this family where the bilinearity is broken is discussed in Heston and Nandi (2000, Review of Financial Studies):

$$S_{k} = S_{k-1}\exp \left\{ \sigma Y_{k}+\left( r+\lambda \sigma -\frac{1}{2}\sigma _{k}^{2}\right) \right\}$$

$$\sigma _{k}^{2} = \alpha +\beta \sigma _{k-1}^{2}+\gamma (Y_{k-1}-\theta)^{2}$$

One can further generalize these features, including jumps and state dependent long-run attractors [t.b.a.], with

$$S_{k} = S_{k-1}\exp \left\{ \sigma Y_{k}+\left( r+\lambda \sigma -\frac{1}{2}\sigma _{k}^{2}\right)+\mathcal{J}_k(\mathbf{x}_k) \right\}$$

$$\sigma _{k}^{2} = \alpha(\mathbf{x}_k) +\beta \sigma _{k-1}^{2}+\gamma (Y_{k-1}-\theta)^{2}$$

Such models are more tractable computationally. In fact the resulting option pricing formulae are similar to the Black-Scholes one, given by

$$C=S_{0}\Pi _{1}-Ke^{-r\tau}\Pi _{2}$$   .

where $\Pi_1$ is the options delta, while $\Pi_2$ is the probability of the option to end up in the money [and therefore to be exercised]. This result has been presented in a slightly different form in Geman, El Karoui and Rochet (1995, Journal of Applied Probability) and Bakshi and Madan (2000, Journal of Financial Economics).