Derivation of the Duan (1995) GARCH and NGARCH Model Under the Risk-Neutral Measure
The key result of Duan's (1995) derivation, applied to the NGARCH model, is the set of risk-neutral dynamics ($\mathbb{Q}$) necessary for option pricing. The core transformation involves enforcing the no-arbitrage condition, which sets the risk-neutral drift to $\mu_{t}^{\mathbb{Q}} = r_{f} - \frac{1}{2}h_{t}$, and subsequently adjusting the NGARCH variance equation to absorb the equity risk premium ($\lambda$) into the leverage parameter ($\gamma$), yielding the risk-neutral conditional variance:
$$\text{ln}(\frac{S_{t}}{S_{t-1}})=r_{f}-\frac{1}{2}h_{t}+\sqrt{h_{t}}z_{t}^{*}$$
$$h_{t}=\omega+\beta h_{t-1}+\alpha(z_{t-1}^{*}-(\lambda+\gamma)\sqrt{h_{t-1}})^{2}$$
where $z_{t}^{*} \sim \mathcal{N}(0,1)$ under $\mathbb{Q}$.
A step-by-step calibration algorithm and an illustrative example are included in Derivation of the Duan (1995) GARCH and NGARCH Model Under the Risk-Neutral Measure.
Methods
Derivatives Markets
- Stock Options
Including Exotic Options - Interest Rate Derivatives
Short Rate Model - Credit Derivatives
structural Models and Reduced Models