Definition
GARCH option pricing values options when the underlying's return volatility follows a discrete-time GARCH process, by specifying a change of measure from the physical distribution P to a risk-neutral distribution Q under which discounted prices are martingales. Duan's (1995) locally risk-neutral valuation relationship (LRNVR) is the standard device for that change of measure; Zhang-Zhang (2020) modify it so the model can capture the variance risk premium.
Key Ideas
- Duan's (1995) LRNVR. The original GARCH option-pricing measure change keeps the conditional variance process and its parameters identical under P and Q — only the mean is adjusted so the discounted stock is a Q-martingale. Simple and widely used, but it forces the physical and risk-neutral variance dynamics to coincide.
- The variance risk premium (VRP). Investors pay to hedge volatility, so the risk-neutral expectation of future variance exceeds the physical one; the gap is the VRP. Empirically the CBOE VIX (a one-month variance-swap rate) is a risk-neutral expected variance and sits above physical/realized-variance forecasts — a feature Duan's LRNVR, with identical variance under both measures, cannot reproduce.
- The modified LRNVR (mLRNVR). Zhang-Zhang let the conditional variances differ across measures, making the variance process more persistent under Q than under P. This wedge in persistence is the mechanism that generates a variance risk premium inside the GARCH option-pricing model.
- Why it matters empirically. Under Duan's LRNVR the GARCH models under-price the SPX one-month variance-swap rate (VIX) by ~10% (Hao-Zhang 2013); under the mLRNVR the same models price the VIX accurately, so Zhang-Zhang recommend the mLRNVR for GARCH option pricing.
Why It Matters
- Discrete-time counterpart to stochastic-volatility pricing. GARCH option pricing gives a fully estimable, discrete-time alternative to continuous-time stochastic-volatility and Heston models, fit directly to return data.
- Ties pricing to the VIX. By targeting the variance-swap rate, it connects the GARCH volatility literature to the market price of variance risk and to VIX-based products.
- A measure-change lesson. The LRNVR→mLRNVR refinement shows that matching only the mean under Q is not enough; the volatility dynamics themselves must be re-specified under Q to reproduce observed option/variance prices.
Open Questions
- The precise form and identification of the physical-vs-risk-neutral persistence wedge, and its stability across regimes.
- Joint calibration to the full option surface (not just the one-month VIX) and to VIX term structure.
- Relationship between the discrete-time VRP wedge and continuous-time variance-risk-premium specifications.
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