Amin-Ng (1993) Option Valuation with Systematic Stochastic Volatility

Summary

Amin and Ng (1993) develop an equilibrium discrete-time option pricing framework in which the asset's conditional variance has a systematic component proportional to aggregate consumption variance and an idiosyncratic residual. Extending Rubinstein (1976) and Brennan (1979), they show that under constant proportional risk aversion (CPRA) preferences and bivariate conditional lognormality of asset-consumption returns, when the variance process is predictable the European call price equals an expectation of Black-Scholes values over future cumulative variance — a preference-free formula with no explicit risk-aversion parameters. The model endogenizes the riskless rate through consumption variance, generating procyclical interest rates, and extends to systematic Poisson jump risk.

Key Claims

• Proposition 1 (General): Under CPRA utility $E_0[\sum \rho^t C_t^{1-b}/(1-b)]$ and bivariate conditional lognormality of asset-consumption returns, the European call price equals a preference-dependent weighted expectation (eq. 15); the weights depend on the covariance of the stochastic discount factor with the cumulative asset return.
• Proposition 2 (Predictable-process, preference-free): If the volatility process $\{h_{s,t}\}$ is predictable (in the investor's time-0 information set), the call price reduces to $\pi(\text{Call}) = E_0[C_\text{BS}(\sigma^2_{0T})]$ — an expectation over Black-Scholes values at the cumulative variance $\sigma^2_{0T} = \sum_{t=1}^T h_{s,t}$ (eq. 16). No preference parameters appear.
• Variance decomposition (eq. 3): $h_{s,t} = \beta^2 h_{c,t} + h_{d,t}$, where $h_{c,t}$ is the conditional variance of log consumption growth (systematic component) and $h_{d,t}$ is the idiosyncratic component; $\beta$ is the sensitivity of log stock returns to log consumption growth.
• Endogenous riskless rate (eq. 11): Under predictable consumption variance, $r_t = -\ln\rho + b\mu_{c,t+1} - \tfrac{1}{2}b(1+b)h_{c,t+1}$ — declining in consumption variance, consistent with a flight-to-quality channel.
• Risk-neutral form (eq. 18): $\pi(\text{Call}) = S_0 N(d_1) - KB_0(T) N(d_2)$ where $d_1$ and $d_2$ use cumulative variance $\sigma^2_{0T}$, and $B_0(T)$ is the price of a pure discount bond.
• Systematic jump formula (eq. 27): When the systematic variance includes a compound Poisson jump component, the call price becomes a weighted sum of Black-Scholes terms $\sum_{n=0}^\infty [e^{-\lambda T}(\lambda T)^n / n!] \cdot C_\text{BS}(\sigma^2_{0T} + n\sigma_J^2)$, analogous to Merton (1976) but with jump risk priced via the CPRA kernel.
• Simulation results: With U.S. equity parameters ($\beta \approx 0.7$, $h_{c,t}$ following an estimated generalized autoregressive conditional heteroscedasticity (GARCH) process), the systematic stochastic volatility (SV) model produces option prices significantly above the Black-Scholes value at mean variance, especially for longer maturities and near-the-money options.

Concepts Introduced or Extended

• Stochastic Volatility
• Black-Scholes Option Pricing
• Jump-Diffusion Model

Entities Mentioned

• Kaushik I. Amin
• Victor K. Ng
• Mark Rubinstein

Quotes

"We find that when systematic volatility risk is significant, option prices computed using the standard Black-Scholes model using mean volatility can be substantially below the true option price."

My Take

Amin-Ng (1993) occupies a distinctive niche: it provides an equilibrium foundation for the expected-Black-Scholes option pricing formula (which Ball-Roma (1994) derive in a no-arbitrage setting) within a pure discrete-time CPRA framework, without continuous-time mathematics. The preference-free Proposition 2 is elegant but requires predictability of the variance process — empirically a stronger assumption than GARCH but weaker than pure deterministic volatility. The systematic/idiosyncratic variance decomposition directly anticipates factor GARCH models. The interest rate endogeneity is unusual among SV option pricing models and adds macroeconomic discipline at the cost of requiring calibration of consumption dynamics.