Stochastic Volatility Option Pricing

Summary

Ball and Roma unify the Hull-White (1987) (H-W) power series and Stein-Stein (1991) (S-S) Fourier inversion approaches to stochastic volatility option pricing, showing both are applications of the moment generating function (MGF) of average variance $AV = (1/T)\int\sigma^2(s)\,ds$. They extend both methods to the Cox-Ingersoll-Ross (CIR)/Heston (1993) square-root volatility model using the CIR bond pricing formula, derive a formal analytical explanation for the volatility smile, and correct errors in Stein-Stein's analysis — specifically their misuse of $\bar\sigma^2$ as the Black-Scholes (BS) benchmark instead of $E[AV]$.

Key Claims

• MGF unification: H-W and S-S are two uses of the same MGF $K(\lambda) = E[\exp(-\lambda\int\sigma^2(s)\,ds)]$. H-W differentiates $K$ at $\lambda=0$ for moments → power series; S-S inverts $K$ as a Fourier transform → exact density.
• Square-root model MGF: For Heston (1993) variance process $dv = a(m-v)\,dt + \zeta\sqrt{v}\,dz$, the MGF of average variance is $I^*(\lambda) = \exp(N^*(T) + M^*(T)\cdot v_0)$, where $\gamma = \sqrt{a^2 + 2\lambda\zeta^2/T}$, $g(T) = 2\gamma + (a-\gamma)(1-e^{-\gamma T})$, $N^*(T) = (am/\zeta^2)\cdot\ln[2\gamma/g(T)]$, $M^*(T) = -2(1-e^{-\gamma T})/g(T)$. Read directly from CIR (1985) bond pricing formula — far simpler than S-S's sinh/cosh expressions.
• Power series accuracy: option price $\text{OP} \approx \text{BS}(E[AV]) + \tfrac{1}{2}\cdot\frac{\partial^2\text{BS}}{\partial AV^2}\cdot\text{Var}(AV) + \tfrac{1}{6}\cdot\frac{\partial^3\text{BS}}{\partial AV^3}\cdot\text{Skew}(AV)$. Second-order truncation (OP2) accurate to the penny for all plausible parameter values. Third-order (OP3) can be unstable; OP2 recommended.
• Volatility smile derivation: Implied variance under the false BS assumption is approximately $E[AV] + \varepsilon$, where $\varepsilon \approx \tfrac{1}{2}\cdot\tfrac{\text{Var}(AV)}{E[AV]}\cdot(-(c_2/2)^2 + (c_1/c_2)^2 - 1)$, with $c_1 = \ln(P/Ke^{-rT})$ and $c_2 = \sqrt{E[AV]\cdot T}$. $\varepsilon$ is quadratic in log-moneyness $c_1$, minimized at-the-money (ATM, $c_1=0$) and rising for out-of-the-money/in-the-money (OTM/ITM) options — the U-shaped volatility smile follows analytically from Jensen's inequality applied to the BS price's convexity in variance.
• S-S misdiagnosis corrected: S-S reported systematic upward option pricing bias from SV. Error: they compared their SV price against BS at variance $\bar\sigma^2$ (Ornstein-Uhlenbeck, OU, long-run mean of $\sigma$ squared). Correct benchmark is $E[AV] = \bar\sigma^2 + k^2/2\delta + \text{transient terms} > \bar\sigma^2$ (Jensen's inequality). With correct benchmark, SV gives downward bias ATM and upward bias OTM/ITM.
• $|\sigma|$ vs. reflected OU: S-S claimed their arithmetic OU model is equivalent to a reflecting barrier at $\sigma=0$. In fact they model $|\sigma(t)|$ (absolute value of OU), whose transition density is $p(x,y;t)+p(x,-y;t)$ — not the reflecting-barrier density satisfying the Kolmogorov boundary condition at 0. The absolute-value process has a higher long-run mean than $\bar\sigma$, imparting upward bias to $E[AV]$.
• Expected average variance (OU model): $E[AV] = \bar\sigma^2 + k^2/2\delta + (2\delta(\sigma_0-\bar\sigma)-k^2(1-e^{-\delta T}))/(\delta^2 T)$. The limiting average variance ($T\to\infty$) converges to $\bar\sigma^2 + k^2/2\delta > \bar\sigma^2$, confirming the upward bias.
• Zero correlation constraint: All methods require $dz_1 \perp dz_2$. With nonzero correlation, $P$ conditional on $AV$ is no longer lognormal; both power series and Fourier-via-AV methods break down. Heston (1993) handles nonzero correlation via characteristic functions of log-price directly.

Concepts Introduced or Extended

• Stochastic Volatility

Entities Mentioned

• Clifford Ball
• Antonio Roma

Quotes

"The core technical problem, tackled by these two papers, is the characterization of the distribution of the average variance of the underlying asset, which may be achieved through its moment generating function."

"Not only are the option prices in close agreement but the stochastic volatility price is below the BS counterpart. This behavior was not demonstrated in the S&S paper."

"The implied variance of the BS option price when the true price process is subject to stochastic volatility is approximately quadratic in $c_1$, the 'out-ness-of-the-money.'"

My Take

The paper's lasting contribution is the MGF-of-average-variance framework: it clarifies that all zero-correlation SV option pricing methods are equivalent in principle and differ only in how they use the MGF. The closed-form connection to CIR bond pricing is elegant and makes the Heston model tractable without requiring the full characteristic-function machinery Heston used. The S-S correction is important pedagogically — the error (using $\bar\sigma^2$ instead of $E[AV]$ as the BS benchmark) is easy to make and was influential. The smile derivation (Section V) is clean and intuitive, though it only works for the uncorrelated case; the empirically observed skew (asymmetric smile) requires nonzero leverage, which is outside this paper's scope.