A Closed-Form Solution for Options with Stochastic Volatility

Source: Heston, S. L. (1993) · Review of Financial Studies 6(2), 327–343 · DOI: 10.1093/rfs/6.2.327

TL;DR

Heston prices a European call when volatility is itself random. The asset follows geometric

Brownian motion while its variance follows a mean-reverting square-root (CIR-type) process, with

the two Wiener processes allowed arbitrary correlation ρ. Using a new technique based on

characteristic functions and Fourier inversion, he obtains a closed-form price of the Black-Scholes

form C = S·P1 − K·e^{−rT}·P2, where P1 and P2 are recovered by inverting characteristic functions.

The correlation ρ controls return skewness (and the strike-price/smile bias) while the volatility of

volatility σ controls kurtosis. He extends the model to stochastic interest rates, and thus to bond

and currency options.

What it prices

A European call option on a spot asset whose volatility is stochastic and may be correlated with the

spot return — the case that uncorrelated-volatility approaches (e.g., averaging Black-Scholes over

volatility paths) cannot handle. The same technique is then applied to options under stochastic

interest rates, including bond options and foreign-currency options. The goal is a tractable

generalization of Black-Scholes that reproduces the observed skewness and strike-price biases.

Setup & assumptions

The spot asset follows the diffusion

dS(t) = μ S dt + √v(t) · S dz₁(t)

and the variance v(t) follows the familiar square-root process [as in Cox, Ingersoll, and Ross (1985)]

dv(t) = κ(θ − v) dt + σ √v(t) dz₂(t)

with

κ = speed of mean reversion, θ = long-run variance, σ = volatility of volatility,

corr(dz₁, dz₂) = ρ (arbitrary correlation between volatility and spot return).

Heston notes that if volatility instead followed an Ornstein–Uhlenbeck process (as in Stein and

Stein 1991), Itô's lemma yields this square-root process for the variance. Interest rate r is

constant in the base model. Because there is a second source of risk (volatility), pricing requires a

price of volatility risk λ(S,v,t), assumed proportional to v (motivated by Breeden's (1979)

consumption model and the CIR equilibrium), though the pricing results follow from arbitrage. Any

asset U(S,v,t) must satisfy a two-state-variable PDE.

Key result

Guessing a solution of Black-Scholes form in terms of the log spot price,

C = S · P1 − K · e^{−r(T−t)} · P2

where each Pⱼ (j = 1,2) satisfies its own PDE with terminal condition the in-the-money indicator, so

Pⱼ is interpreted as a "risk-neutralized" probability that the option finishes in the money (P2 is the

risk-neutral exercise probability; P1 is the probability under the spot-numeraire measure). These

probabilities are not available in closed form, but their characteristic functions fⱼ(x, v, T; φ)

are, taking an exponential-affine form f = exp{C(τ;φ) + D(τ;φ)v + iφx} (the source of the later

"affine" terminology). The probabilities are then obtained by Fourier inversion:

Pⱼ = ½ + (1/π) ∫₀^∞ Re[ e^{−iφ ln K} fⱼ / (iφ) ] dφ.

Simulations (Figures using Table 1 parameters) show:

ρ → skewness: the correlation parameter positively affects skewness of spot returns; positive

ρ fattens the right tail (high variance when the asset rises), negative ρ does the opposite —

generating the strike-price biases relative to Black-Scholes.

σ → kurtosis: when σ = 0 the spot return is normal; otherwise σ raises kurtosis, and (with ρ ≈ 0)

σ increases kurtosis without inducing skewness.

Inputs & implementation

Inputs: spot S, strike K, maturity T, interest rate r, current variance v, and the four risk-neutral

parameters κ, θ, σ, ρ (plus, with stochastic rates, bond-price/rate-process parameters). Implementation:

evaluate the two single-dimensional Fourier-inversion integrals for P1 and P2 numerically (one

integral per probability) using the closed-form characteristic functions — far cheaper than solving

the two-dimensional PDE required by earlier stochastic-volatility models, which makes repeated

calibration to a cross-section of option prices feasible. For stochastic interest rates, Heston

modifies Equation (1) and the characteristic function retains the same exponential-affine form, so the

same inversion machinery prices bond and currency options. Author notes characteristic functions

always exist (Kendall and Stuart 1977).

Limitations

No jumps: a pure diffusion underfits the very short-maturity smile; later work (e.g., Bates 1996)

adds jumps on top of Heston dynamics.

Volatility risk premium / calibration: the price of volatility risk must be specified; the

risk-neutral long-run variance θ* implied by option prices need not equal the physical variance

because of the volatility risk premium, and parameters can be hard to identify.

Single variance factor: one square-root factor cannot match the full term structure of the

volatility surface or of volatility-of-volatility.

Key references

Heston, S. L. (1993) — A Closed-Form Solution for Options with Stochastic Volatility — Review of Financial Studies

Black, F. & Scholes, M. (1973) — The Pricing of Options and Corporate Liabilities — Journal of Political Economy

Cox, J., Ingersoll, J. & Ross, M. (1985) — A Theory of the Term Structure of Interest Rates (square-root process) — Econometrica

Stein, E. & Stein, J. (1991) — Stock Price Distributions with Stochastic Volatility — Review of Financial Studies

Bates, D. (1996) — Jumps and Stochastic Volatility: Exchange Rate Processes Implicit in Deutsche Mark Options — Review of Financial Studies

Provenance: verified/generated from the paper's full text.