Stochastic volatility model

A stochastic volatility model gives an asset a separate random volatility or variance process. For example, a stock can have pricing dynamics $d{S}_t/S_t=rdt+\sqrt{v_t}d{W}_t$, with random $v_t$, instead of the constant variance of the Black-Scholes model. The Heston model uses a CIR model for $v$; an Ornstein-Uhlenbeck process can instead model a signed volatility whose square is the variance.

The extra state lets current uncertainty affect future uncertainty, and correlated asset and variance shocks change the distribution of future returns and option values. This differs from a local volatility model, where the coefficient is a deterministic function of time and current spot alone.

With two independent sources of uncertainty and only a stock and bank account, the variance risk is generally not fully hedgeable. A physical variance model then does not uniquely determine its pricing dynamics: a volatility-risk premium or additional traded instruments must be specified. Heston's original model makes the joint asset/variance system tractable through an affine characteristic-function calculation.^[ref]

For log-price dynamics $dX=-{\sigma }^{2}dt/2+\sigma d{W}^X$ and volatility dynamics $d\sigma =A(\sigma )dt+B(\sigma )d{W}^{\sigma }$, with correlation $\rho$, the exponential payoff ansatz $U=e^{\theta X}V$ reduces the backward partial differential equation to the displayed one-dimensional equation. The terminal value is $V(T,\sigma )=1$. Correlation changes the volatility drift, while the log-price drift supplies the negative $\theta$ term.

For Ornstein-Uhlenbeck process volatility, matching powers of $\sigma$ reduces the transform PDE to a scalar Riccati equation for $R$, a linear equation for $Q$ and an integral for $P$. All coefficients start at zero. The representation exists up to the Riccati moment-explosion horizon; it need not remain finite on every horizon for arbitrary exponential powers.

The maximal time on which the coefficient solution of the exponential-quadratic transform stays finite. For $R^{\prime }=2R^2-2R+1$, $R(0)=0$, the solution is $R=(1+\tan (\tau -\pi /4))/2$ and explodes at $3\pi /4$. Thus local solvability of the coefficient ordinary differential equations does not imply an unrestricted global moment formula. For payoff exponent $0\le \theta \le 1$, the Riccati forcing is nonpositive and a negative equilibrium bounds the solution.

Spot volatility is the instantaneous diffusion coefficient multiplying relative price increments in a stochastic differential equation for an asset. In a stochastic volatility model it is itself a stochastic process.

The Heston model couples a positive stock price to a square-root variance process. Under a chosen risk-neutral measure, a standard specification with continuous dividend yield $\delta$ iswith $d[W^S,W^v{]}_t=\rho dt$, $\kappa ,\theta ,\eta >0$ and $|\rho |\le 1$. Here $\theta$ is long-run variance, $\kappa$ the mean-reversion speed, and $\eta$ the scale of variance shocks. The parameters under the pricing measure need not equal those under the physical measure.

The variance is a CIR model. It remains nonnegative; for $v_0>0$, $2\kappa \theta \ge {\eta }^2$ makes zero inaccessible. Correlation $\rho$ couples price changes to subsequent uncertainty, a mechanism absent from constant-volatility Black-Scholes model.

The joint process is affine: its conditional log-price characteristic function has the form $\exp (iu\log S_t+C(\tau ,u)+D(\tau ,u)v_t)$, with coefficients solving Riccati equations. Fourier inversion then prices European options. The familiar closed-form description includes a one-dimensional numerical integral, rather than a purely elementary call-price expression.^[ref]