Stochastic volatility model

A stochastic volatility model treats an asset's instantaneous variance as a random process rather than a constant parameter.

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 geometric asset price to a square-root variance diffusion. Its affine structure gives explicit transforms of the logarithmic asset price, which can be inverted to price European options.