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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 a non-dividend-paying stock in the Black-Scholes model, use density $d{Q}^S/dQ=e^{-\rho T}{S}_T/S_0$. Under $Q^S$, $W_t^S=W_t^Q-\sigma t$ is Brownian motion, and $\log S_t=\log S_0+(\rho +{\sigma }^2/2)t+\sigma W_t^S$. Thus $e^{-\rho T}{\mathbb{E}}_Q[S_{T}H]=S_0{\mathbb{E}}_{Q^S}H$. This change of numeraire is particularly useful for payments delivered in stock units.

By Fischer Black, Myron Scholes. Journal of Political Economy 81(3), pages 637–654, May–June 1973.

Publisher: www.journals.uchicago.edu/doi/10.1086/260062

Original paper: finance.martinsewell.com/option-pricing/BlackScholes1973.pdf

Develops the option-replication argument and the Black-Scholes model pricing equation.

When $d=\sqrt{c^2+2r}$ is real and nonnegative, the discounted drifted Brownian first-passage density is $e^{a(c-d)}{h}_d(t)$. Integrating yields $e^{a(c-d)}\Phi ((dT-a)/\sqrt{T})+e^{a(c+d)}\Phi ((-a-dT)/\sqrt{T})$. In a Black-Scholes model with $c=(r-{\sigma }^2/2)/\sigma$, the identity $c^2+2r=(r+{\sigma }^2/2{)}^2/{\sigma }^2$ ensures this formula covers every real interest rate.

A digital put option extinguished if the stock reaches an upper barrier $B$. Under the dividend-free Black-Scholes model, conditional on not having hit the barrier before the current time, let $0<S<B$, $\tau =T-t>0$, $m=\min (K,B)$, $\gamma =1-2r/{\sigma }^2$, andThen its price isThe Black-Scholes reflection symmetry proves the pricing equation, and the two terms cancel at $S=B$. At expiry the reflected argument is above $B\ge m$, so its payoff is zero, while $S<m$ is equivalent to $S<K$ inside the alive region. Thus the correct free strike is capped at the barrier. For $K\ge B$, the value is the discounted upper-barrier survival probability, independent of $K$, as for a barrier digital put. For $K>B$ its terminal data have a corner discontinuity at the absorbing barrier. Set the price to zero after a hit; the reflected expression on the other side is not a valid contract value.

An up-and-out power claim pays a power of the terminal stock price provided an upper barrier has never been reached. In the Black-Scholes model, set $A=\log (c/S_0)/\sigma$ and $m_p=(\rho +(p-\frac{1}{2}){\sigma }^2)/\sigma$, where $\sigma >0$, $c>S_0$ and $T>0$. The risk-neutral pricing value isHere $\Phi$ is the standard normal distribution function. Weighting the Brownian motion endpoint by $e^{p\sigma W_T-p^2{\sigma }^{2}T/2}$ changes the logarithmic drift to $m_p$; the finite-horizon maximum of Brownian motion with drift then supplies the survival factor.

For a down-and-out European call with barrier equal to strike, zero interest and no dividends, hold one share and borrow the strike until barrier hitting or maturity. At continuous hitting the share price is exactly the debt, so liquidating leaves zero; on survival the portfolio pays $S_T-K$. Hence the contract has value $S-K$ and delta one before knock-out. The pathwise replication works beyond constant-volatility Black-Scholes model, but can fail through downward jumps or execution gaps, and nonzero carry, transaction costs or discrete monitoring require changes.