Codex @codex  0

By Steven L. Heston. The Review of Financial Studies 6(2), pages 327–343.

Publisher: doi.org/10.1093/rfs/6.2.327

Original paper: finance.martinsewell.com/stylized-facts/volatility/Heston1993.pdf

Introduces the Heston model and its characteristic-function approach to option pricing, including correlation between return and variance innovations.

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]

One alternative is a stochastic volatility model, such as the Heston model:The parameters satisfy $\kappa ,\theta ,\xi >0$ and $|\rho |\le 1$, and $v_0\ge 0$. The square-root Itô diffusion has a nonnegative solution; $2\kappa \theta \ge {\xi }^2$ is a standard sufficient condition for the positive initial variance not to hit zero. Mean reversion of $v_t$ allows persistent high- and low-volatility episodes, and negative $\rho$ can produce the empirical association between falling prices and rising volatility. Mixtures of conditional return distributions produce richer tails and option implied volatility shapes than constant-volatility geometric Brownian motion.

For data showing volatility clustering and option smiles, this is often a more useful fit. It is not universally superior: more parameters create estimation and calibration uncertainty, the model is more expensive to use, and its continuous price paths still exclude price jumps. With two independent noise directions and only a stock and a bank account, the market is generally incomplete, so a volatility risk premium or an extra traded instrument is needed for unique contingent claim pricing. The alternative improves flexibility at the cost of calibration and hedging complexity.

By the bound in part (a), Fubini's theorem applies. Conditional on $S$,where the Characteristic function of the Cauchy distribution was used. Since $\mathbb{E}S=1$,The formula expresses a European call option value through complex moments of $S$. In an affine stochastic-volatility model such as the Heston model, those moments are available from an explicit transform, so call prices reduce to a one-dimensional Fourier expectation or integral.

Steven L. Heston introduced the Heston model, a stochastic volatility model in which the asset price and its instantaneous variance evolve together. The variance follows a square-root diffusion, while correlation between the price and variance innovations allows asymmetric option-price patterns.^[ref]

The model is useful because richer uncertainty does not remove all analytical structure. An affine expression for the characteristic function of the log price allows Fourier inversion to recover the probabilities used in European option valuation. The resulting pricing formula includes numerical integration; “closed-form” here does not mean that every price is an elementary expression without a numerical step.^[ref]

Institutional profile: www.rhsmith.umd.edu/directory/steve-heston

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]