Heston model
The Heston model is a mathematical framework in financial modeling that describes the joint dynamics of an asset's price and its instantaneous variance (volatility squared) as correlated stochastic processes, enabling the pricing of derivative securities such as European call options. Introduced by Steven L. Heston in 1993, it assumes the asset price follows a geometric Brownian motion driven by stochastic volatility, while the variance process adheres to a mean-reverting square-root diffusion to ensure non-negativity, with a correlation parameter capturing the leverage effect where negative asset returns coincide with volatility increases.[1] The model's core equations specify the asset price dynamics as dS_t = \mu S_t dt + \sqrt{v_t} S_t dZ_{1t} under the physical measure, where \mu is the drift, and the variance as dv_t = \kappa (\theta - v_t) dt + \sigma \sqrt{v_t} dZ_{2t}, with \kappa > 0 as the speed of mean reversion, \theta > 0 as the long-term variance mean, \sigma > 0 as the volatility of variance, and \rho as the correlation between the Wiener processes dZ_{1t} and dZ_{2t}.[1] Under the risk-neutral measure, the drift \mu shifts to the risk-free rate r, facilitating a semi-closed-form solution for option prices via the characteristic function and Fourier inversion: the call price is C(S_0, K, T) = S_0 P_1 - K e^{-rT} P_2, where P_1 and P_2 are risk-neutral probabilities computed numerically.[1] This formulation addresses limitations of the constant-volatility Black-Scholes model by generating volatility smiles, skewness, and excess kurtosis in return distributions, which align better with empirical option data.[1] Key advantages include its ability to incorporate arbitrary correlations between spot returns and volatility changes, thus explaining strike-price biases and return asymmetries observed in markets, while remaining computationally tractable for calibration to observed prices.[1] Originally applied to stock options, the model extends to bond and currency options by integrating stochastic interest rates, and it has influenced subsequent extensions like stochastic volatility jump-diffusion models for enhanced realism in volatile markets.[1] Despite requiring numerical methods for the integrals in P_1 and P_2, its affine structure allows efficient Fourier-based implementations, making it a cornerstone for quantitative finance practitioners.[2]Introduction and Background
Overview
The Heston model is a mathematical framework in financial engineering that models the dynamics of an asset price under stochastic volatility, where the instantaneous variance of returns follows a mean-reverting square-root diffusion process known as the Cox-Ingersoll-Ross (CIR) process.[3] This approach allows the volatility to fluctuate randomly over time, providing a more realistic representation of market behavior compared to models with fixed volatility.[3] Developed to improve upon the Black-Scholes model's assumption of constant volatility—which fails to replicate the empirical volatility smile and skew in option prices across different strikes and maturities—the Heston model incorporates stochastic variance to generate these observed patterns through the interplay of mean reversion and volatility clustering.[3] The leverage effect, captured by a negative correlation between asset returns and variance changes, further enhances its ability to fit market-implied volatilities.[3] At its core, the Heston model is governed by coupled stochastic differential equations describing the joint evolution of the asset price and its variance, enabling the derivation of option prices via Fourier transform methods.[3] Key parameters include the initial variance v_0, which sets the starting level of volatility; the long-term variance \theta, representing the equilibrium mean; the mean reversion speed \kappa, controlling how quickly variance returns to its long-term level; and the volatility of volatility \sigma, which measures the randomness in the variance process itself.[3] The model finds broad applications in pricing derivatives across asset classes, including equity options, foreign exchange (FX) contracts, and interest rate instruments, due to its flexibility in capturing volatility dynamics relevant to these markets.[3]Historical Development
The Heston model was introduced by Steven L. Heston in 1993 through his seminal paper titled "A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options," published in The Review of Financial Studies.[4] In this work, Heston proposed a stochastic volatility framework for pricing European call options on assets whose volatility follows a mean-reverting square-root process, deriving a closed-form solution via Fourier transform techniques that express option prices in terms of the characteristic function of the log asset price.[4] The model's development addressed key limitations of the Black-Scholes framework, which assumes constant volatility and fails to capture observed volatility dynamics such as the volatility smile and leverage effects in financial markets.[3] Earlier stochastic volatility models, including the one by Hull and White in 1987, had introduced time-varying volatility but often lacked tractable pricing formulas, relying instead on numerical approximations.[5] Heston's innovation built on these foundations by ensuring the joint process for asset price and variance admits an affine structure, enabling efficient computation while accommodating correlation between asset returns and volatility shocks.[4] Following its publication, the Heston model saw rapid adoption in quantitative finance practice during the late 1990s and beyond, becoming one of the most widely used stochastic volatility models for option pricing and risk management due to its balance of realism and computational feasibility.[6] Its affine diffusion properties also exerted significant influence on the broader class of affine term structure models, providing a foundational example for pricing interest rate derivatives and extending to multi-factor settings as formalized in subsequent theoretical advances.[7] While Heston himself did not publish major updates to the core model, it inspired numerous extensions, such as multifactor volatility variants that incorporate matrix Wishart processes for enhanced flexibility in capturing term structure dynamics.[8]Model Formulation
Asset Price Dynamics
The asset price dynamics in the Heston model are described by the stochastic differential equation (SDE) under the physical probability measure: dS_t = \mu S_t \, dt + \sqrt{v_t} \, S_t \, dW_t^S, where S_t denotes the price of the underlying asset at time t, \mu is the constant drift representing the expected return, v_t is the instantaneous variance process, and W_t^S is a standard Brownian motion driving the asset price innovations.[9] This formulation extends the geometric Brownian motion framework by incorporating stochastic volatility through the term \sqrt{v_t}, which scales the diffusion component.[10] Conditional on the realized path of v_t, the asset returns are log-normally distributed, reflecting the multiplicative nature of the shocks; the stochastic scaling by \sqrt{v_t} allows the model to capture time-varying volatility and empirical features such as volatility clustering in asset returns.[11] The asset price Brownian motion W_t^S is correlated with the Brownian motion W_t^v of the variance process, characterized by the instantaneous correlation \rho = \frac{d\langle W^S, W^v \rangle_t}{dt}, where \rho \in (-1, 1).[10] This correlation parameter introduces the leverage effect, typically with \rho < 0, whereby negative shocks to the asset price tend to coincide with increases in future volatility, consistent with empirical observations in equity markets.[9] The model specifies initial conditions S_0 > 0 for the asset price and v_0 \geq 0 for the initial variance, with \mu calibrated to historical data under the physical measure.[9] The derivation relies on Itô's lemma and properties of stochastic integrals for multivariate diffusions.[9] The variance process v_t exhibits mean reversion, linking the asset dynamics to a separate stochastic process for volatility (detailed in the Volatility Process section).Volatility Process
The volatility process in the Heston model describes the evolution of the instantaneous variance v_t, which follows a Cox–Ingersoll–Ross (CIR) process to capture mean-reverting behavior and ensure non-negative values. The stochastic differential equation (SDE) for v_t is given by dv_t = \kappa (\theta - v_t) \, dt + \sigma \sqrt{v_t} \, dW_t^v, where \kappa > 0 denotes the speed of mean reversion, \theta > 0 is the long-term mean level of variance, \sigma > 0 represents the volatility of volatility, and W_t^v is a standard Brownian motion under the physical measure. This formulation, adapted from the interest rate context, allows volatility to fluctuate realistically while reverting toward its equilibrium.[1][12] The parameters carry clear economic interpretations in modeling financial volatility. The mean-reversion speed \kappa governs the persistence of volatility deviations, with larger \kappa implying quicker decay of shocks and less prolonged volatility regimes. The long-term variance \theta sets the baseline equilibrium level around which volatility oscillates, often calibrated to historical averages. The vol-of-vol parameter \sigma quantifies the intensity of volatility fluctuations, capturing phenomena like clustering where high-volatility periods beget further variability. These interpretations align with empirical observations of equity markets, where volatility exhibits persistence and bursts.[1][13] Key properties of the CIR process ensure its suitability for variance modeling. It guarantees v_t > 0 almost surely when starting from a positive initial value, provided the Feller condition $2\kappa\theta > \sigma^2 holds; under this restriction, the process mean-reverts to \theta without reaching the origin, maintaining strict positivity. The process is ergodic under standard parameter assumptions, converging to a stationary gamma distribution with shape $2\kappa\theta / \sigma^2 and scale \sigma^2 / (2\kappa), which is equivalently a scaled non-central chi-squared distribution for transitional densities. This stationarity reflects long-run stability in volatility levels.[12][14] Regarding boundary behavior, the origin acts as a natural barrier: when the Feller condition is satisfied, zero is unattainable and entrance-regular, meaning the process cannot cross it and reflects instantaneously if approached. If violated ($2\kappa\theta < \sigma^2), zero becomes attainable but instantaneously reflecting, allowing brief touches without absorption, which preserves positivity while permitting extreme low-volatility states observed in calm markets. This behavior is analyzed via scale functions and speed measures in diffusion theory.[12][14] The variance process operates independently of the asset price dynamics except for the correlation \rho between their driving Brownian motions, which introduces leverage effects without altering the marginal law of v_t. In simulations, the square-root diffusion term poses stability challenges, as naive Euler discretization can produce negative values, particularly when the Feller condition fails or time steps are large. To address this, exact simulation methods exploiting the non-central chi-squared transition density ensure positivity and accuracy, while approximate schemes like full truncation or Milstein variants enhance stability for Monte Carlo applications. The asset price process scales its diffusion by \sqrt{v_t} to incorporate this stochastic volatility.[1][15][16]Pricing Framework
Risk-Neutral Measure
In derivative pricing within the Heston model, the risk-neutral measure \mathbb{Q} is employed to ensure no-arbitrage conditions, under which the discounted asset price process is a martingale. This measure adjusts the drifts of the stochastic processes while preserving their diffusion structures, allowing the fair price of a derivative to be the expected value of its discounted payoff under \mathbb{Q}.[1] Under the physical measure \mathbb{P}, the asset price S_t evolves as dS_t = \mu S_t \, dt + \sqrt{v_t} S_t \, dW_t^{S,\mathbb{P}}, where \mu is the expected return, and the instantaneous variance v_t follows the Cox-Ingersoll-Ross (CIR) process dv_t = \kappa (\theta - v_t) \, dt + \sigma \sqrt{v_t} \, dW_t^{v,\mathbb{P}}, with \langle dW_t^{S,\mathbb{P}}, dW_t^{v,\mathbb{P}} \rangle = \rho \, dt. The transition to \mathbb{Q} is achieved via Girsanov's theorem, which redefines the Brownian motions through a change-of-measure density that incorporates the market prices of risk. The stock price risk is priced by shifting the asset Brownian motion by (\mu - r)/\sqrt{v_t}, where r is the risk-free rate, while the volatility risk is priced via a premium \lambda(S_t, v_t, t) = \lambda v_t, leading to a shift of \lambda v_t / (\sigma \sqrt{v_t}) for the volatility Brownian motion. Due to the correlation \rho, the shifts interact, but the affine structure ensures the processes retain their form under \mathbb{Q}. Under \mathbb{Q}, the dynamics simplify to dS_t = r S_t \, dt + \sqrt{v_t} S_t \, dW_t^{S,\mathbb{Q}} and dv_t = \kappa^* (\theta^* - v_t) \, dt + \sigma \sqrt{v_t} \, dW_t^{v,\mathbb{Q}}, where \kappa^* = \kappa + \lambda, \theta^* = \kappa \theta / (\kappa + \lambda), and \langle dW_t^{S,\mathbb{Q}}, dW_t^{v,\mathbb{Q}} \rangle = \rho \, dt. The volatility process remains unchanged in its diffusion term and correlation, as the model is affine, facilitating equivalent risk-neutral pricing.[1] This measure change implies that the forward price of the asset is S_0 e^{rT} = \mathbb{E}^{\mathbb{Q}}[S_T], enabling consistent no-arbitrage valuation of options and other derivatives by taking expectations under \mathbb{Q}.[1]Characteristic Function Derivation
The Heston model exhibits an affine structure, where the log-asset price \log S_T and the variance v_T jointly form an affine diffusion process under the risk-neutral measure. This property implies that the characteristic function of \log S_T, defined as \phi(u) = \mathbb{E}[\exp(i u \log S_T) \mid \log S_t, v_t], admits an exponential-affine form:\phi(u) = \exp\left( C(\tau, u) + D(\tau, u) v_t + i u (\log S_t + (r - q)\tau) \right),
where \tau = T - t is the time to maturity, r is the risk-free rate, and q is the dividend yield.[1] The derivation of this characteristic function proceeds via the Feynman-Kac theorem, which links the expectation to the solution of a partial differential equation (PDE) satisfied by \phi(u). Under the risk-neutral dynamics, the PDE for \phi is obtained from the infinitesimal generator of the joint process for (\log S_t, v_t), incorporating the stochastic differential equations for the asset price and variance. Assuming an affine ansatz for \phi, the PDE reduces to a system of ordinary differential equations (ODEs) for the functions C(\tau, u) and D(\tau, u).[1] The resulting Riccati ODE for D(\tau, u) is
\frac{d D}{d \tau} = \frac{1}{2} \sigma^2 D^2 + (\rho \sigma i u - \kappa^*) D + \frac{1}{2} (u^2 - i u),
with initial condition D(0, u) = 0, where \kappa^* is the risk-neutral mean-reversion speed, \theta^* is the long-term variance under the risk-neutral measure, \sigma > 0 as the volatility of variance, and \rho as the correlation between asset and variance shocks. The ODE for C(\tau, u) is
\frac{d C}{d \tau} = \kappa \theta D - i u (r - q),
with C(0, u) = 0. These equations arise directly from substituting the affine form into the PDE and collecting coefficients of v_t and the constant terms.[1] The Riccati equation for D admits a closed-form solution:
D(\tau, u) = \frac{r_1 - \rho \sigma i u + d}{\sigma^2} \left( 1 - \frac{1 - g e^{d \tau}}{1 - g} \right),
where d = \sqrt{ (\kappa^* - \rho \sigma i u)^2 + \sigma^2 (u^2 - i u) }, r_1 = \kappa + \lambda (with \lambda the market price of volatility risk, often set to zero), and g = \frac{r_1 - \rho \sigma i u + d}{r_1 - \rho \sigma i u - d}. Integrating the ODE for C yields
C(\tau, u) = (r - q) i u \tau + \frac{\kappa \theta}{\sigma^2} \left[ (r_1 - \rho \sigma i u + d) \tau - 2 \log \left( \frac{1 - g e^{d \tau}}{1 - g} \right) \right]. Due to the complex-valued arguments, the expressions involve branch cuts in the complex plane, particularly along the imaginary axis for u, which must be handled carefully in numerical implementations to ensure convergence and avoid discontinuities.[1] In the special case where \sigma = 0, the variance process becomes deterministic, v_t = \theta^* for all t assuming v_0 = \theta^*, and the characteristic function reduces to the Black-Scholes form: \phi(u) = \exp\left( i u (\log S_t + (r - q - \theta^*/2) \tau) - \frac{1}{2} u^2 \theta^* \tau \right). This limit confirms the model's consistency with the constant-volatility case.[1]