Heston Model Characteristic Function¶

The characteristic function (CF) is the Fourier transform of the probability density function. For the Heston stochastic volatility model, the characteristic function has a closed-form expression that makes it invaluable for option pricing via Fourier methods.

Learning Objectives

By the end of this section, you will be able to:

Derive the Heston characteristic function
Understand the role of the non-centrality parameter
Use the CF for density recovery and option pricing

Heston Model Specification¶

The Heston (1993) model for asset price and volatility dynamics under the risk-neutral measure is:

\[ dS(t) = r S(t) dt + \sqrt{v(t)} S(t) dW_x^{\mathbb{Q}}(t) \]

\[ dv(t) = \kappa(\bar{v} - v(t)) dt + \gamma\sqrt{v(t)} dW_v^{\mathbb{Q}}(t) \]

where the two Brownian motions are correlated: \(d W_x^{\mathbb{Q}} dW_v^{\mathbb{Q}} = \rho dt\).

Characteristic Function Formula¶

The characteristic function of \(X_T = \log S(T)\) conditional on the initial state \((t, S, v)\) is:

\[ \varphi(u) := \mathbb{E}^{\mathbb{Q}}\left[e^{iu X_T} \big| \mathcal{F}(t)\right] = e^{r(T-t) + A(\tau,u) + B(\tau,u)X + C(\tau,u)v} \]

where \(\tau = T - t\) is time to maturity, and the components are:

\[ \boxed{ \begin{aligned} A(\tau,u) &= \frac{\kappa\bar{v}\tau}{\gamma^2}\left(\kappa - \gamma\rho iu - D\right) - \frac{2\kappa\bar{v}}{\gamma^2}\log\left(\frac{1 - ge^{-D\tau}}{1-g}\right)\\[0.5em] B(\tau,u) &= iu\\[0.5em] C(\tau,u) &= \frac{1 - e^{-D\tau}}{\gamma^2(1 - ge^{-D\tau})}(\kappa - \gamma\rho iu - D)\\[0.5em] D &= \sqrt{(\kappa - \gamma\rho iu)^2 + (u^2 + iu)\gamma^2}\\[0.5em] g &= \frac{\kappa - \gamma\rho iu - D}{\kappa - \gamma\rho iu + D} \end{aligned} } \]

Key Structural Elements¶

Complex argument \(D\): The term \(D\) encodes the interaction between mean reversion rate \(\kappa\), volatility of volatility \(\gamma\), and correlation \(\rho\). The square root ensures the solution is well-defined for all complex arguments \(u\).

Ratio \(g\): The ratio \(g\) is crucial for stability in numerical computation and controls the asymptotic behavior as \(\tau \to \infty\).

Logarithmic term in \(A\): The log term arises from integrating the CIR-type variance process and captures the mean reversion dynamics.

Transform to Physical Measure¶

If we wish to transition from the risk-neutral measure \(\mathbb{Q}\) to the physical measure \(\mathbb{P}\), we adjust the mean reversion level and the drift:

\[ dS(t) = \mu S(t) dt + \sqrt{v(t)} S(t) dW_x^{\mathbb{P}}(t) \]

\[ dv(t) = \kappa(\bar{v}^{\mathbb{P}} - v(t)) dt + \gamma\sqrt{v(t)} dW_v^{\mathbb{P}}(t) \]

Under Girsanov's theorem, the risk-neutral mean reversion level becomes:

\[ \bar{v} = \bar{v}^{\mathbb{P}} - \frac{\rho\gamma(\mu - r)}{\kappa} \]

This adjustment ensures consistency between physical and risk-neutral dynamics while preserving the Heston model structure.

Properties and Computational Notes¶

Affine structure: The exponent is linear in \(X\) and \(v\), making the Heston model an affine jump-diffusion (without jumps).
Numerical stability: When computing \(D\) for complex \(u\), care must be taken with the square root branch. Typically, the square root is chosen such that \(\text{Im}(D) > 0\).
Decay at infinity: As \(|u| \to \infty\), the CF decays sufficiently fast, allowing Fourier inversion via FFT.
Special case: When \(\rho = 0\), the stock and variance dynamics decouple, simplifying the derivation.

Summary¶

The Heston characteristic function provides a closed-form expression for the joint distribution of log-returns and variance. Its affine structure enables efficient option pricing through Fourier methods (FFT and COS methods) and is the foundation for understanding smile dynamics and variance-of-variance effects in modern derivatives markets.

Exercises¶

Exercise 1. Starting from the Heston PDE for the characteristic function, substitute the exponential-affine ansatz and collect terms independent of \(v\) (giving the \(C\)-equation) and terms proportional to \(v\) (giving the \(D\)-equation). Verify the resulting Riccati system.

Solution to Exercise 1

The Heston PDE for the characteristic function \(\varphi(u, \tau; x, v) = \mathbb{E}[e^{iux_T} \mid x_t = x, v_t = v]\) (with \(\tau = T - t\)) is:

\[ \frac{\partial\varphi}{\partial\tau} = (r - q - \tfrac{1}{2}v)\frac{\partial\varphi}{\partial x} + \kappa(\theta - v)\frac{\partial\varphi}{\partial v} + \tfrac{1}{2}v\frac{\partial^2\varphi}{\partial x^2} + \rho\sigma_v v\frac{\partial^2\varphi}{\partial x\,\partial v} + \tfrac{1}{2}\sigma_v^2 v\frac{\partial^2\varphi}{\partial v^2} \]

Substitute the exponential-affine ansatz \(\varphi = \exp(C(\tau, u) + D(\tau, u)v + iux)\). The relevant partial derivatives are:

\[ \partial_\tau\varphi = (C' + D'v)\varphi, \quad \partial_x\varphi = iu\,\varphi, \quad \partial_{xx}\varphi = -u^2\varphi \]

\[ \partial_v\varphi = D\,\varphi, \quad \partial_{vv}\varphi = D^2\varphi, \quad \partial_{xv}\varphi = iu\,D\,\varphi \]

Substituting and dividing by \(\varphi \neq 0\):

\[ C' + D'v = (r-q-\tfrac{1}{2}v)(iu) + \kappa(\theta - v)D + \tfrac{1}{2}v(-u^2) + \rho\sigma_v v(iu\,D) + \tfrac{1}{2}\sigma_v^2 v\,D^2 \]

Collecting terms independent of \(v\) (the \(C\)-equation):

\[ C' = (r-q)iu + \kappa\theta D \]

Collecting terms proportional to \(v\) (the \(D\)-equation, the Riccati ODE):

\[ D' = \tfrac{1}{2}\sigma_v^2 D^2 + (\rho\sigma_v iu - \kappa)D + \tfrac{1}{2}(iu - u^2) \]

with initial conditions \(C(0, u) = 0\), \(D(0, u) = 0\). This is verified because the matching of \(v^0\) and \(v^1\) coefficients is exact, and the ansatz is consistent with the PDE structure.

Exercise 2. For the special case \(\rho = 0\) and \(\sigma_v = 0\), show that the Heston characteristic function reduces to the Black-Scholes characteristic function \(\exp(iux + (iu(r-q) - \frac{1}{2}u^2 v_0)\tau)\).

Solution to Exercise 2

When \(\rho = 0\) and \(\sigma_v = 0\), the variance \(v_t = v_0\) is constant (no stochastic volatility). The Riccati coefficients become:

\[ a = 0, \quad b = -\kappa, \quad c = \tfrac{1}{2}(iu - u^2) \]

The \(D\)-equation reduces to \(D' = -\kappa D + \frac{1}{2}(iu - u^2)\) with \(D(0) = 0\). Since \(a = 0\), this is a linear ODE with solution:

\[ D(\tau) = \frac{iu - u^2}{2\kappa}(1 - e^{-\kappa\tau}) \]

As \(\kappa \to 0\) (no mean reversion), L'Hopital's rule gives \(D(\tau) \to \frac{1}{2}(iu - u^2)\tau\).

The \(C\)-equation gives \(C(\tau) = (r-q)iu\tau + \kappa\theta\int_0^\tau D(s)\,ds\). With \(\sigma_v = 0\), the variance is identically \(v_0\), so \(\kappa\theta = \kappa v_0\) (at stationarity). In the limit \(\kappa \to 0\):

\[ C(\tau) = (r-q)iu\tau \]

and \(D(\tau)v_0 = \frac{1}{2}(iu - u^2)v_0\tau\).

The characteristic function becomes:

\[ \varphi(u) = \exp\!\left(iux + (r-q)iu\tau + \tfrac{1}{2}(iu - u^2)v_0\tau\right) \]

\[ = \exp\!\left(iu\bigl[x + (r-q)\tau\bigr] - \tfrac{1}{2}u^2 v_0\tau\right) \]

This is exactly the Black-Scholes characteristic function for \(\log S_T\) with constant variance \(v_0\) (i.e., constant volatility \(\sigma = \sqrt{v_0}\)), confirming the reduction.

Exercise 3. The Heston characteristic function enables pricing via \(C = e^{-r\tau}\frac{1}{\pi}\int_0^\infty \operatorname{Re}[\cdots]\,du\). Explain why this integral converges and how the decay rate of \(|\varphi(u)|\) as \(u \to \infty\) affects the number of quadrature points needed.

Solution to Exercise 3

The Fourier inversion integral for a European call takes the form:

\[ C_{\text{call}} = e^{-r\tau}\frac{1}{\pi}\int_0^\infty \operatorname{Re}\!\left[\frac{e^{-iu\ln K}\varphi(u)}{iu}\right] du \]

(or a similar expression using the Gil-Pelaez formula). The integral converges because:

1. Decay of \(|\varphi(u)|\): As \(u \to \infty\), the characteristic function decays to zero. The exponent \(C(\tau, u) + D(\tau, u)v_0\) has a real part that becomes increasingly negative. Specifically, the dominant contribution for large \(u\) is the \(-u^2\) term in the Riccati coefficient \(c = \frac{1}{2}(iu - u^2)\), which drives \(\operatorname{Re}(D) \sim -\alpha u^2\) for some \(\alpha > 0\), so:

\[ |\varphi(u)| = \exp\!\bigl(\operatorname{Re}(C + Dv_0)\bigr) \to 0 \quad \text{as } u \to \infty \]

2. Integrand boundedness: The factor \(1/(iu)\) contributes a \(1/u\) decay, and the oscillatory factor \(e^{-iu\ln K}\) has modulus 1. So the integrand decays at least as fast as \(|\varphi(u)|/u\).

3. Effect on quadrature: The rate of decay of \(|\varphi(u)|\) determines the effective truncation point \(u_{\max}\) of the integral. Faster decay means fewer quadrature points are needed:

High \(\sigma_v\) (large vol-of-vol): the CF decays more slowly because the distribution has fatter tails, requiring more quadrature points (e.g., \(u_{\max} \sim 100\))
Low \(\sigma_v\): the CF decays rapidly (closer to Gaussian), and \(u_{\max} \sim 30\text{--}50\) suffices
A practical rule of thumb is to integrate until \(|\varphi(u_{\max})| < 10^{-8}\)

Exercise 4. Compute \(\varphi(-i, \tau; x, v)\) and show that it equals \(e^{x+(r-q)\tau} = S_t e^{(r-q)\tau}\), confirming the martingale condition \(\mathbb{E}[S_T/S_t] = e^{(r-q)\tau}\).

Solution to Exercise 4

The characteristic function at \(u = -i\) gives \(\varphi(-i) = \mathbb{E}[e^{i(-i)X_T}] = \mathbb{E}[e^{X_T}] = \mathbb{E}[S_T]\).

Using the Heston CF formula, we evaluate at \(u = -i\). The key intermediate quantities are:

Discriminant:

\[ \gamma(-i) = \sqrt{(\kappa - i\rho\gamma(-i))^2 + \gamma^2(i(-i) + (-i)^2)} = \sqrt{(\kappa - \rho\gamma)^2 + \gamma^2(1 - 1)} = |\kappa - \rho\gamma| \]

(Here \(\gamma\) on the left is the discriminant, while \(\gamma\) on the right is the vol-of-vol parameter; using the notation from the file where vol-of-vol is \(\gamma\).) With \(\operatorname{Re}(\kappa - \rho\gamma) > 0\), we get \(\gamma(-i) = \kappa - \rho\gamma\).

Ratio: \(g(-i) = \frac{(\kappa - \rho\gamma) - (\kappa - \rho\gamma)}{(\kappa - \rho\gamma) + (\kappa - \rho\gamma)} = 0\).

\(D\) function: \(D(\tau, -i) = 0\) (the numerator \(\kappa - \rho\gamma - \gamma(-i) = 0\)).

\(C\) function: With \(g = 0\), the log term vanishes, and:

\[ C(\tau, -i) = (r-q)\cdot i(-i)\cdot\tau + 0 = (r-q)\tau \]

Therefore:

\[ \varphi(-i) = e^{C + Dv + i(-i)x} = e^{(r-q)\tau + 0 + x} = S_t\,e^{(r-q)\tau} \]

This confirms \(\mathbb{E}^{\mathbb{Q}}[S_T \mid \mathcal{F}_t] = S_t\,e^{(r-q)\tau}\), the risk-neutral martingale condition for the discounted stock price (after adjusting for dividends).

Exercise 5. Explain qualitatively how each Heston parameter (\(\kappa\), \(\theta\), \(\sigma_v\), \(\rho\)) affects the shape of \(|\varphi(u)|\) as a function of \(u\). Which parameter causes the fastest decay?

Solution to Exercise 5

Each Heston parameter affects \(|\varphi(u)|\) as follows:

Mean-reversion speed \(\kappa\): Higher \(\kappa\) pulls the variance toward \(\theta\) more quickly, reducing the effective randomness of the integrated variance. This makes the log-return distribution closer to Gaussian, so \(|\varphi(u)|\) decays more slowly (Gaussian CFs decay as \(e^{-cu^2}\), which is the slowest among common distributions relative to fat-tailed distributions). However, \(\kappa\) also affects the characteristic time scale \(\gamma \approx \kappa\) for small \(u\), and large \(\kappa\) compresses the transient dynamics.

Long-run variance \(\theta\): Higher \(\theta\) increases the total variance of log-returns, which broadens the distribution. A broader distribution means faster decay of \(|\varphi(u)|\) because \(\operatorname{Re}(D)v + \operatorname{Re}(C)\) becomes more negative for large \(u\). The effect enters through the \(C\) function, which scales with \(\kappa\theta\).

Vol-of-vol \(\sigma_v\): This is the most important parameter for CF decay. Higher \(\sigma_v\) produces heavier tails (higher excess kurtosis), which means \(|\varphi(u)|\) decays more slowly. The decay rate transitions from approximately Gaussian (\(e^{-cu^2}\)) when \(\sigma_v\) is small to a slower rate when \(\sigma_v\) is large. Numerically, doubling \(\sigma_v\) can increase the truncation point \(u_{\max}\) by a factor of 2--4.

Correlation \(\rho\): Negative \(\rho\) introduces asymmetry (skewness) in the distribution. While \(\rho\) affects the phase of \(\varphi(u)\) (causing the real and imaginary parts to oscillate differently), its effect on \(|\varphi(u)|\) is secondary compared to \(\sigma_v\). Extreme \(|\rho|\) values slightly slow the decay because they amplify the coupling between the stock and variance processes.

Which parameter causes the fastest decay? The vol-of-vol \(\sigma_v\) is the dominant parameter. When \(\sigma_v\) is small, the model is close to Black-Scholes and \(|\varphi(u)|\) decays approximately as \(\exp(-\frac{1}{2}u^2 v_0 \tau)\), which is very fast. As \(\sigma_v\) increases, tail heaviness grows and the CF decay slows considerably. Therefore, low \(\sigma_v\) produces the fastest decay, and high \(\sigma_v\) produces the slowest.