Heston Stochastic Volatility Semi-Analytical Pricing via Fourier Inversion

Scenario

In the Heston (1993) stochastic volatility model, the underlying asset $S_t$ and its instantaneous variance $v_t$ follow coupled SDEs under the risk-neutral measure $\mathbb{Q}$: $dS_t = r S_t dt + \sqrt{v_t} S_t dW_t^S$ $dv_t = \kappa(\theta - v_t)dt + \sigma_v \sqrt{v_t} dW_t^v, \quad d\langle W^S, W^v \rangle_t = \rho dt$

Your Task

1. State the Gil-Pelaez Fourier inversion formula decomposing the European call price into risk-neutral probabilities $P_1$ and $P_2$: $C = S_0 P_1 - K e^{-r T} P_2, \quad P_j = \frac{1}{2} + \frac{1}{\pi} \int_0^\infty \text{Re}\left( \frac{e^{-i \phi \ln K} f_j(\ln S_0, v_0, T; \phi)}{i \phi} \right) d\phi$
2. State the analytical characteristic functions $f_1$ and $f_2$.
3. Numerically implement the Fourier integral and compute the call price for parameters: $S_0 = 100, K = 100, r = 0.03, T = 1.0, v_0 = 0.04, \kappa = 2.0, \theta = 0.04, \sigma_v = 0.30, \rho = -0.70$.

Constraints

• Feller condition $2\kappa\theta > \sigma_v^2$ satisfied ($2(2)(0.04) = 0.16 > 0.09$).
• Standard Albrecher rotation formulation for numerical stability of complex branch cuts.

Examples When $\sigma_v = 0$, Heston collapses to Black-Scholes with constant volatility $\sigma = \sqrt{v_0} = 0.20$.