Heston Option Pricing and Complex Characteristic Functions

# Heston stochastic model - Intuition, why a price of a call would involved complex numbers?


# Heston stochastic model - Intuition, why a price of a call would involved complex numbers?












I am new to stochastic volatility and Heston model and I don't understand why would a price of a call option involve complex numbers. I can see technically why but I don't see the intuition.

I was looking at this article http://www.rogerlord.com/complexlogarithmsheston.pdf and I was wondering :

1 - How would one compute numerically the integral in equation (1) in the article?

2 - Would the integral yield a real number surely?

thanks

## Answer by Kevin (score 1, accepted)

https://quant.stackexchange.com/a/51762

In the Heston (1993) model, the stock price is defined by the SDE system \begin{align*} \mathrm{d}S_t&=(r-q) S_t \mathrm{d}t+\sqrt{v_t} S_t \mathrm{d}W_{1,t}, \\ \mathrm{d}v_t&=\kappa(\theta-v_t) \mathrm{d}t+\xi \sqrt{v_t} \mathrm{d}W_{2,t}, \end{align*} where $\mathbb{E}^\mathbb{Q}[\mathrm{d}W_{1,t}\mathrm{d}W_{2,t}]=\rho\mathrm{d}t$. So, I assume all parameter are given under the risk-neutral measure, The characteristic function of the log stock price $\ln(S_t)$ under the risk-neutral measure $\mathbb{Q}$ is given by \begin{align*} \varphi_t^\text{Heston}(u) &= \exp\big( \ln\big(S_0e^{(r-q)t}\big)iu + C_t(u) + D_t(u)\cdot v_0 \big), \end{align*} where \begin{align*} C_t(u) &= \frac{\kappa\theta}{\xi^2} \left(\big( h(u)+d(u)\big) t - 2\ln\left(\frac{1-g(u)e^{d(u)t}}{1-g(u)}\right) \right),\\ D_t(u) &= \frac{h(u)+d(u)}{\xi^2}\cdot\frac{1-e^{d(u)t}}{1-g(u)e^{d(u)t}}, \\ g(u) &= \frac{h(u) + d(u)}{h(u)-d(u)}, \\ h(u) &= \kappa - \rho\xi \cdot i u, \\ d(u) &= \sqrt{h(u)^2+\xi^2\big(i u + u^2\big)}. \end{align*}

There are some numerical issues about the Heston chararcteristic function. Just google ``little Heston trap''. The author of the paper in your question, Roger Lord, alongside Kahl and Jäckel, did some research on this. The simplest case seems to be the adjustment from Albrecher et al. (2007) and Gatheral (2006).