Heston Model Option Price Formula

What is the formula for the vanilla option (Call/Put) price in the Heston model?

I only found the bi-variate system of stochastic differential equations of Heston model but no expression for the option prices.

Answer

In the Heston Model we have

$$\begin{align} C(t\,,{{S}_{t}},{{v}_{t}},K,T)={{S}_{t}}{{P}_{1}}-K\,{{e}^{-r\tau }}{{P}_{2}} \end{align}$$ where, for $j=1,2$

$$\begin{align} & {{P}_{j}}({{x}_{t}}\,,\,{{v}_{t}}\,;\,\,{{x}_{T}},\ln K)=\frac{1}{2}+\frac{1}{\pi }\int\limits_{0}^{\infty }{\operatorname{Re}\left( \frac{{{e}^{-i\phi \ln K}}{{f}_{j}}(\phi ;t,x,v)}{i\phi } \right)}\,d\phi \\ & {{f}_{j}}(\phi \,;{{v}_{t}},{{x}_{t}})=\exp [{{C}_{j}}(\tau ,\phi )+{{D}_{j}}(\tau ,\phi ){{v}_{t}}+i\phi {{x}_{t}}] \\ \end{align}$$

and

$$\begin{align} & {{C}_{j}}(\tau ,\phi )=(r-q)i\phi \,\tau +\frac{a}{{{\sigma }^{2}}}{{\left( ({{b}_{j}}-\rho \sigma i\phi +{{d}_{j}})\,\tau -2\ln \frac{1-{{g}_{j}}{{e}^{{{d}_{j}}\tau }}}{1-{{g}_{j}}} \right)}_{_{_{_{{}}}}}} \\ & {{D}_{j}}(\tau ,\phi )=\frac{{{b}_{j}}-\rho \sigma i\phi +{{d}_{j}}}{{{\sigma }^{2}}}\left( \frac{1-{{e}^{{{d}_{j}}\tau }}}{1-{{g}_{j}}{{e}^{{{d}_{j}}\tau }}} \right) \\ \end{align}$$

where

$$\begin{align} & {{g}_{j}}=\frac{{{b}_{j}}-\rho \sigma i\phi +{{d}_{j}}}{{{b}_{j}}-\rho \sigma i\phi -{{d}_{j}}} \\ & {{d}_{j}}=\sqrt{{{({{b}_{j}}-\rho \sigma i\phi )}^{2}}-{{\sigma }^{2}}(2i{{u}_{j}}\phi -{{\phi }^{2}})} \\ & {{u}_{1}}=\frac{1}{2}\,,\,{{u}_{2}}=-\frac{1}{2}\,,\,a=\kappa \theta \,,\,{{b}_{1}}=\kappa +\lambda -\rho \sigma \,,\,{{b}_{2}}=\kappa +\lambda \,,\ {{i}^{2}}=-1 \\ \end{align}$$ Other representations:

1. Carr-Madan (1999)


2. Lewis (2000)


3. Attari (2004)


4. Gatheral (2006)


5. Albercher (2007)