7
$\begingroup$

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.

$\endgroup$
6
$\begingroup$

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)
$\endgroup$
  • 1
    $\begingroup$ For numerical estimation, the "Re" part of the integral would by approximated by FFT? $\endgroup$ – emcor Jul 5 '15 at 11:22
  • 1
    $\begingroup$ @emcor Yes,the fast Fourier transform was applied by Carr and Madan (1999) to speed up the computation of option prices $\endgroup$ – user16651 Jul 5 '15 at 12:36
  • 1
    $\begingroup$ Thx I posted a follow-up question on the Bates SVJ model here. $\endgroup$ – emcor Aug 5 '15 at 12:47

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.