# Evaluation of the semi-closed Heston pricing formula for call options

I'd like to know, how the integral part of the semi-closed Heston pricing formula for call options can be simulated for a given set of model parameters. Monte Carlo simulations shoud work for this purpose.

It would be great, if someone could provide an example, where this is computed or can give a paper, where an explicit example is given. Maybe a Matlab code can also be helpful in this case.

I am refering to equation (18) on page 331 of the paper

http://elis.sigmath.es.osaka-u.ac.jp/research/Heston-original.pdf

Thanks.

Regards,

Arjen

• does i = sqrt(-1) in the eqn – nathanesau Jul 16 '15 at 9:05
• Did you extract this code somewhere ? Or did you program it yourself ? In case you used a paper, it would be great to know, which paper you used. For the theory, I can use "Monte Carlo Methods in Financial Engineering" by Paul Glasserman. I have the following questions regarding the code: - I assume m stands for the number of simulations, what is then n supposed to be ? - phi is the variable of integration - but what do the "down" and "up" measure ? - "sum1", "sum2" and "epsilon" are also not so clear to me. Thanks for your Matlab code. – user16930 Jul 16 '15 at 15:15
• @Arjen epsilon: error of integration , up : upper bound of integration , down: lower bound of integration, n=number of normal random variable – user16891 Jul 16 '15 at 16:50
• @ Farahvartish You must have transformed the bounds somehow, which should be necessary. But I do not understand how you did that. In equation (18) the bounds are 0 and infinity. Which theory did you use for this ? Thanks very much – user16930 Jul 16 '15 at 17:14
• The probabilities $P_j$ in the Heston (1993) model require the integration domain$(0,\infty)$ so that $a = 0$ and $b=\infty$ are required as the limits of the integral (18).When we use Numerical Methods, however, we must truncate the upper limit.it described by Zhu (2010). Alternatively, Kahl and Jackel (2005) transform the domain of integration to [0, 1], which eliminates the need for truncation altogether. – user16891 Jul 16 '15 at 17:38

## 1 Answer

I have approximate the integrals by Monte Carlo Method but you can use several method such as Newton-Cotes formulas and Gaussian quadrature.

## Function ## Example ## Solutions

Call =

34.0976

Put =

4.8941


Parameters were extracted from Jianwei Zhu(2008),Page 10,Table 4