pricing exotic options in a levy market with stochastic volatility

[smile_for_vol]

Hi,

I am attempting to implement the following paper:

The pricing of exotic options by Monte-Carlo simulations in a Levy market with stochastic volatility
AU - Schoutens, Wim
AU - Symens, Stijn
DO - 10.1142/S0219024903002249
JO - International Journal of Theoretical and Applied Finance

So far I have found a blog which provides some python code to implement the Carr-Madan formula using the FFT:

Fourier

python, option pricing, julien gosme, binomial tree, pde, mc, vellekoop, american option, dividend, brownian motion

gosmej1977.blogspot.com

I have quickly modified the code to create a simple call function:

import numpy as np
import scipy as sp
import scipy.interpolate
import scipy.stats

def FourierST3(S0,K,sigma,T,r,q,N):

   j=complex(0,1)
   #create vector in the real space
   x_min=-7.5
   x_max=7.5
   dx=(x_max-x_min)/(N-1)
   x=np.linspace(x_min,x_max,N)

   #create vector in the fourier space
   w_max=np.pi/dx;
   dw=2.0*w_max/(N);
   w=np.concatenate((np.linspace(0,w_max,N/2+1),np.linspace(-w_max+dw,-dw,N/2-1)))

   # Option payoff
   s = S0*np.exp(x);
   v_call = np.maximum(s-K,0)
   # v_put = np.maximum(K-s,0)

   # FST method
   char_exp_factor = np.exp((j*(r-0.5*sigma**2)*w - 0.5*sigma**2*(w**2)-r)*T) # characteristic function for lognormal density
   VC = np.real(np.fft.ifft(np.fft.fft(v_call)*char_exp_factor))
   # VP = np.real(np.fft.ifft(np.fft.fft(v_put)*char_exp_factor))

   #Interpolate option prices
   tck=sp.interpolate.splrep(s,VC)
   C=sp.interpolate.splev(S0,tck,der=0)
   # tck=sp.interpolate.splrep(s,VP)
   # P=sp.interpolate.splev(S0,tck,der=0)

   return C

FourierST3(1,1,0.25,1,0.02,0,2**18)

Any help with the following would be greatly appreciated:

• How do we reach the values of x_min and x_max? In the above blog, they are -7.5 and 7.5.
• I assume that in order to price the options we need the three characteristic functions that would replace the variable 'char_exp_factor' : VG-CIR, Meixner-CIR and NIG-CIR. I am confused as to how to compute and hence code these.

I am yet to reach the Monte-Carlo simulation and pricing sections, but if you had any material that would help the implementation of them that would be greatly appreciated too!

Thank you for your time.