# 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:

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

python:
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.

Last edited:

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