# -*- coding: iso-8859-1 -*-
# mp_laplace.py
# laplace.py with mpmath
# appropriate for high precision
# Talbot suggested that the Bromwich line be deformed into a contour that begins
# and ends in the left half plane, i.e., z \to \infty at both ends.
# Due to the exponential factor the integrand decays rapidly
# on such a contour. In such situations the trapezoidal rule converge
# extraordinarily rapidly.
# For example here we compute the inverse transform of F(s) = 1/(s+1) at t = 1
#
# >>> error = Talbot(1,24)-exp(-1)
# >>> error
# (3.3306690738754696e-015+0j)
#
# Talbot method is very powerful here we see an error of 3.3e-015
# with only 24 function evaluations
#
# Created by Fernando Damian Nieuwveldt
# email:fdnieuwveldt@gmail.com
# Date : 25 October 2009
#
# Adapted to mpmath and classes by Dieter Kadelka
# email: Dieter.Kadelka@kit.edu
# Date : 27 October 2009
# Automatic precision control by D. Kadelka 2009-11-26
#
# Reference
# L.N.Trefethen, J.A.C.Weideman, and T.Schmelzer. Talbot quadratures
# and rational approximations. BIT. Numerical Mathematics,
# 46(3):653 670, 2006.
try:
import psyco
psyco.full()
except ImportError:
print 'Psyco not installed, the program will just run slower'
from mpmath import mp,mpf,mpc,pi,sin,tan,exp,floor,log10
# testfunction: Laplace-transform of exp(-t)
def F(s):
return 1.0/(s+1.0)
class Talbot(object):
# parameters from
# T. Schmelzer, L.N. Trefethen, SIAM J. Numer. Anal. 45 (2007) 558-571
c1 = mpf('0.5017')
c2 = mpf('0.6407')
c3 = mpf('0.6122')
c4 = mpc('0','0.2645')
# High precision of these parameters not needed
def __init__(self,F=F,shift=0.0,prec=50):
self.F = F
# test = Talbot() or test = Talbot(F) initializes with testfunction F
# Assumption: F realvalued and analytic
self.shift = mpf(shift)
# Shift contour to the right in case there is a pole on the
# positive real axis :
# Note the contour will not be optimal since it was originally devoloped
# for function with singularities on the negative real axis For example
# take F(s) = 1/(s-1), it has a pole at s = 1, the contour needs to be
# shifted with one unit, i.e shift = 1.
# But in the test example no shifting is necessary
self.N = 12
# with double precision this constant N seems to best for the testfunction
# given. For N = 11 or N = 13 the error is larger (for this special
# testfunction).
self.prec = prec
# calculations with prec more digits
def __call__(self,t):
with mp.extradps(self.prec):
t = mpf(t)
if t == 0:
print "ERROR: Inverse transform can not be calculated for t=0"
return ("Error");
N = 2*self.N
# Initiate the stepsize (mit aktueller Präsision)
h = 2*pi/N
# The for loop is evaluating the Laplace inversion at each point theta i
# which is based on the trapezoidal rule
ans = 0.0
for k in range(self.N):
theta = -pi + (k+0.5)*h
z = self.shift + N/t*(Talbot.c1*theta/tan(Talbot.c2*theta) - Talbot.c3 + Talbot.c4*theta)
dz = N/t * (-Talbot.c1*Talbot.c2*theta/sin(Talbot.c2*theta)**2 + Talbot.c1/tan(Talbot.c2*theta)+Talbot.c4)
v1 = exp(z*t)*dz
prec = floor(max(log10(abs(v1)),0))
with mp.extradps(prec):
value = self.F(z)
ans += v1*value
return ((h/pi)*ans).imag
*********************************************************************************
# -*- coding: iso-8859-1 -*-
# asian.py
# Title : Numerical inversion of the Laplace transform for pricing Asian options
# The Geman and Yor model
#
# Numerical inversion is done by Asian's method.
#
################################################################################
## Created by Fernando Damian Nieuwveldt
## Date : 26 October 2009
## email : fdnieuwveldt@gmail.com
## This was part work of my masters thesis (The Asian method not mpmath part)
## in Applied Mathematics at the University of Stellenbosch, South Africa
## Thesis title : A Survey of Computational Methods for Pricing Asian Options
## For reference details contact me via email.
################################################################################
# Example :
# Asian(2,2,1,0,0.1,0.02,100)
# 0.0559860415440030213974642963090994900722---mp.dps = 100
# Asian(2,2,1,0,0.05,0.02,250)
# 0.03394203103227322980773---mp.dps = 150
#
# NB : Computational time increases as the volatility becomes small, because of
# the argument for the hypergeometric function becomes large
#
# H. Geman and M. Yor. Bessel processes, Asian options and perpetuities.
# Mathematical Finance, 3:349 375, 1993.
# L.N.Trefethen, J.A.C.Weideman, and T.Schmelzer. Asian quadratures
# and rational approximations. BIT. Numerical Mathematics,
# 46(3):653 670, 2006.
# adapted to mp_laplace by D. Kadelka 2009-11-17
# Automatic precision control by D. Kadelka 2009-11-26
# email: Dieter.Kadelka@stoch.uni-karlsruhe.de
# Example:
# from asian import Asian
# f = Asian()
# print f
# Pricing Asian options: The Geman and Yor model with
# S = 2, K = 2, T = 1, t = 0, sig = 0.1, r = 0.02
# print f()
# 0.0559860415440029
# f.ch_sig('0.05')
# print f
# Pricing Asian options: The Geman and Yor model with
# S = 2, K = 2, T = 1, t = 0, sig = 0.05, r = 0.02
# print f()
# 0.0345709175410301
# f.N = 100
# print f()
# 0.0339410537085201
# from mpmath import mp
# mp.dps = 50
# f.update()
# 0.033941053708520319031364170122438704213486236188948
try:
import psyco
psyco.full()
except ImportError:
print 'Psyco not installed, the program will just run slower'
from mpmath import mp,mpf,mpc,pi,sin,tan,exp,gamma,hyp1f1,sqrt,log10,floor
from mp_laplace import Talbot
class Asian(object):
def G(self,s): # Laplace-Transform
zz = 2*self.v + 2 + s
mu = sqrt(self.v**2+2*zz)
a = mu/2 - self.v/2 - 1
b = mu/2 + self.v/2 + 2
v1 = (2*self.alp)**(-a)*gamma(b)/gamma(mu+1)/(zz*(zz - 2*(1 + self.v)))
prec = floor(max(log10(abs(v1)),mp.dps))+self.prec
# additional precision needed for computation of hyp1f1
with mp.extradps(prec):
value = hyp1f1(a,mu + 1,self.beta)*v1
return value
def update(self):
# Geman and Yor's variable
# possibly with infinite precision (strings)
self.S = mpf(self.parameter['S'])
self.K = mpf(self.parameter['K'])
self.T = mpf(self.parameter['T'])
self.t = mpf(self.parameter['t'])
self.sig = mpf(self.parameter['sig'])
self.r = mpf(self.parameter['r'])
self.v = 2*self.r/(self.sig**2) - 1
self.alp = self.sig**2/(4*self.S)*self.K*self.T
self.beta = -1/(2*self.alp)
self.f.shift = self.shift
def __init__(self,S=2,K=2,T=1,t=0,sig='0.1',r='0.02',N=50,shift=0.0,prec=0):
# Strings allowed for infinite precision
# prec compensates rounding errors not catched with automatic precision control
# parameters may be changed later
# after changing mp.dps or any of these parameters (except prec, N and t),
# update (v,alp,beta depend on these parameters)
self.N = N
self.shift = shift
self.prec = max(prec,0)
self.parameter = {'S':S,'K':K,'T':T,'t':t,'sig':sig,'r':r}
# input: possibly strings with infinite precision
self.f = Talbot(self.G,shift=self.shift,prec=0)
self.update()
def __call__(self):
# Initialize the stepsize (with actual precision)
self.f.N = self.N
tau = ((self.sig**2)/4)*(self.T - self.t)
# Evaluation of the integral at tau
return 4*exp(tau*(2*self.v+2))*exp(-self.r*(self.T - self.t))*self.S/(self.T*self.sig**2)*self.f(tau)
# Update Parameters
def ch_S(self,S):
self.parameter['S'] = S
self.update()
def ch_K(self,K):
self.parameter['K'] = K
self.update()
def ch_T(self,T):
self.parameter['T'] = T
self.update()
def ch_t(self,t):
self.parameter['t'] = t
self.update()
def ch_r(self,r):
self.parameter['r'] = r
self.update()
def ch_sig(self,sig):
self.parameter['sig'] = sig
self.update()
# Actual Parametes
def __str__(self):
s = 'Pricing Asian options: The Geman and Yor model with\n'
s += " S = %(S)s, K = %(K)s, T = %(T)s, t = %(t)s, sig = %(sig)s, r = %(r)s" % self.parameter
return s
