Python for Finance

Exotic Options
European, American, and Bermuda
options
In Chapter 10, Options and Futures, we have learnt that for a European option, the
option buyer could exercise their right only on maturity dates, while for an American
option buyer, they could exercise their right any time before and on maturity
dates. Thus, an American option would be more valuable than its counterparty of
European option. Bermudan options could be exercised once or several times on a
few predetermined dates. Consequently, the price of a Bermudan option should be
between a European and an American option with the same features, such as the
same maturity dates and the same exercises prices, see the following two inequalities
for call options:
Here is an example for a Bermudan option. Assume that a company issues a 10-year
bond. After seven years, the company could call back, that is, retire, the bond at
the end of each year for the next three years. This callable property is eventually an
embedded Bermudan option with exercise dates in December of years 8, 9, and 10.
First, let's look at the Python program for an American call by using the binomial
model:
def binomialCallAmerican(s,x,T,r,sigma,n=100):
from math import exp,sqrt
import numpy as np
deltaT = T /n
u = exp(sigma * sqrt(deltaT))
d = 1.0 / u
a = exp(r * deltaT)
p = (a - d) / (u - d)
v = [[0.0 for j in np.arange(i + 1)] for i in np.arange(n + 1)]
for j in np.arange(n+1):
v[n][j] = max(s * u**j * d**(n - j) - x, 0.0)
for i in np.arange(n-1, -1, -1):
for j in np.arange(i + 1):
v1=exp(-r*deltaT)*(p*v[i+1][j+1]+(1.0-p)*v[i+1][j])
v2=max(v[i][j]-x,0)
# early exercise
v[i][j]=max(v1,v2)
return v[0][0]
#
s=40. # stock price today
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