The lookback options depend on the paths (history) travelled by the underlying security. Thus, they are also called path-dependent exotic options. One of them is named floating strikes. The payoff function of a call when the exercise price is the minimum price achieved during the life of the option is given as follows:
The Python code for this lookback option is shown as follows:
plt.show()
def lookback_min_price_as_strike(s,T,r,sigma,n_simulation):
n_steps=100
dt=T/n_steps
total=0
for j in range(n_simulation):
min_price=100000. # a very big number
sT=s
for i in range(int(n_steps)):
e=sp.random.normal()
sT*=sp.exp((r-0.5*sigma*sigma)*dt+sigma*e*sp.sqrt(dt))
if sT<min_price:
min_price=sT
#print 'j=',j,'i=',i,'total=',total
total+=p4f.bs_call(s,min_price,T,r,sigma)
return total/n_simulation