# Closed-form equation for geometric asian call option

I'm looking to use the geometric asian option as a control variable for a monte carlo simulation. However, I have an issue with the closed-form equation to get the geometric price.

I'm using the formula from a previous related Q&A on this site:

When I compute it for T=1, it gives me the right price. But when T > 1, the price found is different from the solution that I get from MC.

With the parameters below, I get 9.124 and something around 9.65 with MC (SE = 0.02).

Here is the code I use:

import numpy as np
from scipy.stats import norm
np.random.seed(1)

#### Closed form equation for geometric asian option ####
def BS_geo(S0,K,T,vol,r,n,type):
varbis = vol**2 *   (((n+1)*(2*n+1))/(6*(n**2)))
rbis = (varbis/2) + (r-((vol**2)/2)) * ((n+1)/(2*n))
d1 = (np.log(S0/K) + (  (rbis+0.5*varbis)*T)) / np.sqrt(varbis)*np.sqrt(T)
d2 = d1 - (np.sqrt(varbis)*np.sqrt(T))
if type == "Call":
price = np.exp(-r*T) * (S0*np.exp(rbis*T)*norm.cdf(d1)-K*norm.cdf(d2))
else:
price = np.exp(-r*T) * (K*norm.cdf(-d2) - S0*np.exp(rbis*T)*norm.cdf(-d1))
return price

S0=100
K=105
T=3
vol = 0.25
r = 0.05
n= 100
type = "Call"

print(BS_geo(S0,K,T,vol,r,n,type))


When I compute it for different T between 1 and 20, here is what I got. Orange line is from Monte Carlo and the blue one from the closed-form equation.

• is there a reason you're discounting by np.exp(-rbis*T) rather than np.exp(-r*T)? perhaps that's the source of the discrepancy Aug 5, 2023 at 4:44
• I have slightly modify my question for greater clarity. I've also modify np.exp(-rbisT) into np.exp(-rT) but it is not enough to give me the right answer :/
– Vpaq
Aug 5, 2023 at 7:50
• Exactly where did you get the formula (screenshot) in your question from? Aug 5, 2023 at 9:42
• I took it from here quant.stackexchange.com/questions/68929/… and I saw the same here: macsphere.mcmaster.ca/handle/11375/23088 but if you have another source, I would be grateful :)
– Vpaq
Aug 5, 2023 at 9:44
• @Vpaq Sorry, I don't. In which page(s) in the second source (thesis) is the same formula given? Aug 5, 2023 at 10:58