- We start with the assumption that underlying follow Geometric Brownian Motion (GBM):
- We use Ito’s Lemma with , then we have
- Therefore, the change of between time 0 and future time T, is normally distributed as following:
Thus, the future underlying price can be written as,
is the noise term from standard normal distribution.
Note, we will take , which is the risk free rate. This means investors are risk neutral and requires risk free return on underlying asset. This is to be consistent with the risk neutral probabilities used in simulation.
Correspondingly, we also use risk free rate in the discount factor in step 5.
- So now we can simulate the future underlying price at expiry. With European Call or Put boundary condition to calculate the payoff.
- We then need to discount the future payoff back to present by multiplying a discount factor,
- The above two steps are repeated many times and its expectation is calculated as the final simulation result.
Determine the option price using a Monte Carlo approach.
The log return of underlying follow Normal distribution.
s_T = s_t * exp((r - 1/2 * sig^2) * (T-t) + sig * sqrt(T-t) * sig_Normal)
calc = np.zeros([self.iterations, 2])
rand = np.random.normal(0, 1, [1, self.iterations])
mult = self.spot * np.exp(self.tenor * (self.rate - 0.5 * self.sigma**2))
if self.callPut == 'Call':
calc[:,1] = mult * np.exp(np.sqrt((self.sigma**2)*self.tenor) * rand) - self.strike
elif self.callPut == 'Put':
calc[:,1] = self.strike - mult*np.exp(np.sqrt((self.sigma**2) * self.tenor) * rand)
avgPayOff = np.sum(np.amax(calc, axis=1)) / float(self.iterations)
return np.exp(-self.rate * self.tenor) * avgPayOff
""" Determine the option price using the exact Black-Scholes expression. """
return blackScholesOptionPrice(self.callPut, self.spot, self.strike, self.tenor, self.rate, self.sigma)
We can run the above in Python console：
from option_pricer import EuropeanVanillaPricer
pricer = EuropeanVanillaPricer()
As we can see in the above Monte Carlo simulation, we rely on drawing random numbers,
from a Standard Normal distribution.
Alternatively, we can use random numbers from a Uniform distribution, i.e. equal probability of each random number.
To do this, we combine step 3, 4 and 5, the current option price is obtained by integrating the terminal payoff under the risk neutral measure:
In the first line, function
is just the payoff condition at expiry. As we are integrating with regard to
, which follows Standard Normal distribution, the last term is the probability density function.
In the second line, we just use a new function h of epsilon to make the expression more compact.
In the third line, we do inverse transformation to integrate with regard to the cumulative probability,
So now it becomes an integral of function
over the UNIFORM distribution with range [0, 1].
Now our simulation task becomes taking random number
from the Uniform distribution [0, 1], and then calculate integral of function
using Monte Carlo.
To be more specific, our task has been changed from calculating
When we use Monte Carlo to estimate function integral, we may run into problem of random number clustering, which essentially leads to Convergence rate of
To conquer this issue, instead of using pseudo-random numbers, we can use a deterministic sequence, whose numbers are more equally spaced. And this is exactly what Quasi Monte Carlo (QMC) does. More details on the low-discrepancy sequences can be found in this post