An alternative approach to pricing options and other financial instruments
- Monte Carlo simulation is one of the most important algorithms in quantitative finance
- Monte Carlo simulation can be utilized as an alternative tool to price options ( the most popular option pricing model is based on the Black-Scholes-Merton formula)
Before demonstrating the implementation of the Monte Carlo algorithm, it’s important to fully comprehend the science behind it. Simply put, Monte Carlo simulation generates a series of random variables that have similar properties to the risk factors which the simulation is trying to simulate.
The simulation produces a large number of possible outcomes along with their probabilities. In summary, it’s used to simulate realistic scenarios (stock prices, option prices, probabilities…).
Note: Monte Carlo simulations can get computationally expensive and slow depending on the number of generated scenarios.
Next, I will demonstrate how we can leverage Monte Carlo simulation to price a European call option and implement its algorithm in Python.
Let’s start by looking at the famous Black-Scholes-Merton formula (1973):
S(t) = Stock price at time t
r = Risk free rate
σ = Volatility
Z(t) = Brownian motion
Our goal is to solve the equation above to obtain an explicit formula for S(t).
We utilized Euler Discretization Scheme to solve the stochastic equation above. The solution is given by the expression:
Let’s apply the logarithm function to equation 3–2 above which will allow a faster implementation in Python (the vectorization process using the numpy package in Python would easily ingest the log version of the solution above).
