Python for Finance: Analyze Big Financial Data

Monte Carlo Simulation

Monte Carlo simulation is one of the most important algorithms in finance and numerical

science in general. Its importance stems from the fact that it is quite powerful when it

comes to option pricing or risk management problems. In comparison to other numerical

methods, the Monte Carlo method can easily cope with high-dimensional problems where

the complexity and computational demand, respectively, generally increase in linear

fashion.

The downside of the Monte Carlo method is that it is per se computationally demanding

and often needs huge amounts of memory even for quite simple problems. Therefore, it is

necessary to implement Monte Carlo algorithms efficiently. The example that follows

illustrates different implementation strategies in Python and offers three different

implementation approaches for a Monte Carlo-based valuation of a European option.

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The three approaches are:

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Pure Python

This example sticks with the standard library — i.e., those libraries and packages that

come with a standard Python installation — and uses only built-in Python

capabilities to implement the Monte Carlo valuation.

Vectorized NumPy

This implementation uses the capabilities of NumPy to make the implementation more

compact and much faster.

Fully vectorized NumPy

The final example combines a different mathematical formulation with the

vectorization capabilities of NumPy to get an even more compact version of the same

algorithm.

The examples are again based on the model economy of Black-Scholes-Merton (1973),

where the risky underlying (e.g., a stock price or index level) follows, under risk

neutrality, a geometric Brownian motion with a stochastic differential equation (SDE), as

in Equation 3-5.

Equation 3-5. Black-Scholes-Merton (1973) stochastic differential equation

The parameters are defined as in Equation 3-1 and Z is a Brownian motion. A

discretization scheme for the SDE in Equation 3-5 is given by the difference equation in

Equation 3-6.

Equation 3-6. Euler discretization of SDE