Chapter 16. Simulation of Financial Models
The purpose of science is not to analyze or describe but to make useful models of the world.
Chapter 10 introduces in some detail the Monte Carlo simulation of stochastic processes using
NumPy. This chapter applies the basic techniques presented there to implement simulation classes as a central component of the
DX library. We restrict our attention to three widely used stochastic processes:
Geometric Brownian motion
This is the process that was introduced to the
option pricing literature by the seminal work of
Black and Scholes (1973); it is used several
times throughout this book and still
represents—despite its known shortcomings and
given the mounting empirical evidence from
financial reality—a benchmark process for option
and derivative valuation purposes.
The jump diffusion, as introduced by Merton (1976),
adds a log-normally distributed jump component to
the geometric Brownian motion (GBM); this allows
us to take into account that, for example, short-term out-of-the-money (OTM) options often seem to
have priced in the possibility of large jumps. In
other words, relying on GBM as a financial model
often cannot explain the market values of such OTM
options satisfactorily, while a jump diffusion may
be able to do so.
- The square-root diffusion, popularized for finance by Cox, Ingersoll, and Ross (1985), is used to model mean-reverting quantities like interest rates ...