Derivatives Analytics with Python: Data Analysis, Models, Simulation, Calibration and Hedging

CHAPTER 10 Monte Carlo Simulation

10.1 Introduction

Monte Carlo simulation is among the most important numerical algorithms of the 20th century (cf. Cipra (2000)) and obviously will remain so in the 21st century as well. Its importance for financial applications stems from the fact that it is most flexible in terms of financial products that can be valued. First applied to European option pricing in 1977 by Phelim Boyle (cf. Boyle (1977)), it took until the 21st century for the problem of valuing American options by Monte Carlo simulation to be satisfactorily solved by Francis Longstaff and Eduardo Schwartz (cf. Longstaff and Schwartz (2001)) and others (cf. Chapter 7). Glasserman (2004) provides a comprehensive introduction to Monte Carlo methods for financial engineering and is a standard reference. Kohler (2009) is a survey article of regression-based valuation approaches for American options.

Although quite flexible, Monte Carlo simulation is generally not very fast (relative to alternative approaches) since millions of computations are necessary to value a single option. Consider a simulation run with 100 time intervals (=100 exercise dates) and 100,000 paths for an American put option on a single stock with constant volatility and constant short rate. You need 10 million random numbers, several arrays (i.e. matrices) of size 101 times 100,000 and you have to estimate 100 least-squares regressions over 100,000 pairs of numbers as well as discounting 100 times 100,000 numbers. ...