Chapter 12. Other Numerical Methods for European Options
In this chapter, we look at two numerical methods (Monte Carlo simulation and numerical integration) that can also be used to value European options. They are alternatives to binomial trees for calculating expectation. However, for American options we prefer binomial valuation for standard options.
Monte Carlo simulation is a well-established technique used widely in other fields. However, the ordinary random sampling approach used in Monte Carlo simulation is relatively inefficient for option valuation (especially when compared to binomial tree valuation). For example, to halve the sampling error requires a quadrupling of the number of simulation trials. So, the recent introduction of quasi-random number sequences in finance applications, coupled with enhanced computing power, has boosted the use of Monte Carlo simulation. For valuing path-dependent options, simulation is now considered the best numerical method. Background reading on simulation is in Hull's (2000) text on options, Chapter 16.
In this chapter, we compare the valuation of a European call first using ordinary Monte Carlo simulation, then simulation with antithetic variates and lastly simulation with quasi-random sequences. The aim in using antithetic variables is to improve efficiency of estimation by reducing the standard error of the simulation results. Quasi-random sampling is a further method for effectively controlling the randomness of simulation results. ...