# Monte Carlo simulering: Anvendelse af metoden samt introduktion af de variansreducerende metoder og Heston-modellen

Thomas Aamand Nielsen

Studenteropgave: Kandidatafhandlinger

## Abstrakt

There are three main subjects in this master thesis, which all involves the simulation techniques of the Monte Carlo method. The first subject is to introduce the idea of the most basic variation of the Monte Carlo method. This introduction will later make it possible to model more sophisticated Monte Carlo models, known as Antithetic Variates and Control Variates, which will be the second main subject. These models are known as variance reduction methods, because the main idea is to make the estimate from simulation more precise, by minimizing the standard deviation of the estimated price. The last subject is to take a closer look at how it’s possible to use the Monte Carlo techniques to price options with stochastic volatility. This is done by introducing the Heston-model, which simulates both the variance and the asset at the same time. The first model introduced in this thesis, is the most basic Monte Carlo model, known as the standard model, and its ability to price options will be shown from examples where it prices European call-options. The reason why this kind of option is chosen is that the estimated price can be evaluated against the analytical solution from the Black-Scholes formula, and it’s therefore possible to test the speed in which the estimated price will converge against the analytical solution. The thing about the standard model is, that I take a lot of simulations, to make a precise estimate off the price, and this is the main reason why the variance reduction techniques are introduced. Antithetic Variates is one of the simplest techniques to minimize the standard deviation of the estimate, and the basic idea behind the model is to simulate paths of the asset by using both the random number Z and –Z. The model is simple and takes only a few minutes to add to the original standard model. A more sophisticated model is therefore introduced, and it’s called Control Variates. The idea behind this model is to simulate something you know the value of at the first place, with the same random numbers as you use to simulate the price of the option. By doing this, the model gives you the chance to adjust the estimated price of the option, according to the estimation error of the known value. Both these models will be tested, to see which one is best to price options, with different characteristics. A model outside the world of Black-Scholes is The Heston model, and the model gives some realism to the simulation, by introducing stochastic volatility. The variation reduction techniques will also be tested together with The Heston model, but the main subject is to investigate some of the problems that may be found, by simulating The Heston model with the Euler-scheme. This analysis will be based on the article of Leif Andersen, and the solution is to test a more robust scheme, that handles more stressed scenarios in a better way.

Uddannelser Cand.merc.mat Erhvervsøkonomi og Matematik, (Kandidatuddannelse) Afsluttende afhandling Dansk 2010 115