This paper aims to use Monte Carlo methods to price American call options on equities using the variance reduction technique of control variates and to price American put options using the binomial model. We use this information to form option positions. This project was done a part of the masters capstone course Math 573: Computational Methods of Financial Mathematics.

In computational finance, finite differences are a widely used tool in the valuation of standard derivative contracts. In a lower-dimensional setting, high accuracy and speed often characterize such methods, which gives them a competitive advantage against Monte Carlo methods. For option contracts with discontinuous payoff functions, however, finite differences encounter problems to maintain the order of convergence of the employed finite difference scheme. Therefore the timesteps are often computed in a conservative manner, which might increase the total execution time of the solver more than necessary.     It can be shown that for American call options written on dividend paying stocks, it may be optimal to exercise the option right before a dividend is paid out. The result is that yet another discontinuity is introduced in the solution and the timestep is often reduced to preserve the intrinsic convergence order. However, it is thought that at least in theory the optimal length of the timestep is an increasing function of the time elapsed since the last discontinuity occured. The objective thus becomes that of finding an explicit method for adjusting the timestep both at the dividend instants and between dividend instants. Keeping the discretization in space constant leads to a time non-equidistant finite difference problem.     The aim of this thesis is to propose a time non-equidistant numerical finite difference algorithm for valuation of American call options on stocks with dividends known in advance. In particular, an explicit formula is proposed for computing timesteps at the dividend instants and between dividend payments given a user-specified error tolerance. A portion of the report is also devoted to numerical stabilization techniques that are applied to maintain the convergence order, including Rannacher time-marching and mollification.

In this thesis, important theories in financial mathematics will be explained and derived. These theories will later be used to value financial derivatives. An analytical formula for valuing European call and put option will be derived and European call options will be valued under the Black-Scholes partial differential equation using three different finite difference methods. The Crank-Nicholson method will then be used to value American call options and solve their corresponding free boundary value problem. The optimal exercise boundary can then be plotted from the solution of the free boundary value problem. The algorithm for valuing American call options will then be further developed to solve the stock loan problem. This will be achieved by exploiting a link that exists between American call options and stock loans. The Crank-Nicholson method will be used to value stock loans and their corresponding free boundary value problem. The optimal exit boundary can then be plotted from the solution of the free boundary value problem. The results that are obtained from the numerical calculations will finally be used to discuss how different parameters affect the valuation of American call options and the valuation of stock loans. In the end of the thesis, conclusions about the effect of the different parameters on the optimal prices will be presented. / I det här kandidatexamensarbetet kommer fundamentala teorier inom finansiell matematik förklaras och härledas. Dessa teorier kommer lägga grunden för värderingen av finansiella derivat i detta arbete. En analytisk formel för att värdera europeiska köp- och säljoptioner kommer att härledas. Dessutom kommer europeiska köpoptioner att värderas numeriskt med tre olika finita differensmetoder. Den finita differensmetoden Crank-Nicholson kommer sedan användas för att värdera amerikanska köpoptioner och lösa det fria gränsvärdesproblemet (free boundary value problem). Den optimala omvandlingsgränsen (Optimal Exercise Boundary) kan därefter härledas från det fria gränsvärdesproblemet. Algoritmen för att värdera amerikanska köpoptioner utökas därefter till att värdera lån med aktier som säkerhet. Detta kan åstadkommas genom att utnyttja ett samband mellan amerikanska köpoptioner med lån där aktier används som säkerhet. Den finita differensmetoden Crank-Nicholson kommer dessutom att användas för att värdera lån med aktier som säkerhet. Den optimala avyttringsgränsen (Optimal Exit Boundary) kan därefter härledas från det fria gränsvärdesproblemet. Resultaten från de numeriska beräkningarna kommer slutligen att användas för att diskutera hur olika parametrar påverkar värderingen av amerikanska köpoptioner, samt värdering av lån med aktier som säkerhet. Avslutningsvis kommer slutsatser om effekterna av dessa parametrar att presenteras.