American options are financial contracts that allow exercise at any time until ex- piration. While the pricing of standard American option contracts has been well researched, with a few exceptions no analytical solutions exist. Valuation of more in- volved American option contracts, which include multiple underlying assets or path- dependent payoff, is still to a high degree an uncharted area. Most numerical methods work badly for such options as their time complexity scales exponentially with the number of dimensions. In this Master’s thesis we study valuation methods based on Monte Carlo sim- ulations. Monte Carlo methods don’t suffer from exponential time complexity, but have been known to be difficult to use for American option pricing due to the forward nature of simulations and the backward nature of American option valuation. The studied methods are: Parametrization of exercise rule, Random Tree, Stochastic Mesh and Regression based method with a dual approach. These methods are evaluated and compared for the standard American put option and for the American maximum call option. Where applicable the values are compared with those from deterministic reference methods. The strengths and weaknesses of each method is discussed. The Regression based method essentially reduces the problem to one of selecting suitable basis functions. This choice is empirically evaluated for the following Amer- ican option contracts; standard put, maximum call, basket call, Asian call and Asian call on a basket. The set of basis functions considered include polynomials in the underlying assets, the payoff, the price of the corresponding European contract as well as certain analytic approximation of the latter. Results from the empirical studies show that the regression based method is the best choice when pricing exotic American options. Furthermore, using available analytical approximations for the corresponding European option values as a basis function seems to improve the performance of the method in most cases.

This thesis consists of an introduction and five articles. A common theme in all the articles is optimal timing when acting on a financial market. The main topics are optimal selling of an asset, optimal exercising of an American option, optimal stopping games and optimal strategies in trend following trading. In all the articles, we consider a financial market different from the standard Black-Scholes market. In two of the articles this difference consists in allowing for jumps of the underlying process. In the other three, the difference is that we have incomplete information about the drift of the underlying process. This is a natural assumption in many situations, including the case of a true buyer of an American option, trading in a market which exhibits trends, and optimal liquidation of an asset in the presence of a bubble. These examples are all addressed in this thesis.

This paper describes our work pricing options in the binomial model on leveraged exchange traded funds (ETFs) with three different approaches. A leveraged exchange traded fund attempts to achieve a similar daily return as the index it follows but at a specified positive or negative multiple of the return of the index. We price options on these funds using the leveraged multiple, predetermined by the leveraged ETF, of the volatility of the index. The initial approach is a basic time step approach followed by the standard Cox, Ross, and Rubinstein method. The final approach follows a different format which we will call the Trigeorgis pricing model. We demonstrate the difficulties in pricing these options based off the dynamics of the indices the ETFs follow.

The main contribution of this thesis is to examine binary options within the British payoff mechanism introduced by Peskir and Samee. This includes British cash-or-nothing put, British asset-or-nothing put, British binary call and American barrier binary options. We assume the geometric Brownian motion model and reduce the optimal stopping problems to free-boundary problems under the Markovian nature of the underlying process. With the help of the local time-space formula on curves, we derive a closed form expression for the arbitrage-free price in terms of the rational exercise boundary and show that the rational exercise boundary itself can be characterised as the unique solution to a non-linear integral equation. We begin by investigating the binary options of American-type which are also called `one-touch' binary options. Then we move on to examine the British binary options. Chapter~2 reviews the existing work on all different types of the binary options and sets the background for the British binary options. We price and analyse the American-type (one-touch) binary options using the risk-neutral probability method. In Chapters~3 ~4 and ~5, we present the British binary options where the holder enjoys the early exercise feature of American binary options whereupon his payoff is the `best prediction' of the European binary options payoff under the hypothesis that the true drift equals a contract drift. Based on the observed price movements, if the option holder finds that the true drift of the stock price is unfavourable then he can substitute it with the contract drift and minimise his losses. The key to the British binary option is the protection feature as not only can the option holder exercise at unfavourable stock price to a substantial reimbursement of the original option price (covering the ability to sell in a liquid option market completely endogenously) but also when the stock price movements are favourable he will generally receive high returns. Chapters~3 and~4 focus on the British binary put options and Chapter~5 on call options. We also analyse the financial meaning of the British binary options and show that with the contract drift properly selected the British binary options become very attractive alternatives to the classic European/American options. Chapter~6 extends the binary options into barrier binary options and discusses the application of the optimal structure without a smooth-fit condition in the option pricing. We first review the existing work for the knock-in options and present the main results from the literature. Then we examine the method in \cite{dai2004knock} in the application to the knock-in binary options. For the American knock-out binary options, the smooth-fit property does not hold when we apply the local time-space formula on curves. We transfer the expectation of the local time term into a computational form under the basic properties of Brownian motion. Using standard arguments based on Markov processes, we analyse the properties of the value function.

Pricing different financial derivatives is an essential part of the financial industry. For some derivatives there exists a closed form solution, however the pricing of high-dimensional American-style derivatives is still today a challenging problem. This project focuses on the derivative called option and especially pricing of American-style basket options, i.e. options with both an early exercise feature and multiple underlying assets. In high-dimensional problems, which is definitely the case for American-style options, Monte Carlo methods is advantageous. Therefore, in this thesis, regression-based Monte Carlo has been used to determine early exercise strategies for the option. The well known Least Squares Monte Carlo (LSM) algorithm of Longstaff and Schwartz (2001) has been implemented and compared to Robust Regression Monte Carlo (RRM) by C.Jonen (2011). The difference between these methods is that robust regression is used instead of least square regression to calculate continuation values of American style options. Since robust regression is more stable against outliers the result using this approach is claimed by C.Jonen to give better estimations of the option price. It was hard to compare the techniques without the duality approach of Andersen and Broadie (2004) therefore this method was added. The numerical tests then indicate that the exercise strategy determined using RRM produces a higher lower bound and a tighter upper bound compared to LSM. The difference between upper and lower bound could be up to 4 times smaller using RRM. Importance sampling and Quasi Monte Carlo have also been used to reduce the variance in the estimation of the option price and to speed up the convergence rate. / Prissättning av olika finansiella derivat är en viktig del av den finansiella sektorn. För vissa derivat existerar en sluten lösning, men prissättningen av derivat med hög dimensionalitet och av amerikansk stil är fortfarande ett utmanande problem. Detta projekt fokuserar på derivatet som kallas option och särskilt prissättningen av amerikanska korg optioner, dvs optioner som både kan avslutas i förtid och som bygger på flera underliggande tillgångar. För problem med hög dimensionalitet, vilket definitivt är fallet för optioner av amerikansk stil, är Monte Carlo metoder fördelaktiga. I detta examensarbete har därför regressions baserad Monte Carlo använts för att bestämma avslutningsstrategier för optionen. Den välkända minsta kvadrat Monte Carlo (LSM) algoritmen av Longstaff och Schwartz (2001) har implementerats och jämförts med Robust Regression Monte Carlo (RRM) av C.Jonen (2011). Skillnaden mellan metoderna är att robust regression används istället för minsta kvadratmetoden för att beräkna fortsättningsvärden för optioner av amerikansk stil. Eftersom robust regression är mer stabil mot avvikande värden påstår C.Jonen att denna metod ger bättre skattingar av optionspriset. Det var svårt att jämföra teknikerna utan tillvägagångssättet med dualitet av Andersen och Broadie (2004) därför lades denna metod till. De numeriska testerna indikerar då att avslutningsstrategin som bestämts med RRM producerar en högre undre gräns och en snävare övre gräns jämfört med LSM. Skillnaden mellan övre och undre gränsen kunde vara upp till 4 gånger mindre med RRM. Importance sampling och Quasi Monte Carlo har också använts för att reducera variansen i skattningen av optionspriset och för att påskynda konvergenshastigheten.

Doctor of Philosophy (PhD) / For many practitioners and market participants, the valuation of financial derivatives is considered of very high importance as its uses range from a risk management tool, to a speculative investment strategy or capital enhancement. A developing market requires efficient but accurate methods for valuing financial derivatives such as American options. A closed form analytical solution for American options has been very difficult to obtain due to the different boundary conditions imposed on the valuation problem. Following the method of solving the American option as a free boundary problem in the spirit of the "no-arbitrage" pricing framework of Black-Scholes, the option price and hedging parameters can be represented as an integral equation consisting of the European option value and an early exercise value dependent upon the optimal free boundary. Such methods exist in the literature and along with risk-neutral pricing methods have been implemented in practice. Yet existing methods are accurate but inefficient, or accuracy has been compensated for computational speed. A new numerical approach to the valuation of American options by cubic splines is proposed which is proven to be accurate and efficient when compared to existing option pricing methods. Further comparison is made to the behaviour of the American option's early exercise boundary with other pricing models.

Optimal stopping and mathematical finance are intimately connected since the value of an American option is given as the solution to an optimal stopping problem. Such a problem can be viewed as a game in which we are trying to maximize an expected reward. The solution involves finding the best possible strategy, or equivalently, an optimal stopping time for the game. Moreover, the reward corresponding to this optimal time should be determined. It is also of interest to know how the solution depends on the model parameters. For example, when pricing and hedging an American option, the volatility needs to be estimated and it is of great practical importance to know how the price and hedging portfolio are affected by a possible misspecification. The first paper of this thesis investigates the performance of the delta hedging strategy for a class of American options with non-convex payoffs. It turns out that an option writer who overestimates the volatility will obtain a superhedge for the option when using the misspecified hedging portfolio. In the second paper we consider the valuation of a so-called stock loan when the lender is allowed to issue a margin call. We show that the price of such an instrument is equivalent to that of an American down-and-out barrier option with a rebate. The value of this option is determined explicitly together with the optimal repayment strategy of the stock loan. The third paper considers the problem of how to optimally stop a Brownian bridge. A finite horizon optimal stopping problem like this can rarely be solved explicitly. However, one expects the value function and the optimal stopping boundary to satisfy a time-dependent free boundary problem. By assuming a special form of the boundary, we are able to transform this problem into one which does not depend on time and solving this we obtain candidates for the value function and the boundary. Using stochastic calculus we then verify that these indeed satisfy our original problem. In the fourth paper we consider an investor wanting to take advantage of a mispricing in the market by purchasing a bull spread, which is liquidated in case of a market downturn. We show that this can be formulated as an optimal stopping problem which we then, using similar techniques as in the third paper, solve explicitly. In the fifth and final paper we study convexity preservation of option prices in a model with jumps. This is done by finding a sufficient condition for the no-crossing property to hold in a jump-diffusion setting.

In this study, we use the McKean's integral equation to evaluate the American option price for constant jump di

In this master thesis we have examined the possibility of pricing multiple American options, on an underlying asset with discrete dividends, with a finite difference method. We have found a good and stable way to price one American option by solving the BSM PDE backwards, while also calculating the Greeks of the option with automatic differentiation. The list of Greeks for an option is quite extensive since we have been using a local volatility surface. We have also tried to find a way to price several American options simultaneously by solving a forward PDE. Unfortunately, we haven't found any previous work that we could use with our local volatility surface, while still keeping down the computational time. The closest we got was to calculate the value of a compound option in a forward mode, but in order to use this to value an American option, we needed to go through an iterative process which calculated a forward or backward European PDE in every step.

The purpose of this thesis is to look into the difficulty of valuing American options, put as well as call, on an asset that pays continuous dividends. The authors are willing to demonstrate how mentioned above securities can be priced using a simple approximation of the Kim integral equations by quadrature formulas. This approach is compared with closed form American Option price formula proposed by Bjerksund-Stenslands in 2002. The results obtained by Bjerksund-Stenslands method are numerically compared by authors to the Kim’s. In Joon Kim’s approximation seems to be more accurate and closer to the chosen “true” value of an American option, however, Bjerksund-Stenslands model is demonstrating a higher speed in calculations.