This thesis sets out to examine if and how private equity funds should hedge foreign exchange exposure. To our knowledge the field of foreign exchange hedging within private equity, from the private equity firms’ point of view, is vastly unexplored scientifically. The subject is important since foreign ex-change risk has a larger impact on private equity returns now than historically due to increased competition, cross-boarder investments and foreign exchange volatility. In order to answer the research question a simulation model is constructed and implemented under different scenarios. Foreign exchange rates are simulated and theoretical private equity funds are investigated and com-pared under different performance measures. The underlying mathematical theory originates from the work of Black and Scholes. The main result of this thesis is that private equity funds cannot achieve a higher internal rate of return on average through hedging of foreign exchange exposure independent of the slope of the foreign exchange forward curve. However, hedging strategies yielding the same mean internal rate of return but performing better in terms of performance measures accounting for volatility of returns have been found. Furthermore, we found that the conclusions are independent of whether the current or forward foreign exchange rate is a better approximation for the future foreign exchange rate. / Uppsatsens syfte är att undersöka om och i sådana fall hur private equity fonder ska hedgea valutaexponering. Ämnet är såvitt vi vet ej tidigare undersökt inom vetenskaplig forskning ur private equity företagens synvinkel. Ämnet är viktigt eftersom valutarisk har fått en större påverkan på private equity företagens avkastning jämfört med hur det har sett ut historiskt på grund av högre konkurrens, mer internationella investeringar samt ökad volatilitet i valutakurser. En simuleringsmodell har konstruerats och implementerats under olika scenarier för att besvara forskningsfrågan. Valutakurser simuleras och teoretiska private equity fonder undersöks samt jämförs utefter olika nyckeltal. Den underliggande matematiska modelleringen härstammar från Black och Scholes forskning. Uppsatsens viktigaste resultat är att private equity fonder inte kan uppnå en högre avkastning genom att hedgea valutaexponering oavsett lutningen av den förväntade valutautvecklingskurvan. Vi har dock funnit att det existerar hedgingstrategier som ger samma avkastning med lägre volatilitet. Vidare är slutsatserna oberoende av om nuvarande eller förväntad framtida valutakurs är den bästa approximationen av den framtida valutakursen.

During the last few years, financial derivatives have been growing in trading volume. There seem to be a high demand and supply of derivatives on the market and one common derivative is the option contract. The option contract is frequently the subject of studies and many different pricing models have been created for options. One popular model is the Black-Scholes model, that prices European call options. This project will look closer at the Black-Scholes model and its assumption that volatility remains constant. The project will try to establish what predictive power the implied volatility has for the sign of the price changes in the option’s daily closing price. The implied volatility is defined as the value of volatility that can be used in an option pricing formula to yield the current market price and goes against the assumption of constant volatility. The model consists of a dependent variable that is the binary variable for the sign of the price changes, 1 if positive and 0 if negative. The independent variables are implied volatility, volume, price of the underlying, and VIX. The models used for testing are logistic regression, CART, random forest and XGBoost. The research question is: Can the sign of option price jumps be predicted with the implied volatility? The answer to the research question is that there are indications for the implied volatility having predictive power when predicting the sign of the price changes in the option, even though the results are not conclusive across all models. / Under de senaste åren har finansiella derivat ökat i handelsvolym. Det verkar finnas en hög efterfrågan och tillgång på derivat generellt på marknaden och ett vanligt sådant derivat är optionskontraktet. Optioner är ofta föremål för forskning och många olika prissättningsmodeller har skapats för optioner. En populär modell är Black-Scholes modellen som prissätter europeiska köpoptioner. Detta projekt kommer att titta närmare på Black-Scholes modellen och dess antagande om att volatiliteten förblir konstant. Projektet kommer att försöka fastställa vilken prediktiv kraft den implicita volatiliteten har för tecknet på prisförändringarna i optionens dagliga stängningskurs. Den implicita volatiliteten definieras som värdet av volatiliteten som kan användas i en optionsprissättningsformel för att ge det aktuella marknadspriset och går emot antagandet om konstant volatilitet. Modellen består av en beroende variabel som är en binär variabel för tecknet på prisförändringarna, 1 om positivt och 0 om negativt. De oberoende variablerna är implicit volatilitet, volym, pris på det underliggande instrumentet och VIX. Modellerna som används för att genomföra testen är logistisk regression, CART, random forest och XGBoost. Projektets frågeställning är: Kan tecknet på en options prisförändringar förutsägas med den implicita volatiliteten? Svaret som projektet kommit fram till är att det finns indikationer på att den implicita volatiliteten har prediktiv kraft när det gäller att förutsäga tecknet på prisförändringarna i optionen, även om resultaten inte är helt överensstämmande för alla modeller.

Diffusion equations arise in areas such as fluid mechanics, cellular biology, weather forecasting, electronics, mechanical engineering, atomic physics, environmental science, medicine, etc. This dissertation considers equations of this type that arise in mathematical finance. For over 40 years traders in financial markets around the world have used Black-Scholes equations for valuing financial options. These equations need to be solved quickly and accurately so that the traders can make prompt and accurate investment decisions. One way to do this is to use parallel numerical algorithms. This dissertation develops and evaluates algorithms of this kind that are based on the Laplace transform, numerical inversion algorithms and finite difference methods. Laplace transform-based algorithms have faced a legitimate criticism that they are ill-posed i.e. prone to instability. We demonstrate with reference to the Black-Scholes equation, contrary to the received wisdom, that the use of the Laplace transform may be used to produce reasonably accurate solutions (i.e. to two decimal places), in a fast and reliable manner when used in conjunction with standard PDE techniques. To set the scene for the investigations that follow, the reader is introduced to financial options, option pricing and the one-dimensional and two-dimensional linear and nonlinear Black-Scholes equations. This is followed by a description of the Laplace transform method and in particular, four widely used numerical algorithms that can be used for finding inverse Laplace transform values. Chapter 4 describes methodology used in the investigations completed i.e. the programming environment used, the measures used to evaluate the performance of the numerical algorithms, the method of data collection used, issues in the design of parallel programs and the parameter values used. To demonstrate the potential of the Laplace transform based approach, Chapter 5 uses existing procedures of this kind to solve the one-dimensional, linear Black-Scholes equation. Chapters 6, 7, 8, and 9 then develop and evaluate new Laplace transform-finite difference algorithms for solving one-dimensional and two-dimensional, linear and nonlinear Black-Scholes equations. They also determine the optimal parameter values to use in each case i.e. the parameter values that produce the fastest and most accurate solutions. Chapters 7 and 9 also develop new, iterative Monte Carlo algorithms for calculating the reference solutions needed to determine the accuracy of the LTFD solutions. Chapter 10 identifies the general patterns of behaviour observed within the LTFD solutions and explains them. The dissertation then concludes by explaining how this programme of work can be extended. The investigations completed make significant contributions to knowledge. These are summarised at the end of the chapters in which they occur. Perhaps the most important of these is the development of fast and accurate numerical algorithms that can be used for solving diffusion equations in a variety of application areas.

Mémoire numérisé par la Direction des bibliothèques de l'Université de Montréal.

The famous Black-Scholes formula provided the first mathematically sound mechanism to price financial options. It is based on the assumption, that daily random stock returns are identically normally distributed and hence stock prices follow a stochastic process with a constant volatility. Observed prices, at which options trade on the markets, don¡¯t fully support this hypothesis. Options corresponding to different strike prices trade as if they were driven by different volatilities. To capture this so-called volatility smile, we need a more sophisticated option-pricing model assuming that the volatility itself is a random process. The price we have to pay for this stochastic volatility model is that such models are computationally extremely intensive to simulate and hence difficult to fit to observed market prices. This difficulty has severely limited the use of stochastic volatility models in the practice. In this project we propose to overcome the obstacle of computational complexity by executing the simulations in a massively parallel fashion on the graphics processing unit (GPU) of the computer, utilizing its hundreds of parallel processors. We succeed in generating the trillions of random numbers needed to fit a monthly options contract in 3 hours on a desktop computer with a Tesla GPU. This enables us to accurately price any derivative security based on the same underlying stock. In addition, our method also allows extracting quantitative measures of the riskiness of the underlying stock that are implied by the views of the forward-looking traders on the option markets.

This diploma thesis deals with problem of option pricing with stochastic volatility. At first, the Black-Scholes model is derived and then its biases are discussed. We explain shortly the concept of volatility. Further, we introduce three pricing models with stochastic volatility- Hull-White model, Heston model and Stein-Stein model. At the end, these models are reviewed.

Dans une première partie, après avoir rappelé les principales caractéristiques statistiques des séries financières, en particulier l'existence de corrélations non linéaires à longue portée et d'une asymétrie fortement persistante, nous mettons en évidence la pertinence des processus multifractals pour la modélisation de ces faits stylisés. Les constructions récemment proposées dans la littérature demeurent cependant exclusivement symétriques et nous montrons comment introduire de l'asymétrie dans ces modèles sans sacrifier leurs propriétés d'échelle. Il est alors possible de rendre compte du phénomène de smile de volatilité. Dans une deuxième partie, nous proposons une méthode numérique pour la valorisation et la couverture d'options en marché incomplet. Notre algorithme peut en outre être généralisé sans difficulté pour tenir compte d'autres imperfections du marché comme les frais de transaction.

<p>This study examines whether the performance of the Black-Scholes model to price stock index options is influenced by the general conditions of the financial markets. For this purpose we calculated the theoretical values of 5814 options (3366 put option price observations and 2448 call option price observations) under the Black-Scholes assumptions. We compared these theoretical values with the real market prices in order to put the degree of deviations in two different time windows built around the bankruptcy of Lehman Brothers (September 15th 2008) to the test. We find clear evidences to state that the Black-Scholes model performed differently in the period after Lehman Brothers than in the period before; therefore we are able to blame this event for our findings.</p>

This study examines whether the performance of the Black-Scholes model to price stock index options is influenced by the general conditions of the financial markets. For this purpose we calculated the theoretical values of 5814 options (3366 put option price observations and 2448 call option price observations) under the Black-Scholes assumptions. We compared these theoretical values with the real market prices in order to put the degree of deviations in two different time windows built around the bankruptcy of Lehman Brothers (September 15th 2008) to the test. We find clear evidences to state that the Black-Scholes model performed differently in the period after Lehman Brothers than in the period before; therefore we are able to blame this event for our findings.

There are some nonlinear models for pricing financial derivatives which can improve the linear Black-Scholes model introduced by Black, Scholes and Merton. In these models volatility is not constant anymore, but depends on some extra variables. It can be, for example, transaction costs, a risk from a portfolio, preferences of a large trader, etc. In this thesis we focus on these models. In the first chapter we introduce some important theory of financial derivatives. The second chapter is devoted to the volatility models. We derive three models concerning transaction costs (RAPM, Leland's  and Barles-Soner's model) and Frey's model which assumes a large (dominant) trader on the market. In the third and in the forth chapter we derive portfolio and make numerical experiments with a free boundary. We use the first order additive and the second order Strang splitting methods. We also use approximations of Barles-Soner's model using the identity function and introduce an approximation with the logarithm function of Barles-Soner's model. These models we finally compare with models where the volatility includes constant transaction costs.