In this thesis, we propose two continuous time stochastic volatility models with long memory that generalize two existing models. More importantly, we provide analytical formulae that allow us to study option prices numerically, rather than by means of simulation. We are not aware about analytical results in continuous time long memory case. In both models, we allow for the non-zero correlation between the stochastic volatility and stock price processes. We numerically study the effects of long memory on the option prices. We show that the fractional integration parameter has the opposite effect to that of volatility of volatility parameter in short memory models. We also find that long memory models have the potential to accommodate the short term options and the decay of volatility skew better than the corresponding short memory stochastic volatility models.

The thesis is mapping current situation on the market in warrants as nonstandardised option contracts. Particular attention is paid to the special issues of warrants modified into the form of so called "access products". The function of such financial instrument is to open extraordinary investment opportunity to certain kind of investors, who otherwise are not allowed to make direct investment into the underlying asset itself. Access products can be considered as particular financial innovation, which spread out on Central European capital markets. The most familiar form of access products are low strike price covered call warrants, the characteristics of which are being analyzed and compared to standard options and warrants. The author also focuses on the process of issuing warrant securities on Czech capital market according to Czech law and problems linked to it, which the issuer encounters. The thesis proves on the example of Fondul Proprietatea Warrants, which were in 2010 successfully placed to third market of Vienna Stock Exchange by Czech entity, that such warrant issue is possible despite of material complications stemming from the absence of clear definition of warrants in Czech law.

Student Number : 0003819T - MSc dissertation - School of Computational and Applied Mathematics - Faculty of Science / The over-the-counter (OTC) interest rate derivative market is large and rapidly developing. In March 2005, the Bank for International Settlements published its “Triennial Central Bank Survey” which examined the derivative market activity in 2004 (http://www.bis.org/publ/rpfx05.htm). The reported total gross market value of OTC derivatives stood at $6.4 trillion at the end of June 2004. The gross market value of interest rate derivatives comprised a massive 71.7% of the total, followed by foreign exchange derivatives (17.5%) and equity derivatives (5%). Further, the daily turnover in interest rate option trading increased from 5.9% (of the total daily turnover in the interest rate derivative market) in April 2001 to 16.7% in April 2004. This growth and success of the interest rate derivative market has resulted in the introduction of exotic interest rate products and the ongoing search for accurate and efficient pricing and hedging techniques for them. Interest rate caps and (European) swaptions form the largest and the most liquid part of the interest rate option market. These vanilla instruments depend only on the level of the yield curve. The market standard for pricing them is the Black (1976) model. Caps and swaptions are typically used by traders of interest rate derivatives to gamma and vega hedge complex products. Thus an important feature of an interest rate model is not only its ability to recover an arbitrary input yield curve, but also an ability to calibrate to the implied at-the-money cap and swaption volatilities. The LIBOR market model developed out of the market’s need to price and hedge exotic interest rate derivatives consistently with the Black (1976) caplet formula. The focus of this dissertation is this popular class of interest rate models. The fundamental traded assets in an interest rate model are zero-coupon bonds. The evolution of their values, assuming that the underlying movements are continuous, is driven by a finite number of Brownian motions. The traditional approach to modelling the term structure of interest rates is to postulate the evolution of the instantaneous short or forward rates. Contrastingly, in the LIBOR market model, the discrete forward rates are modelled directly. The additional assumption imposed is that the volatility function of the discrete forward rates is a deterministic function of time. In Chapter 2 we provide a brief overview of the history of interest rate modelling which led to the LIBOR market model. The general theory of derivative pricing is presented, followed by a exposition and derivation of the stochastic differential equations governing the forward LIBOR rates. The LIBOR market model framework only truly becomes a model once the volatility functions of the discrete forward rates are specified. The information provided by the yield curve, the cap and the swaption markets does not imply a unique form for these functions. In Chapter 3, we examine various specifications of the LIBOR market model. Once the model is specified, it is calibrated to the above mentioned market data. An advantage of the LIBOR market model is the ability to calibrate to a large set of liquid market instruments while generating a realistic evolution of the forward rate volatility structure (Piterbarg 2004). We examine some of the practical problems that arise when calibrating the market model and present an example calibration in the UK market. The necessity, in general, of pricing derivatives in the LIBOR market model using Monte Carlo simulation is explained in Chapter 4. Both the Monte Carlo and quasi-Monte Carlo simulation approaches are presented, together with an examination of the various discretizations of the forward rate stochastic differential equations. The chapter concludes with some numerical results comparing the performance of Monte Carlo estimates with quasi-Monte Carlo estimates and the performance of the discretization approaches. In the final chapter we discuss numerical techniques based on Monte Carlo simulation for pricing American derivatives. We present the primal and dual American option pricing problem formulations, followed by an overview of the two main numerical techniques for pricing American options using Monte Carlo simulation. Callable LIBOR exotics is a name given to a class of interest rate derivatives that have early exercise provisions (Bermudan style) to exercise into various underlying interest rate products. A popular approach for valuing these instruments in the LIBOR market model is to estimate the continuation value of the option using parametric regression and, subsequently, to estimate the option value using backward induction. This approach relies on the choice of relevant, i.e. problem specific predictor variables and also on the functional form of the regression function. It is certainly not a “black-box” type of approach. Instead of choosing the relevant predictor variables, we present the sliced inverse regression technique. Sliced inverse regression is a statistical technique that aims to capture the main features of the data with a few low-dimensional projections. In particular, we use the sliced inverse regression technique to identify the low-dimensional projections of the forward LIBOR rates and then we estimate the continuation value of the option using nonparametric regression techniques. The results for a Bermudan swaption in a two-factor LIBOR market model are compared to those in Andersen (2000).

In this project, we study an asymptotic expansion method for solving stochastic volatility European option pricing problems. We explain the backgrounds and details associated with the method. Specifically, we present in full detail the arguments behind the derivation of the pricing PDEs and detailed calculation in deriving asymptotic option pricing formulas using our own model specifications. Finally, we discuss potential difficulties and problems in the implementation of the methods.

The project investigates the prices of barrier options from the constant underlying volatility in the Black-Scholes model to stochastic volatility model in SABR framework. The constant volatility assumption in derivative pricing is not able to capture the dynamics of volatility. In order to resolve the shortcomings of the Black-Scholes model, it becomes necessary to find a model that reproduces the smile effect of the volatility. To model the volatility more accurately, we look into the recently developed SABR model which is widely used by practitioners in the financial industry. Pricing a barrier option whose payoff to be path dependent intrigued us to find a proper numerical method to approximate its price. We discuss the basic sampling methods of Monte Carlo and several popular variance reduction techniques. Then, we apply Monte Carlo methods to simulate the price of the down-and-out put barrier options under the Black-Scholes model and the SABR model as well as compare the features of these two models.

The use of forward-looking information from option prices attracted a lot of attention after the 2008 financial crisis, which highlighting the difficulty of using historical data to predict extreme events. Although a considerable number of papers investigate extraction of forward-information from cross-sectional option prices, Figlewski (2008) argues that it is still an open question and none of the techniques is clearly superior. This thesis focuses on getting information from option prices and investigates two broad topics: applying machine learning in extracting state price density and recovering natural probability from option prices. The estimation of state price density (often described as risk-neutral density in the option pricing litera- ture) is of considerable importance since it contains valuable information about investors' expectations and risk preferences. However, this is a non-trivial task due to data limitation and complex arbitrage-free constraints. In this thesis, I develop a more efficient linear programming support vector machine (L1-SVM) estimator for state price density which incorporates no-arbitrage restrictions and bid-ask spread. This method does not depend on a particular approximation function and framework and is, therefore, universally applicable. In a parallel empirical study, I apply the method to options on the S&P 500, showing it to be comparatively accurate and smooth. In addition, since the existing literature has no consensus about what information is recovered from The Recovery Theorem, I empirically examine this recovery problem in a continuous diffusion setting. Using the market data of S&P 500 index option and synthetic data generated by Ornstein-Uhlenbeck (OU) process, I show that the recovered probability is not the real-world probability. Finally, to further explain why The Recovery Theorem fails and show the existence of associated martingale component, I demonstrate a example bivariate recovery.

We will begin with a review of key financial topics and outline many of the crucial ideas utilized in the latter half of the paper. Formal notation for important variables will also be established. Then, a derivation of the Black-Scholes equation will lead to a discussion of its shortcomings, and the introduction of stochastic volatility models. Chapter 2 will focus on a variation of the CIR Model using stock price in the volatility driving process, and its behavior to a greater degree. The key area of discussion will be to approximate a hedging function for European call option prices by Taylor Expansion. We will apply this estimation to real data, and analyze the behavior of the price correction. Then make conclusions about whether stock price has any positive effects on the model.

Increasingly used in online auctions, buyout prices allow bidders to instantly purchase the item listed. We distinguish two types: a temporary buyout option disappears if a bid above the reserve price is made; a permanent one remains throughout the auction or until it is exercised. In a model featuring time-sensitive bidders with uniform valuations and Poisson arrivals but endogenous bidding times, we focus on finding temporary and permanent buyout prices maximizing the seller's discounted revenue, and examine the relative benefit of using each type of option in various environments. We characterize equilibrium bidder strategies in both cases and then solve the problem of maximizing seller's utility by simulation. Our numerical experiments suggest that buyout options may significantly increase a seller’s revenue. Additionally, while a temporary buyout option promotes early bidding, a permanent option gives an incentive to the bidders to bid late, thus leading to concentrated bids near the end of the auction. / Singapore-MIT Alliance (SMA)

The cost of construction projects depends on their size, complexity, and duration. Construction management applies effective management techniques to the planning, design, and construction of a project from conception to completion for the purpose of controlling time, cost and quality. A real options approach in construction projects, improves strategic thinking by helping planners recognize, design and use flexible alternatives to manage dynamic uncertainty. In order to manage uncertainty using this approach, it is necessary to value the real options. Real option models assume independence of option holder and the impacts of underlying uncertainties on performance and value. The current work proposes and initially tests whether project management reduces the value of real options. The example of resource allocation is used to test this hypothesis. Based on the results, it is concluded that project management reduces the value of real options by reducing variance of the exercise signal and the difference between exercise conditions and the mean exercise signal.

In this thesis, we propose two continuous time stochastic volatility models with long memory that generalize two existing models. More importantly, we provide analytical formulae that allow us to study option prices numerically, rather than by means of simulation. We are not aware about analytical results in continuous time long memory case. In both models, we allow for the non-zero correlation between the stochastic volatility and stock price processes. We numerically study the effects of long memory on the option prices. We show that the fractional integration parameter has the opposite effect to that of volatility of volatility parameter in short memory models. We also find that long memory models have the potential to accommodate the short term options and the decay of volatility skew better than the corresponding short memory stochastic volatility models.