This thesis applies the concept of cost e ciency to the design of executive compensation. In a classical Black-Scholes framework, we are able to express the cost e cient counterpart of the Asian Executive Option explicitly, and design a payo that has the same distribution as the Asian Executive Indexed Option but comes at a cheaper price. The cost e cient counterpart of the latter option is not analytically tractable, but we are able to simulate its price. Furthermore, we extend the study of these two types of options in the presence of stochastic interest rates modeled by a Vasicek process. We are able to derive new closedform pricing formulas for these options. A framework for crafting the state price process is introduced. From here, an explicit expression for the state process is given and its distribution is derived. Using the pricing formulas and the state price process, we are then able to simulate the prices of the corresponding cost e cient counterparts in a stochastic interest rate environment. We conclude with some avenues for future research.

Asian option is an option which payoff depends on the average underlying price over some some specific time period. Although there is no closed form solution of asian option, appropriate change of variable and Num¡¦eraire would reduce some terms of equation satisfies the Asian call price function. This thesis presents asian option¡¦s properties and process of reduction terms.

This thesis applies the concept of cost e ciency to the design of executive compensation. In a classical Black-Scholes framework, we are able to express the cost e cient counterpart of the Asian Executive Option explicitly, and design a payo that has the same distribution as the Asian Executive Indexed Option but comes at a cheaper price. The cost e cient counterpart of the latter option is not analytically tractable, but we are able to simulate its price. Furthermore, we extend the study of these two types of options in the presence of stochastic interest rates modeled by a Vasicek process. We are able to derive new closedform pricing formulas for these options. A framework for crafting the state price process is introduced. From here, an explicit expression for the state process is given and its distribution is derived. Using the pricing formulas and the state price process, we are then able to simulate the prices of the corresponding cost e cient counterparts in a stochastic interest rate environment. We conclude with some avenues for future research.

Thesis (MSc (Mathematics))--University of Stellenbosch, 2010. / ENGLISH ABSTRACT: In this thesis, we discuss and apply the Monte Carlo and integral transform methods in pricing options. These methods have proved to be very e ective in the valuation of options especially when acceleration techniques are introduced. By rst pricing European call options we have motivated the use of these methods in pricing arithmetic Asian options which have proved to be di cult to price and hedge under the Black􀀀Scholes framework. The arithmetic average of the prices in this framework, is a sum of correlated lognormal distributions whose distribution does not admit a simple analytic expression. However, many approaches have been reported in the academic literature for pricing these options. We provide a hedging strategy by manipulating the results by Geman and Yor [42] for continuous xed strike arithmetic Asian call options. We then derive a double Laplace transform formula for pricing continuous Asian call options following the approach by Fu et al. [39]. By applying the multi-Laguerre and iterated Talbot inversion techniques for Laplace transforms to the resulting pricing formula we obtain the option prices. Finally, we discuss the shortcomings of using the Laplace transform in pricing options. / AFRIKAANSE OPSOMMING: In hierdie tesis bespreek ons Monte Carlo- en integraaltransform metodes om die pryse van nansi ele opsies te bepaal. Hierdie metodes is baie e ektief, veral wanneer versnellingsmetodes ingevoer word. Ons bepaal eers die pryse van Europese opsies as motivering, voordat ons die bostaande metodes gebruik vir prysbepaling van Asiatiese opsies met rekenkundige gemiddeldes, wat baie moeiliker is om te hanteer in die Black􀀀Scholes raamwerk. Die rekenkundige gemiddelde van batepryse in hierdie raamwerk is 'n som van gekorreleerde lognormale distribusies wie se distribusie nie oor 'n eenvoudige analitiese vorm beskik nie. Daar is egter talle benaderings vir die prysbepaling van hierdie opsies in die akademiese literatuur. Ons bied 'n verskansingsstrategie vir Asiatiese opsies in kontinue tyd met 'n vaste trefprys aan deur die resultate van Geman en Yor [42] te manipuleer. Daarna volg ons Fu et al. [39] om 'n dubbele Laplace transform formule vir die pryse af te lei. Deur toepassing van multi-Laguerre en herhaalde Talbotinversie tegnieke vir Laplace transforms op hierdie formule, bepaal ons dan die opsiepryse. Ons sluit af met 'n bespreking van die tekortkominge van die gebruik van die Laplace transform vir prysbepaling.

We study stockpiling problems under uncertain economic and physical factors, and investigate the valuation and optimisation of storage systems where the availability and spot price of the underlying are both subject to stochasticity. Following a Real Options valuation approach, we first study financial derivatives linked to Asian options. A comprehensive set of boundary conditions is compiled, and an alternative (and novel) similarity reduction for fixed-strike Asian options is derived. Hybrid semi-Lagrangian methods for numerically solving the related partial differential equations (PDEs) are implemented, and we assess the accuracy of the valuations thus obtained with respect to results from classical finite-difference valuation methods and with respect to high precision calculations for valuing Asian options with spectral expansion theory techniques. Next we derive a PDE model for valuing the storage of electricity from a wind farm, with an attached back-up battery, that operates by trading electricity in a volatile market in order to meet a contracted fixed rate of energy generation; this system comprises two diffusive-type (stochastic) variables, namely the energy production and the electricity spot price, and two time-like (deterministic) variables, specifically the battery state and time itself. An efficient and novel semi-Lagrangian alternating-direction implicit (SLADI) methodology for numerically solving advection-diffusion problems is developed: here a semi-Lagrangian approach for hyperbolic problems of advection is combined with an alternating-direction implicit method for parabolic problems involving diffusion. Efficiency is obtained by solving (just) tridiagonal systems of equations at every time step. The results are compared to more standard semi-Lagrangian Crank-Nicolson (SLCN) and semi-Lagrangian fully implicit (SLFI) methods. Once he have established our PDE model for a storage-upgraded wind farm, a system that depends heavily on the highly stochastic nature of wind and the volatile market where electricity is sold, we derive a Hamilton-Jacobi-Bellman (HJB) equation for optimally controlling charging and discharging rates of the battery in time, and we assess a series of operation regimes. The solution of the related PDE models is approached numerically using our SLADI methodology to efficiently treat this mixed advection and diffusion problem in four dimensions. Extensive numerical experimentation confirms our SLADI methodology to be robust and yields highly accurate solutions and efficient computations, we also explore effects from correlation between stochastic electricity generation and random prices of electricity as well as effects from a seasonal electricity spot price. Ultimately, the objective of approximating optimal storage policies for a system under uncertain economic and physical factors is accomplished. Finally we examine the steady-state solution of a stochastic storage problem under uncertain electricity market prices and fixed demand. We use a HJB formulation for optimally controlling charging and discharging rates of the storage device with respect to the electricity spot price. A projected successive over-relaxation coupled with the semi-Lagrangian method is implemented, and we explore the use of boundary-fitted coordinates techniques.

>Magister Scientiae - MSc / This dissertation studies the computation methods of pricing of Asian options. Asian options are options in which the underlying variable is the average price over a period of time. Because of this, Asian options have a lower volatility and this render them cheaper relative to their European counterparts. Asian options belong to the so-called path-dependent derivatives; they are among the most difficult to price and hedge both analytically and numerically. In practice, it is only discrete Asian options that are traded, however continuous Asian options are used for studying purposes. Several approaches have been proposed in the literature, including Monte Carlo simulations, tree-based methods, Taylor’s expansion, partial differential equations, and analytical ap- proximations among others. When using partial differential equations for pricing of continuous time Asian options, the high dimensionality is problematic. In this dissertation we focus on the discrete time methods. We start off by explaining the binomial tree method, and our last chapter presents the very exciting and relatively simple method of Tsao and Huang, using Taylor approximations. The main papers that are used in this dissertation are articles by Jan Vecer (2001); LCG Rogers (1995); Eric Benhamou (2001); Gianluca Fusai (2007); Kamizono, Kariya and Nakatsuma (2006) and Tsao and Huang (2007). The author has provided computations, including graphs and tables dispersed over the different chapters, to demonstrate the utility of the methods. We observe various parameters of influence such as correlation, volatility, strike, etc. A further contribution by the author of this dissertation is, in particular, in Chapter 5, in the presentation of the work of Tsao et al. Here we have provided slightly more detailed explanations and again some further computational tables.

In this thesis a general framework and accompanying guidelines for how to classify financial instruments within the fair value hierarchy (included within IFRS 13) is presented. IFRS 13 introduces a broad and loosely defined regulation of how to classify a financial instrument which leaves room for misinterpretation and uncertainties. In this thesis the pricing of financial instruments and behaviour of the market data used as inputs in the models has been investigated. This is to give better insight into what is classified as significant market data, how it is used and how it is approximated. Instruments that have been investigated are autocalls, swaps, European options and Asian options. The result is presented as general recommendations for how to classify the specified instruments with clearer boarders introduced between the levels in the hierarchy. Methods and deductions introduced in the thesis could also further be implemented in classification of closely related financial instruments but has been limited in this thesis due to time restrictions.     Nyckelord på svenska IFRS, Finansiella instrument, Klassificering, Fair value, Fair value hierarchy, Autocall, Swap, Europeisk option, Asiatisk option, Implicit volatilitet, Korrelation, Marknadsaktivitet, Räntesatser / I denna uppsats är ett generellt ramverk och medföljande riktlinjer för hur man klassificerar finansiella instrument inom fair value hierarkin (inkluderad i IFRS 13) presenterat. IFRS 13 introducerar en bred och löst definierad regulation om hur klassificering finansiella instrument ska gå till som lämnar rum för feltolkningar och oklarheter. I denna uppsats har prissättningen av finansiella instrument och beteende av marknadsdata som används i modellerna undersökts. Detta ger en bättre inblick i vad som klassificeras som signifikant marknadsdata, hur den används och hur den kan approximeras. Instrument som har undersökts är autocalls, swaps, europeiska optioner och asiatiska optioner. Resultatet presenteras som allmänna rekommendationer för hur man klassificerar de angivna instrumenten med tydligare gränser som införts mellan nivåerna i hierarkin. Metoder och slutsatser som är presenterade i uppsatsen kan även vidare användas vid klassifikation av liknande finansiella instrument men har i denna avhandling begränsats på grund av tidsskäl.

爲規避利率上升風險，市場上有很多避險工具，諸如遠期利率協定、利率交換、我國期交所於2004年1月2日所推出的債券期貨（或稱利率期貨）、歐元期貨契約。本論文所要探討的是平均利率上限選擇權之評價，使用的方法是建構Forward LIBOR Tree之利率樹，再使用Timothy. R. Klassen（2001）評價亞式選擇權的方法來評價平均利率上限選擇權。

90年代由於世界各地巨災頻傳，導致再保險人的承保能量嚴重不足，甚至於威脅到再保險人的清償能力，由於保險市場的容量已無法足夠涵蓋巨災的損失，再保險人開始尋找其他的風險移轉工具，因而在1992年芝加哥交易所保險證券化商品問世---巨災保險期貨。 本篇文章主要在介紹保險證券化最典型的兩個商品：巨災債券及巨災選擇權。首先針對巨災債券及巨災選擇權的商品內容加以描述；然後我們採用1970~2000年全球前四十大巨災損失作為巨災損失指數，透過定價模型對巨災債券及巨災選擇權作評價；最後分別對巨災債券及巨災選擇權的價格作敏感度分析，了解相關變數對巨災債券及巨災選擇權的價格的影響性。 在敏感度分析中，可看出巨災債券及巨災選擇權之價格具下列特性： (1) 在巨災債券方面，巨災債券價格和損失基準點、利息收回比率、本金收回比率、報酬率標準差呈正向關係，和報酬率、巨災發生次數呈反向關係； (2) 在巨災選擇權方面，損失資料分為理論巨災損失指數及實際損失指數，選擇權型態採用亞式選擇權，故選擇權價格有平均價格式及平均履約價格式兩種。 <1>在理論巨災損失指數選擇權方面，平均價格式選擇權價格和履約價、報酬率呈反向關係，和巨災發生次數呈正向關係，和標準差無關；平均履約價格式選擇權價格和報酬率呈反向關係，和巨災發生次數呈正向關係，和履約價、標準差無關； <2>在實際巨災損失指數選擇權方面，平均價格式選擇權價格和履約價、報酬率呈反向關係，和標準差無關；平均履約價格式選擇權價格和履約價、報酬率及標準差都呈反向關係。 希望藉由此篇論文之撰寫，將保險證券化的概念帶進國內保險領域，並提供保險業者了解另一種風險管理的方法。 / In the 1990s, many catastrophes occurred around the world, leading to a lack in reinsurers’ underwriting capacity and even, in some cases, threatening their solvency. Because the insurance market as a whole was unable to provide sufficient coverage for the catastrophe losses, reinsurers started looking for other risk transfer and management tools. In 1992, the first insurance securitization product was traded on the Chicago Board of Trade---Catastrophe Insurance Futures. This article aims to introduce two typical products: Catastrophe Bond (Cat Bond) and Catastrophe Option (Cat Option). First, we describe merchandise contents of Cat Bond and Cat Option. We then adopt global catastrophe losses as the catastrophe losses index for the period 1970-2000 and price models to evaluate Cat Bond and Cat Option. Finally, we conduct a sensitive analysis of Cat Bond and Cat Option prices. This allows us to understand the variables related to influencing the prices of Cat Bond and Cat Option. From the sensitive analysis, we realize that Cat Bond and Cat Option prices have the following characteristics: (1) From the Cat Bond aspect, the Cat Bond price has positive relationships with the loss trigger level, the receivable coupon ratio, the receivable principal ratio and the deviation error of return rate. It has negative relationships with the return rate and the times of catastrophe occurrences. (2) From the Cat Option aspect, we can distinguish between loss data to the theoretical catastrophe loss index and the actual catastrophe loss index. We adopt Asian Option as the option type, so that there are two types of option prices: the average price type of option price and the average exercise price type of option price. <1> From the aspect of the theoretical catastrophe loss index option, the average price type of option price has negative relationships with the exercise price and the return rate. It has a positive relationship with the times of catastrophe occurrence and has no relationships with the deviation error of the return rate. The average exercise price type of option price has negative relationships with the return rate. It has positive relationships with the times of catastrophe occurrence and has no relationships with the exercise price and the deviation error of the return rate. <2> From the aspect of the actual catastrophe loss index option, the average price type of option price has negative relationships with the exercise price and the return rate. It has no relationship with the deviation error of the return rate. The average exercise price type of option price has negative relationships with the exercise price, the return rate and the deviation error of the return rate. We hope that by writing this thesis, we can bring insurance securitization knowledge to the domestic insurance industry and can offer insurers an understanding of another risk management tool.

In this thesis, we study numerical analysis for random processes and fields. We investigate the behavior of the approximation accuracy for specific linear methods based on a finite number of observations. Furthermore, we propose techniques for optimizing performance of the methods for particular classes of random functions. The thesis consists of an introductory survey of the subject and related theory and four papers (A-D). In paper A, we study a Hermite spline approximation of quadratic mean continuous and differentiable random processes with an isolated point singularity. We consider a piecewise polynomial approximation combining two different Hermite interpolation splines for the interval adjacent to the singularity point and for the remaining part. For locally stationary random processes, sequences of sampling designs eliminating asymptotically the effect of the singularity are constructed. In Paper B, we focus on approximation of quadratic mean continuous real-valued random fields by a multivariate piecewise linear interpolator based on a finite number of observations placed on a hyperrectangular grid. We extend the concept of local stationarity to random fields and for the fields from this class, we provide an exact asymptotics for the approximation accuracy. Some asymptotic optimization results are also provided. In Paper C, we investigate numerical approximation of integrals (quadrature) of random functions over the unit hypercube. We study the asymptotics of a stratified Monte Carlo quadrature based on a finite number of randomly chosen observations in strata generated by a hyperrectangular grid. For the locally stationary random fields (introduced in Paper B), we derive exact asymptotic results together with some optimization methods. Moreover, for a certain class of random functions with an isolated singularity, we construct a sequence of designs eliminating the effect of the singularity. In Paper D, we consider a Monte Carlo pricing method for arithmetic Asian options. An estimator is constructed using a piecewise constant approximation of an underlying asset price process. For a wide class of Lévy market models, we provide upper bounds for the discretization error and the variance of the estimator. We construct an algorithm for accurate simulations with controlled discretization and Monte Carlo errors, andobtain the estimates of the option price with a predetermined accuracy at a given confidence level. Additionally, for the Black-Scholes model, we optimize the performance of the estimator by using a suitable variance reduction technique.