The study investigates whether the main capital market theories and pricing models provide a reasonably accurate description of the working and efficiency of capital markets, of the pricing of shares and options and the effect the risk/return relationship has on investor behaviour. The capital market theories and pricing models included in the study are Portfolio Theory, the Efficient Market Hypothesis (EMH), the Capital Asset Pricing Model (CAPM), the Arbitrage Pricing Theory (APT), Options Theory and the BlackScholes (8-S) Option Pricing Model. The main conclusion of the study is that the main capital market theories and pricing models, as reviewed in the study, do provide a reasonably accurate description of reality, but a number of anomalies and controversial issues still need to be resolved. The main recommendation of the study is that research into these theories and models should continue unabated, while the specific recommendations in a South African context are the following: ( 1) the benefits of global diversification for South African investors should continue to be investigated; (2) the level and degree of efficiency of the JSE Securities Exchange SA (JSE) should continue to be monitored, and it should be established whether alternative theories to the EMH provide complementary or better descriptions of the efficiency of the South African market; (3) both the CAPM and the APT should continue to be tested, both individually and jointly, in order to better understand the pricing mechanism of, and risk/return relationship on the JSE; (4) much South African research still needs to be conducted on the efficiency of the relatively new options market and the application of the B-S Option Pricing Model under South African conditions. / Financial Accounting / M. Com. (Accounting)

Este trabalho analisa três estratégias de hedge de opção, buscando identificar a importância da escolha da estratégia para a obtenção de um bom desempenho do hedge. O conceito de hedge é analisado de forma retrospectiva e uma teoria geral de hedge é apresentada. Em seguida são descritos alguns estudos comparativos de desempenho de estratégias de hedge de opção e suas metodologias de implementação. Para esta análise comparativa são selecionadas três estratégias de hedge de opção de compra do tipo européia: a primeira utiliza o modelo Black-Scholes-Merton de precificação de opções, a segunda utiliza uma solução de programação dinâmica para hedge dinâmico multiperiódico e a terceira utiliza um modelo GARCH para precificação de opções. As estratégias são comentadas e comparadas do ponto de vista de suas premissas teóricas e por meio de testes comparativos de desempenho. O desempenho das estratégias é comparado sob uma perspectiva dinâmicamente ajustada, multiperiódica e autofinanciável. Os dados para comparação de desempenho são gerados por simulação e o desempenho é avaliado pelos erros absolutos médios e erros quadráticos médios, resultantes na carteira de hedge. São feitas ainda considerações a respeito de alternativas de estimação e suas implicações no desempenho das estratégias. / This work analyzes three option hedging strategies, to identify the importance of choosing a strategy in order to achieve a good hedging performance. A retrospective analysis of the concept of hedging is conducted and a general hedging theory is presented. Following, some comparative papers of hedging performance and their implementation methodologies are described. For the present comparative analysis, three hedging strategies for European options have been selected: the first one based on the Black-Scholes-Merton model for option pricing, the second one based on a dynamic programming solution for dynamic multiperiod hedging and the third one based on a GARCH model for option pricing. The strategies are compared under their theoric premisses and through comparative performance testes. The performances of the strategies are compared under a dynamically adjusted multiperiodic and self-financing perspective. Data for performance comparison are generated by simulation and performance is evaluated by mean absolute errors and mean squared errors resulting on the hedging portfolio. An analysis is also done regarding estimation approaches and their implications over the performance of the strategies.

Optioner säljs och köps idag flitigt av många olika anledningar. En av dessa kan vara spekulation kring framtida händelser för aktiepriser där optioner har fördelar jämfört med aktier i form av en hävstångseffekt. En annan anledning för optionshandel är för att hedga (säkra) risker vilket ställer krav på att innehavet av optionen ska kompensera den negativa effekt som riskerna bidrar till. Med andra ord, om det finns en risk för ett negativt framtida scenario som man inte vill riskera att utsätta sig för så kan optioner vara rätt verktyg att använda sig av.   Risker finns idag överallt, i olika former, vilket har bidragit till att efterfrågan av optioner har ökat enormt de senaste årtiondena. Dock kan risker vara både komplexa och varierande vilket har lett till att mer komplexa optioner har utvecklats för att mätta den efterfrågan som utvecklats på marknaden. Dessa, mer komplexa optioner, kallas exotiska optioner och de skiljer sig från de vanliga europeiska och amerikanska köp- och säljoptionerna. Däribland hittar vi bland annat lookback-optioner i form av bland annat köpoptioner på maximum och kvantiloptioner vilka är två av de huvudsakliga optionerna som diskuteras i uppsatsen.   Det har länge varit känt hur man prissätter europeiska köp- och säljoptioner via Black-Scholes-Mertons modell men desto fler komplexa optioner som tillkommer på marknaden desto mer komplicerade prissättningsmodeller utvecklas. Till skillnad från europeiska köp- och säljoptioner vars utdelning beror på aktiepriset på lösendagen så är lookback-optioner beroende av aktieprisets rörelse under hela kontraktstiden. Detta medför att prissättningen av dessa beror av fler parametrar än i Black-Scholes-Mertons modell, bland annat ockupationstiden för den stokastiska process som beskriver aktiepriset, vilket bidrar till andra prissättningsmodeller.   Uppsatsen har som syfte att redogöra för modellen som används vid prissättningen av kvantiloptioner samt presentera hur deras egenskaper förhåller sig till andra typer av lookback-optioners egenskaper. Det presenteras i rapporten att kvantiloptioner liknar vissa typer av lookback-optioner, mer bestämt köpoptioner på maximum, och att kvantiloptioners egenskaper faktiskt konvergerar mot köpoptioner på maximums egenskaper då kvantilen närmar sig 1. Utifrån detta resonemang så kan det finnas fördelar i att använda kvantiloptioner snarare än köpoptioner på maximum vilket investerare bör ta i hänsyn när, och om, kvantiloptioner introduceras på marknaden. / Options are today used by investors for multiple reasons. One of these are speculation about future market movements, here ownership of options is advantageous over usual ownership of shares in the underlying stock in terms of a leverage effect. Furthermore, investors use options to hedge different kinds of risks that they are exposed to, this demands that the option compensates the possible negative effect that the risk brings to the table. In other words, if there is a risk of a future negative scenario which the investor is risk averse to, then owning specific options which neutralize this risk could be the perfect tool to use.   Risks are today seen all over the market in different shapes which have created a great demand for options over the last decades. However, since risks can be both complex and range over multiple business areas, investors have demanded more complex options which can neutralize the risk exposures. These, more complex options, are called exotic options, and they differ from the regular American and European options in the way they behave with respect to the underlying stock. Amongst these exotic options, we can find different kind of lookback options as well as quantile options which are two of the main options that are discussed in this thesis.   It has been known for a while how to price European call and put options by the Black-Scholes-Merton model. However, with more complex options also comes more complex pricing models and unlike the European options’ payoff which depend on the underlying stock price at time of maturity, the lookback option’s and quantile option’s payoff depend on the stock price movement over the total life span of the option contract. Hence, the pricing of these options depends on more variables than the classic Black-Scholes-Merton model include. One of these variables is the occupation time of the stochastic process which describes the stock price movement, this leads to a more complex and extensive pricing model than the general Black-Scholes-Merton’s model.   The objective of this thesis is to derive the pricing model that is used for quantile options and prove that the properties of quantile options are advantageous when compared to some specific lookback options, viz. call options on maximum.  It is concluded in the thesis that quantile options in fact converges to the call option on maximum for quantiles approaching 1. However, quantile options come with some different properties which potentially makes them a good substitute for the call option on maximum. This is a relevant factor for investors to consider when, and if, quantile options are introduced to the market.

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.

This thesis deals with regularization parameter selection methods in the context of Tikhonov-type regularization with Poisson distributed data, in particular the reconstruction of images, as well as with the identification of the volatility surface from observed option prices. In Part I we examine the choice of the regularization parameter when reconstructing an image, which is disturbed by Poisson noise, with Tikhonov-type regularization. This type of regularization is a generalization of the classical Tikhonov regularization in the Banach space setting and often called variational regularization. After a general consideration of Tikhonov-type regularization for data corrupted by Poisson noise, we examine the methods for choosing the regularization parameter numerically on the basis of two test images and real PET data. In Part II we consider the estimation of the volatility function from observed call option prices with the explicit formula which has been derived by Dupire using the Black-Scholes partial differential equation. The option prices are only available as discrete noisy observations so that the main difficulty is the ill-posedness of the numerical differentiation. Finite difference schemes, as regularization by discretization of the inverse and ill-posed problem, do not overcome these difficulties when they are used to evaluate the partial derivatives. Therefore we construct an alternative algorithm based on the weak formulation of the dual Black-Scholes partial differential equation and evaluate the performance of the finite difference schemes and the new algorithm for synthetic and real option prices.

This thesis deals with regularization parameter selection methods in the context of Tikhonov-type regularization with Poisson distributed data, in particular the reconstruction of images, as well as with the identification of the volatility surface from observed option prices. In Part I we examine the choice of the regularization parameter when reconstructing an image, which is disturbed by Poisson noise, with Tikhonov-type regularization. This type of regularization is a generalization of the classical Tikhonov regularization in the Banach space setting and often called variational regularization. After a general consideration of Tikhonov-type regularization for data corrupted by Poisson noise, we examine the methods for choosing the regularization parameter numerically on the basis of two test images and real PET data. In Part II we consider the estimation of the volatility function from observed call option prices with the explicit formula which has been derived by Dupire using the Black-Scholes partial differential equation. The option prices are only available as discrete noisy observations so that the main difficulty is the ill-posedness of the numerical differentiation. Finite difference schemes, as regularization by discretization of the inverse and ill-posed problem, do not overcome these difficulties when they are used to evaluate the partial derivatives. Therefore we construct an alternative algorithm based on the weak formulation of the dual Black-Scholes partial differential equation and evaluate the performance of the finite difference schemes and the new algorithm for synthetic and real option prices.

The main goal of this dissertation was to obtain an understanding as to the true economic nature of employee share options and the problems surrounding the accounting thereof. The main conclusion of this study is that employee share options should be expensed in the income statement as and when the employee's services are performed. The reason is that employee share options are valuable financial instruments which the employer has used to compensate the employee for his services. It was also concluded that exercise date accounting and classification of outstanding employee share options as liabilities on the balance sheet is the most appropriate accounting treatment. Such accounting treatment trues up the accounting of employee share options with that of cash-settled share appreciation rights, which are economically equivalent transactions. The measurement of employee share options should be based on their fair value using an option-pricing model adapted for the specific features of employee share options. / Accounting / Thesis (M. Com. (Accounting Science))

一. 研究動機： 最近幾年，台灣股市在需求與供給互動之下，呈現一股狂熱現象，造成此現象一個不 可忽視的原因即為過多的游資追求有限籌碼，解決此一失衡狀態開放多元的投資管道 成為必然的越勢。以台灣地下金融市場的活絡及財政部目前正著手研擬的期貨市場開 放準則，都證實之民間與政府對多元投資管道需求迫切的認同。 二. 研究目的： 目的一、：介紹一種在國外行之有年的投資管道一選擇權市場，對國內大多數的投資 者而言，選擇權市場或許是一個陌生的名詞，但隨著國際金融市場的漸次轉移亞洲地 區，進而帶動台灣金融市場與國際金融的互動結果，將使選擇權市場對迫切需求投資 管道的我國提供另一項投資發展空間，即成為另一金融市場有利的投資交易避險工具 。 目的二、：修正 Black-Scholes實證模型，即外匯選擇權定價模型以估算此模型的準 確性。經由測試外匯選擇權模型之準確性後，我們可探討該模型在台灣使用之可行性 ，並希望對台灣未來成立之選擇權市場做一些政策上的建議。 三、資料來源： 本論文所需要的資料除利率及變異數外皆取自華街日報費城每日外匯收盤盤價，利率 取自“London Financal Times”， 而變異數的估算方法則使用Robert L.Welsh及D- avid M.Chen 於“Advances and Options Research” 一文中的隱含性變異數推估而 得。期限則從民國77年11月至78年10月，為日資料，共260 筆，所使用的外幣有英鎊 、馬克、瑞士法朗、日圓等四種。 四、實證方法： C=se N(d1)-xe N(d1- )d1= 理論模型–外匯選擇權的定價方程為： C： 權利金，S： 外匯價格，X： 執行價格，r.r ：本國及外國利率，G： 變異數， T： 到期日，N ()：機率密度函數。 Black-Scholes 模型主要應用於股票選擇權市場，股票為非孳息債券，而擁有外匯可 同時擁有國外的利息報酬，所以 B-S模型與外匯定價模型主要的差異為國外利率折現 部份。 本論文利用此外匯模型估算出模型價格，再與市場價格做比較，以測試模型的準確性 ，或在任何情況下模型可能高估或估市場價格。 五、實證預估結果： (1) 當美國利率高於國外利率時，美式買入選擇權不會提早執行其權利所以其價格與 歐式買入選擇權價格一致，所以利用此外匯模型因能準確估算其結果。當美國利率低 於國外種率時，美式買入選擇權不會提早執行其權利，所以其價格會高於歐式買入選 擇權，而此外匯模型將可能產生低估現象。從資料知除英鎊利率高於美元利率，餘皆 小於美元利率，所以可能發生英鎊低估現象。 (2) 利用此模型測試市場是否具效率，得相對於模型價格市場價格低估較現值有利的 選擇權，而高估較現值不利的選擇權。

Title: Stochastic Models in Financial Mathematics Author: Bc. Oliver Waczulík Department: Department of Probability and Mathematical Statistics Supervisor: doc. RNDr. Jan Hurt, CSc., Department of Probability and Mathe- matical Statistics Abstract: This thesis looks into the problems of ordinary stochastic models used in financial mathematics, which are often influenced by unrealistic assumptions of Brownian motion. The thesis deals with and suggests more sophisticated alternatives to Brownian motion models. By applying the fractional Brownian motion we derive a modification of the Black-Scholes pricing formula for a mixed fractional Bro- wnian motion. We use Lévy processes to introduce subordinated stable process of Ornstein-Uhlenbeck type serving for modeling interest rates. We present the calibration procedures for these models along with a simulation study for estima- tion of Hurst parameter. To illustrate the practical use of the models introduced in the paper we have used real financial data and custom procedures program- med in the system Wolfram Mathematica. We have achieved almost 90% decline in the value of Kolmogorov-Smirnov statistics by the application of subordinated stable process of Ornstein-Uhlenbeck type for the historical values of the monthly PRIBOR (Prague Interbank Offered Rate) rates in...

Magister Scientiae - MSc / We investigate some simulation-based approaches for the valuing of the employee stock options. The mathematical models that deal with valuation of such options include the work of Jennergren and Naeslund [L.P Jennergren and B. Naeslund, A comment on valuation of executive stock options and the FASB proposal, Accounting Review 68 (1993) 179-183]. They used the Black and Scholes [F. Black and M. Scholes, The pricing of options and corporate liabilities, Journal of Political Economy 81(1973) 637-659] and extended partial differential equation for an option that includes the early exercise. Some other major relevant works to this mini thesis are Hemmer et al. [T Hemmer, S. Matsunaga and T Shevlin, The influence of risk diversification on the early exercise of employee stock options by executive officers, Journal of Accounting and Economics 21(1) (1996) 45-68] and Baril et al. [C. Baril, L. Betancourt, J. Briggs, Valuing employee stock options under SFAS 123 R using the Black-Scholes-Merton and lattice model approaches, Journal of Accounting Education 25 (1-2) (2007) 88-101]. The underlying assets are studied under the GARCH (generalized autoregressive conditional heteroskedasticity) effects. Particular emphasis is made on the American style employee stock options. / South Africa