Doctor Scientiae - DSc / Robust Spectral Methods for Solving Option Pricing Problems by Edson Pindza PhD thesis, Department of Mathematics and Applied Mathematics, Faculty of Natural Sciences, University of the Western Cape Ever since the invention of the classical Black-Scholes formula to price the financial derivatives, a number of mathematical models have been proposed by numerous researchers in this direction. Many of these models are in general very complex, thus closed form analytical solutions are rarely obtainable. In view of this, we present a class of efficient spectral methods to numerically solve several mathematical models of pricing options. We begin with solving European options. Then we move to solve their American counterparts which involve a free boundary and therefore normally difficult to price by other conventional numerical methods. We obtain very promising results for the above two types of options and therefore we extend this approach to solve some more difficult problems for pricing options, viz., jump-diffusion models and local volatility models. The numerical methods involve solving partial differential equations, partial integro-differential equations and associated complementary problems which are used to model the financial derivatives. In order to retain their exponential accuracy, we discuss the necessary modification of the spectral methods. Finally, we present several comparative numerical results showing the superiority of our spectral methods.

The purpose of this thesis is to compare option pricing models. We have investigated the constant volatility models Black-Scholes-Merton (BSM) and Merton’s Jump Diffusion (MJD) as well as the stochastic volatility models Heston and Bates. The data used were option prices from Microsoft, Advanced Micro Devices Inc, Walt Disney Company, and the S&P 500 index. The data was then divided into training and testing sets, where the training data was used for parameter calibration for each model, and the testing data was used for testing the model prices against prices observed on the market. Calibration of the parameters for each model were carried out using the nonlinear least-squares method. By using the calibrated parameters the price was calculated using the method of Carr and Madan. Generally it was found that the stochastic volatility models, Heston and Bates, replicated the market option prices better than both the constant volatility models, MJD and BSM for most data sets. The mean average relative percentage error for Heston and Bates was found to be 2.26% and 2.17%, respectively. Merton and BSM had a mean average relative percentage error of 6.90% and 5.45%, respectively. We therefore suggest that a stochastic volatility model is to be preferred over a constant volatility model for pricing options. / Syftet med denna tes är att jämföra prissättningsmodeller för optioner. Vi har undersökt de konstanta volatilitetsmodellerna Black-Scholes-Merton (BSM) och Merton’s Jump Diffusion (MJD) samt de stokastiska volatilitetsmodellerna Heston och Bates. Datat vi använt är optionspriser från Microsoft, Advanced Micro Devices Inc, Walt Disney Company och S&P 500 indexet. Datat delades upp i en träningsmängd och en test- mängd. Träningsdatat användes för parameterkalibrering med hänsyn till varje modell. Testdatat användes för att jämföra modellpriser med priser som observerats på mark- naden. Parameterkalibreringen för varje modell utfördes genom att använda den icke- linjära minsta-kvadratmetoden. Med hjälp av de kalibrerade parametrarna kunde priset räknas ut genom att använda Carr och Madan-metoden. Vi kunde se att de stokastiska volatilitetsmodellerna, Heston och Bates, replikerade marknadens optionspriser bättre än båda de konstanta volatilitetsmodellerna, MJD och BSM för de flesta dataseten. Medelvärdet av det relativa medelvärdesfelet i procent för Heston och Bates beräknades till 2.26% respektive 2.17%. För Merton och BSM beräknades medelvärdet av det relativa medelvärdesfelet i procent till 6.90% respektive 5.45%. Vi anser därför att en stokastisk volatilitetsmodell är att föredra framför en konstant volatilitetsmodell för att prissätta optioner.

In this thesis, an optimal investment problem is studied for an investor who can only invest in a financial market modelled by an Itô-Lévy process; with one risk free (bond) and one risky (stock) investment possibility. We present the dynamic programming method and the associated Hamilton-Jacobi-Bellman (HJB) equation to explicitly solve this problem. It is shown that with purification and simplification to the standard jump diffusion process, closed form solutions for the optimal investment strategy and for the value function are attainable. It is also shown that, an explicit solution can be obtained via a finite training of a neural network using Stochastic gradient descent (SGD) for a specific case.

>Magister Scientiae - MSc / Warrant pricing has become very crucial in the present market scenario. See, for example, M. Hanke and K. Potzelberger, Consistent pricing of warrants and traded options, Review Financial Economics 11(1) (2002) 63-77 where the authors indicate that warrants issuance affects the stock price process of the issuing company. This change in the stock price process leads to subsequent changes in the prices of options written on the issuing company's stocks. Another notable work is W.G. Zhang, W.L. Xiao and C.X. He, Equity warrant pricing model under Fractional Brownian motion and an empirical study, Expert System with Applications 36(2) (2009) 3056-3065 where the authors construct equity warrants pricing model under Fractional Brownian motion and deduce the European options pricing formula with a simple method. We study this paper in details in this mini-thesis. We also study some of the mathematical models on warrant pricing using the Black-Scholes framework. The relationship between the price of the warrants and the price of the call accounts for the dilution effect is also studied mathematically. Finally we do some numerical simulations to derive the value of warrants.

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In this paper, Monte Carlo simulation for CCR (Counterparty Credit Risk) modeling is investigated. A jump-diffusion model, Bates' model, is used to describe the price process of an asset, and the counterparty default probability is described by a stochastic intensity model with constant intensity. In combination with Monte Carlo simulation, the variance reduction technique importance sampling is used in an attempt to make the simulations more efficient. Importance sampling is used for simulation of both the asset price and, for CVA (Credit Valuation Adjustment) estimation, the default time. CVA is simulated for both European and Bermudan options. It is shown that a significant variance reduction can be achieved by utilizing importance sampling for asset price simulations. It is also shown that a significant variance reduction for CVA simulation can be achieved for counterparties with small default probabilities by employing importance sampling for the default times. This holds for both European and Bermudan options. Furthermore, the regression based method least squares Monte Carlo is used to estimate the price of a Bermudan option, resulting in CVA estimates that lie within an interval of feasible values. Finally, some topics of further research are suggested. / I denna rapport undersöks Monte Carlo-simuleringar för motpartskreditrisk. En jump-diffusion-modell, Bates modell, används för att beskriva prisprocessen hos en tillgång, och sannolikheten att motparten drabbas av insolvens beskrivs av en stokastisk intensitetsmodell med konstant intensitet. Tillsammans med Monte Carlo-simuleringar används variansreduktionstekinken importance sampling i ett försök att effektivisera simuleringarna. Importance sampling används för simulering av både tillgångens pris och, för estimering av CVA (Credit Valuation Adjustment), tidpunkten för insolvens. CVA simuleras för både europeiska optioner och Bermuda-optioner. Det visas att en signifikant variansreduktion kan uppnås genom att använda importance sampling för simuleringen av tillgångens pris. Det visas även att en signifikant variansreduktion för CVA-simulering kan uppnås för motparter med små sannolikheter att drabbas av insolvens genom att använda importance sampling för simulering av tidpunkter för insolvens. Detta gäller både europeiska optioner och Bermuda-optioner. Vidare, används regressionsmetoden least squares Monte Carlo för att estimera priset av en Bermuda-option, vilket resulterar i CVA-estimat som ligger inom ett intervall av rimliga värden. Slutligen föreslås några ämnen för ytterligare forskning.

This thesis treats a range of stochastic methods with various applications, most notably in finance. It is comprised of five articles, and a summary of the key concepts and results these are built on. The first two papers consider a jump-to-default model, which is a model where some quantity, e.g. the price of a financial asset, is represented by a stochastic process which has continuous sample paths except for the possibility of a sudden drop to zero. In Paper I prices of European-type options in this model are studied together with the partial integro-differential equation that characterizes the price. In Paper II the price of a perpetual American put option in the same model is found in terms of explicit formulas. Both papers also study the parameter monotonicity and convexity properties of the option prices. The third and fourth articles both deal with valuation problems in a jump-diffusion model. Paper III concerns the optimal level at which to exercise an American put option with finite time horizon. More specifically, the integral equation that characterizes the optimal boundary is studied. In Paper IV we consider a stochastic game between two players and determine the optimal value and exercise strategy using an iterative technique. Paper V employs a similar iterative method to solve the statistical problem of determining the unknown drift of a stochastic process, where not only running time but also each observation of the process is costly.

建立適當的資本投資決策，對於企業未來的發展具有深遠的影響。如何能擬定出適合的資本預算計畫，以增加公司的成長機會與競爭能力，便是當前重要的課題。本論文以三個階段探討企業投資歷程中所具有的實質選擇權評價：包括對於計畫案擬定之初期，進行投資機會價值評估的實質成長選擇權。以及針對投資計畫開始進行時，管理者所擁有的各種管理彈性，如遞延、擴張、縮減與暫停投資的決策彈性，進行多重實質選擇權的價值評估。最後，針對未能順利成功的計畫案，管理者擁有將其永遠放棄，以收回投資成本殘值的實質放棄選擇權價值進行評估。 　　對於第一階段的成長選擇權價值評估，本文已建立出同時考量標的資產與投資成本隨機變動，以及標的資產存在不連續跳躍特性下的選擇權評價封閉解，結果可用來評估計畫方案擬定初期的實質成長選擇權價值。若將評價模式中的參數進行限制，則本模型將會分別退化至Black and Scholes(1973), Merton(1976), Fischer(1978), Margrabe(1978), McDonald and Siegel(1985)等重要的選擇權評價文獻，可知本文已獲致較一般化的評價模型。 　　在第二階段的多重實質選擇權價值評估，本文採用Trigeorgis(1991)所建立的對數轉換二項評價模式，再加入跳躍模型的考量，以符合科技產業所具有的創新、競爭特性，期較能合理評估其價值，也獲得了較一般化的評價模式。再者，本文以模擬方式對於管理者在投資計畫的進行過程中所擁有的遞延、擴張、縮減以及暫停投資等彈性決策價值進行評估，以彰顯出利用實質選擇權評價方法進行彈性決策價值評估的必要性。由數值分析的結果得到，當多個實質選擇權同時存在時，其間將產生不同程度的交互作用，因此並不能直接將個別價值予以加總來求算整體的實質選擇權價值。不過，每項管理彈性的加入對於整體價值的增加皆具有正向貢獻。 　　對於第三階段的放棄選擇權價值評估，本文建立同時存在多項投資方案下的實質放棄選擇權評價模型，結果可用來評估研發計畫方案未能成功時的實質放棄選擇權價值。此外，本文進一步對於此評價模型進行數值分析，並將所得到的結果歸納如下：(1)方案間價值變動相關係數對於實質放棄選擇權價值的影響上，有相關係數越高時，實質放棄選擇權的價值就越高的現象。(2)殘值回收比率較高時，若採取較多的投資計畫方案，將可以獲致較高的實質放棄選擇權價值，此結果可作偽管理者在選擇備抵方案數目時的參考。(3)對於敏威性分析的探討，發現到當殘值增加、利率下降以及剩餘期間較長時，實質放棄選擇權的價值是增加的，此現象與賣權特性結果一致。 　　因此，本文針對企業投資歷程中所具有的實質選擇權評價進行深入探討，分別建立選擇權評價模型，也獲致了較以往模型更一般化的評價結果。並於各評價模型建構完成後，輔以數值模擬與敏感性分析，以進一步說明本文所建構模型之一般性與合理性。最後，希望此結果有助於日後企業對於投資價值評估時之參考，並可彌補此類研究文獻的不足。 / This dissertation presents three essays, each provides a general real option pricing model. In the first essay, we derive a generalized option pricing formula for the case of the underlying asset and exercise price both being driven by a mixture of continuous and jump diffusion processes. Our pricing model is a generalized version of Black and Scholes(1973), Merton(1976), Fischer(1978), Margrabe(1978), and McDonald-Siegel(l 985). And each of the historical model is shown to be a special case of ours. We then use the model developed in this article to evaluate real growth options where the underlying assets follow jump diffusion processes. The second essay develops a multi-option pricing model incorporating jump characteristics. The model we provide here can be used to value various types of flexibilities, including the option to defer, the option to shut down, the option to contract, and the option to expand. Based on our numerical results, we find that the model can deal with the interactions among these options. The third essay considers an abandonment option on the maximizing value of several investment projects. Here we develop a model to evaluate R&D projects that may not be accomplished. We show that both Black-Scholes's model and Stuiz's model are special cases of ours under certain restrictions on parameters. From the simulation results, we find a positive relation between the correlation of project value changes and the value of the real abandonment options. Furthermore, our simulation results show that the higher the percentage of recovering salvage value, the more number of investment projects should be carried out. The result we find can help managers to choose the better backup projects. Our sensitivity analysis shows that the value of the real abandonment options increase when the riskless interest rate decreases, and at the same time the salvage value and the time to maturity increase.

保險業及再保險業以往對於巨災危險的風險管理方式大部份都佼給全世界的再保險承保能量去承擔。然而從1995年開始，美國芝加哥交易所(CBOT)與產物損失部門(PCS)共同推出巨災保險選擇權，提供保險人以及再保險人利用國際金融市場移轉核保業務上所承擔之巨災危險的管道。此種業務上的巨災危險提供保險業處理巨災損失的新管道，例如產險業因為天然災害或是人為疏失所導致的鉅額核保損失以及壽險業的團體保險和健康保險的鉅額損失。巨災保險選擇權是一種新的衍生性金融商品，其交易標的物是專門針對保險業所承保的業務（尤其是巨災），因此如果運用得當，除了能有效的分散核保風險之外，更可以避免傳統的再保險契約所衍生的問題。 本研究在第一章首先說明台灣地區是地震、颱風以及水患等天然災害頗為集中的地區，因為傳統再保險的分散風險方式有其成本較高以及資訊不對稱的問題，所以保業以及再保險業應該考慮其他類型的危險管理策略。第二章以巨災保險選擇權評價的相關基礎理論為主要的架構，並且探討美國PCS所開發的巨災保險選擇權，並說明如何利用此種金融工具移轉保險與再保險人因地理上的核保因素所產生的風險。 第三章以及第四章討論模擬方法與分析模擬所得的結果，我們並利用情境分析的方式，探討在單位時間內，平均跳躍次數對於每一個模型中假設，交易標的物為損失指數時的影響，以及依此損失指數所得對於巨災保險選擇權價格之變化幅度。第五章則是歸納本研究所得的結果並且提出後續研究的建議。 / The insurance and reinsurance industries traditionally transfer their insurance risk of catastrophe disasters through the international reinsurance market. Since the capacity of the international reinsurance market is not always available to cover the entire risks. In 1995, CBOT (Chicago Board of Trade) and PCS (Property Claims Service) have begun trading the PCS catastrophe options Through the catastrophe options, the insures and reinsures could hedging their operating risks in the international financial market. These risks consist of large amount of underwriting losses from the natural disasters, personal default in property insurance, inflation of claims amount and the large claims in group insurance and health insurance. The loss ratios of the insured business are trading through the catastrophe options. Hedging the operating risks of the insures and reinsures in the financial market could effectively reduce the costs and avoid the complexity from the reinsurance contracts. In this study, we have reviewed the development of the catastrophe option. Asian style call options are illustrated to monitor the process of option pricing. The trading loss ratios are modeled through lognormal distribution based on the claim experience collected from 1970-1996. The methodology of pricing the modified options based on pure jump model proposed by Cox, et al (1976) and the jump diffusion model proposed by Merton (1976) are discussed. Computer simulations and scenario analysis are performed to investigate the pricing of Asian style catastrophe option under various proposed models. Sensitivity analysis is also completed at various parameters in the jump process. Finally, comments on future works and the limitation of the proposed risk-transfer mechanism using catastrophe options are discussed.

As ruin theory evolves in recent years, there has been a variety of quantities pertaining to an insurer's bankruptcy at the centre of focus in the literature. Despite the fact that these quantities are distinct from each other, it was brought to our attention that many solution methods apply to nearly all ruin-related quantities. Such a peculiar similarity among their solution methods inspired us to search for a general form that reconciles those seemingly different ruin-related quantities. The stochastic approach proposed in the thesis addresses such issues and contributes to the current literature in three major directions. (1) It provides a new function that unifies many existing ruin-related quantities and that produces more new quantities of potential use in both practice and academia. (2) It applies generally to a vast majority of risk processes and permits the consideration of combined effects of investment strategies, policy modifications, etc, which were either impossible or difficult tasks using traditional approaches. (3) It gives a shortcut to the derivation of intermediate solution equations. In addition to the efficiency, the new approach also leads to a standardized procedure to cope with various situations. The thesis covers a wide range of ruin-related and financial topics while developing the unifying stochastic approach. Not only does it attempt to provide insights into the unification of quantities in ruin theory, the thesis also seeks to extend its applications in other related areas.