With the exception of holders of default-free instruments, a key risk run by investors is credit risk. To meet the need of investors to hedge this risk, the market uses credit derivatives. The South African credit derivatives market is still in its infancy and only the very simplistic instruments are traded. One of the reasons is due to the technical sophistication required in pricing these instruments. This dissertation introduces the key concepts of risk neutral probabilities, arbitrage free pricing, martingales, default probabilities, survival probabilities, hazard rates and forward spreads. These mathematical concepts are then used as a building block to develop pricing formulae which can be used to infer valuations to the most popular credit derivatives in the South African financial markets. / Operations Research / M.Sc. (Operations Research)

The submitted work deals with option pricing. Mathematical approach is immediately followed by an economic interpretation. The main problem is to model the underlying uncertainities driving the stock price. Using two well-known valuation models, binomial model and Black-Scholes model, we explain basic principles, especially risk neutral pricing. Due to the empirical biases new models have been developped, based on pure jump process. Variance gamma process and its special symmetric case are presented.

Brigo與Mercurio提出了三種新的資產價格過程，分別是位移CEV過程、位移對數常態過程與混合對數常態過程。在這三種過程中，資產價格的波動度不再是一個固定的常數，而是時間與資產價格的明確函數。而由這三種過程所推導出來的歐式選擇權評價公式，將會導致隱含波動度曲線呈現傾斜曲線或是微笑曲線，且提供了參數讓我們能夠配適市場的波動度結構。本文利用台指買權來實證Brigo與Mercurio所提出的三種歐式選擇權評價公式，我們發現校準結果以混合對數常態過程優於位移CEV過程，而位移CEV過程則稍優於位移對數常態過程。因此，在實務校準時，我們建議以混合對數常態過程為台指買權的評價模型，以達到較佳的校準結果。 / Brigo and Mercurio proposed three types of asset-price dynamics which are shifted-CEV process, shifted-lognormal process and mixture-of-lognormals process respectively. In these three processes, the volatility of the asset price is no more a constant but a deterministic function of time and asset price. The European option pricing formulas derived from these three processes lead respectively to skew and smile in the term structure of implied volatilities. Also, the pricing formula provides several parameters for fitting the market volatility term structure. The thesis applies Taiwan’s call option to verifying these three pricing formulas proposed by Brigo and Mercurio. We find that the calibration result of mixture-of-lognormals process is better than the result of shifted-CEV process and the calibration result of shifted-CEV process is a little better than the result of shifted-lognormal process. Therefore, we recommend applying the pricing formula derived from mixture-of-lognormals process to getting a better calibration.

Thèse numérisée par la Division de la gestion de documents et des archives de l'Université de Montréal.

台灣特殊的兩黨對立政治環境及幾乎每年都會有的固定選舉，使得政治的不確定性深深的影響著國內的投資環境及投資人心態。本研究便是要探討，2002/1/1~2006/1/16 研究期間台灣的投資人在選舉前後的投資行為，是否真如大家所預期的，會受到台灣選舉事件的影響。 本研究首先利用適當的機率密度函數模型及選擇權市場資訊來導出隱含的風險中立密度值。再利用這些風險中立密度值，求出各個選舉事件相對應的機率分配圖形，並透過其機率分配圖形及波動率指數等統計值於投票日前後的變化來觀察某一選舉事件前後投資者的反應。 研究結果發現：1. 選舉事件的發生確實會影響投資者的心理，且投資者會透過選擇權市場有效率的反應預期的未來股價指數分佈情況。2. 越大型、越具爭議且全國性的選舉結果，其選舉期間機率分配圖形及波動率指數具有較高的波動性。3. 一般而言，選舉過後市場不確定因素降低，將使投資者對於股市的預期較為一致和樂觀。而若這個選舉結果使投資者感到意外，因而增加了市場的不確定性，則選後機率分配圖形及波動率指數的改變反而會更為明顯。4. 在此研究下對數常態混合法比傳統的 Black-Scholes 方法產生較低的誤差值，因此就實證的分析上能提供更好的配適。 / This research examines the behavior of investors during election periods from January 1st 2002 to January 6th 2006 in Taiwan. The research includes a few steps. First, we adopted a proper probability density function composed of stock index options data to construct the implied distribution. Then, when changing the whole shape of the risk-neutral implied distribution, the volatility indexes, and the statistics of the implied distribution, we observed investors' response around a specific election event. According to the empirical results, we found that: 1. An election event would influence investors’ behavior, and investors tend to reflect their expectation of future stock index in the option market in an efficient way. 2. The result of a large-scale and more disputed nationwide election will cause a higher fluctuation in both the implied distribution and the volatility index. 3. In general, the factor resulting from investors’ uncertainty of the market is likely to reduce after the election, which makes investors’ relatively unanimous and optimistic expectation of the stock market. However, if this election result surprises investors, their uncertainty of the market will increase, and thus the changes of the implied distribution and the volatility index become quite obvious. 4. The in-sample performance of the lognormal mixtures method employed in the research is considerably better than that of the traditional Black-Scholes model by having a lower root mean squared error.

波動度對於金融市場影響甚多，同時為金融資產定價的重要參數以及市場穩 定度的衡量指標，尤其在金融危機發生時，波動度指數的驟升反映資產價格震盪。 本篇論文嘗試捕捉S&P500 指數實現波動度與VIX變動率未來之動態，並將VIX、 VIX 選擇權與VVIX 納入預測模型中，探討其資訊內涵。透過研究S&P500 指數 實現波動度，能夠預測S&P500 指數未來之波動度與報酬，除了能夠觀察市場變 動，亦能使未來選擇權定價更為準確；而藉由模型預測VIX，能夠藉由VIX 選 擇權或VIX 期貨，提供避險或投資之依據。文章採用2006 年至2011 年之S&P500 指數、VIX、VIX 選擇權與VVIX 資料。 在 S&P500 指數之實現波動度預測當中，本篇論文的模型改良自先前文獻， 結合實現波動度、隱含波動度與S&P500 指數選擇權之風險中立偏態，所構成之 異質自我回歸模型(HAR-RV-IV-SK model)。論文額外加入VIX 變動率以及VIX指數選擇權之風險中立偏態作為模型因子，預測未來S&P500 指數實現波動度。 研究結果表示，加入VIX 變動率作為S&P500 指數實現波動度預測模型變數後， 可增加S&P500 指數實現波動度預測模型之準確性。 在 VIX 變動率預測模型之中，論文採用動態轉換模型，作為高低波動度之 下，區分預測模型的方法。以VIX 過去的變動率、VIX 選擇權之風險中立動差 以及VIX 之波動度指數(VVIX)作為變數，預測未來VIX 變動率。結果顯示動態 轉換模型能夠提升VIX 預測模型的解釋能力，並且在動態轉換模型下，VVIX 與 VIX 選擇權之風險中立動差，對於VIX 預測具有相當之資訊隱涵於其中。 / This paper tries to capture the future dynamic of S&P 500 index realized volatility and VIX. We add the VIX change rate and the risk neutral skewness of VIX options into the Heterogeneous Autoregressive model of Realized Volatility, Implied Volatility and Skewness (HAR-RV-IV-SK) model to forecast the S&P 500 realized volatility. Also, this paper uses the regime switching model and joins the VIX, risk neutral moments of VIX options and VVIX variables to raise the explanatory ability in the VIX forecasting. The result shows that the VIX change rate has additional information on the S&P 500 realized volatility. By using the regime switching model, the VVIX and the risk neutral moments of VIX options variables have information contents in VIX forecasting. These models can be used for hedging or investment purposes.

Thèse numérisée par la Division de la gestion de documents et des archives de l'Université de Montréal

本論文提出線性規劃的方法以還原隱藏於選擇權市場價格中的風險中立機率測度，並利用該機率測度計算選擇權的合理價格。模型中假設選擇權對應同一標的資產與到期日，資產價格於到期日的狀態為離散點且個數有限，當市場不具任何套利機會時，以極小化市場價格與合理價格之離差總和作為挑選風險中立機率測度的準則。最後，以臺指選擇權(TXO)的交易資料做為實證對象。實證中發現，加入平滑限制式與離差權重之線性規劃模型在評價歐式選擇權合理價格的效能最為優異。 / The thesis proposes a liner programming to recover the risk-neutral probability distribution of an underlying asset price from its associated market option prices, and we evaluate the fair prices of options via the resulting risk-neutral probability distribution. Assume that we face a series of European options with different exercise prices on the same maturity and underlying asset in this linear programming model. The criterion of choosing a risk-neutral probability distribution is minimizing the sum of total deviations subject to requiring that the fair prices of options are consistent with observed market option prices. Finally, we take the trading data of TXO as an empirical study. The empirical study indicates that the model with smooth constraints and weighted deviations has the best performance in pricing the rational price of European options.

Topic of the master thesis is practice of interest rate models. Literature dedicated to the interest rate models usually presents theory in very general form. Theory presented in general form leads to a gap between theory and practice. Author tries to fill this gap. Thesis describes basic theory and presents practical computations, which are relevant to generating interest rate scenarios. Contribution is given by derivation of formulas and computational methods in form directly applicable for implementation of presented models. It is common practice to validate quality of interest rate scenarios. Author presents several tests and implements them in programming language Python. Tests are implemented as application with graphical user interface.

Dans cette thèse, j’exploite le cadre d’analyse de données fonctionnelles et développe l’analyse d’inférence et de prédiction, avec une application à des sujets sur les marchés financiers. Cette thèse est organisée en trois chapitres. Le premier chapitre est un article co-écrit avec Marine Carrasco. Dans ce chapitre, nous considérons un modèle de régression linéaire fonctionnelle avec une variable prédictive fonctionnelle et une réponse scalaire. Nous effectuons une comparaison théorique des techniques d’analyse des composantes principales fonctionnelles (FPCA) et des moindres carrés partiels fonctionnels (FPLS). Nous déterminons la vitesse de convergence de l’erreur quadratique moyen d’estimation (MSE) pour ces méthodes. Aussi, nous montrons cette vitesse est sharp. Nous découvrons également que le biais de régularisation de la méthode FPLS est plus petit que celui de FPCA, tandis que son erreur d’estimation a tendance à être plus grande que celle de FPCA. De plus, nous montrons que le FPLS surpasse le FPCA en termes de prédiction avec moins de composantes. Le deuxième chapitre considère un modèle autorégressif entièrement fonctionnel (FAR) pour prèvoir toute la courbe de rendement du S&P 500 a la prochaine journée. Je mène une analyse comparative de quatre techniques de Big Data, dont la méthode de Tikhonov fonctionnelle (FT), la technique de Landweber-Fridman fonctionnelle (FLF), la coupure spectrale fonctionnelle (FSC) et les moindres carrés partiels fonctionnels (FPLS). La vitesse de convergence, la distribution asymptotique et une stratégie de test statistique pour sélectionner le nombre de retard sont fournis. Les simulations et les données réelles montrent que les méthode FPLS performe mieux les autres en terme d’estimation du paramètre tandis que toutes ces méthodes affichent des performances similaires en termes de prédiction. Le troisième chapitre propose d’estimer la densité de neutralité au risque (RND) dans le contexte de la tarification des options, à l’aide d’un modèle fonctionnel. L’avantage de cette approche est qu’elle exploite la théorie d’absence d’arbitrage et qu’il est possible d’éviter toute sorte de paramétrisation. L’estimation conduit à un problème d’inversibilité et la technique fonctionnelle de Landweber-Fridman (FLF) est utilisée pour le surmonter. / In this thesis, I exploit the functional data analysis framework and develop inference, prediction and forecasting analysis, with an application to topics in the financial market. This thesis is organized in three chapters. The first chapter is a paper co-authored with Marine Carrasco. In this chapter, we consider a functional linear regression model with a functional predictor variable and a scalar response. We develop a theoretical comparison of the Functional Principal Component Analysis (FPCA) and Functional Partial Least Squares (FPLS) techniques. We derive the convergence rate of the Mean Squared Error (MSE) for these methods. We show that this rate of convergence is sharp. We also find that the regularization bias of the FPLS method is smaller than the one of FPCA, while its estimation error tends to be larger than that of FPCA. Additionally, we show that FPLS outperforms FPCA in terms of prediction accuracy with a fewer number of components. The second chapter considers a fully functional autoregressive model (FAR) to forecast the next day’s return curve of the S&P 500. In contrast to the standard AR(1) model where each observation is a scalar, in this research each daily return curve is a collection of 390 points and is considered as one observation. I conduct a comparative analysis of four big data techniques including Functional Tikhonov method (FT), Functional Landweber-Fridman technique (FLF), Functional spectral-cut off (FSC), and Functional Partial Least Squares (FPLS). The convergence rate, asymptotic distribution, and a test-based strategy to select the lag number are provided. Simulations and real data show that FPLS method tends to outperform the other in terms of estimation accuracy while all the considered methods display almost the same predictive performance. The third chapter proposes to estimate the risk neutral density (RND) for options pricing with a functional linear model. The benefit of this approach is that it exploits directly the fundamental arbitrage-free equation and it is possible to avoid any additional density parametrization. The estimation problem leads to an inverse problem and the functional Landweber-Fridman (FLF) technique is used to overcome this issue.