The pricing of exotic derivatives continues to attract much attention from academics and practitioners alike. Despite the overwhelming interest, the task of finding a robust methodology that could derive closed-form solutions for exotic derivatives remains a difficult challenge. In addition, the level of sophistication is greatly enhanced when options are priced in a more realistic framework. This includes, but not limited to, utilising jump-diffusion models with mean-reversion, stochastic volatility, and/or stochastic jump intensity. More pertinently, these inclusions allow the resulting asset price process to capture the various empirical features, such as heavy tails and asymmetry, commonly observed in financial data. However, under such a framework, the density function governing the underlying asset price process is generally not available. This leads to a breakdown of the classical risk-neutral option valuation method via the discounted expectation of the final payoff. Furthermore, when an analytical expression for the option pricing formula becomes available, the solution is often complex and in semi closed-form. Hence, a substantial amount of computational time is required to obtain the value of the option, which may not satisfy the efficiency demanded in practice. Such drawbacks may be remedied by utilising numerical integration techniques to price options more efficiently in the Fourier domain instead, since the associated characteristic functions are more readily available. This thesis is concerned primarily with the efficient and accurate pricing of exotic derivatives under the aforementioned framework. We address the research opportunity by exploring the valuation of exotic options with numerical integration techniques once the associated characteristic functions are developed. In particular, we advocate the use of the novel Fourier-cosine (COS) expansions, and the more recent Shannon wavelet inverse Fourier technique (SWIFT). Once the option prices are obtained, the efficiency of the two techniques are benchmarked against the widely-acclaimed fast Fourier transform (FFT) method. More importantly, we perform extensive numerical experiments and error analyses to show that, under our proposed framework, not only is the COS and SWIFT methods more efficient, but are also highly accurate with exponential rate of error convergence. Finally, we conduct a set of sensitivity analyses to evaluate the models’ consistency and robustness under different market conditions

Recently, the maximum drawdown (MD) has been proposed as an alternative risk measure ideal for capturing downside risk. Furthermore, the maximum drawdown is associated with a Pain ratio and therefore may be a desirable insurance product. This thesis focuses on the pricing of the discrete maximum drawdown option under jump-diffusion by solving the associated partial integro differential equation (PIDE). To achieve this, a finite difference method is used to solve a set of one-dimensional PIDEs and appropriate observation conditions are applied at a set of observation dates. We handle arbitrary strikes on the option for both the absolute and relative maximum drawdown and then show that a similarity reduction is possible for the absolute maximum drawdown with zero strike, and for the relative maximum drawdown with arbitrary strike. We present numerical tests of validation and convergence for various grid types and interpolation methods. These results are in agreement with previous results for the maximum drawdown and indicate that scaled grids using a tri-linear interpolation achieves the best rate of convergence. A comparison with mutual fund fees is performed to illustrate a possible rationalization for why investors continue to purchase such funds, with high management fees.

Recently, the maximum drawdown (MD) has been proposed as an alternative risk measure ideal for capturing downside risk. Furthermore, the maximum drawdown is associated with a Pain ratio and therefore may be a desirable insurance product. This thesis focuses on the pricing of the discrete maximum drawdown option under jump-diffusion by solving the associated partial integro differential equation (PIDE). To achieve this, a finite difference method is used to solve a set of one-dimensional PIDEs and appropriate observation conditions are applied at a set of observation dates. We handle arbitrary strikes on the option for both the absolute and relative maximum drawdown and then show that a similarity reduction is possible for the absolute maximum drawdown with zero strike, and for the relative maximum drawdown with arbitrary strike. We present numerical tests of validation and convergence for various grid types and interpolation methods. These results are in agreement with previous results for the maximum drawdown and indicate that scaled grids using a tri-linear interpolation achieves the best rate of convergence. A comparison with mutual fund fees is performed to illustrate a possible rationalization for why investors continue to purchase such funds, with high management fees.

Philosophiae Doctor - PhD / In this thesis, we design and implement a class of numerical methods (based on integral transforms) to solve PDEs for pricing a variety of financial derivatives. Our approach is based on spectral discretization of the spatial (asset) derivatives and the use of inverse Laplace transforms to solve the resulting problem in time. The conventional spectral methods are further modified by using piecewise high order rational interpolants on the Chebyshev mesh within each sub-domain with the boundary domain placed at the strike price where the discontinuity is located. The resulting system is then solved by applying Laplace transform method through deformation of a contour integral. Firstly, we use this approach to price plain vanilla options and then extend it to price options described by a jump-diffusion model, barrier options and the Heston’s volatility model. To approximate the integral part in the jump-diffusion model, we use the Gauss-Legendre quadrature method. Finally, we carry out extensive numerical simulations to value these options and associated Greeks (the measures of sensitivity). The results presented in this thesis demonstrate the spectral accuracy and efficiency of our approach, which can therefore be considered as an alternative approach to price these class of options.

Machine learning models can produce balanced financial portfolios through a variety of methods. Genetic algorithms are one such method that can optimally combine different funds that may occupy a portfolio. This study introduces a genetic algorithm model that finds optimal combinations of funds for a portfolio through a new approach to fitness formula calculation. Each fund in a given population has a base fitness score consisting of the sum of several technical analysis indicators. Each indicator chosen measures a different performance aspect of a fund, allowing for a balanced fitness score. Additionally, each fund has multiple category variables that determine diversity when combined into a portfolio. The base fitness score for each portfolio is the sum of its funds' individual fitness scores. Portfolio fitness scores adjust based on the included funds' category variable diversity. Portfolios that consist of funds with largely similar categories receive lower adjusted fitness scores and do not cross over. This process encourages strong and diversified portfolios to reproduce. This model creates diverse portfolios that outperform market benchmarks and demonstrates future potential as a diversification-aware investment strategy.

An option is a financial instrument which derives its value from an underlying asset. There are a wide range of options traded today. Some are simple and plain, like the European options, while others are very difficult to evaluate. Both buyers and sellers continue to look for efficient algorithms and faster technology to price options for profit. In this thesis, I will first map the PSO parameters to the parameters in the option pricing problem. Then, I extend this to study pricing of complex chooser option. Further, I design a parallel algorithm that avails of the inherent concurrency in PSO while searching for a optimum solution. For implementation of my algorithm I used graphics processor unit (GPU). Analyzing the characteristics of PSO and option pricing, I propose a strategy to normalize some of the PSO parameters that helps in better understanding the sensitivity of various parameters on option pricing results.

An option is a financial instrument which derives its value from an underlying asset. There are a wide range of options traded today. Some are simple and plain, like the European options, while others are very difficult to evaluate. Both buyers and sellers continue to look for efficient algorithms and faster technology to price options for profit. In this thesis, I will first map the PSO parameters to the parameters in the option pricing problem. Then, I extend this to study pricing of complex chooser option. Further, I design a parallel algorithm that avails of the inherent concurrency in PSO while searching for a optimum solution. For implementation of my algorithm I used graphics processor unit (GPU). Analyzing the characteristics of PSO and option pricing, I propose a strategy to normalize some of the PSO parameters that helps in better understanding the sensitivity of various parameters on option pricing results.

The valuation of weather derivatives is complex since the underlying temperature process has no negotiable price. This thesis introduces a selection of models for the valuation of weather derivative contracts, governed by a stochastic underlying temperature process. We then present a new weather pricing model, which is used to determine the fair hedging price of a weather derivative under the assumptions of mean self-financing. This model is then extended to incorporate a compensation (or market price of risk) awarded to investors who hold undiversifiable risks. This results in the derivation of a non-linear two-dimensional PDE, for which the numerical evaluation cannot be performed using standard finite-difference techniques. The numerical techniques applied in this thesis are based on a broad range of lattice based schemes, including enhancements to finite-differences, quadrature methods and binomial trees. Furthermore simulations of temperature processes are undertaken that involves the development of Monte Carlo based methods.

Philosophiae Doctor - PhD / Conventional partial differential equations under the classical Black-Scholes approach have been extensively explored over the past few decades in solving option pricing problems. However, the underlying Efficient Market Hypothesis (EMH) of classical economic theory neglects the effects of memory in asset return series, though memory has long been observed in a number financial data. With advancements in computational methodologies, it has now become possible to model different real life physical phenomenons using complex approaches such as, fractional differential equations (FDEs). Fractional models are generalised models which based on literature have been found appropriate for explaining memory effects observed in a number of financial markets including the stock market. The use of fractional model has thus recently taken over the context of academic literatures and debates on financial modelling. / 2023-12-02

Cette these propose un cadre mathematique pour la modelisation du risque endogene dans les marches financiers. Le risque endogene designe le risque genere, et amplifie, au sein du systeme financier lui-meme, par les differents acteurs economiques et leurs interactions, par opposition au risque exogene, genere par des chocs exterieurs au systeme financier. Notre etude est motivee par l'observation des differentes crises financieres passees, qui montre le role central du risque endogene dans les marches financiers. Ainsi, les periodes de crises sont souvent associees a des phenomenes de liquidation/ventes 'eclair ('fire sales'), qui g'en'erent, de mani'ere endog'ene, une importante volatilit'e pour les actifs financiers et des pics inattendus de corr'elations entre les rendements de ces actifs, entraˆınant de fortes pertes pour les investisseurs. Alors que la structure de d'ependance entre les rendements d'actifs financiers est traditionnellement mod'elis'ee de mani'ere exog'ene, les faits d'ecrits pr'ec'edemment sugg'erent qu'une telle mod'elisation exog'ene ne peut rendre compte du risque endog'ene observ'e dans les march'es financiers. L'id'ee principale de cette th'ese est de distinguer entre deux origines pour la corr'elation entre actifs. La premi'ere est exog'ene et refl'ete une corr'elation fondamentale. La seconde est endog'ene et trouve son origine dans l'offre et la demande syst'ematiques g'en'er'ees par les grandes institutions financi'eres. Nous mod'elisons la dynamique en temps discret des prix d'actifs financiers d'un march'e multi-actifs par une chaˆıne de Markov dans lequel le rendement de chaque actif, 'a chaque p'eriode k, se d'ecompose en un terme al'eatoire - qui repr'esente les fondamentaux de l'actif et est independant du passe - et un terme d'offre/demande syst'ematique, genere de maniere endogene. Dans chaque chapitre, nous caracterisons mathematiquement cette offre endog'ene et son impact sur les prix d'actifs financiers. Nous exhibons des conditions sous lesquelles la chaıne de Markov converge faiblement, lorsque le pas de temps du modele discret tend vers zero, vers la solution d'une equation differentielle stochastique dont nous donnons le drift et la volatilite multi-dimensionnels. L''etude du processus de covariation quadratique de la limite diffusive nous permet de quantifier l'impact de l'offre et demande endogenes systematiques sur la structure de dependance entre actifs. Enfin, nous developpons des outils statistiques et econometriques visant a resoudre le probleme inverse d'identification et d'estimation des parametres de notre modele a partir de donnees de prix d'actifs financiers. Le Chapitre 1 'etudie le risque endogene genere par un fonds sujet a des ventes forcees en raison d'investisseurs qui sortent de leurs positions lorsque le fonds sous-performe et que sa valeur passe en-dessous d'un seuil. Nous modelisons l'offre et la demande provenant de telles ventes forcees en introduisant une 'fonction de liquidation' f qui mesure la vitesse 'a laquelle les investisseurs sortent de leurs positions dans le fonds. Nous supposons que l'offre en exces due au fonds sur chaque actif impacte le rendement de l'actif de maniere lineaire et nous explicitons les conditions pour que la dynamique de prix en temps discret soit une chaˆıne de Markov dans (R∗ )n, ou' n est le nombre d'actifs dans le marche. Nous exhibons des conditions sous lesquelles le modele discret converge faiblement vers une diffusion en temps continu et calculons les drifts et volatilites multi- 7 dimensionnels de la dynamique de prix en temps continu. L'etude de la covariation quadratique de la limite diffusive permet d'expliciter l'impact des ventes forcees dans le fonds sur la structure de d'ependance entre les actifs financiers. En particulier, nous montrons que la matrice de covariance realisee s''ecrit comme la somme d'une matrice de covariance fondamentale et une matrice de covariance en exces, qui depend des positions du fonds, de la liquidite des actifs et de la trajectoire passee des prix et qui est nulle lorsqu'il n'y a pas de ventes forcees. Nous prouvons alors que cet impact endogene augmente la volatilite du fonds en question, exactement dans les scenarios ou' le fonds subit des pertes. Nous calculons 'egalement son impact sur la volatilit'e d'autres fonds investissant dans les memes actifs et prouvons l'existence d'une relation d'orthogonalite entre les positions du fonds de reference et d'un autre fonds telle que, si cette relation d'orthogonalite est verifiee, des ventes forcees dans le fonds de reference n'affectent pas la volatilit'e de l'autre fonds. Le Chapitre 2 etend les resultats du Chapitre 1 au cas de plusieurs fonds et d'un impact quelconque (pas necessairement lineaire) de l'offre aggregee en exces provenant de ventes forcees dans ces fonds, sur les rendements de prix d'actifs. Nous exhibons des conditions sous lesquelles le mod'ele discret converge faiblement vers une diffusion en temps continu. La fonction de volatilite de la limite diffusive ne depend de la fonction de d'impact qu''a travers sa derivee premiere en zero, montrant qu'un modele de d'impact lineaire capture completement l'impact des effets de retroaction dus aux ventes forcees dans les differents fonds sur la structure de dependance entre actifs. Nous calculons la matrice de covariance realisee, en fonction des positions liquid'ees, en particulier dans un cas simple ou' les liquidations ont lieu 'a vitesse constante, dans un intervalle de temps fix'e et nous donnons des conditions assurant que cette relation peut etre inversee et les volumes de liquidations identifi'es. Nous construisons alors un estimateur du volume de liquidation dans chaque actif, dont nous prouvons la consistance, et pour lequel nous d'erivons un th'eor'eme central limite, qui nous permet de construire un test statistique testant si, pendant une periode donnee, des liquidations ont eu lieu. Nous illustrons notre procedure d'estimation avec deux exemples empiriques: le 'quant event' d'aouˆt 2007 et les liquidations suivant la faillite de Lehman Brothers en Automne 2008. Le Chapitre 3 etudie l'impact d'un investisseur institutionnel investissant une portion constante de sa richesse dans chaque actif (strategie fixed-mix). Pour un vecteur d'allocations donne, nous prouvons l'existence d'une unique strategie fixed-mix autofinancante. A chaque p'eriode, le prix des n actifs et la valeur du fonds fixed-mix sont obtenus comme la solution d'un probl'eme de point fixe. Nous montrons que, sous certaines conditions que nous explicitons, le modele discret converge vers une limite diffusive, pour laquelle nous calculons la covariance et la correlation realisee 'a l'ordre un en liquidit'e. Nos r'esultats montrent que la presence d'investisseurs institutionnels peut modifier les correlations de facon significative. Nous calculons les vecteurs propres et valeurs propres de la matrice de correlation realisee (a' l'ordre un en liquidite). L''etude des drifts de la limite continue nous permet de calculer les rendements esperes des actifs et montre qu'en raison de la presence de l'investisseur institutionnel, les rendements esp'er'es des actifs avec grand (resp. faible) drift fondamental, compares au rendement fondamental du fonds, diminuent (resp. augmentent). Nous calculons, dans un exemple simple, la strategie efficiente pour un critere moyenne-variance et montrons qu'elle est differente de la strategie optimale fondamentale (sans le fonds). L''etude de la frontiere optimale dans cet exemple montre qu'un investisseur prenant en compte l'impact de l'investisseur institutionnel peut ameliorer son rendement pour un niveau de risque donne.