High dimensional American options

Pricing single asset American options is a hard problem in mathematical finance. There are no closed form solutions available (apart from in the case of the perpetual option), so many approximations and numerical techniques have been developed. Pricing multi–asset (high dimensional) American options is still more difficult. We extend the method proposed theoretically by Glasserman and Yu (2004) by employing regression basis functions that are martingales under geometric Brownian motion. This results in more accurate Monte Carlo simulations, and computationally cheap lower and upper bounds to the American option price. We have implemented these models in QuantLib, the open–source derivatives pricing library. The code for many of the models discussed in this thesis can be downloaded from quantlib.org as part of a practical pricing and risk management library. We propose a new type of multi–asset option, the “Radial Barrier Option” for which we find analytic solutions. This is a barrier style option that pays out when a barrier, which is a function of the assets and their correlations, is hit. This is a useful benchmark test case for Monte Carlo simulations and may be of use in approximating multi–asset American options. We use Laplace transforms in this analysis which can be applied to give analytic results for the hitting times of Bessel processes. We investigate the asymptotic solution of the single asset Black–Scholes–Merton equation in the case of low volatility. This analysis explains the success of some American option approximations, and has the potential to be extended to basket options

Computational Methods for Pricing and Hedging Derivatives

In this thesis, we propose three new computational methods to price financial derivatives and construct hedging strategies under several underlying asset price dynamics. First, we introduce a method to price and hedge European basket options under two displaced processes with jumps, which are capable of accommodating negative skewness and excess kurtosis. The new approach uses Hermite polynomial expansion of a standard normal variable to match the first m moments of the standardised basket return. It consists of Black-and-Scholes type formulae and its improvement on the existing methods is twofold: we consider more realistic asset price dynamics and we allow more flexible specifications for the basket. Additionally, we propose two methods for pricing and hedging American options: one quasi-analytic and one numerical method. The first approach aims to increase the accuracy of almost any existing quasi-analytic method for American options under the geometric Brownian motion dynamics. The new method relies on an approximation of the optimal exercise price near the beginning of the contract combined with existing pricing approaches. An extensive scenario-based study shows that the new method improves the existing pricing and hedging formulae, for various maturity ranges, and, in particular, for long-maturity options where the existing methods perform worst. The second method combines Monte Carlo simulation with weighted least squares regressions to estimate the continuation value of American-style derivatives, in a similar framework to the one of the least squares Monte Carlo method proposed by Longstaff and Schwartz. We justify the introduction of the weighted least squares regressions by numerically and theoretically demonstrating that the regression estimators in the least squares Monte Carlo method are not the best linear unbiased estimators (BLUE) since there is evidence of heteroscedasticity in the regression errors. We find that the new method considerably reduces the upward bias in pricing that affects the least squares Monte Carlo algorithm. Finally, the superiority of our new two approaches for American options are also illustrated over real financial data by considering S&P 100 options and LEAPS®, traded from 15 February 2012 to 10 December 2014

The Effect of Non-Smooth Payoffs on the Penalty Approximation of American Options

This article combines various methods of analysis to draw a comprehensive picture of penalty approximations to the value, hedge ratio, and optimal exercise strategy of American options. While convergence of the penalised solution for sufficiently smooth obstacles is well established in the literature, sharp rates of convergence and particularly the effect of gradient discontinuities (i.e., the omni-present `kinks' in option payoffs) on this rate have not been fully analysed so far. This effect becomes important not least when using penalisation as a numerical technique. We use matched asymptotic expansions to characterise the boundary layers between exercise and hold regions, and to compute first order corrections for representative payoffs on a single asset following a diffusion or jump-diffusion model. Furthermore, we demonstrate how the viscosity theory framework in [Jakobsen, 2006] can be applied to this setting to derive upper and lower bounds on the value. In a small extension to [Bensoussan & Lions, 1982], we derive weak convergence rates also for option sensitivities for convex payoffs under jump-diffusion models. Finally, we outline applications of the results, including accuracy improvements by extrapolation.Comment: 34 Pages, 10 Figure

Essays on option pricing under alternative one-dimensional diffusions

Essays on Option Pricing under Alternative One-Dimensional Diffusions Given its analytical attractiveness, the process most commonly used in the financial and real options literature is the geometric Brownian motion. However, this assumption embodies some unrealistic implications for the dynamical behavior of the underlying asset price. To overcome this issue, alternative stochastic processes have been considered in the valuation of financial and real options. This thesis examines financial and real options using alternative one-dimensional diffusions, namely the constant elasticity of variance (CEV) and mean-reverting CEV diffusions. This thesis has two main purposes. First, it derives closed-form solutions for computing Greeks of European-style options under both the CEV and CIR (Cox, Ingersoll and Ross) models. Second, it analyzes the optimal entry and exit policy of a firm in the presence of output price uncertainty and costly reversibility of investment under a generalized class of one-dimensional diffusions and shows how the hysteretic band is affected; **** Resumo: Ensaios sobre a Avaliação de Opções sob Difusões Unidimensionais Alternativas Na avaliação de opções financeiras e reais, o processo mais utilizado na literatura ´e o movimento Browniano geométrico. Contudo, esta suposição ao incorpora algumas implicações irrealistas para o comportamento dinâmico do preço do activo subjacente. Para ultrapassar estas limitações, têm sido considerados processos estocásticos alternativos para a avaliação de opções financeiras e reais. Esta tese analisa opções, financeiras e reais, utilizando difusões unidimensionais alternativas, nomeadamente as difusões elasticidade constante da vari ˆancia (CEV - constant elasticity of variance) e CEV com reversão `a média. Esta tese tem dois objectivos principais. Primeiro, derivar soluções analíticas para calcular as letras gregas de opções de tipo Europeu para os modelos CEV e CIR (Cox, Ingersoll e Ross). Segundo, analisar a política óptima de entrada e de saída de uma empresa na presença de incerteza no prec¸o do output e de reversibilidade dos custos de investimento, para uma classe generalizada de difusões unidimensionais, e mostrar a influência sobre a banda de histerese económica

Optimal capital structure with endogenous bankruptcy : payouts, tax benefits asymetry and volatility risk

La thèse concerne la modélisation du risque de crédit en suivant l'approche de modèles structurels. La thèse se compose de trois articles dans lesquels nous nous appuyons sur la structure du capital d'une entreprise proposée par Leland et nous étudions différentes extensions de son article fondateur dans le but d'obtenir des résultats plus conformes aux normes historiques et évidence empiriques, en étudiant en détail tous les aspects mathématiques. On analyse la modélisation du risque de crédit en suivant l'approche de modèles structurels avec défaut endogène. Nous prolongeons le cadre classique de Leland dans trois directions principales pour obtenir des résultats plus conformes aux données empiriques. Nous introduisons des dividendes et une asymétrie dans la déduction fiscale : les résultats numériques montrent que ces modifications conduisent à des ratios de levier proches des normes observées, grâce à leur influence conjointe sur la structure optimale du capital. Enfin, nous introduisons un risque de volatilité. En suivant les suggestions de Leland, nous proposons un cadre dans lequel l'hypothèse de volatilité constante pour l'évolution de la valeur de l'entreprise est supprimé. En analysant les dérivés sujets au risque de faillite impliqués dans la structure du capital de l'entreprise, on obtient leur prix corrigé dans une classe assez large de modèles à volatilité stochastique en appliquant la théorie des perturbations singulières. En considérant la structure du capital optimal, la volatilité stochastique semble être un modèle efficace pour améliorer les résultats dans le sens de 'spreads' plus élevés et des ratios de levier, plus faibles de façon significative.The dissertation deals with modeling credit risk through a structural model approach. The thesis consists of three papers in which we build on the capital structure of a firm proposed by Leland and we study different extensions of his seminal paper with the purpose of obtaining results more in line with historical norms and empirical evidence, studying in details all mathematical aspects. The thesis analyses credit risk modelling following a structural model approach with endogenous default. We extend the classical Leland framework in three main directions with the aim at obtaining results more in line with empirical evidence. We introduce payouts and then also consider corporate tax rate asymmetry : numerical results show that these lead to predicted leverage ratios closer to historical norms, through their joint influence on optimal capital structure. Finally, we introduce volatility risk. Following Leland suggestions we consider a framework in which the assumption of constant volatility in the underlying firm's assets value stochastic evolution is removed. Analyzing defaultable claims involved in the capital structure of the firm we derive their corrected prices under a fairly large class of stochastic volatility framework seems to be a robus way to improve results in the direction of both higher spreads and lower leverage ratios in a quantitatively significant way

Valuation of dynamic fund protection under levy processes.

Lam, Ka Wai.Thesis (M.Phil.)--Chinese University of Hong Kong, 2008.Includes bibliographical references (leaves 51-55).Abstracts in English and Chinese.Chapter 1 --- Introduction --- p.1Chapter 2 --- Levy Processes --- p.6Chapter 2.1 --- Definition --- p.6Chapter 2.2 --- Levy-Khinchine formula --- p.7Chapter 2.3 --- Applications of Levy Processes in Finance --- p.10Chapter 2.4 --- Option pricing under Levy Processes --- p.12Chapter 2.4.1 --- Black-Scholes Formula with Characteristic Function --- p.12Chapter 2.4.2 --- Fast Fourier Transform --- p.14Chapter 2.4.3 --- Other Payoff Functions --- p.16Chapter 3 --- Dynamic Fund Protection --- p.19Chapter 3.1 --- Discrete Dynamic Fund Protection --- p.20Chapter 3.2 --- Link DFP to Discrete Lookback Options --- p.22Chapter 4 --- Spitzer´ةs Identity --- p.25Chapter 4.1 --- Applications of Spitzer's Identity --- p.25Chapter 4.2 --- Discrete Lookback Options --- p.29Chapter 5 --- Pricing Discrete DFP --- p.32Chapter 5.1 --- Girsanov´ةs Theorem --- p.32Chapter 5.2 --- Equivalent Martingale Measure in DFP --- p.34Chapter 5.3 --- Pricing DFP at any Time Points --- p.36Chapter 5.4 --- The Main Algorithm --- p.38Chapter 6 --- Numerical Results --- p.40Chapter 6.1 --- Simulation of Discrete DFP --- p.40Chapter 6.2 --- Numerical Implementation --- p.42Chapter 7 --- Conclusion --- p.50Bibliography --- p.5

Asymptotic techniques and stochastic volatility in option pricing problems

This thesis investigates the use of asymptotic techniques and stochastic volatility models in option pricing problems. For the Heston stochastic volatility model, a fast mean reverting asymptotic approach, similar to Fouque et al. (2000) is taken. The asymptotic solution derived extends on their most recent work, with the solution presented expanded out to four terms. The worthiness and robustness of the asymptotic solution is then tested by applying it to the theory of locally risk minimizing hedges. The asymptotic approach is then further developed by applying it to a real options framework, allowing for a better understanding of what the asymptotic solution actually reflects under this model, and in particular, how it affects the optimal investment threshold, a key component in real options theory. Asian options with general call type payoffs are then investigated and equivalency theorems derived linking them to Australian options under both a Black-Scholes model and a Heston stochastic volatility model. Examining Asian options from this ‘Australian’ perspective gives a new angle on how one can approach the pricing of Asian options under stochastic volatility. Advances are made in areas such as the PDE pricing equation, and Monte Carlo simulations. Finally, an asymptotic solution under a low volatility assumption in the Black-Scholes model for an Australian call option is derived. This extends the work of Dewynne and Shaw (2008), to cater for Australian options. It is argued that this can be used as an alterative to existing approximations under a low volatility regime, for both pricing general Australian call options and general Asian options through the equivalency theorems. Aside from the over arching theme of asymptotic techniques and stochastic volatility, this thesis looks at how each of the newly presented solutions and/or methods, can be of benefit to the pricing of their respective option types. In particular, focus will be placed on the usage, accuracy and computational efficiency of these techniques. In all cases, the new solutions provide a high level of accuracy compared to the true solution, and/or are much more computationally efficient than existing methodologies. The simplicity and advantages of these solutions make a valuable contribution to current option pricing techniques