In this thesis, we consider the pricing problem of an American put option. We introduce a new market model for the evolution of the underlying asset price. Our model adds a new parameter to the well known Heston model. Hence we name our model the extended Heston model. To solve the American put pricing problem we adapt the idea developed by Fouque et al. (2000) to derive the asymptotic formula. We then connect the idea developed by Medvedev and Scaillet (2010) to provide an asymptotic solution for the leading order term P0. We do numerical analysis to gain insight into the accuracy and validity of our asymptotic approximation formula.

We consider tail empirical processes for long memory stochastic volatility models with heavy tails and leverage. We show a dichotomous behaviour for the tail empirical process with fixed levels, according to the interplay between the long memory parameter and the tail index; leverage does not play a role. On the other hand, the tail empirical process with random levels is not affected by either long memory or leverage. The tail empirical process with random levels is used to construct a family of estimators of the tail index, including the famous Hill estimator and harmonic moment estimators. The limiting behaviour of these estimators is not affected by either long memory or leverage. Furthermore, we consider estimators of risk measures such as Value-at-Risk and Expected Shortfall. In these cases, the limiting behaviour is affected by long memory, but it is not affected by leverage. The theoretical results are illustrated by simulation studies.

The Heston stochastic volatility model is one extension of the Black-Scholes model which describes the money markets more accurately so that more realistic prices for derivative products are obtained. From the stochastic differential equation of the underlying financial product a partial differential equation (p.d.e.) for the value function of an option can be derived. This p.d.e. can be solved with the finite difference method (f.d.m.). The stability and consistency of the method is examined. Furthermore a boundary condition is proposed to reduce the numerical error. Finally a non uniform structured grid is derived which is fairly optimal for the numerical result in the most interesting point. / Das stochastische Volatilitaetsmodell von Heston ist eines der Erweiterungen des Black-Scholes-Modells. Von der stochastischen Differentialgleichung fuer den unterliegenden Prozess kann eine partielle Differentialgleichung fuer die Wertfunktion einer Option abgeleitet werden. Es wird die Loesung mittels Finiter Differenzenmethode untersucht (Konsistenz, Stabilitaet). Weiterhin wird eine Randbedingung und ein spezielles nicht-uniformes Netz vorgeschlagen, was zu einer starken Reduzierung des numerischen Fehlers der Wertfunktion in einem ganz bestimmten Punkt fuehrt.

Derivatives have a large and significant role on the financial markets today and the popularity of options has increased. This has also increased the demand of finding a suitable option pricing model, since the ground-breaking model developed by Black & Scholes (1973) have poor pricing performance. Practitioners and academics have over the years developed different models with the assumption of non-constant volatility, without reaching any conclusions regarding which model is more suitable to use. This thesis examines four different models, the first model is the Practitioners Black & Scholes model proposed by Christoffersen & Jacobs (2004b). The second model is the Heston´s (1993) continuous time stochastic volatility model, a modification of the model is also included, which is called the Strike Vector Computation suggested by Kilin (2011). The last model is the Heston & Nandi (2000) Generalized Autoregressive Conditional Heteroscedasticity type discrete model. From a practical point of view the models are evaluated, with the goal of finding the model with the best pricing performance and the most practical usage. The model´s robustness is also tested to see how the models perform in out-of-sample during a high respectively low implied volatility market. All the models are effected in the robustness test, the out-sample ability is negatively affected by a high implied volatility market. The results show that both of the stochastic volatility models have superior performances in the in-sample and out-sample analysis. The Generalized Autoregressive Conditional Heteroscedasticity type discrete model shows surprisingly poor results both in the in-sample and out-sample analysis. The results indicate that option data should be used instead of historical return data to estimate the model’s parameters. This thesis also provides an insight on why overnight-index-swap (OIS) rates should be used instead of LIBOR rates as a proxy for the risk-free rate.

The Heston stochastic volatility model is one extension of the Black-Scholes model which describes the money markets more accurately so that more realistic prices for derivative products are obtained. From the stochastic differential equation of the underlying financial product a partial differential equation (p.d.e.) for the value function of an option can be derived. This p.d.e. can be solved with the finite difference method (f.d.m.). The stability and consistency of the method is examined. Furthermore a boundary condition is proposed to reduce the numerical error. Finally a non uniform structured grid is derived which is fairly optimal for the numerical result in the most interesting point. / Das stochastische Volatilitaetsmodell von Heston ist eines der Erweiterungen des Black-Scholes-Modells. Von der stochastischen Differentialgleichung fuer den unterliegenden Prozess kann eine partielle Differentialgleichung fuer die Wertfunktion einer Option abgeleitet werden. Es wird die Loesung mittels Finiter Differenzenmethode untersucht (Konsistenz, Stabilitaet). Weiterhin wird eine Randbedingung und ein spezielles nicht-uniformes Netz vorgeschlagen, was zu einer starken Reduzierung des numerischen Fehlers der Wertfunktion in einem ganz bestimmten Punkt fuehrt.

Spread options are derivative securities, which are written on the difference between the values of two underlying market variables. They are very important tools to hedge the correlation risk. American style spread options allow the holder to exercise the option at any time up to and including maturity. Although they are widely used to hedge and speculate in financial market, the valuation of the American spread option is very challenging. Because even under the classic assumptions that the underlying assets follow the log-normal distribution, the resulting spread doesn't have a distribution with a simple closed formula. In this dissertation, we investigate the American spread option pricing problem. Several approaches for the geometric Brownian motion model and the stochastic volatility model are developed. We also implement the above models and the numerical results are compared among different approaches.

This thesis focuses on comparing an online parameter estimator to an offline estimator, both based on the PaRIS-algorithm, when estimating parameter values for a stochastic volatility model. By modeling the stochastic volatility model as a hidden Markov model, estimators based on particle filters can be implemented in order to estimate the unknown parameters of the model. The results from this thesis implies that the proposed online estimator could be considered as a superior method to the offline counterpart. The results are however somewhat inconclusive, and further research regarding the subject is recommended. / Detta examensarbetefokuserar på att jämföra en online och offline parameter-skattare i stokastiskavolatilitets modeller. De två parameter-skattarna som jämförs är båda baseradepå PaRIS-algoritmen. Genom att modellera en stokastisk volatilitets-model somen dold Markov kedja, kunde partikelbaserade parameter-skattare användas föratt uppskatta de okända parametrarna i modellen. Resultaten presenterade idetta examensarbete tyder på att online-implementationen av PaRIS-algorimen kanses som det bästa alternativet, jämfört med offline-implementationen.Resultaten är dock inte helt övertygande, och ytterligare forskning inomområdet

京都大学 / 新制・課程博士 / 博士(経済学) / 甲第24164号 / 経博第658号 / 新制||経||302(附属図書館) / 京都大学大学院経済学研究科経済学専攻 / (主査)教授 西山 慎一, 准教授 高橋 修平, 准教授 安井 大真 / 学位規則第4条第1項該当 / Doctor of Economics / Kyoto University / DFAM

The calibration of model parameters is a crucial step in the process of valuation of complex derivatives. It consists of choosing the model parameters that correspond to the implied market data especially the call and put prices. We discuss in this thesis the calibration strategy for the Heston model, one of the most used stochastic volatility models for pricing complex derivatives. The main problem with this model is that the asset price does not have a known probability distribution function. Thus we use either Fourier expansions through its characteristic function or Monte Carlo simulations to have access to it. We hence discuss the approximation induced by these methods and elaborate a calibration strategy with a focus on the choice of the objective function and the choice of inputs for the calibration. We assess that the put option prices are a better input than the call prices for the optimization function. Then through a set of experiments on simulated put prices, we find that the sum of squared error performs better choice of the objective function for the differential evolution optimization. We also establish that the put option prices where the Black Scholes delta is equal to 10\%, 25\%, 50\% 75\% and 90\% gives enough in formations on the implied volatility surface for the calibration of the Heston model. We then implement this calibration strategy on real market data of Eurostoxx50 Index and observe the same distribution of errors as in the set of experiments. / Kalibreringen av modellparametrar är ett viktigt steg i värderingen av komplexa derivat. Den består av att välja modellparametrar som motsvarar de implicita marknadsdata, särskilt köp- och säljpriserna. I denna avhandling diskuterar vi kalibreringsstrategin för Hestonmodellen, en av de mest använda modellerna för stokastisk volatilitet för prissättning av komplexa derivat. Huvudproblemet med denna modell är att tillgångspriset inte har en känd sannolikhetsfördelningsfunktion. Därför använder vi antingen Fourier-expansioner genom dess karakteristiska funktion eller Monte Carlo-simuleringar för att få tillgång till den. Vi diskuterar därför den approximation som dessa genereras av dessa metoder och utarbetar en kalibreringsstrategi med fokus på valet av målfunktion och valet av indata för kalibreringen. Vi bedömer att säljoptionspriserna är en bättre input än samtalspriserna för differentialutvecklingsoptimeringsfunktionen. Genom flera experiment med simulerade säljpriser finner vi sedan att summan av kvadratfel ger bättre val av objektivfunktionen för differentialutvecklingsoptimering. Vi konstaterar också att säljoptionspriserna där Black Scholes deltat är lika med 10\%, 25\%, 50\%, 75\% och 90\% ger tillräcklig information om den implicita volatilitetsytan för kalibrering av Hestonmodellen. Vi tillämpar sedan denna kalibreringsstrategi på verkliga marknadsdata för Eurostoxx50-indexet och observerar samma felfördelning som i experimenten.

The Almost-Exact Scheme (AES), as proposed by Oosterlee and Grzelak, has been applied to the Heston stochastic volatility model to show improved error convergence for small time-steps, as opposed to the classical Euler-Maruyama (EM) scheme, in European option pricing. This idea has been extended to the double Heston stochastic volatility model, to show similar improved results for Bermudan options. In this thesis, we extend this idea even further and develop an Almost-Exact Scheme to the Gatheral double mean reverting (DMR) model, to show improved error convergence for American put options. We illustrate that, because of the complexity of the dynamics of our model, a direct application of the AES is not possible, and therefore derive a diffusion trick, so we can instead use a partial implementation of the AES. Our partial implementation has two variants. In the first variant, we implement the AES on the long-run mean process combined with the Milstein scheme on the variance process. In the second variant, the Milstein scheme is replaced by a second order refinement. We name these two schemes AEMS and AEMS-SOR respectively. We conduct extensive simulation studies to evaluate the proposed schemes. The results indicate improved error convergence of the proposed scheme for small time-steps when time-to-maturity is equal to half a year, but does not seem to differ much from the EM scheme for a shorter time-to-maturity.