In this thesis, we propose two continuous time stochastic volatility models with long memory that generalize two existing models. More importantly, we provide analytical formulae that allow us to study option prices numerically, rather than by means of simulation. We are not aware about analytical results in continuous time long memory case. In both models, we allow for the non-zero correlation between the stochastic volatility and stock price processes. We numerically study the effects of long memory on the option prices. We show that the fractional integration parameter has the opposite effect to that of volatility of volatility parameter in short memory models. We also find that long memory models have the potential to accommodate the short term options and the decay of volatility skew better than the corresponding short memory stochastic volatility models.

The focus of this dissertation is modeling volatility in option pricing by the Black-Scholes formula. A major drawback of the formula is that the returns from assets are assumed to have constant volatility over time. The empirical evidence is overwhelmingly against it. In this dissertation, we allow random volatility for estimating call option prices by Black-Scholes formula and by Monte Carlo simulation. The Black-Scholes formula follows from an assumption that assets evolve according to a Geometric Brownian Motion with constant volatility. This dissertation allows time-varying random volatility in the Geometric Brownian Motion to outline a proof of the formula, thus addressing this drawback. To estimate option prices with the Black-Scholes, the dissertation considers its expectation with respect to two potential probability models of random volatility. Unfortunately, a closed form expression of the expectation of the formula for computing the option prices is intractable. Then the dissertation settles with using an approximation which to its credit incorporates in it the kurtosis of the probability model of random volatility. To our knowledge, option pricing methods in literature do not incorporate kurtosis information. The option pricing with random volatility is pursued for two stochastic volatility models. One model is a member of generalized auto regressive conditional heteroscedasticity (GARCH). The second is a member of Stochastic Volatility models. For each model, estimation of their parameters is outlined. Two real financial series data are then used to illustrate estimation of the option prices, and compared them with those from the Black-Scholes formula with constant volatility. Motivated by a Monte Carlo procedure in the literature for option pricing when the volatility follows a GARCH model, this dissertation lays a foundation for future research to simulate option prices when the random volatility is assumed to follow a Stochastic Volatility model instead of GARCH. / Statistics

This article investigates several variance reduction techniques in Monte Carlo simulation applied in option pricing. It first shows how Monte Carlo simulation could be leveraged in the field of option pricing by demonstrating the quality of Monte Carlo methods and properties of stock options. Then the articles simulate stock price trajectories to infer the optimal option price by averaging the payoff at maturity. The article shows in depth the effect of control variates and antithetic variates, and importance sampling in reducing variance. The last part of the article shows how the same variance reduction techniques could be used in more exotic options such as Asian and Bermuda options. In these cases, their closed-form expressions are more difficult to derive compared to the European options, and thus simulation is widely practiced in the industry.

The high pace at which many of the world's energy markets have gradually been opened to competition have generated a significant amount of new financial activity. Both academicians and practitioners alike recently started to develop the tools of energy derivatives pricing/hedging as a quantitative topic of its own. The energy contract structures as well as their underlying asset properties set the energy risk management industry apart from its more standard equity and fixed income counterparts. This thesis naturaly contributes to these broad market developments in participating to the advances of the mathematical tools aiming at a better theory of energy contingent claim pricing/hedging. We propose many realistic two-factor and three-factor models for spot and forward price processes that generalize some well known and standard modeling assumptions. We develop the associated pricing methodologies and propose stable calibration algorithms that motivate the application of the relevant modeling schemes.

The high pace at which many of the world's energy markets have gradually been opened to competition have generated a significant amount of new financial activity. Both academicians and practitioners alike recently started to develop the tools of energy derivatives pricing/hedging as a quantitative topic of its own. The energy contract structures as well as their underlying asset properties set the energy risk management industry apart from its more standard equity and fixed income counterparts. This thesis naturaly contributes to these broad market developments in participating to the advances of the mathematical tools aiming at a better theory of energy contingent claim pricing/hedging. We propose many realistic two-factor and three-factor models for spot and forward price processes that generalize some well known and standard modeling assumptions. We develop the associated pricing methodologies and propose stable calibration algorithms that motivate the application of the relevant modeling schemes.

Suppose <em>X</em> is a diffusion process, possibly multivariate, and suppose that there are various segments of the components of <em>X</em> that are missing. This happens, for example, if <em>X</em> is the price of various assets and these prices are only observed at specific discrete trading times. Imputation (or conditional simulation) of the missing pieces of the sample paths of <em>X</em> is discussed in several settings. When <em>X</em> is a Brownian motion the conditioned process is a tied down Brownian motion or a Brownian bridge process. In the special case of Gaussian stochastic processes the problem is simplified since the conditional finite dimensional distributions of the process are multivariate Normal. For more general diffusion processes, including those with jump components, an acceptance-rejection simulation algorithm is introduced which enables one to sample from the exact conditional distribution without appealing to approximate time step methods such as the popular Euler or Milstein schemes. The method is referred to as <em>pathwise imputation</em>. Its practical implementation relies only on the basic elements of simulation while its theoretical justification depends on the pathwise properties of stochastic processes and in particular Girsanov's theorem. The method allows for the complete characterization of the bridge paths of complicated diffusions using only Brownian bridge interpolation. The imputation methods discussed are applied to estimation, variance reduction and exotic option pricing.

Suppose <em>X</em> is a diffusion process, possibly multivariate, and suppose that there are various segments of the components of <em>X</em> that are missing. This happens, for example, if <em>X</em> is the price of various assets and these prices are only observed at specific discrete trading times. Imputation (or conditional simulation) of the missing pieces of the sample paths of <em>X</em> is discussed in several settings. When <em>X</em> is a Brownian motion the conditioned process is a tied down Brownian motion or a Brownian bridge process. In the special case of Gaussian stochastic processes the problem is simplified since the conditional finite dimensional distributions of the process are multivariate Normal. For more general diffusion processes, including those with jump components, an acceptance-rejection simulation algorithm is introduced which enables one to sample from the exact conditional distribution without appealing to approximate time step methods such as the popular Euler or Milstein schemes. The method is referred to as <em>pathwise imputation</em>. Its practical implementation relies only on the basic elements of simulation while its theoretical justification depends on the pathwise properties of stochastic processes and in particular Girsanov's theorem. The method allows for the complete characterization of the bridge paths of complicated diffusions using only Brownian bridge interpolation. The imputation methods discussed are applied to estimation, variance reduction and exotic option pricing.

The Black-Scholes option pricing model has several well recognized deficiencies, one of which is its assumption of a constant and time-homogeneous stock return volatility term. The implied volatility smile has been studied by subsequent researchers and various models have been developed in an attempt to reproduce this phenomenon from within the models. However, few of these models yield closed-form pricing formulas that are easy to implement in practice. In this thesis, we study a Mixture Lognormal model (MLN) for European option pricing, which assumes that future stock prices are conditionally described by a mixture of lognormal distributions. The ability of mixture models in generating volatility smiles as well as delivering pricing improvement over the traditional Black-Scholes framework have been much researched under multi-component mixtures for many derivatives and high-volatility individual stock options. In this thesis, we investigate the performance of the model under the simplest two-component mixture in a market characterized by relative tranquillity and over a relatively stable period for broad-based index options. A careful interpretation is given to the model and the results obtained in the thesis. This di erentiates our study from many previous studies on this subject. Throughout the thesis, we establish the unique advantage of the MLN model, which is having closed-form option pricing formulas equal to the weighted mixture of Black-Scholes option prices. We also propose a robust calibration methodology to fit the model to market data. Extreme market states, in particular the so-called crash-o-phobia effect, are shown to be well captured by the calibrated model, albeit small pricing improvements are made over a relatively stable period of index option market. As a major contribution of this thesis, we extend the MLN model to price exotic options including binary, Asian, and barrier options. Closed-form formulas are derived for binary and continuously monitored barrier options and simulation-based pricing techniques are proposed for Asian and discretely monitored barrier options. Lastly, comparative results are analysed for various strike-maturity combinations, which provides insights into the formulation of hedging and risk management strategies.

In this paper we present a finite volume method for a two-dimensional Black-Scholes equation with stochastic volatility governing European option pricing. In this work, we first formulate the Black-Scholes equation with a tensor (or matrix) diffusion coefficient into a conversative form. We then present a finite volume method for the resulting equation, based on a fitting technique proposed for a one-dimensional Black-Scholes equation. We show that the method is monotone by proving that the system matrix of the discretized equation is an M-matrix. Numerical experiments, performed to demonstrate the usefulness of the method, will be presented.

The first option has been publicly traded for more than 30 years. With the progress of time, despite the European option is still the exchange-traded option. But evolved through the years, the European option has not meet people's needs, so exotic option was born. Similarly, the pricing model, from the traditional closed-form solution (under the Black-Scholes assumption), now commonly used binomial trees, finite difference, or by using the Monte Carlo simulation. The main impact of the following factors: the first, with the complexity of the option contract - from single asset to multi-assets, from the plain vanilla option to the path-dependent option, it is more difficult to find the closed-form solution of the option. Second, with the development of personal computers, making numerical computing is no longer a difficult task. It is precisely these two front reason, there will be the birth of this article. Himalaya option is also an exotic options. With the multi-assets and path dependent features, we want to find a closed-form solution is very difficult. Under multi-assets situation, the binomial tree and finite difference will be time-consuming calculation. Therefore, this paper is using Monte Carlo simulation of reasons. In this paper, we use Monte Carlo simulation to pricing Himalaya option, which includes several variance reduction techniques used to reduce sample variance. Finally, when pricing completed, we try to do a simple study to option hedging.