Pricing Discretely Monitored Asian Options by Maturity Randomization

We present methodologies to price discretely monitored Asian options when the underlying evolves according to a generic Levy process. For geometric Asian options we provide closed-form solutions in terms of the Fourier transform and we study in particular these formulas in the Levy-stable case. For arithmetic Asian options we solve the valuation problem by recursive integration and derive a recursive theoretical formula for the moments to check the accuracy of the results. We compare the implementation of our method to Monte Carlo simulation implemented with control variates and using different parametric Levy processes. We also discuss model-risk issues

Pricing discretely monitored Asian options under Levy processes

We present methodologies to price discretely monitored Asian options when the underlying evolves according to a generic Lévy process. For geometric Asian options we provide closed-form solutions in terms of the Fourier transform and we study in particular these formulas in the Lévy-stable case. For arithmetic Asian options we solve the valuation problem by recursive integration and derive a recursive theoretical formula for the moments to check the accuracy of the results. We compare the implementation of our method to Monte Carlo simulation implemented with control variates and using different parametric Lévy processes. We also discuss model risk issues

American options under stochastic volatility: control variates, maturity randomization & multiscale asymptotics

American options are actively traded worldwide on exchanges, thus making their accurate and efficient pricing an important problem. As most financial markets exhibit randomly varying volatility, in this paper we introduce an approximation of American option price under stochastic volatility models. We achieve this by using the maturity randomization method known as Canadization. The volatility process is characterized by fast and slow scale fluctuating factors. In particular, we study the case of an American put with a single underlying asset and use perturbative expansion techniques to approximate its price as well as the optimal exercise boundary up to the first order. We then use the approximate optimal exercise boundary formula to price American put via Monte Carlo. We also develop efficient control variates for our simulation method using martingales resulting from the approximate price formula. A numerical study is conducted to demonstrate that the proposed method performs better than the least squares regression method popular in the financial industry, in typical settings where values of the scaling parameters are small. Further, it is empirically observed that in the regimes where scaling parameter value is equal to unity, fast and slow scale approximations are equally accurate

Efficient valuation of exotic derivatives with path-dependence and early exercise features

The main objective of this thesis is to provide effective means for the valuation of popular financial derivative contracts with path-dependence and/or early-exercisable provisions. Starting from the risk-neutral valuation formula, the approach we propose is to sequentially compute convolutions of the value function of the contract at a monitoring date with the transition density between two dates, to provide the value function at the previous monitoring date, until the present date. A rigorous computational algorithm for the convolutions is then developed based on transformations to the Fourier domain. In the first part of the thesis, we deal with arithmetic Asian options, which, due to the growing popularity they enjoy in the financial marketplace, have been researched signicantly over the last two decades. Although few remarkable approaches have been proposed so far, these are restricted to the market assumptions imposed by the standard Black-Scholes-Merton paradigm. Others, although in theory applicable to Lévy models, are shown to suffer a non-monotone convergence when implemented numerically. To solve the Asian option pricing problem, we initially propose a flexible framework for independently distributed log-returns on the underlying asset. This allows us to generalize firstly in calculating the price sensitivities. Secondly, we consider an extension to non-Lévy stochastic volatility models. We highlight the benefits of the new scheme and, where relevant, benchmark its performance against an analytical approximation, control variate Monte Carlo strategies and existing forward convolution algorithms for the recovery of the density of the underlying average price. In the second part of the thesis, we carry out an analysis on the rapidly growing market of convertible bonds (CBs). Despite the vast amount of research which has been undertaken yet. This is due to the need for proper modelling of the CBs composite payout structure and the multi factor modelling arising in the CB valuation. Given the dimensional capacity of the convolution algorithm, we are now able to introduce a new jump diffusion structural approach in the CB literature, towards more realistic modelling of the default risk, and further include correlated stochastic interest rates. This aims at fixing dimensionality and convergence limitations which previously have been restricting the range of applicability of popular grid- based, lattice and Monte Carlo methods. The convolution scheme further permits flexible handling of real-world CB specications; this allows us to properly model the call policy and investigate its impact on the computed CB prices. We illustrate the performance of the numerical scheme and highlight the effects originated by the inclusion of jumps

Numerical methods for Lévy processes

We survey the use and limitations of some numerical methods for pricing derivative contracts in multidimensional geometric Lévy model

On Kolmogorov equations for anisotropic multivariate Lévy processes

For d-dimensional exponential Lévy models, variational formulations of the Kolmogorov equations arising in asset pricing are derived. Well-posedness of these equations is verified. Particular attention is paid to pure jump, d-variate Lévy processes built from parametric, copula dependence models in their jump structure. The domains of the associated Dirichlet forms are shown to be certain anisotropic Sobolev spaces. Singularity-free representations of the Dirichlet forms are given which remain bounded for piecewise polynomial, continuous functions of finite element type. We prove that the variational problem can be localized to a bounded domain with explicit localization error bounds. Furthermore, we collect several analytical tools for further numerical analysi

Stochastic time-changed Lévy processes with their implementation

Includes bibliographical references.We focus on the implementation details for Lévy processes and their extension to stochastic volatility models for pricing European vanilla options and exotic options. We calibrated five models to European options on the S&P500 and used the calibrated models to price a cliquet option using Monte Carlo simulation. We provide the algorithms required to value the options when using Lévy processes. We found that these models were able to closely reproduce the market option prices for many strikes and maturities. We also found that the models we studied produced different prices for the cliquet option even though all the models produced the same prices for vanilla options. This highlighted a feature of model uncertainty when valuing a cliquet option. Further research is required to develop tools to understand and manage this model uncertainty. We make a recommendation on how to proceed with this research by studying the cliquet option’s sensitivity to the model parameters

Stochastic models in finance

Stochastic models for pricing financial securities are developed. First, we consider the Black Scholes model, which is a classic example of a complete market model and finally focus on Lévy driven models. Jumps may render the market incomplete and are induced in a model by inclusion of a Poisson process. Lévy driven models are more realistic in modelling of asset price dynamics than the Black Scholes model. Martingales are central in pricing, especially of derivatives and we give them the desired attention in the context of pricing. There are an increasing number of important pricing models where analytical solutions are not available hence computational methods come in handy, see Broadie and Glasserman (1997). It is also important to note that computational methods are also applicable to models with analytical solutions. We computationally value selected stochastic financial models using C++. Computational methods are also used to value or price complex financial instruments such as path dependent derivatives. This pricing procedure is applied in the computational valuation of a stochastic (revenue based) loan contract. Derivatives with simple pay of functions and models with analytical solutions are considered for illustrative purposes. The Black-Scholes P.D.E is complex to solve analytically and finite difference methods are widely used. Explicit finite difference scheme is considered in this thesis for computational valuation of derivatives that are modelled by the Black-Scholes P.D.E. Stochastic modelling of asset prices is important for the valuation of derivatives: Gaussian, exponential and gamma variates are simulated for the valuation purposes

The History of the Quantitative Methods in Finance Conference Series. 1992-2007

This report charts the history of the Quantitative Methods in Finance (QMF) conference from its beginning in 1993 to the 15th conference in 2007. It lists alphabetically the 1037 speakers who presented at all 15 conferences and the titles of their papers.