Pricing European convertible bonds: geometric brownian motion vs. CGMY

Paper presented at Strathmore International Math Research Conference on July 23 - 27, 201

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

Computational Methods for Game Options

Game options are American-type options with the additional property that the seller of the option has the right the cancel the option at any time prior to the buyer exercise or the expiration date of the option. The cancelation by the seller can be achieved through a payment of an additional penalty to the exercise payoff or using a payoff process greater than or equal to the exercise value. The main contribution of this thesis is a numerical framework for computing the value of such options with finite maturity time as well as in the perpetual setting. This framework employs the theory of weak solutions of parabolic and elliptic variational inequalities. These solutions will be computed using finite element methods. The computational advantage of this framework is that it allows the user to go from one type of process to another by changing the stiffness matrix in the algorithm. Several types of Levy processes will be used to show the functionality of this method. The processes considered are of pure diffusion type (Black-Scholes model), the CGMY process as a pure jump model and a combination of the two for the case of jump diffusion. Computational results of the option prices as well as exercise, hold and cancelation regions are shown together with numerical estimates of the error convergence rates with respect to the L2 norm and the energy norm

Option Pricing Under the Variance Gamma Process

In this dissertation we price European and American vanilla and barrier options assuming that the underlying follows the variance gamma process. We solve numerically the problem implementing a finite difference algorithm and we present numerical experiments on the option pricing. This dissertation includes detailed algorithms as well as programming code in C++ to price European and American vanilla and barrier options under variance gamma.Variance Gamma Process; Option Pricing Under Variance Gamma; Numerical Solution of Option Prices Under Variance Gamma; Numerical Solution of Variance Gamma PIDE; Numerical Solutions of Variance Gamma Partial Differential Equation; Programming Code for Variance Gamma Option Pricing

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

Pricing Convertible Bonds with Credit Risk under Regime Switching and Numerical Solutions

This paper discusses the convertible bonds pricing problem with regime switching and credit risk in the convertible bond market. We derive a Black-Scholes-type partial differential equation of convertible bonds and propose a convertible bond pricing model with boundary conditions. We explore the impact of dilution effect and debt leverage on the value of the convertible bond and also give an adjustment method. Furthermore, we present two numerical solutions for the convertible bond pricing model and prove their consistency. Finally, the pricing results by comparing the finite difference method with the trinomial tree show that the strength of the effect of regime switching on the convertible bond depends on the generator matrix or the regime switching strength

Option Pricing Under the Variance Gamma Process

In this dissertation we price European and American vanilla and barrier options assuming that the underlying follows the variance gamma process. We solve numerically the problem implementing a finite difference algorithm and we present numerical experiments on the option pricing. This dissertation includes detailed algorithms as well as programming code in C++ to price European and American vanilla and barrier options under variance gamma

Option Pricing Under the Variance Gamma Process

In this dissertation we price European and American vanilla and barrier options assuming that the underlying follows the variance gamma process. We solve numerically the problem implementing a finite difference algorithm and we present numerical experiments on the option pricing. This dissertation includes detailed algorithms as well as programming code in C++ to price European and American vanilla and barrier options under variance gamma

Unified moment-based modelling of integrated stochastic processes

In this paper we present a new method for simulating integrals of stochastic processes. We focus on the nontrivial case of time integrals, conditional on the state variable levels at the endpoints of a time interval, through a moment-based probability distribution construction. We present different classes of models with important uses in finance, medicine, epidemiology, climatology, bioeconomics and physics. The method is generally applicable in well-posed moment problem settings. We study its convergence, point out its advantages through a series of numerical experiments and compare its performance against existing schemes