A volatility decomposition control variate technique for Monte Carlo simulations of Heath Jarrow Morton models

The aim of this work is to develop a simulation approach to the yield curve evolution in the Heath, Jarrow and Morton [Econometrica 60 (1) (1992) 77] framework. The stochastic quantities considered as affecting the forward rate volatility function are the spot rate and the forward rate. A decomposition of the volatility function into a Hull and White [Rev. Financial Stud. 3 (1990) 573] volatility and a remainder allows us to develop an efficient Control Variate Method that makes use of the closed form solution of the Hull and White call option. This technique considerably speeds up the simulation algorithm to approximate call option values with Monte Carlo simulation. © 2003 Elsevier B.V. All rights reserved

Pricing swaptions on amortising swaps

In this dissertation, two efficient approaches for pricing European options on amortising swaps are explored. The first approach is to decompose the pricing of a European amortising swaption into a series of discount bond options, with an assumption that the interest rate follows a one-factor affine model. The second approach is using a one-dimensional numerical integral technique to approximate the price of European amortising swaption, with an assumption that the interest rate follows an additive two-factor affine model. The efficacy of the two methods was tested by making a comparison with the prices generated using Monte Carlo methods. Two methods were used to accelerate the convergence rate of the Monte Carlo model, a variance reduction method, namely the control variates technique and a method of using deterministic low-discrepancy sequences (also called quasi-Monte Carlo methods)

Monte Carlo Simulation with Asymptotic Method (Published in "Journal of Japan Statistical Society", Vol.35-2, 171-203, 2005. )

We shall propose a new computational scheme with the asymptotic method to achieve variance reduction of Monte Carlo simulation for numerical analysis especially in finance. We not only provide general scheme of our method, but also show its effectiveness through numerical examples such as computing optimal portfolio and pricing an average option. Finally, we show mathematical validity of our method.

Bias-Free Joint Simulation of Multi-Factor Short Rate Models and Discount Factor

This dissertation explores the use of single- and multi-factor Gaussian short rate models for the valuation of interest rate sensitive European options. Specifically, the focus is on deriving the joint distribution of the short rate and the discount factor, so that an exact and unbiased simulation scheme can be derived for risk-neutral valuation. We see that the derivation of the joint distribution remains tractable when working with the class of Gaussian short rate models. The dissertation compares three joint and exact simulation schemes for the short rate and the discount factor in the single-factor case; and two schemes in the multifactor case. We price European floor options and European swaptions using a twofactor Gaussian short rate model and explore the use of variance reduction techniques. We compare the exact and unbiased schemes to other solutions available in the literature: simulating the short rate under the forward measure and approximating the discount factor using quadrature

The LIBOR Market Model

Student Number : 0003819T - MSc dissertation - School of Computational and Applied Mathematics - Faculty of ScienceThe over-the-counter (OTC) interest rate derivative market is large and rapidly developing. In March 2005, the Bank for International Settlements published its “Triennial Central Bank Survey” which examined the derivative market activity in 2004 (http://www.bis.org/publ/rpfx05.htm). The reported total gross market value of OTC derivatives stood at $6.4 trillion at the end of June 2004. The gross market value of interest rate derivatives comprised a massive 71.7% of the total, followed by foreign exchange derivatives (17.5%) and equity derivatives (5%). Further, the daily turnover in interest rate option trading increased from 5.9% (of the total daily turnover in the interest rate derivative market) in April 2001 to 16.7% in April 2004. This growth and success of the interest rate derivative market has resulted in the introduction of exotic interest rate products and the ongoing search for accurate and efficient pricing and hedging techniques for them. Interest rate caps and (European) swaptions form the largest and the most liquid part of the interest rate option market. These vanilla instruments depend only on the level of the yield curve. The market standard for pricing them is the Black (1976) model. Caps and swaptions are typically used by traders of interest rate derivatives to gamma and vega hedge complex products. Thus an important feature of an interest rate model is not only its ability to recover an arbitrary input yield curve, but also an ability to calibrate to the implied at-the-money cap and swaption volatilities. The LIBOR market model developed out of the market’s need to price and hedge exotic interest rate derivatives consistently with the Black (1976) caplet formula. The focus of this dissertation is this popular class of interest rate models. The fundamental traded assets in an interest rate model are zero-coupon bonds. The evolution of their values, assuming that the underlying movements are continuous, is driven by a finite number of Brownian motions. The traditional approach to modelling the term structure of interest rates is to postulate the evolution of the instantaneous short or forward rates. Contrastingly, in the LIBOR market model, the discrete forward rates are modelled directly. The additional assumption imposed is that the volatility function of the discrete forward rates is a deterministic function of time. In Chapter 2 we provide a brief overview of the history of interest rate modelling which led to the LIBOR market model. The general theory of derivative pricing is presented, followed by a exposition and derivation of the stochastic differential equations governing the forward LIBOR rates. The LIBOR market model framework only truly becomes a model once the volatility functions of the discrete forward rates are specified. The information provided by the yield curve, the cap and the swaption markets does not imply a unique form for these functions. In Chapter 3, we examine various specifications of the LIBOR market model. Once the model is specified, it is calibrated to the above mentioned market data. An advantage of the LIBOR market model is the ability to calibrate to a large set of liquid market instruments while generating a realistic evolution of the forward rate volatility structure (Piterbarg 2004). We examine some of the practical problems that arise when calibrating the market model and present an example calibration in the UK market. The necessity, in general, of pricing derivatives in the LIBOR market model using Monte Carlo simulation is explained in Chapter 4. Both the Monte Carlo and quasi-Monte Carlo simulation approaches are presented, together with an examination of the various discretizations of the forward rate stochastic differential equations. The chapter concludes with some numerical results comparing the performance of Monte Carlo estimates with quasi-Monte Carlo estimates and the performance of the discretization approaches. In the final chapter we discuss numerical techniques based on Monte Carlo simulation for pricing American derivatives. We present the primal and dual American option pricing problem formulations, followed by an overview of the two main numerical techniques for pricing American options using Monte Carlo simulation. Callable LIBOR exotics is a name given to a class of interest rate derivatives that have early exercise provisions (Bermudan style) to exercise into various underlying interest rate products. A popular approach for valuing these instruments in the LIBOR market model is to estimate the continuation value of the option using parametric regression and, subsequently, to estimate the option value using backward induction. This approach relies on the choice of relevant, i.e. problem specific predictor variables and also on the functional form of the regression function. It is certainly not a “black-box” type of approach. Instead of choosing the relevant predictor variables, we present the sliced inverse regression technique. Sliced inverse regression is a statistical technique that aims to capture the main features of the data with a few low-dimensional projections. In particular, we use the sliced inverse regression technique to identify the low-dimensional projections of the forward LIBOR rates and then we estimate the continuation value of the option using nonparametric regression techniques. The results for a Bermudan swaption in a two-factor LIBOR market model are compared to those in Andersen (2000)

Bias-Free Joint Simulation of Multi-Factor Short Rate Models and Discount Factor

This dissertation explores the use of single- and multi-factor Gaussian short rate models for the valuation of interest rate sensitive European options. Specifically, the focus is on deriving the joint distribution of the short rate and the discount factor, so that an exact and unbiased simulation scheme can be derived for risk-neutral valuation. We see that the derivation of the joint distribution remains tractable when working with the class of Gaussian short rate models. The dissertation compares three joint and exact simulation schemes for the short rate and the discount factor in the single-factor case; and two schemes in the multifactor case. We price European floor options and European swaptions using a twofactor Gaussian short rate model and explore the use of variance reduction techniques. We compare the exact and unbiased schemes to other solutions available in the literature: simulating the short rate under the forward measure and approximating the discount factor using quadrature

A Hybrid Asymptotic Expansion Scheme: an Application to Long-term Currency Options ( Revised in April 2008, January 2009 and April 2010; forthcoming in "International Journal of Theoretical and Applied Finance". )

This paper develops a general approximation scheme, henceforth called a hybrid asymptotic expansion scheme for the valuation of multi-factor European path-independent derivatives. Specifically, we apply it to pricing long-term currency options under a market model of interest rates and a general diffusion stochastic volatility model with jumps of spot exchange rates. Our scheme is very effective for a type of models in which there exist correlations among all the factors whose dynamics are not necessarily affine nor even Markovian so long as the randomness is generated by Brownian motions. It can also handle models that include jump components under an assumption of their independence of the other random variables when the characteristic functions for the jump parts can be analytically obtained. Moreover, the hybrid scheme develops Fourier transform method with an asymptotic expansion to utilize closed-form characteristic functions obtainable in parts of a model. Our scheme also introduces a characteristic-function-based Monte Carlo simulation method with the asymptotic expansion as a control variable in order to make full use of analytical approximations by the asymptotic expansion and of closed-form characteristic functions. Finally, a series of numerical examples shows the validity of our scheme.

Essays on option pricing, with applications on interest rates, equities and credit derivatives

JEL Classification: G13This thesis is devoted to option pricing, with applications on interest rates, equities and credit derivatives, and is comprised of three separate and self-contained essays: A Pricing Swaptions under Multifactor Gaussian HJM Models Several approximations have been proposed in the literature for the pricing of European-style swaptions under multifactor term structure models. However, none of them provides an estimate for the inherent approximation error. Until now, only the Edgeworth expansion technique of Collin-Dufresne and Goldstein (2002) is able to characterize the order of the approximation error. Under a multifactor Heath, Jarrow, and Morton (1992) Gaussian framework, this paper proposes a new approximation for European-style swaptions, which is able to set bounds on the magnitude of the approximation error and is based on the conditioning approach initiated by Curran (1994) and Rogers and Shi (1995). All the proposed pricing bounds will arise as a simple by-product of the Nielsen and Sandmann (2002) setup, and will be shown to provide a better accuracy-efficiency trade-off than all the approximations already proposed in the literature. B Pricing of European-style Barrier Options under Stochastic Interest Rates This paper offers an extremely fast and accurate novel methodology for the pricing of (long-term) European-style single barrier options on underlying spot prices driven by a geometric Brownian motion and under the stochastic interest rates framework of Vasiček (1977). The proposed valuation methodology extends the stopping time approach of Kuan and Webber (2003) to a more general setting, and expresses the price of a European-style barrier option in terms of the first passage time density of the underlying asset price to the barrier level. Using several model parameter constellations and option maturities, our numerical results show that the proposed pricing approach is much more accurate and faster than the two-dimensional extended Fortet method of Bernard et al. (2008). C Pricing Credit and Equity Default Swaps under the Jump to Default Extended CEV Model This paper offers a novel methodology for the pricing of credit and equity default swaps under the jump to default extended constant elasticity of variance (JDCEV) model of Carr and Linetsky (2006). The proposed method extends the stopping time approach of Kuan and Webber (2003), and expresses the value of the building blocks of both contracts in terms of the first passage time density of the underlying asset price to the contract triggering level. The numerical results show that the proposed pricing methodology is extremely accurate and much faster than the Laplace transform approach of Mendoza-Arriaga and Linetsky (2011).Esta tese dedica-se ao tema da avaliação de opções, com aplicações a taxas de juro, ações e derivados de crédito, e é composta por três artigos distintos: A Pricing Swaptions under Multifactor Gaussian HJM Models Várias aproximações foram já propostas na literatura para a avaliação de swaptions de estilo Europeu, no âmbito de modelos de taxa de juro multi-fator. Contudo, nenhuma delas fornece uma estimativa para o erro de aproximação subjacente. Até agora, apenas a Edgeworth expansion technique de Collin-Dufresne e Goldstein (2002) é capaz de caracterizar a ordem do erro de aproximação. No âmbito de um modelo Heath, Jarrow e Morton (1992) Gaussiano multi-fator, este artigo propõe uma nova aproximação para swaptions de estilo Europeu, que é capaz de estabelecer limites para a magnitude do erro de aproximação e é baseada na conditioning approach iniciada por Curran (1994) e Rogers e Shi (1995). Todos os limites de preço propostos surgirão como um simples sub-produto da estrutura de Nielsen e Sandmann (2002), e será demonstrado que estes proporcionam um melhor equilíbrio entre precisão e eficiência do que todas as aproximações já propostas na literatura. B Pricing of European-style Barrier Options under Stochastic Interest Rates Este artigo oferece uma nova metodologia, extremamente rápida e precisa, para a avaliação de opções de estilo Europeu com barreira sobre ativos subjacentes caracterizados por um geometric Brownian motion e no âmbito do modelo de taxas de juro estocásticas de Vasiček (1977). A metodologia de avaliação proposta estende a stopping time approach de Kuan e Webber (2003) a uma configuração mais geral, e expressa o preço de uma opção de estilo Europeu com barreira em termos da densidade de probabilidade do primeiro tempo de passagem do preço do ativo subjacente pelo nível da barreira. Utilizando várias configurações de parâmetros e maturidades de opções, os nossos resultados numéricos mostram que a metodologia de avaliação proposta é muito mais precisa e rápida do extended Fortet method bi-dimensional de Bernard et al. (2008). C Pricing Credit and Equity Default Swaps under the Jump to Default Extended CEV Model Este artigo oferece uma nova metodologia para a avaliação de credit e equity default swaps no âmbito do modelo jump to default extended constant elasticity of variance (JDCEV) de Carr e Linetsky (2006). A abordagem proposta estende a stopping time approach de Kuan e Webber (2003), e expressa o valor das componentes de ambos os contratos em termos da densidade de probabilidade do primeiro tempo de passagem do preço do ativo subjacente pelo nível de acionamento do contrato. Os resultados numéricos mostram que a abordagem de avaliação proposta é precisa e muito mais rápida do que a Laplace transform approach de Mendoza-Arriaga e Linetsky (2011)

Valuation of derivative securities using stochastic analytic and numerical methods

This thesis details methods and procedures to compute prices and hedging strategies for derivative securities in financial mathematics using stochastic analytic, numerical and variance reduction techniques. Results are obtained on explicit hedge ratio representations for non-smooth payoff functionals and mult idimensional diffusion processes with stopping boundaries. These methods are used to determine hedge ratios for the maximum of several assets and lookback options. A number of powerful variance reduction techniques are described. These include the use of measure transformations, discrete versions of importance sampling estimators, control variates based on Ito integral representations, stratified sampling and quasi Monte Carlo. For many of these techniques explicit formulas for the variance of the resulting estimators are obtained. Pricing and hedging procedures are developed for a class of foreign exchange barrier options under stochastic volatility. These procedures are applied to the calculation of down-and-out call options for the Heston model. A general methodology for pricing discount bonds and options on discount bonds for multifactor term structure models is established. This approach is used for both European and American style securities for a version of the two-factor Fong and Vasicek model, extended to include time dependent parameters. For American pricing an exact representation of the early exercise premium is derived for a class of one-factor models. This enables both American option prices and the corresponding two-dimensional critical exercise boundary to be computed for the extended Fong and Vasicek model

Application of Malliavin Calculus and Wiener chaos to option pricing theory.

This dissertation provides a contribution to the option pricing literature by means of some recent developments in probability theory, namely the Malliavin Calculus and the Wiener chaos theory. It concentrates on the issue of faster convergence of Monte Carlo and Quasi-Monte Carlo simulations for the Greeks, on the topic of the Asian option as well as on the approximation for convexity adjustment for fixed income derivatives. The first part presents a new method to speed up the convergence of Monte- Carlo and Quasi-Monte Carlo simulations of the Greeks by means of Malliavin weighted schemes. We extend the pioneering works of Fournie et al. (1999), (2000) by deriving necessary and sufficient conditions for a function to serve as a weight function and by providing the weight function with minimum variance. To do so, we introduce its generator defined as its Skorohod integrand. On a numerical example, we find evidence of spectacular efficiency of this method for corridor options, especially for the gamma calculation. The second part brings new insights on the Asian option. We first show how to price discrete Asian options consistent with different types of underlying densities, especially non-normal returns, by means of the Fast Fourier Transform algorithm. We then extends Malliavin weighted schemes to continuous time Asian options. In the last part, we first prove that the Black Scholes convexity adjustment (Brotherton-Ratcliffe and Iben (1993)) can be consistently derived in a martingale framework. As an application, we examine the convexity bias between CMS and forward swap rates. However, for more complicated term structures assumptions, this approach does not hold any more. We offer a solution to this, thanks to an approximation formula, in the case of multi-factor lognormal zero coupon models, using Wiener chaos theory