Pricing Interest Rate Exotics in Multi-Factor Gaussian Interest Rate Models

For many interest rate exotic options, for example options on the slope of the yield curve or American featured options, a one factor assumption for term structure evolution is inappropriate. These options derive their value from changes in the slope or cuvature of the yield curve and hence are more realistically priced with multiple factor models. However, efficient construction of short rate trees becomes computationally intractable as we increase the number of factors and in particular as we move to non-Markovian models. In this paper we describe a general framework for pricing a wide range of interest rate exotic options under a very general family of multi-factor Gaussian interest rate models. Our framework is based on a computationally efficient implementation of Monte Carlo integration utilising analytical approximations as control variates. These techniques extend the analysis of Clewlow, Pang and Strickland [1997] for pricing interest rate caps and swaptions.

Markov Functional Market Model nd Standard Market Model

The introduction of so called Market Models (BGM) in 1990s has developed the world of interest rate modelling into a fresh period. The obvious advantages of the market model have generated a vast amount of research on the market model and recently a new model, called Markov functional market model, has been developed and is becoming increasingly popular. To be clearer between them, the former is called standard market model in this paper. Both standard market models and Markov functional market models are practically popular and the aim here is to explain theoretically how each of them works in practice. Particularly, implementation of the standard market model has to rely on advanced numerical techniques since Monte Carlo simulation does not work well on path-dependent derivatives. This is where the strength of the Longstaff-Schwartz algorithm comes in. The successful application of the Longstaff-Schwartz algorithm with the standard market model, more or less, adds another weight to the fact that the Longstaff-Schwartz algorithm is extensively applied in practice

A Comparison of Single Factor Markov-Functional and Multi Factor Market Models

We compare single factor Markov-functional and multi factor market models for hedging performance of Bermudan swaptions. We show that hedging performance of both models is comparable, thereby supporting the claim that Bermudan swaptions can be adequately riskmanaged with single factor models. Moreover, we show that the impact of smile can be much larger than the impact of correlation. We propose a new method for calculating risk sensitivities of callable products in market models, which is a modification of the least-squares Monte Carlo method. The hedge results show that this new method enables proper functioning of market models as risk-management tools

Pricing Models for Bermudan-Style Interest Rate Derivatives

Bermuda-stijl rente derivaten vormen een belangrijke klasse van opties. Veel bancaire en verzekeringsproducten, zoals hypotheken, vervroegd aflosbare obligaties, en levensverzekeringen, bevatten Bermuda rente opties, die een gevolg zijn van de mogelijkheid tot vervroegde terugbetaling of stopzetting van het contract. Het veel voorkomen van deze opties maakt duidelijk dat het belangrijk is, voor banken en verzekeraars, om de waarde en risico van deze producten op de juiste manier in te schatten. Het juist inschatten van het risico maakt het mogelijk om markt risico af te dekken met onderliggende en regelmatig verhandelde waardes en opties. Waarderingsmodellen moeten arbitrage-vrij zijn, en dienen gekalibreerd te zijn aan prijzen van actief verhandelde onderliggende opties. De dynamica van de modellen moet overeen komen met de geobserveerde dynamica van de rente-termijnstructuur, zoals bijvoorbeeld correlatie tussen rentestanden. Bovendien moeten waarderingsalgoritmes efficiënt zijn: Financiële beslissingen gebaseerd op derivaten waarderingsberekeningen worden veeleer binnen enkele seconden genomen, dan binnen uren of dagen. In recente jaren is een succesvolle klasse van modellen naar voren gekomen, genaamd markt modellen. Dit proefschrift breidt de theorie van markt modellen uit, door: (i) een nieuwe, efficiënte en meer nauwkeurige benaderende waarderingstechniek te introduceren, (ii) twee nieuwe en snelle algoritmes voor correlatie-kalibratie te presenteren, (iii) nieuwe modellen te ontwikkelen die een efficiënte kalibratie toestaan voor een hele nieuwe klasse van derivaten, zoals vaste-looptijd Bermuda rente opties, en (iv) nieuwe empirische vergelijkingen te presenteren van bestaande kalibratie technieken en modellen, in termen van reductie van risico.Bermudan-style interest rate derivatives are an important class of options. Many banking and insurance products, such as mortgages, cancellable bonds, and life insurance products, contain Bermudan interest rate options associated with early redemption or cancellation of the contract. The abundance of these options makes evident that their proper valuation and risk measurement are important to banks and insurance companies. Risk measurement allows for offsetting market risk by hedging with underlying liquidly traded assets and options. Pricing models must be arbitrage-free, and calibrated to prices of actively traded underlying options. Model dynamics need be consistent with observed dynamics of the term structure of interest rates, e.g., correlation. Moreover, valuation algorithms need be efficient: Derivatives pricing calculations need be performed in seconds, rather than hours or days. Recently, a successful class of models appeared in the literature known as market models. This thesis extends market model theory: (i) it introduces a new, efficient, and more accurate approximate pricing technique, (ii) it presents two new fast algorithms for correlation-calibration, (iii) it develops new models enabling efficient calibration for a new range of derivatives, such as fixed-maturity Bermudan swaptions, and (iv) it presents novel empirical comparisons of hedge performance of existing calibrations and models.Raoul Pietersz was born on 12 June 1978 in Rotterdam, The Netherlands. In 2000, he obtained a Certificate of Advanced Studies in Mathematics (Mathematical Tripos Part III), with distinction, from the University of Cambridge. Over the academic year 1999-2000, he was awarded a title of Cambridge European Trust Scholar, and a retrospective title of Scholar at Peterhouse, Cambridge. In the summer of 2000, he completed internships at UBS Warburg and Dresdner Kleinwort Wasserstein, in London. In 2001, he obtained a first class M.Sc. degree in Mathematics from Leiden University. His Master’s thesis entitled “The LIBOR market model”was completed during an internship at ABN AMRO Bank, in Amsterdam. Over the period 1997-2001, he was awarded the Shell International Scholarship for undergraduate studies. His Ph.D. research, under supervision of Antoon Pelsser and Ton Vorst, focuses on the efficient valuation and risk management of interest rate derivatives. He has published articles in The Journal of Computational Finance, The Journal of Derivatives, Quantitative Finance, Risk Magazine and Wilmott Magazine. He has presented his research at various international conferences. His teaching experience includes lecturing taught Master courses on derivatives at the Rotterdam School of Management. Since the start of the Ph.D. period, he has held a part-time position at ABN AMRO Bank, initially at Quantitative Risk Analytics, Risk Management. Since July 2004, he is a Senior Derivatives Researcher, developing front-office pricing models for interest rate derivatives, at Product Development Group, Quantitative Analytics, as part of Structured Derivatives

Markov-functional and stochastic volatility modelling

In this thesis, we study two practical problems in applied mathematical fi nance. The first topic discusses the issue of pricing and hedging Bermudan swaptions within a one factor Markov-functional model. We focus on the implications for hedging of the choice of instantaneous volatility for the one-dimensional driving Markov process of the model. We find that there is a strong evidence in favour of what we term \parametrization by time" as opposed to \parametrization by expiry". We further propose a new parametrization by time for the driving process which takes as inputs into the model the market correlations of relevant swap rates. We show that the new driving process enables a very effective vega-delta hedge with a much more stable gamma profile for the hedging portfolio compared with the existing ones. The second part of the thesis mainly addresses the topic of pricing European options within the popular stochastic volatility SABR model and its extension with mean reversion. We investigate some effcient approximations for these models to be used in real time. We first derive a probabilistic approximation for three different versions of the SABR model: Normal, Log-Normal and a displaced diffusion version for the general constant elasticity of variance case. Specifically, we focus on capturing the terminal distribution of the underlying process (conditional on the terminal volatility) to arrive at the implied volatilities of the corresponding European options for all strikes and maturities. Our resulting method allows us to work with a variety of parameters which cover long dated options and highly stress market condition. This is a different feature from other current approaches which rely on the assumption of very small total volatility and usually fail for longer than 10 years maturity or large volatility of volatility. A similar study is done for the extension of the SABR model with mean reversion (SABR-MR). We first compare the SABR model with this extended model in terms of forward volatility to point out the fundamental difference in the dynamics of the two models. This is done through a numerical example of pricing forward start options. We then derive an effcient probabilistic approximation for the SABRMR model to price European options in a similar fashion to the one for the SABR model. The numerical results are shown to be still satisfactory for a wide range of market conditions

Irregular grid methods for pricing high-dimensional American options

This thesis proposes and studies numerical methods for pricing high-dimensional American options; important examples being basket options, Bermudan swaptions and real options. Four new methods are presented and analysed, both in terms of their application to various test problems, and in terms of their theoretical stability and convergence properties. A method using matrix roots (Chapter 2) and a method using local consistency conditions (Chapter 4) are found to be stable and to give accurate solutions, in up to ten dimensions for the latter case. A method which uses local quadratic functions to approximate the value function (Chapter 3) is found to be vulnerable to instabilities in two dimensions, and thus not suitable for high-dimensional problems. A proof of convergence related to these methods is provided in Chapter 6. Finally, a method based on interpolation of the value function (Chapter 5) is found to be effective in pricing Bermudan swaptions.

CALLABLE SWAPS, SNOWBALLS AND VIDEOGAMES

Although economically more meaningful than the alternatives, short rate models have been dismissed for financial engineering applications in favor of market models as the latter are more flexible and best suited to cluster computing implementations. In this paper, we argue that the paradigm shift toward GPU architectures currently taking place in the high performance computing world can potentially change the situation and tilt the balance back in favor of a new generation of short rate models. We find that operator methods provide a natural mathematical framework for the implementation of realistic short rate models that match features of the historical process such as stochastic monetary policy, calibrate well to liquid derivatives and provide new insights on complex structures. In this paper, we show that callable swaps, callable range accruals, target redemption notes (TARNs) and various flavors of snowballs and snowblades can be priced with methods numerically as precise, fast and stable as the ones based on analytic closed form solutions by means of BLAS level-3 methods on massively parallel GPU architectures.Interest Rate Derivatives; stochastic monetary policy; callable swaps; snowballs; GPU programming; operator methods