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Papers by Dariusz Gatarek

We consider an optimal stopping problem for a Hilbert-space valued diffusion. We prove that the v... more We consider an optimal stopping problem for a Hilbert-space valued diffusion. We prove that the value function of the problem is the unique viscosity solution of an obstacle problem for the associated parabolic partial differential equation in the Hilbert space. The results are applied to investigate the pricing of American interest rate options in the lognormal Heath-Jarrow-Morton model of yield curve dynamics. Key words. Optimal stopping, obstacle problems, viscosity solutions, option pricing. AMS Subject Classification. 35R20, 49L25, 90A09, 49J15, 60H10. 1 Introduction Optimal stopping problems in finite dimensional domains and obstacle partial differential equations associated with them have been studied extensively in the past. Classical results Andrzej ' Swi¸ech was partially supported by NSF grant DMS-9706760 on the subject can be found in [2],[1]. In recent years the field of stochastic optimal control has developed significantly due to the introduction of the notion of...

Bermudan swaptions are options on interest rate swaps which can be exercised on one or more dates... more Bermudan swaptions are options on interest rate swaps which can be exercised on one or more dates before the final maturity of the swap. Because the exercise boundary between the continuation area and stopping area is inherently complex and multi-dimensional for interest rate products, there is an inherent “tug of war” between the pursuit of calibration and pricing precision, tractability, and implementation efficiency. After reviewing the main ideas and implementation techniques underlying both single- and multi-factor models, we offer our own approach based on dimension reduction via Markovian projection. Specifically, on the theoretical side, we provide a reinterpretation and extension of the classic result due to Gyongy which covers non-probabilistic, discounted, distributions relevant in option pricing. Thus, we show that for purposes of swaption pricing, a potentially complex and multidimensional process for the underlying swap rate can be collapsed to a one-dimensional one. T...

Control and Cybernetics, 2015

Abstract: In this paper we present a new arbitrage-free bottomup model of correlated defaults, ba... more Abstract: In this paper we present a new arbitrage-free bottomup model of correlated defaults, based on a special approach to systematic and idiosyncratic risks for individual obligors. The model admits several attractive features, like consistency with currency and interest rate models, as well as numerical tractability and flexibility, making it capable to fit the market for practically all self-consistent CDO tranche prices. Its background is rather remote from other approaches, like copulas and point processes, so our presentation is detailed.

The Journal of Derivatives, 2019

The authors propose a unified approach to local volatility modeling, encompassing all asset class... more The authors propose a unified approach to local volatility modeling, encompassing all asset classes, with straightforward application to equity and interest rate underlyings. Specifically, they consider a local volatility model for asset-for-asset or Margrabe (1978) options under general conditions that underlying dynamics follow Itô processes and derive a closed-form non-parametric local volatility formula. They then show that many standard contracts—European equity, FX, and interest rate options—can be seen as particular examples of the Margrabe-type payoff, which allows them to analyze equity and interest rate instruments, for example, as special cases of the same general local volatility model, rather than two separate models. They then derive a Markovian projection for the general model, with an approximate local volatility diffusion for the Margrabe option underlying. Finally, they discuss a specific application of the model to swaptions qua asset-for-asset options, where they consider the Markovian projection with some frozen parameters as a minimal “poor man’s” model, featuring equity-like dynamics for the swap rate with its own “short rate” and the “dividend” implied from the term structure of interest rates. Using a number of numerical examples, they compare the minimal model to a fully fledged Cheyette local volatility model and the market benchmark Hull–White one-factor model (Hull and White 1990), demonstrating the adequacy of the “poor man’s” model for pricing European and Bermudan payoffs. TOPICS: Options, statistical methods

The Journal of Computational Finance, 2017

We propose a nonparametric local volatility Cheyette model and apply it to pricing interest rate ... more We propose a nonparametric local volatility Cheyette model and apply it to pricing interest rate swaptions. Concretely, given market prices of swaptions, we show how to construct a unique diffusion process consistent with these prices. We then link the resulting local volatility to the dynamics of the entire interest rate curve. The model preserves completeness and allows consistent pricing of illiquid, out-of-the-money and exotic interest rate products. The model is relatively straightforward to implement and calibrate and less involved than stochastic volatility approaches.

Stochastics and Stochastic Reports, 1995

We consider a Burgers equation perturbed by a multiplicative white noise. We prove the existence ... more We consider a Burgers equation perturbed by a multiplicative white noise. We prove the existence and uniqueness of the global solution as well as the strong Feller property and irreducibility for the corresponding transition semigroup. We close the paper by proving existence and uniqueness of an invariant measure

The Journal of Derivatives, 2015

Credit risk exposure of a cash provider in a repo transaction is limited to 'double default e... more Credit risk exposure of a cash provider in a repo transaction is limited to 'double default events' when the counterparty and the issuer of the underlying collateral asset both default in a short period of time. This article presents a new and intuitive model for modeling correlated defaults, which are the key drivers of residual credit risk in repo portfolios. In the model, default times of counterparties and collateral issuers are determined by idiosyncratic and systematic factors, whereby a name defaults if it is struck by either factor for the first time. The novelty of the approach lies in representing systematic factors as increasing sequences of random variables. Such a setting allows us to precisely capture the clustering of defaults in time and build a rich dependence structure that is free of the flaws inherent in the Gaussian copula-based approaches still widely used for portfolio credit risk applications. Thanks to its general formulation, the model can be applied not only to repos, but also more broadly to pricing and risk-managing any default-correlation-sensitive instruments, e.g., credit default swaps, default swaptions, and CDOs.

Systems & Control Letters, 1982

SSRN Electronic Journal, 2014

SSRN Electronic Journal, 2013

SSRN Electronic Journal, 2003

There exist two classes of interest rate models. Short rate models (HW, CIR, BDT), easy in pricin... more There exist two classes of interest rate models. Short rate models (HW, CIR, BDT), easy in pricing and tough in calibration and forward rate models (HJM, BGM), easy in calibration and tough in pricing. Parameters in short rate models have no natural interpretation in terms of market volatility but many options can be priced on recombining trees. We find particularly inconvenient the procedure of fitting the initial yield curve - necessary for many short rate models. Parameters of forward rate models (especially BGM) have direct link to market volatility but there exists a common prejudice that recombining trees cannot be applied to forward rate models. This paper is an attempt to construct a model allowing both recombining trees and "calibration without programming". We would like to call both presented models "simplest possible term structure models" - at least we do not know any simpler model.