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Abstract:
Callable Libor exotics is a class of single-currency interest-rate contracts that are Bermuda-style exercisable into underlying contracts consisting of fixed-rate, floating-rate and option legs. Bermuda swaptions, callable inverse floaters and callable range accruals are all examples of callable Libor exotics. It is commonly agreed that these instruments are best modeled using forward Libor models. There are many problems, both technical and conceptual, that arise when applying forward Libor models to callable Libor exotics. These problems span calibration, valuation and computation of risk sensitivities. This paper, to the best of our knowledge, is the first comprehensive overview of calibration, pricing and Greeks calculation techniques for callable Libor exotics in forward Libor models. Many technical results and practical methods presented in the paper are original. Others are adaptations, generalizations and extensions of known approaches. Among the technical contributions of this paper are the recommendations for basis functions for the Longstaff-Schwartz valuation algorithm, the extension of the pathwise differentiation method to callable Libor exotics and elegant Greeks formulas that result, novel smoothing techniques for Monte-Carlo, application of Markovian approximations and PDE methods to the problem of variance reduction, and practical algorithms for obtaining vegas in forward Libor models. In addition, strategies for calibrating forward Libor models for callable Libor exotics are discussed at length.

Bermuda-style derivatives, Bermudan swaptions, callable Libor exotics, callable range accruals, callable inverse floaters, hedging, Greeks, deltas, vegas, gammas, Monte-Carlo, market model, forward Libor model, Libor market model, LMM, BGM, pathwise deltas, Markov approximation, variance reduction, control variate, smoothing

Abstract:
Volatility smiles of European swaptions of various expiries and maturities typically have different slopes. This important feature of interest rate markets has not been incorporated in any of the practical interest rate models available to date. In this paper, we build a model that treats the swaption skew matrix as a market input and is calibrated to it. The model is constructed as an extension of a Stochastic Volatility Forward Libor model, with local volatility functions imposed upon forward Libor rates being time-dependent and Libor-rate specific. The focus of the paper is on deriving efficient European swaption approximation formulas that allow calibration of the model to all European swaptions across all expiries, maturities and strikes. The main conceptual contribution of the paper is its focus on recovering all available market volatility skew information across a full swaption grid within a consistent model. The model we develop has a potential to change the way skew calibration is approached, in the same way the introduction of the log-normal forward Libor model had changed the way volatility calibration is approached. The main technical contribution of the paper is a formula for the "effective" skew in a stochastic volatility model, a formula that relates a total amount of skew generated by the model over a given time period to the time-dependent slope of the instantaneous local volatility function. A new "effective" volatility approximation for stochastic volatility models with time-dependent volatility functions is also derived. The formulas we obtain are simple and intuitive; their applicability goes beyond interest rate modeling.

Market model, forward Libor model, Libor market model, LMM, BGM, stochastic volatility, volatility smile, volatility calibration, skew calibration, interest rate models, time-dependent local volatility, effective volatility, effective skew, average skew, homogenization, averaging

Abstract:
The paper develops a multi-currency model with FX skew for power-reverse dual-currency (PRDC) swaps, with a particular emphasis on model calibration to FX options across different maturities and strikes. New theoretical results on locally-optimal Markovian projections are obtained. When combined with powerful skew averaging techniques, a fast and robust calibration method is developed. The impact of the FX skew on cancellable and knockout PRDC swaps is analyzed.

FX hybrids, Power-reverse dual-currency notes, PRDC, FX volatility skew, three-factor model, multi-currency model

Abstract:
We present the Markovian projection method, a method to obtain closed-form approximations to European option prices on various underlyings that, in principle, is applicable to any (diffusive) model. Successful applications of the method have already appeared in the literature, in particular for interest rate models (short rate and forward Libor models with stochastic volatility), and interest rate/FX hybrid models with FX skew. The purpose of this note is thus not to present other instances where the Markovian projection method is applicable (even though more examples are indeed given) but to distill the essence of the method into a conceptually simple plan of attack, a plan that anyone who wants to obtain European option approximations can follow.

Local volatility, stochastic volatility, Markovian projection, parameter averaging, Dupire's local volatility, index options, basket options, spread options

Abstract:
We present new theoretical results for risk sensitivities of Bermuda swaptions, and derive new representations for them. We apply these results to the problem of risk sensitivities computation and derive algorithms that perform the task much faster and more accurately than the traditional approach. Computation of risk sensitivities to market and model parameters (deltas, gammas, vegas) is one of the most important applications for any model. In most practical situations, the Greeks are computed numerically by shocking appropriate inputs and revaluing the instrument. The time needed to execute such a scheme grows linearly with the number of Greeks required. Our approach allows one to compute any number of Greeks for a Bermuda swaption in nearly constant time. Computational advantages versus the standard approach are significant, with time needed to compute a large number of sensitivities reduced by orders of magnitude. Our approach explores symmetries in the structure of Bermuda swaptions, and is essentially model-independent. The approach is based on a newly discovered set of recursive relations between different sensitivities. The recursive relations allow us to represent sensitivities in a number of interesting ways, in particular as integrals over the "survival" density. The survival density is obtained as a solution to a forward Kolmogorov equation. This representation is the basis for practical applications of our approach.

Bermudan swaptions, fast greeks, risk sensitivities, interest rate derivatives valuation and hedging, BGM, Cheyette, PDE methods

Abstract:
A formula that explicitly incorporates volatility smile, as well as a realistic correlation structure of forward rates, in computing Eurodollar futures convexity adjustments is derived. The effect of volatility smile on convexity adjustments is studied and is found significant.

Eurodollar convexity adjustment, stochastic volatility, volatility smile, forward Libor models

Abstract:
In this paper, we demonstrate that many stochastic volatility models have the undesirable property that moments of order higher than one can become infinite in finite time. As arbitrage-free price computation for a number of important fixed income products involves forming expectations of functions with super-linear growth, such lack of moment stability is of significant practical importance. For instance, we demonstrate that reasonably parameterized models can produce infinite prices for Eurodollar futures and for swaps with floating legs paying either Libor-in-arrears or a constant maturity swap (CMS) rate. We systematically examine the moment explosion property across a spectrum of stochastic volatility models. Related properties such as the failure of the martingale property, and asymptotics of the volatility smile are also considered.

Stochastic volatility models, CEV model, displaced diffusion, moment stability, martingale property, integrability, volatility smile asymptotics

Abstract:
Callable Libor exotics is a class of single-currency interest-rate contracts that are Bermuda-style exercisable into underlying contracts consisting of fixed-rate, floating-rate and option legs. The most common callable Libor exotic is a Bermuda swaption. Other, more complicated examples include callable inverse floaters and callable range accruals. Because of their non-trivial dependence on the volatility structure of interest rates, these instruments need a flexible multi-factor model, such as a forward Libor model, for pricing. Only Monte-Carlo based methods are generally available for such models. Being able to obtain risk sensitivities from a model is a prerequisite for its successful application to a given class of products. Computing risk sensitivities in a Monte-Carlo simulation is a difficult task. Monte-Carlo valuation is generally quite slow and noisy. Additionally, numerical noise is amplified when computing risk sensitivities by a "bump-and-revalue" method. Various methods have been proposed to improve accuracy and speed of risk sensitivity calculations in Monte-Carlo for European-type options. Building on previous work in this area, most notably Glasserman and Zhao's (GZ99), we propose a novel extension of some of these methods to the problem of computing deltas of Bermuda-style callable Libor exotics. The method we develop is based on a representation of deltas of a callable Libor exotic as functionals of the optimal exercise time and deltas of the underlying coupons. This representation is obtained by deriving a recursion for the deltas of "nested" Bermuda-style options.

The proposed method saves computational effort by computing all deltas at once in the same simulation in which the value is computed. In addition, it produces significantly more stable and less noisy deltas using only a fraction of the number of paths required by the standard "bump and revalue" approach.

Bermudan swaptions, callable Libor exotics, callable range accruals, callable inverse floaters, hedging, Greeks, deltas, Monte-Carlo, market model, forward Libor model, Libor market model, LMM, BGM, pathwise deltas

Abstract:
The idea of using a weighted average of derivative security prices computed using di?erent "simple" models (the so-called "mixture of models", or "ensemble of models", approach) has been put forth recently by a number of authors. Some view it as a simple way to add stochastic volatility to virtually any model, and others advocate it on the grounds that it provides a simple and tractable method for capturing certain market characteristics, most importantly volatility smile. Ease of calibration to market prices of vanilla and exotic instruments is also cited as the approach's redeeming quality. While not disputing the fact that such "models" are easy to calibrate, we explain that these models are under-specified (leading to multiple possible prices of derivatives). We also demonstrate that the "weighted average" valuation formula, the main selling point of the "mixture of models" approach, is self-inconsistent and cannot be used for valuation.

Mixtures of models, model ensembles, stochastic volatility, lognormal mixture

Abstract:
We develop a systematic approach to the reduction of dimensionality of smile-enabled models by projecting them onto a displaced version of the two-dimensional Heston process. The projection is the key for deriving efficient, analytical approximations to European option prices in such models. This is a further development of the method of Markovian projection previously used for projecting on the displaced-diffusion process (with skew but without smile). The method is derived in a generic form and has a wide range of suitable applications. Examples for spread and basket options are given.

Markovian projection, stochastic volatility, Shifted Heston model, Gyongy lemma, index options, Heston basket options, Heston spread options

Abstract:
We present a new technique for pricing PATH DEPENDENT American-style options where path-dependence is of special kind which we term "weak path dependence".

American options, path dependence, PDE, lattice, accreting notionals, Bermuda swaptions

Abstract:
Callable Libor exotics are a class of single-currency interest rate-derivative securities that includes many important types of instruments such as Bermuda swaptions, callable inverse floaters, callable capped floaters, callable range accruals, and the like. These derivatives exhibit complex dependence on the structure of interest rate volatilities, requiring the most sophisticated and flexible models developed so far, forward Libor models, for valuation and risk management. Despite significant practical interest in applications of forward Libor models to callable Libor exotics, a thorough theoretical analysis of problems that arise in such applications has not yet been performed, and this is a gap that is filled in this paper. We present a comprehensive theoretical framework covering the valuation and computation of risk sensitivities. For valuation, the standard Longstaff-Schwartz algorithm for pricing Bermuda swaptions in a Monte Carlo simulation is significantly expanded to include all callable Libor exotics. Importantly, a collection of effective basis functions is constructed. The problem of computing risk sensitivities (Greeks) is given the most attention. A number of new methods for improving the accuracy and computation speed are presented. These include a special Monte Carlo smoothing technique, a very effective control variate method based on a low-dimensional Markovian approximation, and a robust vega computation method.

Libor, callable Libor exotics, forward Libor models, interest rate derivatives