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Abstract:
Stochastic volatility models are increasingly important in practical derivatives pricing applications, yet relatively little work has been undertaken in the development of practical Monte Carlo simulation methods for this class of models. This paper considers several new algorithms for time-discretization and Monte Carlo simulation of Heston-type stochastic volatility models. The algorithms are based on a careful analysis of the properties of affine stochastic volatility diffusions, and are straightforward and quick to implement and execute. Tests on realistic model parameterizations reveal that the computational efficiency and robustness of the simulation schemes proposed in the paper compare very favorably to existing methods.

Heston model, Monte Carlo simulation, SDE discretization, bias reduction, affine square-root models

Abstract:
This paper considers the pricing of Bermuda-style swaptions in the Libor market model (Brace et al (1997), Jamshidian (1997), Miltersen et al (1997)) and its extensions (Andersen and Andreasen (1998)). Due to its large number of state variables, application of lattice methods to this model class is generally not feasible, and we instead focus on a simple technique to incorporate early exercise features into the Monte Carlo method. Our approach involves a direct search for an early exercise boundary parametrized in intrinsic value and the values of still-alive swaptions. We compare results of the proposed algorithm against prices obtained from Markov Chain approximations and finite difference methods. The proposed algorithm is fast and robust, and produces a lower bound on Bermuda swaption prices that appears to be very tight for many realistic structures. The paper contains several numerical results against which other methods can be tested.

Abstract:
This paper considers extensions of the Libor market model (Brace et al (1997), Jamshidian (1997), Miltersen et al (1997)) to markets with volatility skews in observable option prices. We expand the family of forward rate processes to include diffusions with non-linear forward rate dependence and discuss efficient techniques for calibration to quoted prices of caps and swaptions. Special emphasis is put on generalized CEV processes for which exact closed-form expressions for cap prices are derived. We also discuss modifications of the CEV process which exhibit appealing growth and boundary characteristics. The proposed models are investigated numerically through Crank-Nicholson finite difference schemes and Monte Carlo simulations.

Abstract:
The standard approach (e.g. Dupire (1994) and Rubinstein (1994)) to fitting stock processes to observed option prices models the underlying stock price as a one-factor diffusion process with state- and time-dependent volatility. While this approach is attractive in the sense that market completeness is maintained, the resulting model is often highly non-stationary, difficult to fit to steep volatility smiles, and generally is not well supported by empirical evidence. In this paper, we attempt to overcome some of these problems by overlaying the diffusion dynamics with a jump-process, effectively assuming that a large part of the observed volatility smiles can be explained by fear of sudden large market movements ("crash-o-phobia"). The first part of this paper derives a forward PIDE (Partial Integro-Differential Equation) satisfied by European call option prices and demonstrates how the resulting equation can be used to fit the model to the observed volatility smile/skew. In the second part of the paper, we discuss efficient methods of applying the calibrated model to the pricing of contingent claims. In particular, we develop an ADI (Alternating Directions Implicit) finite difference method that is shown to be unconditionally stable and, if combined with FFT (Fast Fourier Transform) methods, computationally efficient. The paper also discusses the usage of Monte Carlo methods, and contains several detailed examples from the S&P500 market. We compare pricing results obtained by the jump-diffusion approach with those of pure diffusion, and find significant differences for a range of popular contracts.

Abstract:
While convertible bond models recently have come to rest on solid theoretical foundation, issues in model calibration and numerical implementation still remain. This paper highlights and quantifies a number of such issues, demonstrating, among other things, that naïve calibration approaches can lead to highly significant pricing biases. We suggest a number of techniques to resolve such biases. In particular, we demonstrate how applications of the Fokker-Planck PDE allows for efficient joint calibration to debt and option markets, and also discuss volatility smile effects and the derivation of forward PDEs to embed such information into model calibration. Throughout, we rely on modern finite difference techniques, rather than the binomial or trinomial trees that so far have dominated much of the literature.

Abstract:
This paper introduces stochastic volatility to the Libor market model of interest rate dynamics. As in Andersen and Andreasen (2000a) we allow for nonparametric volatility structures with freely specifiable level dependence (such as, but not limited to, the CEV formulation), but now also include a multiplicative perturbation of the forward volatility surface by a general mean-reverting stochastic volatility process. The resulting model dynamics allow for modeling of non-monotonic volatility smiles while explicitly allowing for control of the stationarity properties of the resulting model dynamics. We examine a number of parameterizations of the model, paying particular attention to the development of computationally efficient pricing formulas for calibration of the model to European option prices. Monte Carlo schemes for general pricing applications are proposed and examined.

Stochastic volatility, Libor market model, Andersen and Andreasen, nonparametic, CEV Formulation, mean-reverting, European option prices, Monte Carlo

Abstract:
This paper introduces stochastic volatility to the Libor market model of interest rate dynamics. As in Andersen and Andreasen (2000a) we allow for non-parametric volatility structures with freely specifiable level dependence (such as, but not limited to, the CEV formulation), but now also include a multiplicative perturbation of the forward volatility surface by a general mean-reverting stochastic volatility process. The resulting model dynamics allow for modeling of non-monotonic volatility smiles while explicitly allowing for control of the stationarity properties of the resulting model dynamics. Using asymptotic expansion techniques, we provide closed-form pricing formulas for caps and swaptions that are robust, accurate, and well-suited for both model calibration and general mark-to-market of plain-vanilla instruments. Monte Carlo schemes for the proposed model are proposed and examined.

Volatility smiles, stochastic volatility, Libor market model, asymptotic expansions, ADI finite differences, Monte Carlo simulation

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:
This paper presents a number of new theoretical results for replication of barrier options through a static portfolio of European put and call options. Our results are valid for options with completely general knock-out/knock-in sets, and allow for time- and state-dependent volatility as well as discontinuous asset dynamics. We illustrate the theory with numerical examples and discuss the practical implementation.

Abstract:
This paper investigates the effect of interest rate correlation in the pricing of Bermudan swaptions. Investigating both Gaussian Markov models and Libor Market models, we find that Bermudan swaption prices depend only weakly on the number of factors in the underlying interest rate model. Moreover, we find that prices of standard Bermudan swaptions typically decrease slightly in the number of factors, primarily a consequence of effects on the time evolution of volatility induced by calibration of the model dynamics. Our findings are markedly different from those of Longstaff, Schwarz, and Santa-Clara (1999) who conclude that single-factor interest rate models significantly undervalue Bermudan swaptions. We argue that the conclusions of Longstaff, Schwarz, and Santa-Clara are due to non-standard choices of model dynamics and calibration methodology. Our study highlights the importance of using a reasonable set of calibration instruments when applying and comparing interest rate models.

Abstract:
Polynomial splines are popular in the estimation of discount bond term structures, but suffer from well-documented problems with spurious inflection points, excessive convexity, and lack of locality in the effects of input price perturbations. In this paper, we address these issues through the use of shape-preserving splines from the class of generalized tension splines. Our primary focus is on the classical hyperbolic tension spline which we derive non-parametrically from a penalized least squares criterion, but extensions to generalized tension splines - such as rational splines and exponential splines - are also covered. Our methodology allows both for best-fitting of noisy bonds and for the construction of an exact interpolatory term structure to a set of liquid instruments. Throughout, we work with a local tension B-spline basis, and support both fully non-parametric and user-imposed knot location strategies.

tension splines, term structure of interest rates, yield curve, bond pricing, swap pricing

Abstract:
This objective of this paper is to develop a generic, yet practical framework for construction of Markov models for commodity derivatives. We aim for sufficient richness to permit applications to a broad variety of commodity markets, including those that are characterized by seasonality and by spikes in the spot process. In the first, largely theoretical, part of the paper we derive a series of useful results about low-dimensional Markov representation of the dynamics of an entire term structure of futures prices. Extending previous results in the literature, we cover jump-diffusive models with stochastic volatility as well as several classes of regime-switch models. To demonstrate the process of building models for a specific commodity market, the second part of the paper applies a selection of our theoretical results to the exercise of constructing and calibrating derivatives trading models for USD natural gas. Special attention is paid to the incorporation of empirical seasonality effects in futures prices, in implied volatilities and their smile, and in correlations between futures contracts of different maturities. European option pricing in our proposed gas model is closed-form and of the same complexity as the Black-Scholes formula.

commodity futures, natural gas, seasonality, jump-diffusion, stochastic volatility, regime-switching, Markov model

Abstract:
This paper considers the pricing of European options on assets that follow a stochastic differential equation with a quadratic volatility term. We correct errors in the existing literature, extend the pricing formulas to arbitrary root configurations, and list alternative representations of option pricing formulas to improve computational performance. Our exposition is based entirely on probabilistic arguments, adding a fresh perspective and new intuition to the existing PDE-dominated literature on the subject. Our main tools are martingale methods and shift of probability measure; the fact that the underlying process is typically a strict local martingale is carefully considered throughout the paper.

Quadratic SDE, option pricing, local martingale, volatility smiles

Abstract:
In an influential series of papers, V. Piterbarg demonstrates how to perform time-averaging of parameters in a class of diffusion models with linear local volatility and orthogonal stochastic volatility. In this paper, we consider how to extend the applicability of parameter-averaging techniques to a setting where i) the local volatility function has non-zero convexity; and ii) the correlation between the stochastic volatility process and the underlying asset is non-zero and deterministic. These extension are based on classical small-noise SDE expansions and are of practical use in a number of markets -- foreign exchange being a good example -- where empirical observations of volatility smile moves indicate the presence of non-linear local volatility. For efficient calibration of the time-averaged model, we also derive accurate call option pricing approximations for assets with constant-parameter quadratic local volatility overlaid with (correlated) Heston-type stochastic volatility. Several numerical tests probe the accuracy of the parameter-averaging techniques and the various option pricing approximations.

Time-averaging, local-stochastic volatility, quadratic local volatility, Heston process

Abstract:
Models with systematic factors are popular in the modeling of CDOs, mainly owing to their simplicity and tractability. In this small note we provide a general framework which we use to survey a number of CDO models that have appeared in the literature so far. We suggest extensions and also briefly discuss a select number of issues with factor models, ranging from calibration against CDO market data (ie, base correlation skews) to credit spread hedging and maturity extrapolation. We highlight a number of inherent limitations of factor models and also discuss certain idiosyncracies of popular model-independent approaches to computation of spread hedges.

CDOs, CDO models, calibration against CDO market data, nase correlation skews, model-independent approaches

Abstract:
This paper presents two new models of portfolio default loss that extend the standard Gaussian copula model yet preserve tractability and computational efficiency. In one extension, we randomize recovery rates, explicitly allowing for the empirically well-established effect of inverse correlation between recovery rates and default frequencies. In another extension, we build into the model random systematic factor loadings, effectively allowing default correlations to be higher in bear markets than in bull markets. In both extensions, special cases of the models are shown to be as tractable as the Gaussian copula model and to allow efficient calibration to market credit spreads. We demonstrate that the models - even in their simplest versions - can generate highly significant pricing effects such as fat tails and a correlation "skew" in synthetic CDO tranche prices. When properly calibrated, the skew effect of random recovery is quite minor, but the extension with random factor loadings can produce correlation skews similar to the steep skews observed in the market. We briefly discuss two alternative skew models, one based on the Marshall-Olkin copula, the other on a spread-dependent correlation specification for the Gaussian copula.

Gaussian copula, copulas, portfolio default loss, copula models, random factor loadings