Showing papers in "Journal of Computational Finance in 2008"

Pricing options on realized variance in the Heston model with jumps in returns and volatility

Abstract: We develop analytical methodology for pricing and hedging options on the realized variance under the Heston stochastic variance model (1993) augmented with jumps in asset returns and variance. By employing generalized Fourier transform we obtain analytical solutions (up to numerical inversion of Fourier integral) for swaps on the realized volatility and variance and for options on these swaps. We also extend our framework for pricing forward-start options on the realized variance and volatility, including options on the VIX. Our methodology allows us to consistently unify pricing and risk managing of different volatility options. We provide an example of model parameters estimation using both time series of the VIX and the VIX options data and find that the proposed model is in agreement with both historical and implied market data. Finally, we derive a log-normal approximation to the density of the realized variance in the Heston model and obtain accurate approximate solution for volatility options with longer maturities.

Fourier space time-stepping for option pricing with Lévy models

Abstract: Jump-diffusion and Levy models have been widely used to partially alleviate some of the biases inherent in the classical Black-Scholes-Merton model. Unfortunately, the resulting pricing problem requires solving a more difficult partial-integro differential equation (PIDE), and although several approaches for solving the PIDE have been suggested in the literature, none are entirely satisfactory. We present an efficient algorithm, based on transform methods, which symmetrically treats the diffusive and integrals terms, is applicable to a wide class of pathdependent options (such as Bermudan, American and barrier options) and options on multiple assets, and naturally extends to regime-switching Levy models. Furthermore, we introduce a penalty method to improve the convergence of pricing American options.

BSLP: Markovian Bivariate Spread-Loss Model for Portfolio Credit Derivatives

Abstract: BSLP is a two-dimensional dynamic model of interacting portfolio-level loss and spread (more exactly, loss intensity) processes. The model is similar to the top-down HJM-like frameworks developed by Schonbucher (2005) and Sidenius-Peterbarg-Andersen (SPA) (2005), however is constructed as a Markovian, short-rate intensity model. This property of the model enables fast lattice methods for pricing various portfolio credit derivatives such as tranche options, forward-starting tranches, leveraged super-senior tranches etc. A non-parametric model specification is used to achieve nearly perfect calibration to liquid tranche quotes across strikes and maturities. A non-dynamic version of the model obtained in the zero volatility limit of stochastic intensity is useful on its own as an arbitrage-free interpolation model to price non-standard index tranches off the standard ones.

Multi-asset option pricing using a parallel Fourier-based technique

Abstract: In this paper we present and evaluate a Fourier-based sparse grid method for pricing multi-asset options. This involves computing multidimensional integrals efficiently and we do it by the Fast Fourier Transform. We also propose and evaluate ways to deal with the curse of dimensionality by means of parallel partitioning of the Fourier transform and by incorporating a parallel sparse grids method. Finally, we test the presented method by solving pricing equations for options dependent on up to seven underlying assets.

Modeling correlated defaults: first passage model under stochastic volatility

Abstract: Default dependency structure is crucial in pricing multi-name credit derivatives as well as in credit risk management. In this paper, we extend the first passage model for one name with stochastic volatility (Fouque-Sircar-Solna, Applied Mathematical Finance 2006) to the multi-name case. Correlation of defaults is generated by correlation between the Brownian motions driving the individual names as well as through common stochastic volatility factors. A numerical example for the loss distribution of a portfolio of defaultable bonds is examined after stochastic volatility is incorporated.

Pricing k-th to default swaps under default contagion: The matrix analytic approach

Abstract: We study a model for default contagion in intensity-based credit risk and its consequences for pricing portfolio credit derivatives. The model is specified through default intensities which are assumed to be constant between defaults, but which can jump at the times of defaults. The model is translated into a Markov jump process which represents the default status in the credit portfolio. This makes it possible to use matrix-analytic methods to derive computationally tractable closed-form expressions for single-name credit default swap spreads and k-to-default swap spreads. We ”semicalibrate” the model for portfolios (of up to 15 obligors) against market CDS spreads and compute the corresponding k-to-default spreads. In a numerical study based on a synthetic portfolio of 15 telecom bonds we study a number of questions: how spreads depend on the amount of default interaction; how the values of the underlying market CDS-prices used for calibration influence k-th-to default spreads; how a portfolio with inhomogeneous recovery rates compares with a portfolio which satisfies the standard assumption of identical recovery rates; and, finally, how well k-th-to default spreads in a nonsymmetric portfolio can be approximated by spreads in a symmetric portfolio.

Finite element valuation of swing options

Abstract: In this paper an algorithm based on Finite Element Methods is presented to value American type of swing contracts with multiple exercise rights. Thereby the reduction of multiple stopping time problems to a cascade of single stopping time problems is utilized. The numerical results obtained with the proposed algorithm show a smooth and stable behavior. This allows an interpretation of the swing options’ optimal exercise boundaries and an analysis of the dependence of swing option prices on the initial spot prices. A comparison of the Finite Element algorithm to Monte Carlo and lattice methods demonstrates the strengths of the proposed numerical algorithm.

PDE methods for maximum drawdown

Abstract: Maximum drawdown is a risk measure that plays an important role in portfolio management. In this paper, we address the question of computing the expected value of the maximum drawdown using a partial dierential equation (PDE) approach. First, we derive a two-dimensional convection-diusi on pricing equation for the maximum drawdown in the Black-Scholes framework. Due to the properties of the maximum drawdown, this equation has a nonstandard boundary condition. We apply an alternating direction implicit method to solve the equation numerically. We also discuss stability and convergence of the numerical method.

Partial Proxy Simulation Schemes for Generic and Robust Monte-Carlo Greeks

Abstract: We consider a generic framework which allows to calculate robust Monte-Carlo sensitivities seamlessly through simple finite difference approximation. The method proposed is a generalization and improvement of the proxy simulation scheme method (Fries and Kampen, 2005). As a benchmark we apply the method to the pricing of digital caplets and target redemption notes using LIBOR and CMS indices under a LIBOR Market Model. The framework is generic in the sense that it is model and almost product independent. The only product dependent part is the specification of the proxy constraint. This allows for an elegant implementation, where new products may be included at small additional costs.

Gaussian and Poisson approximation: applications to CDOs tranche pricing

Abstract: This article describes a new numerical method, based on Stein’s method and zero bias transformation, to compute CDO tranche prices. We propose rst order correction terms for both Gauss and Poisson approximations and the approximation errors are discussed. We then combine the two approximations to price CDOs tranches in the conditionally independent framework using a realistic local correlation structure. Numerical tests show that the method provides robust results with a very low computational burden.

An adaptive procedure for estimating coherent risk measures based on generalized scenarios

Abstract: Coherent risk measures based on generalized scenarios can be viewed as estimating the maximum expected value from among a collection of simulated "systems." We present a procedure for generating a fixed-width confidence interval for this coherent risk measure. The procedure improves upon previous methods by being reliably efficient for simulation of generalized scenarios and portfolios with heterogeneous characteristics.

A swaption volatility model using Markov regime switching

Abstract: We present a parsimonious, financially motivated model that provides a good description of the swaption volatility matrix. The core model consists of a hidden Markov chain with two volatility states: normal and excited. Each state has its own curve for the instantaneous forward volatility. The volatilities in the swaption matrix result from averaging over all possible paths along the Markov chain. We provide a fast, accurate, analytic method for calculating the swaption matrix from this model. With this procedure we show dramatic improvements over the Rebonato approach (Rebonato (2005)) in the quality of fits when the market appears to be excited.