Finite Difference Approximation for Linear Stochastic Partial Differential Equations with Method of Lines

A stochastic partial differential equation, or SPDE, describes the dynamics of a stochastic process defined on a space-time continuum. This paper provides a new method for solving SPDEs based on the method of lines (MOL). MOL is a technique that has largely been used for numerically solving deterministic partial differential equations (PDEs). MOL works by transforming the PDE into a system of ordinary differential equations (ODEs) by discretizing the spatial dimension of the PDE. The resulting system of ODEs is then solved by application of either a finite difference or a finite element method. This paper provides a proof that the MOL can be used to provide a finite difference approximation of the boundary value solutions for two broad classes of linear SPDEs, the linear elliptic and parabolic SPDEs.Finite difference approximation; linear stochastic partial differential equations (SPDEs); the method of lines (MOL)

Representation of homothetic forward performance processes in stochastic factor models via ergodic and infinite horizon BSDE

In an incomplete market, with incompleteness stemming from stochastic factors imperfectly correlated with the underlying stocks, we derive representations of homothetic (power, exponential and logarithmic) forward performance processes in factor-form using ergodic BSDE. We also develop a connection between the forward processes and infinite horizon BSDE, and, moreover, with risk-sensitive optimization. In addition, we develop a connection, for large time horizons, with a family of classical homothetic value function processes with random endowments.Comment: 34 page

The History of the Quantitative Methods in Finance Conference Series. 1992-2007

This report charts the history of the Quantitative Methods in Finance (QMF) conference from its beginning in 1993 to the 15th conference in 2007. It lists alphabetically the 1037 speakers who presented at all 15 conferences and the titles of their papers.

Fast drift approximated pricing in the BGM model

This paper shows that the forward rates process discretized by a single time step together with a separability assumption on the volatility function allows for representation by a low-dimensional Markov process. This in turn leads to e±cient pricing by for example finite differences. We then develop a discretization based on the Brownian bridge especially designed to have high accuracy for single time stepping. The scheme is proven to converge weakly with order 1. We compare the single time step method for pricing on a grid with multi step Monte Carlo simulation for a Bermudan swaption, reporting a computational speed increase of a factor 10, yet pricing sufficiently accurate.BGM model, predictor-corrector, Brownian bridge, Markov processes, separability, Feynman-Kac, Bermudan swaption

Extended Libor Market Models with Affine and Quadratic Volatility

The market model of interest rates specifies simple forward or Libor rates as lognormally distributed, their stochastic dynamics has a linear volatility function. In this paper, the model is extended to quadratic volatility functions which are the product of a quadratic polynomial and a level-independent covariance matrix. The extended Libor market models allow for closed form cap pricing formulae, the implied volatilities of the new formulae are smiles and frowns. We give examples for the possible shapes of implied volatilities. Furthermore, we derive a new approximative swaption pricing formula and discuss its properties. The model is calibrated to market prices, it turns out that no extended model specification outperforms the others. The criteria for model choice should thus be theoretical properties and computational efficiency.forward Libor rates, Libor market model, affine volatility, quadratic volatility, dervatives pricing, closed form solutions, LMM, BGM

Pricing Currency Options with a Market Model of Interest Rates under Jump-Diffusion Stochastic Volatility Processes of Spot Exchange Rates

This paper proposes a pricing method of currency options with a market model of interest rates. Using a simple approximation and a Fourier transform method, we derive a formula of the option pricing under jump-diffusion stochastic volatility processes of spot exchange rates. As an application, we apply the formula to the calibration of volatility smiles in the JPY/USD currency option market. Moreover, using the approximate prices as a control variate, we achieve substantial variance reduction in Monte Carlo simulation.

On cross-currency models with stochastic volatility and correlated interest rates

We construct multi-currency models with stochastic volatility and correlated stochastic interest rates with a full matrix of correlations. We frst deal with a foreign exchange (FX) model of Heston-type, in which the domestic and foreign interest rates are generated by the short-rate process of Hull-White [HW96]. We then extend the framework by modeling the interest rate by a stochastic volatility displaced-diffusion Libor Market Model [AA02], which can model an interest rate smile. We provide semi-closed form approximations which lead to effcient calibration of the multi-currency models. Finally, we add a correlated stock to the framework and discuss the construction, model calibration and pricing of equity- FX-interest rate hybrid payoffs.Foreign-exchange (FX); stochastic volatility; Heston model; stochastic interest rates; interest rate smile; forward characteristic function; hybrids; affne diffusion; effcient calibration.

"Pricing Currency Options with a Market Model of Interest Rates under Jump-Diffusion Stochastic Volatility Processes of Spot Exchange Rates"

This paper proposes a pricing method of currency options with a market model of interest rates. Using a simple approximation and a Fourier transform method, we derive a formula of the option pricing under jump-diffusion stochastic volatility processes of spot exchange rates. As an application, we apply the formula to the calibration of volatility smiles in the JPY/USD currency option market. Moreover, using the approximate prices as a control variate, we achieve substantial variance reduction in Monte Carlo simulation.