2,851 research outputs found
Application of Operator Splitting Methods in Finance
Financial derivatives pricing aims to find the fair value of a financial
contract on an underlying asset. Here we consider option pricing in the partial
differential equations framework. The contemporary models lead to
one-dimensional or multidimensional parabolic problems of the
convection-diffusion type and generalizations thereof. An overview of various
operator splitting methods is presented for the efficient numerical solution of
these problems.
Splitting schemes of the Alternating Direction Implicit (ADI) type are
discussed for multidimensional problems, e.g. given by stochastic volatility
(SV) models. For jump models Implicit-Explicit (IMEX) methods are considered
which efficiently treat the nonlocal jump operator. For American options an
easy-to-implement operator splitting method is described for the resulting
linear complementarity problems.
Numerical experiments are presented to illustrate the actual stability and
convergence of the splitting schemes. Here European and American put options
are considered under four asset price models: the classical Black-Scholes
model, the Merton jump-diffusion model, the Heston SV model, and the Bates SV
model with jumps
Reduced Order Models for Pricing European and American Options under Stochastic Volatility and Jump-Diffusion Models
European options can be priced by solving parabolic partial(-integro)
differential equations under stochastic volatility and jump-diffusion models
like Heston, Merton, and Bates models. American option prices can be obtained
by solving linear complementary problems (LCPs) with the same operators. A
finite difference discretization leads to a so-called full order model (FOM).
Reduced order models (ROMs) are derived employing proper orthogonal
decomposition (POD). The early exercise constraint of American options is
enforced by a penalty on subset of grid points. The presented numerical
experiments demonstrate that pricing with ROMs can be orders of magnitude
faster within a given model parameter variation range
Asymptotics for Exponential Levy Processes and their Volatility Smile: Survey and New Results
Exponential L\'evy processes can be used to model the evolution of various
financial variables such as FX rates, stock prices, etc. Considerable efforts
have been devoted to pricing derivatives written on underliers governed by such
processes, and the corresponding implied volatility surfaces have been analyzed
in some detail. In the non-asymptotic regimes, option prices are described by
the Lewis-Lipton formula which allows one to represent them as Fourier
integrals; the prices can be trivially expressed in terms of their implied
volatility. Recently, attempts at calculating the asymptotic limits of the
implied volatility have yielded several expressions for the short-time,
long-time, and wing asymptotics. In order to study the volatility surface in
required detail, in this paper we use the FX conventions and describe the
implied volatility as a function of the Black-Scholes delta. Surprisingly, this
convention is closely related to the resolution of singularities frequently
used in algebraic geometry. In this framework, we survey the literature,
reformulate some known facts regarding the asymptotic behavior of the implied
volatility, and present several new results. We emphasize the role of
fractional differentiation in studying the tempered stable exponential Levy
processes and derive novel numerical methods based on judicial
finite-difference approximations for fractional derivatives. We also briefly
demonstrate how to extend our results in order to study important cases of
local and stochastic volatility models, whose close relation to the L\'evy
process based models is particularly clear when the Lewis-Lipton formula is
used. Our main conclusion is that studying asymptotic properties of the implied
volatility, while theoretically exciting, is not always practically useful
because the domain of validity of many asymptotic expressions is small.Comment: 92 pages, 15 figure
Numerical methods for Lévy processes
We survey the use and limitations of some numerical methods for pricing derivative contracts in multidimensional geometric Lévy model
Parametric and Nonparametric Volatility Measurement
Volatility has been one of the most active areas of research in empirical finance and time series econometrics during the past decade. This chapter provides a unified continuous-time, frictionless, no-arbitrage framework for systematically categorizing the various volatility concepts, measurement procedures, and modeling procedures. We define three different volatility concepts: (i) the notional volatility corresponding to the ex-post sample-path return variability over a fixed time interval, (ii) the ex-ante expected volatility over a fixed time interval, and (iii) the instantaneous volatility corresponding to the strength of the volatility process at a point in time. The parametric procedures rely on explicit functional form assumptions regarding the expected and/or instantaneous volatility. In the discrete-time ARCH class of models, the expectations are formulated in terms of directly observable variables, while the discrete- and continuous-time stochastic volatility models involve latent state variable(s). The nonparametric procedures are generally free from such functional form assumptions and hence afford estimates of notional volatility that are flexible yet consistent (as the sampling frequency of the underlying returns increases). The nonparametric procedures include ARCH filters and smoothers designed to measure the volatility over infinitesimally short horizons, as well as the recently-popularized realized volatility measures for (non-trivial) fixed-length time intervals.
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