240 research outputs found
Stability of central finite difference schemes for the Heston PDE
This paper deals with stability in the numerical solution of the prominent
Heston partial differential equation from mathematical finance. We study the
well-known central second-order finite difference discretization, which leads
to large semi-discrete systems with non-normal matrices A. By employing the
logarithmic spectral norm we prove practical, rigorous stability bounds. Our
theoretical stability results are illustrated by ample numerical experiments
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
Pricing and Hedging GLWB in the Heston and in the Black-Scholes with Stochastic Interest Rate Models
Valuing Guaranteed Lifelong Withdrawal Benefit (GLWB) has attracted
significant attention from both the academic field and real world financial
markets. As remarked by Forsyth and Vetzal the Black and Scholes framework
seems to be inappropriate for such long maturity products. They propose to use
a regime switching model. Alternatively, we propose here to use a stochastic
volatility model (Heston model) and a Black Scholes model with stochastic
interest rate (Hull White model). For this purpose we present four numerical
methods for pricing GLWB variables annuities: a hybrid tree-finite difference
method and a hybrid Monte Carlo method, an ADI finite difference scheme, and a
standard Monte Carlo method. These methods are used to determine the
no-arbitrage fee for the most popular versions of the GLWB contract, and to
calculate the Greeks used in hedging. Both constant withdrawal and optimal
withdrawal (including lapsation) strategies are considered. Numerical results
are presented which demonstrate the sensitivity of the no-arbitrage fee to
economic, contractual and longevity assumptions
Instabilities of Super-Time-Stepping Methods on the Heston Stochastic Volatility Model
This note explores in more details instabilities of explicit
super-time-stepping schemes, such as the Runge-Kutta-Chebyshev or
Runge-Kutta-Legendre schemes, noticed in the litterature, when applied to the
Heston stochastic volatility model. The stability remarks are relevant beyond
the scope of super-time-stepping schemes
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