14,328 research outputs found
Sequential Monte Carlo pricing of American-style options under stochastic volatility models
We introduce a new method to price American-style options on underlying
investments governed by stochastic volatility (SV) models. The method does not
require the volatility process to be observed. Instead, it exploits the fact
that the optimal decision functions in the corresponding dynamic programming
problem can be expressed as functions of conditional distributions of
volatility, given observed data. By constructing statistics summarizing
information about these conditional distributions, one can obtain high quality
approximate solutions. Although the required conditional distributions are in
general intractable, they can be arbitrarily precisely approximated using
sequential Monte Carlo schemes. The drawback, as with many Monte Carlo schemes,
is potentially heavy computational demand. We present two variants of the
algorithm, one closely related to the well-known least-squares Monte Carlo
algorithm of Longstaff and Schwartz [The Review of Financial Studies 14 (2001)
113-147], and the other solving the same problem using a "brute force" gridding
approach. We estimate an illustrative SV model using Markov chain Monte Carlo
(MCMC) methods for three equities. We also demonstrate the use of our algorithm
by estimating the posterior distribution of the market price of volatility risk
for each of the three equities.Comment: Published in at http://dx.doi.org/10.1214/09-AOAS286 the Annals of
Applied Statistics (http://www.imstat.org/aoas/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Pricing American Options by Exercise Rate Optimization
We present a novel method for the numerical pricing of American options based
on Monte Carlo simulation and the optimization of exercise strategies. Previous
solutions to this problem either explicitly or implicitly determine so-called
optimal exercise regions, which consist of points in time and space at which a
given option is exercised. In contrast, our method determines the exercise
rates of randomized exercise strategies. We show that the supremum of the
corresponding stochastic optimization problem provides the correct option
price. By integrating analytically over the random exercise decision, we obtain
an objective function that is differentiable with respect to perturbations of
the exercise rate even for finitely many sample paths. The global optimum of
this function can be approached gradually when starting from a constant
exercise rate.
Numerical experiments on vanilla put options in the multivariate
Black-Scholes model and a preliminary theoretical analysis underline the
efficiency of our method, both with respect to the number of
time-discretization steps and the required number of degrees of freedom in the
parametrization of the exercise rates. Finally, we demonstrate the flexibility
of our method through numerical experiments on max call options in the
classical Black-Scholes model, and vanilla put options in both the Heston model
and the non-Markovian rough Bergomi model
A unified pricing of variable annuity guarantees under the optimal stochastic control framework
In this paper, we review pricing of variable annuity living and death
guarantees offered to retail investors in many countries. Investors purchase
these products to take advantage of market growth and protect savings. We
present pricing of these products via an optimal stochastic control framework,
and review the existing numerical methods. For numerical valuation of these
contracts, we develop a direct integration method based on Gauss-Hermite
quadrature with a one-dimensional cubic spline for calculation of the expected
contract value, and a bi-cubic spline interpolation for applying the jump
conditions across the contract cashflow event times. This method is very
efficient when compared to the partial differential equation methods if the
transition density (or its moments) of the risky asset underlying the contract
is known in closed form between the event times. We also present accurate
numerical results for pricing of a Guaranteed Minimum Accumulation Benefit
(GMAB) guarantee available on the market that can serve as a benchmark for
practitioners and researchers developing pricing of variable annuity
guarantees.Comment: Keywords: variable annuity, guaranteed living and death benefits,
guaranteed minimum accumulation benefit, optimal stochastic control, direct
integration metho
Nonlinear Parabolic Equations arising in Mathematical Finance
This survey paper is focused on qualitative and numerical analyses of fully
nonlinear partial differential equations of parabolic type arising in financial
mathematics. The main purpose is to review various non-linear extensions of the
classical Black-Scholes theory for pricing financial instruments, as well as
models of stochastic dynamic portfolio optimization leading to the
Hamilton-Jacobi-Bellman (HJB) equation. After suitable transformations, both
problems can be represented by solutions to nonlinear parabolic equations.
Qualitative analysis will be focused on issues concerning the existence and
uniqueness of solutions. In the numerical part we discuss a stable
finite-volume and finite difference schemes for solving fully nonlinear
parabolic equations.Comment: arXiv admin note: substantial text overlap with arXiv:1603.0387
HMM based scenario generation for an investment optimisation problem
This is the post-print version of the article. The official published version can be accessed from the link below - Copyright @ 2012 Springer-Verlag.The Geometric Brownian motion (GBM) is a standard method for modelling financial time series. An important criticism of this method is that the parameters of the GBM are assumed to be constants; due to this fact, important features of the time series, like extreme behaviour or volatility clustering cannot be captured. We propose an approach by which the parameters of the GBM are able to switch between regimes, more precisely they are governed by a hidden Markov chain. Thus, we model the financial time series via a hidden Markov model (HMM) with a GBM in each state. Using this approach, we generate scenarios for a financial portfolio optimisation problem in which the portfolio CVaR is minimised. Numerical results are presented.This study was funded by NET ACE at OptiRisk Systems
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