105 research outputs found
A machine learning approach to portfolio pricing and risk management for high-dimensional problems
We present a general framework for portfolio risk management in discrete
time, based on a replicating martingale. This martingale is learned from a
finite sample in a supervised setting. The model learns the features necessary
for an effective low-dimensional representation, overcoming the curse of
dimensionality common to function approximation in high-dimensional spaces. We
show results based on polynomial and neural network bases. Both offer superior
results to naive Monte Carlo methods and other existing methods like
least-squares Monte Carlo and replicating portfolios.Comment: 30 pages (main), 10 pages (appendix), 3 figures, 22 table
Markov cubature rules for polynomial processes
We study discretizations of polynomial processes using finite state Markov
processes satisfying suitable moment matching conditions. The states of these
Markov processes together with their transition probabilities can be
interpreted as Markov cubature rules. The polynomial property allows us to
study such rules using algebraic techniques. Markov cubature rules aid the
tractability of path-dependent tasks such as American option pricing in models
where the underlying factors are polynomial processes.Comment: 29 pages, 6 Figures, 2 Tables; forthcoming in Stochastic Processes
and their Application
Unspanned Stochastic Volatility in the Multi-factor CIR Model
Empirical evidence suggests that fixed income markets exhibit unspanned
stochastic volatility (USV), that is, that one cannot fully hedge volatility
risk solely using a portfolio of bonds. While [1] showed that no two-factor
Cox-Ingersoll-Ross (CIR) model can exhibit USV, it has been unknown to date
whether CIR models with more than two factors can exhibit USV or not. We
formally review USV and relate it to bond market incompleteness. We provide
necessary and sufficient conditions for a multi-factor CIR model to exhibit
USV. We then construct a class of three-factor CIR models that exhibit USV.
This answers in the affirmative the above previously open question. We also
show that multi-factor CIR models with diagonal drift matrix cannot exhibit
USV
Conditional Density Models for Asset Pricing
We model the dynamics of asset prices and associated derivatives by
consideration of the dynamics of the conditional probability density process
for the value of an asset at some specified time in the future. In the case
where the price process is driven by Brownian motion, an associated "master
equation" for the dynamics of the conditional probability density is derived
and expressed in integral form. By a "model" for the conditional density
process we mean a solution to the master equation along with the specification
of (a) the initial density, and (b) the volatility structure of the density.
The volatility structure is assumed at any time and for each value of the
argument of the density to be a functional of the history of the density up to
that time. In practice one specifies the functional modulo sufficient
parametric freedom to allow for the input of additional option data apart from
that implicit in the initial density. The scheme is sufficiently flexible to
allow for the input of various types of data depending on the nature of the
options market and the class of valuation problem being undertaken. Various
examples are studied in detail, with exact solutions provided in some cases.Comment: To appear in International Journal of Theoretical and Applied
Finance, Volume 15, Number 1 (2012), Special Issue on Financial Derivatives
and Risk Managemen
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