Download Journal Review: Mathematical Finance (Oct/July/April/Jan 2006

Journal Review:
Mathematical Finance
(Oct/July/April/Jan 2006)
Zhao, Lu(Matthew)
Department of Math & Stats
University of Calgary
Calgary, AB, Canada
Oct 3rd, 2006
Mathematical Finance
October 2006 - Vol. 16 Issue 4 Page 589-694
RISK MEASURES AND CAPITAL REQUIREMENTS FOR
PROCESSES
Marco Frittelli1 and Giacomo Scandolo1
In this paper we propose a generalization of the concepts of convex
and coherent risk measures to a multiperiod setting, in which
payoffs are spread over different dates. To this end, a careful
examination of the axiom of translation invariance and the related
concept of capital requirement in the one-period model is
performed. These two issues are then suitably extended to the
multiperiod case, in a way that makes their operative financial
meaning clear. A characterization in terms of expected values is
derived for this class of risk measures and some examples are
presented.
A MULTINOMIAL APPROXIMATION FOR AMERICAN OPTION
PRICES IN LÉVY PROCESS MODELS
Ross A. Maller1
David H. Solomon2 Alex Szimayer3
This paper gives a tree-based method for pricing American options
in models where the stock price follows a general exponential Lévy
process. A multinomial model for approximating the stock price
process, which can be viewed as generalizing the binomial model
of Cox, Ross, and Rubinstein (1979) for geometric Brownian
motion, is developed. Under mild conditions, it is proved that the
stock price process and the prices of American-type options on the
stock, calculated from the multinomial model, converge to the
corresponding prices under the continuous time Lévy process
model. Explicit illustrations are given for the variance gamma model
and the normal inverse Gaussian process when the option is an
American put, but the procedure is applicable to a much wider class
of derivatives including some path-dependent options. Our
approach overcomes some practical difficulties that have
previously been encountered when the Lévy process has infinite
activity.
LIFTING QUADRATIC TERM STRUCTURE MODELS TO
INFINITE DIMENSION
Jirô Akahori1 and Keisuke Hara1
We introduce an infinite dimensional generalization of quadratic
term structure models of interest rates, aiming that the lift will give
us a deeper understanding of the classical models. We show that it
preserves some of the favorable properties of the classical
quadratic models.
ASSET ALLOCATION AND ANNUITY-PURCHASE
STRATEGIES TO MINIMIZE THE PROBABILITY OF FINANCIAL
RUIN
Moshe A. Milevsky1
Kristen S. Moore2
Virginia R. Young3
In this paper, we derive the optimal investment and annuitization
strategies for a retiree whose objective is to minimize the
probability of lifetime ruin, namely the probability that a fixed
consumption strategy will lead to zero wealth while the individual is
still alive. Recent papers in the insurance economics literature have
examined utility-maximizing annuitization strategies. Others in the
probability, finance, and risk management literature have derived
shortfall-minimizing investment and hedging strategies given a
limited amount of initial capital. This paper brings the two strands of
research together. Our model pre-supposes a retiree who does not
currently have sufficient wealth to purchase a life annuity that will
yield her exogenously desired fixed consumption level. She seeks
the asset allocation and annuitization strategy that will minimize the
probability of lifetime ruin. We demonstrate that because of the
binary nature of the investor's goal, she will not annuitize any of her
wealth until she can fully cover her desired consumption with a life
annuity. We derive a variational inequality that governs the ruin
probability and the optimal strategies, and we demonstrate that the
problem can be recast as a related optimal stopping problem which
yields a free-boundary problem that is more tractable. We
numerically calculate the ruin probability and optimal strategies and
examine how they change as we vary the mortality assumption and
parameters of the financial model. Moreover, for the special case of
exponential future lifetime, we solve the (dual) problem explicitly.
As a byproduct of our calculations, we are able to quantify the
reduction in lifetime ruin probability that comes from being able to
manage the investment portfolio dynamically and purchase
annuities.
PRICING SWAPTIONS AND COUPON BOND OPTIONS IN
AFFINE TERM STRUCTURE MODELS
David F. Schrager1
Antoon A. J. Pelsser2
We propose an approach to find an approximate price of a
swaption in affine term structure models. Our approach is based on
the derivation of approximate swap rate dynamics in which the
volatility of the forward swap rate is itself an affine function of the
factors. Hence, we remain in the affine framework and well-known
results on transforms and transform inversion can be used to obtain
swaption prices in similar fashion to zero bond options (i.e.,
caplets). The method can easily be generalized to price options on
coupon bonds. Computational times compare favorably with other
approximation methods. Numerical results on the quality of the
approximation are excellent. Our results show that in affine models,
analogously to the LIBOR market model, LIBOR and swap rates
are driven by approximately the same type of (in this case affine)
dynamics.
July 2006 - Vol. 16 Issue 3 Page 469-588
PRICING A CLASS OF EXOTIC OPTIONS VIA MOMENTS AND
SDP RELAXATIONS
J. B. Lasserre1
T. Prieto-Rumeau2
M. Zervos3
We present a new methodology for the numerical pricing of a class
of exotic derivatives such as Asian or barrier options when the
underlying asset price dynamics are modeled by a geometric
Brownian motion or a number of mean-reverting processes of
interest. This methodology identifies derivative prices with
infinite-dimensional linear programming problems involving the
moments of appropriate measures, and then develops suitable
finite-dimensional relaxations that take the form of semidefinite
programs (SDP) indexed by the number of moments involved. By
maximizing or minimizing appropriate criteria, monotone
sequences of both upper and lower bounds are obtained.
Numerical investigation shows that very good results are obtained
with only a small number of moments. Theoretical convergence
results are also established.
HEDGING WITH ENERGY
Francesco Corielli1
In the setting of diffusion models for price evolution, we suggest an
easily implementable approximate evaluation formula for
measuring the errors in option pricing and hedging due to volatility
misspecification. The main tool we use in this paper is a (suitably
modified) classical inequality for the L2 norm of the solution, and the
derivatives of the solution, of a partial differential equation (the
so-called "energy" inequality). This result allows us to give bounds
on the errors implied by the use of approximate models for option
valuation and hedging and can be used to justify formally some
"folk" belief about the robustness of the Black and Scholes model.
Surprisingly enough, the result can also be applied to improve
pricing and hedging with an approximate model. When statistical or
a priori information is available on the "true" volatility, the error
measure given by the energy inequality can be minimized w.r.t. the
parameters of the approximating model. The method suggested in
this paper can help in conjugating statistical estimation of the
volatility function derived from flexible but computationally
cumbersome statistical models, with the use of analytically
tractable approximate models calibrated using error estimates.
MODEL UNCERTAINTY AND ITS IMPACT ON THE PRICING OF
DERIVATIVE INSTRUMENTS
Rama Cont1
Uncertainty on the choice of an option pricing model can lead to
"model risk" in the valuation of portfolios of options. After
discussing some properties which a quantitative measure of model
uncertainty should verify in order to be useful and relevant in the
context of risk management of derivative instruments, we introduce
a quantitative framework for measuring model uncertainty in the
context of derivative pricing. Two methods are proposed: the first
method is based on a coherent risk measure compatible with
market prices of derivatives, while the second method is based on
a convex risk measure. Our measures of model risk lead to a
premium for model uncertainty which is comparable to other risk
measures and compatible with observations of market prices of a
set of benchmark derivatives. Finally, we discuss some implications
for the management of "model risk."
NEWS-GENERATED DEPENDENCE AND OPTIMAL
PORTFOLIOS FOR n STOCKS IN A MARKET OF
BARNDORFF-NIELSEN AND SHEPHARD TYPE
Carl Lindberg1
We consider Merton's portfolio optimization problem in a Black and
Scholes market with non-Gaussian stochastic volatility of
Ornstein–Uhlenbeck type. The investor can trade in n stocks and a
risk-free bond. We assume that the dependence between stocks
lies in that they partly share the Ornstein–Uhlenbeck processes of
the volatility. We refer to these as news processes, and interpret
this as that dependence between stocks lies solely in their
reactions to the same news. The model is primarily intended for
assets that are dependent, but not too dependent, such as stocks
from different branches of industry. We show that this dependence
generates covariance, and give statistical methods for both the
fitting and verification of the model to data. Using dynamic
programming, we derive and verify explicit trading strategies and
Feynman–Kac representations of the value function for power
utility.
NO ARBITRAGE UNDER TRANSACTION COSTS, WITH
FRACTIONAL BROWNIAN MOTION AND BEYOND
Paolo Guasoni1
We establish a simple no-arbitrage criterion that reduces the
absence of arbitrage opportunities under proportional transaction
costs to the condition that the asset price process may move
arbitrarily little over arbitrarily large time intervals.
We show that this criterion is satisfied when the return process is
either a strong Markov process with regular points, or a continuous
process with full support on the space of continuous functions. In
particular, we prove that proportional transaction costs of any
positive size eliminate arbitrage opportunities from geometric
fractional Brownian motion for H
continuous deterministic drift.
(0, 1) and with an arbitrary
A COMMENT ON MARKET FREE LUNCH AND FREE LUNCH
Irene Klein1
Frittelli (2004) introduced a market free lunch depending on the
preferences of the agents in the market. He characterized no
arbitrage and no free lunch with vanishing risk in terms of no
market free lunch (the difference comes from the class of utility
functions determining the market free lunch). In this note we
complete the list of characterizations and show directly (using the
theory of Orlicz spaces) that no free lunch is equivalent to the
absence of market free lunch with respect to monotone concave
utility functions.
April 2006 - Vol. 16 Issue 2 Page 237-467
VALUATION OF FLOATING RANGE NOTES IN LÉVY
TERM-STRUCTURE MODELS
Ernst Eberlein1 and Wolfgang Kluge1
Turnbull (1995) as well as Navatte and Quittard-Pinon (1999)
derived explicit pricing formulae for digital options and range notes
in a one-factor Gaussian Heath–Jarrow–Morton (henceforth HJM)
model. Nunes (2004) extended their results to a multifactor
Gaussian HJM framework. In this paper, we generalize these
results by providing explicit pricing solutions for digital options and
range notes in the multivariate Lévy term-structure model of
Eberlein and Raible (1999), that is, an HJM-type model driven by a
d-dimensional (possibly nonhomogeneous) Lévy process. As a
byproduct, we obtain a pricing formula for floating range notes in
the special case of a multifactor Gaussian HJM model that is
simpler than the one provided by Nunes (2004).
PRICING EQUITY DERIVATIVES SUBJECT TO BANKRUPTCY
Vadim Linetsky1
We solve in closed form a parsimonious extension of the
Black–Scholes–Merton model with bankruptcy where the hazard
rate of bankruptcy is a negative power of the stock price.
Combining a scale change and a measure change, the model
dynamics is reduced to a linear stochastic differential equation
whose solution is a diffusion process that plays a central role in the
pricing of Asian options. The solution is in the form of a spectral
expansion associated with the diffusion infinitesimal generator. The
latter is closely related to the Schrödinger operator with Morse
potential. Pricing formulas for both corporate bonds and stock
options are obtained in closed form. Term credit spreads on
corporate bonds and implied volatility skews of stock options are
closely linked in this model, with parameters of the hazard rate
specification controlling both the shape of the term structure of
credit spreads and the slope of the implied volatility skew. Our
analytical formulas are easy to implement and should prove useful
to researchers and practitioners in corporate debt and equity
derivatives markets.
PORTFOLIO OPTIMIZATION WITH DOWNSIDE CONSTRAINTS
Peter Lakner1
Lan Ma Nygren2
We consider the portfolio optimization problem for an investor
whose consumption rate process and terminal wealth are subject to
downside constraints. In the standard financial market model that
consists of d risky assets and one riskless asset, we assume that
the riskless asset earns a constant instantaneous rate of interest,
r> 0, and that the risky assets are geometric Brownian motions.
The optimal portfolio policy for a wide scale of utility functions is
derived explicitly. The gradient operator and the Clark–Ocone
formula in Malliavin calculus are used in the derivation of this
policy. We show how Malliavin calculus approach can help us get
around certain difficulties that arise in using the classical "delta
hedging" approach.
MULTIDIMENSIONAL PORTFOLIO OPTIMIZATION WITH
PROPORTIONAL TRANSACTION COSTS
Kumar Muthuraman1
Sunil Kumar2
We provide a computational study of the problem of optimally
allocating wealth among multiple stocks and a bank account, to
maximize the infinite horizon discounted utility of consumption. We
consider the situation where the transfer of wealth from one asset
to another involves transaction costs that are proportional to the
amount of wealth transferred. Our model allows for correlation
between the price processes, which in turn gives rise to interesting
hedging strategies. This results in a stochastic control problem with
both drift-rate and singular controls, which can be recast as a free
boundary problem in partial differential equations. Adapting the
finite element method and using an iterative procedure that
converts the free boundary problem into a sequence of fixed
boundary problems, we provide an efficient numerical method for
solving this problem. We present computational results that
describe the impact of volatility, risk aversion of the investor, level
of transaction costs, and correlation among the risky assets on the
structure of the optimal policy. Finally we suggest and quantify
some heuristic approximations.
NONPARAMETRIC KERNEL-BASED SEQUENTIAL
INVESTMENT STRATEGIES
László Györfi1
Gábor Lugosi2 and Frederic Udina2
The purpose of this paper is to introduce sequential investment
strategies that guarantee an optimal rate of growth of the capital,
under minimal assumptions on the behavior of the market. The new
strategies are analyzed both theoretically and empirically. The
theoretical results show that the asymptotic rate of growth matches
the optimal one that one could achieve with a full knowledge of the
statistical properties of the underlying process generating the
market, under the only assumption that the market is stationary and
ergodic. The empirical results show that the performance of the
proposed investment strategies measured on past nyse and
currency exchange data is solid, and sometimes even spectacular.
OPTIMAL STATIC–DYNAMIC HEDGES FOR BARRIER
OPTIONS
Aytaç İlhan1
Ronnie Sircar2
We study optimal hedging of barrier options, using a combination of
a static position in vanilla options and dynamic trading of the
underlying asset. The problem reduces to computing the
Fenchel–Legendre transform of the utility-indifference price as a
function of the number of vanilla options used to hedge. Using the
well-known duality between exponential utility and relative entropy,
we provide a new characterization of the indifference price in terms
of the minimal entropy measure, and give conditions guaranteeing
differentiability and strict convexity in the hedging quantity, and
hence a unique solution to the hedging problem. We discuss
computational approaches within the context of Markovian
stochastic volatility models.
PORTFOLIO INSURANCE AND VOLATILITY REGIME
SWITCHING
Joel M. Vanden1
A new equilibrium model of portfolio insurance is presented in order
to study the volatility effects of dynamic insurance strategies. While
prior research has focused on the relationship between portfolio
insurance and the overall level of market volatility, this article
shows that the salient feature of portfolio insurance is volatility
regime switching. Regime switching is shown to be a necessary
condition for portfolio insurance, which provides a new explanation
for the pervasive volatility clustering effect that is found in most
equity markets. The equilibrium involves a free boundary and the
local time of the equilibrium price process at the free boundary
plays an important role in solving the model.
DISTRIBUTION-INVARIANT RISK MEASURES, INFORMATION,
AND DYNAMIC CONSISTENCY
Stefan Weber1
In the first part of the paper, we characterize distribution-invariant
risk measures with convex acceptance and rejection sets on the
level of distributions. It is shown that these risk measures are
closely related to utility-based shortfall risk.
In the second part of the paper, we provide an axiomatic
characterization for distribution-invariant dynamic risk measures of
terminal payments. We prove a representation theorem and
investigate the relation to static risk measures. A key insight of the
paper is that dynamic consistency and the notion of "measure
convex sets of probability measures" are intimately related. This
result implies that under weak conditions dynamically consistent
dynamic risk measures can be represented by static utility-based
shortfall risk.
DISUTILITY, OPTIMAL RETIREMENT, AND PORTFOLIO
SELECTION
Kyoung Jin Choi1
Gyoocheol Shim2
We study the optimal retirement and consumption/investment
choice of an infinitely-lived economic agent with a time-separable
von Neumann–Morgenstern utility. A particular aspect of our
problem is that the agent has a retirement option. Before retirement
the agent receives labor income but suffers a utility loss from labor.
By retiring, he avoids the utility loss but gives up labor income. We
show that the agent retires optimally if his wealth exceeds a certain
critical level. We also show that the agent consumes less and
invests more in risky assets when he has an option to retire than he
would in the absence of such an option.
An explicit solution can be provided by solving a free boundary
value problem. In particular, the critical wealth level and the optimal
consumption and portfolio policy are provided in explicit forms.
January 2006 - Vol. 16 Issue 1 Page iii-236
MORE ON MINIMAL ENTROPY–HELLINGER MARTINGALE
MEASURE
Tahir Choulli1
Christophe Stricker2
This paper extends our recent paper (Choulli and Stricker 2005) to
the case when the discounted stock price process may be
unbounded and may have predictable jumps. In this very general
context, we provide mild necessary conditions for the existence of
the minimal entropy–Hellinger local martingale density and we give
an explicit description of this extremal martingale density that can
be determined by pointwise solution of equations in
depending only on the local characteristics of the discounted price
process S. The uniform integrability and other integrability
properties are investigated for this extremal density, which lead to
the conditions of the existence of the minimal entropy–Hellinger
martingale measure. Finally, we illustrate the main results of the
paper in the case of a discrete-time market model, where the
relationship of the obtained optimal martingale measure to a
dynamic risk measure is discussed.
APPROXIMATING GARCH-JUMP MODELS, JUMP-DIFFUSION
PROCESSES, AND OPTION PRICING
Jin-Chuan Duan1
Peter Ritchken2
Zhiqiang Sun3
This paper considers the pricing of options when there are jumps in
the pricing kernel and correlated jumps in asset prices and
volatilities. We extend theory developed by Nelson (1990) and
Duan (1997) by considering the limiting models for our
approximating GARCH Jump process. Limiting cases of our
processes consist of models where both asset price and local
volatility follow jump diffusion processes with correlated jump sizes.
Convergence of a few GARCH models to their continuous time
limits is evaluated and the benefits of the models explored.
A NOTE ON SEMIVARIANCE
Hanqing Jin1
Harry Markowitz2
Xun Yu Zhou3
In a recent paper (Jin, Yan, and Zhou 2005), it is proved that
efficient strategies of the continuous-time mean–semivariance
portfolio selection model are in general never achieved save for a
trivial case. In this note, we show that the mean–semivariance
efficient strategies in a single period are always attained
irrespective of the market condition or the security return
distribution. Further, for the below-target semivariance model the
attainability is established under the arbitrage-free condition.
Finally, we extend the results to problems with general downside
risk measures.
CHARACTERIZATION OF OPTIMAL STOPPING REGIONS OF
AMERICAN ASIAN AND LOOKBACK OPTIONS
Min Dai1
Yue Kuen Kwok2
A general framework is developed to analyze the optimal stopping
(exercise) regions of American path-dependent options with either
the Asian feature or lookback feature. We examine the
monotonicity properties of the option values and stopping regions
with respect to the interest rate, dividend yield, and time. From the
ordering properties of the values of American lookback options and
American Asian options, we deduce the corresponding nesting
relations between the exercise regions of these American options.
We illustrate how some properties of the exercise regions of the
American Asian options can be inferred from those of the American
lookback options.
OPTIMAL LOT SOLUTION TO CARDINALITY CONSTRAINED
MEAN–VARIANCE FORMULATION FOR PORTFOLIO
SELECTION
Duan Li1
Xiaoling Sun2
Jun Wang3
The pioneering work of the mean–variance formulation proposed
by Markowitz in the 1950s has provided a scientific foundation for
modern portfolio selection. Although the trade practice often
confines portfolio selection with certain discrete features, the
existing theory and solution methodologies of portfolio selection
have been primarily developed for the continuous solution of the
portfolio policy that could be far away from the real integer
optimum. We consider in this paper an exact solution algorithm in
obtaining an optimal lot solution to cardinality constrained
mean–variance formulation for portfolio selection under concave
transaction costs. Specifically, a convergent Lagrangian and
contour-domain cut method is proposed for solving this class of
discrete-feature constrained portfolio selection problems by
exploiting some prominent features of the mean–variance
formulation and the portfolio model under consideration.
Computational results are reported using data from the Hong Kong
stock market.
CONSTRAINED OPTIMIZATION WITH RESPECT TO
STOCHASTIC DOMINANCE: APPLICATION TO PORTFOLIO
INSURANCE
Nicole El Karoui1 and Asma Meziou1
We are concerned with a classic portfolio optimization problem
where the admissible strategies must dominate a floor process on
every intermediate date (American guarantee). We transform the
problem into a martingale, whose aim is to dominate an obstacle, or
equivalently its Snell envelope. The optimization is performed with
respect to the concave stochastic ordering on the terminal value, so
that we do not impose any explicit specification of the agent's utility
function. A key tool is the representation of the supermartingale
obstacle in terms of a running supremum process. This is illustrated
within the paper by an explicit example based on the geometric
Brownian motion.
A UNIVERSAL OPTIMAL CONSUMPTION RATE FOR AN
INSIDER
Bernt Øksendal1
We consider a cash flow X(c) (t) modeled by the stochastic equation
where B(·) and
are a Brownian motion and a Poissonian random measure,
respectively, and c(t) ≥ 0 is the consumption/dividend rate. No
assumptions are made on adaptedness of the coefficients μ, σ, θ,
and c, and the (possibly anticipating) integrals are interpreted in the
forward integral sense. We solve the problem to find the
consumption rate c(·), which maximizes the expected discounted
utility given by
Here δ(t) ≥ 0 is a given measurable stochastic process
representing a discounting exponent and τ is a random time with
values in (0, ∞), representing a terminal/default time, while γ≥ 0 is a
known constant.
A BENCHMARK APPROACH TO FINANCE
Eckhard Platen1
This paper derives a unified framework for portfolio optimization,
derivative pricing, financial modeling, and risk measurement. It is
based on the natural assumption that investors prefer more rather
than less, in the sense that given two portfolios with the same
diffusion coefficient value, the one with the higher drift is preferred.
Each such investor is shown to hold an efficient portfolio in the
sense of Markowitz with units in the market portfolio and the
savings account. The market portfolio of investable wealth is shown
to equal a combination of the growth optimal portfolio (GOP) and
the savings account. In this setup the capital asset pricing model
follows without the use of expected utility functions, Markovianity,
or equilibrium assumptions. The expected increase of the
discounted value of the GOP is shown to coincide with the
expected increase of its discounted underlying value. The
discounted GOP has the dynamics of a time transformed squared
Bessel process of dimension four. The time transformation is given
by the discounted underlying value of the GOP. The squared
volatility of the GOP equals the discounted GOP drift, when
expressed in units of the discounted GOP. Risk-neutral derivative
pricing and actuarial pricing are generalized by the fair pricing
concept, which uses the GOP as numeraire and the real-world
probability measure as pricing measure. An equivalent risk-neutral
martingale measure does not exist under the derived minimal
market model.
UTILITY MAXIMIZATION IN AN INSIDER INFLUENCED
MARKET
Arturo Kohatsu-Higa1
Agnès Sulem2
We study a controlled stochastic system whose state is described
by a stochastic differential equation with anticipating coefficients.
This setting is used to model markets where insiders have some
influence on the dynamics of prices. We give a characterization
theorem for the optimal logarithmic portfolio of an investor with a
different information flow from that of the insider. We provide
explicit results in the partial information case that we extend in
order to incorporate the enlargement of filtration techniques for
markets with insiders. Finally, we consider a market with an insider
who influences the drift of the underlying price asset process. This
example gives a situation where it makes a difference for a small
agent to acknowledge the existence of an insider in the market.
CLASSICAL AND IMPULSE STOCHASTIC CONTROL FOR THE
OPTIMIZATION OF THE DIVIDEND AND RISK POLICIES OF AN
INSURANCE FIRM
Abel Cadenillas1 and Tahir Choulli1
Michael Taksar2 Lei Zhang3
This paper deals with the dividend optimization problem for a
financial or an insurance entity which can control its business
activities, simultaneously reducing the risk and potential profits. It
also controls the timing and the amount of dividends paid out to the
shareholders. The objective of the corporation is to maximize the
expected total discounted dividends paid out until the time of
bankruptcy. Due to the presence of a fixed transaction cost, the
resulting mathematical problem becomes a mixed
classical-impulse stochastic control problem. The analytical part of
the solution to this problem is reduced to quasivariational
inequalities for a second-order nonlinear differential equation. We
solve this problem explicitly and construct the value function
together with the optimal policy. We also compute the expected
time between dividend payments under the optimal policy.
MARKOWITZ'S PORTFOLIO OPTIMIZATION IN AN
INCOMPLETE MARKET
Jianming Xia1 and Jia-An Yan1
In this paper, for a process S, we establish a duality relation
between Kp, the
- closure of the space of claims in
, which are attainable by "simple" strategies, and
, all signed martingale measures
with
, where p≥ 1, q≥ 1 and
. If there exists a
with
a.s., then Kp consists precisely of the random variables
such that is predictable S-integrable and
for all
. The duality relation corresponding to the case p=q= 2 is used to
investigate the Markowitz's problem of mean–variance portfolio
optimization in an incomplete market of semimartingale model via
martingale/convex duality method. The duality relationship between
the mean–variance efficient portfolios and the variance-optimal
signed martingale measure (VSMM) is established. It turns out that
the so-called market price of risk is just the standard deviation of
the VSMM. An illustrative example of application to a geometric
Lévy processes model is also given.
STOCK LIQUIDATION VIA STOCHASTIC APPROXIMATION
USING NASDAQ DAILY AND INTRA-DAY DATA
G. Yin1 Q. Zhang2
F. Liu3
R. H. Liu4
Y. Cheng5
By focusing on computational aspects, this work is concerned with
numerical methods for stock selling decision using stochastic
approximation methods. Concentrating on the class of decisions
depending on threshold values, an optimal stopping problem is
converted to a parametric stochastic optimization problem. The
algorithms are model free and are easily implementable on-line.
Convergence of the algorithms is established, second moment
bound of estimation error is obtained, and escape probability from a
neighborhood of the true parameter is also derived. Numerical
examples using both daily closing prices and intra-day data are
provided to demonstrate the performance of the algorithms.