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## Strategic asset allocation under a fractional hidden markov model

Fonte: Kluwer Academic Publishers
Publicador: Kluwer Academic Publishers

Tipo: Artigo de Revista Científica

Publicado em //2014
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#Strategic asset allocation#Long memory#Hidden Markov models#Fractional Gaussian VAR process#Mean-variance utility#91B28#91B70

Strategic asset allocation is discussed in a discrete-time economy, where the rates of return from asset classes are explained in terms of some observable and hidden factors. We extend the existing models by incorporating long-term memory in the rates of return and observable economic factors, which have been documented in the empirical literature. Hidden factors are described by a discrete-time, finite-state, hidden Markov chain noisily observed in a fractional Gaussian process. The strategic asset allocation problem is discussed in a mean-variance utility framework. Filtering and parameter estimation are also considered in the hybrid model.; Robert J. Elliott, Tak Kuen Siu

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## Compatibility between pricing rules and risk measures: the CCVaR

Fonte: Universidade Carlos III de Madrid
Publicador: Universidade Carlos III de Madrid

Tipo: info:eu-repo/semantics/acceptedVersion; info:eu-repo/semantics/article

Publicado em /06/2009
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#Risk measure#Pricing rule#Unbounded optimization problem#Compatibility#Compatible conditional value at risk#91B28#91B30

This paper has considered a risk measure? and a (maybe incomplete and/or imperfect) arbitrage-free market with pricing rule p. They are said to be compatible if there are no reachable strategies y such that p (y) remains bounded and ?(y) is close to - 8. We show that the lack of compatibility leads to meaningless situations in financial or actuarial applications. The presence of compatibility is characterized by properties connecting the Stochastic Discount Factor of p and the sub-gradient of ?. Consequently, several examples pointing out that the lack of compatibility may occur in very important pricing models are yielded. For instance the CVaR and the DPT are not compatible with the Black and Scholes model or the CAPM. We prove that for a given incompatible couple (p,?) we can construct a minimal risk measure ?p compatible with p and such that ?p = ? . This result is particularized for the CVaR and the CAPM and the Black and Scholes model. Therefore we construct the Compatible Conditional Value at Risk (CCVaR). It seems that the CCVaR preserves the good properties of the CVaR and overcomes its shortcomings.; Research partially supported by “RD Sistemas SA”, “Comunidad Autónoma de
Madrid” (Spain), Grant s-0505/tic/000230...

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## Time Consistent Dynamic Limit Order Books Calibrated on Options

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 22/09/2008
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#Quantitative Finance - Trading and Market Microstructure#Mathematics - Probability#46A22#60G44#91B24#91B28#91B70

In an incomplete financial market, the axiomatic of Time Consistent Pricing
Procedure (TCPP), recently introduced, is used to assign to any financial asset
a dynamic limit order book, taking into account both the dynamics of basic
assets and the limit order books for options.
Kreps-Yan fundamental theorem is extended to that context. A characterization
of TCPP calibrated on options is given in terms of their dual representation.
In case of perfectly liquid options, these options can be used as the basic
assets to hedge dynamically. A generic family of TCPP calibrated on option
prices is constructed, from cadlag BMO martingales.

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## Viscosity Solutions and American Option Pricing in a Stochastic Volatility Model of the Ornstein-Uhlenbeck Type

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 12/12/2008
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In this paper, we study the valuation of American type derivatives in the
stochastic volatility model of Barndorff-Nielsen and Shephard (2001). We
characterize the value of such derivatives as the unique viscosity solution of
an integral-partial differential equation when the payoff function satisfies a
Lipschitz condition.

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## On the Ruin Probability of the Generalised Ornstein-Uhlenbeck Process in the Cram\'er Case

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 05/01/2011
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For a bivariate \Levy process $(\xi_t,\eta_t)_{t\ge 0}$ and initial value
$V_0$ define the Generalised Ornstein-Uhlenbeck (GOU) process \[
V_t:=e^{\xi_t}\Big(V_0+\int_0^t e^{-\xi_{s-}}\ud \eta_s\Big),\quad t\ge0,\] and
the associated stochastic integral process \[Z_t:=\int_0^t e^{-\xi_{s-}}\ud
\eta_s,\quad t\ge0.\] Let $T_z:=\inf\{t>0:V_t<0\mid V_0=z\}$ and
$\psi(z):=P(T_z<\infty)$ for $z\ge 0$ be the ruin time and infinite horizon
ruin probability of the GOU. Our results extend previous work of Nyrhinen
(2001) and others to give asymptotic estimates for $\psi(z)$ and the
distribution of $T_z$ as $z\to\infty$, under very general, easily checkable,
assumptions, when $\xi$ satisfies a Cram\'er condition.

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## Binomial Approximations for Barrier Options of Israeli Style

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 23/07/2009
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We show that prices and shortfall risks of game (Israeli) barrier options in
a sequence of binomial approximations of the Black--Scholes (BS) market
converge to the corresponding quantities for similar game barrier options in
the BS market with path dependent payoffs and the speed of convergence is
estimated, as well. The results are new also for usual American style options
and they are interesting from the computational point of view, as well, since
in binomial markets these quantities can be obtained via dynamical programming
algorithms. The paper continues the study of [11]and [7] but requires
substantial additional arguments in view of pecularities of barrier options
which, in particular, destroy the regularity of payoffs needed in the above
papers.

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## Local well-posedness of Musiela's SPDE with L\'evy noise

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

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We determine sufficient conditions on the volatility coefficient of Musiela's
stochastic partial differential equation driven by an infinite dimensional
L{\'e}vy process so that it admits a unique local mild solution in spaces of
functions whose first derivative is square integrable with respect to a weight.; Comment: Final version

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## Bubbles, convexity and the Black--Scholes equation

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 31/08/2009
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A bubble is characterized by the presence of an underlying asset whose
discounted price process is a strict local martingale under the pricing
measure. In such markets, many standard results from option pricing theory do
not hold, and in this paper we address some of these issues. In particular, we
derive existence and uniqueness results for the Black--Scholes equation, and we
provide convexity theory for option pricing and derive related ordering results
with respect to volatility. We show that American options are convexity
preserving, whereas European options preserve concavity for general payoffs and
convexity only for bounded contracts.; Comment: Published in at http://dx.doi.org/10.1214/08-AAP579 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org)

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## Optimal reinsurance/investment problems for general insurance models

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 31/08/2009
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#Mathematics - Probability#Quantitative Finance - Portfolio Management#91B28, 91B30 (Primary) 60H10, 93G20 (Secondary)

In this paper the utility optimization problem for a general insurance model
is studied. The reserve process of the insurance company is described by a
stochastic differential equation driven by a Brownian motion and a Poisson
random measure, representing the randomness from the financial market and the
insurance claims, respectively. The random safety loading and stochastic
interest rates are allowed in the model so that the reserve process is
non-Markovian in general. The insurance company can manage the reserves through
both portfolios of the investment and a reinsurance policy to optimize a
certain utility function, defined in a generic way. The main feature of the
problem lies in the intrinsic constraint on the part of reinsurance policy,
which is only proportional to the claim-size instead of the current level of
reserve, and hence it is quite different from the optimal
investment/consumption problem with constraints in finance. Necessary and
sufficient conditions for both well posedness and solvability will be given by
modifying the ``duality method'' in finance and with the help of the
solvability of a special type of backward stochastic differential equations.; Comment: Published in at http://dx.doi.org/10.1214/08-AAP582 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org)

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## Utility maximization in incomplete markets

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 24/08/2005
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#Mathematics - Probability#Quantitative Finance - Computational Finance#60H10, 91B28 (Primary) 60G44, 91B70, 91B16, 60H20, 93E20.
(Secondary)

We consider the problem of utility maximization for small traders on
incomplete financial markets. As opposed to most of the papers dealing with
this subject, the investors' trading strategies we allow underly constraints
described by closed, but not necessarily convex, sets. The final wealths
obtained by trading under these constraints are identified as stochastic
processes which usually are supermartingales, and even martingales for
particular strategies. These strategies are seen to be optimal, and the
corresponding value functions determined simply by the initial values of the
supermartingales. We separately treat the cases of exponential, power and
logarithmic utility.; Comment: Published at http://dx.doi.org/10.1214/105051605000000188 in the
Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute
of Mathematical Statistics (http://www.imstat.org)

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## Dynamic exponential utility indifference valuation

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 25/08/2005
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#Mathematics - Probability#Quantitative Finance - Computational Finance#91B28, 60H10, 91B16, 60G48 (Primary)

We study the dynamics of the exponential utility indifference value process
C(B;\alpha) for a contingent claim B in a semimartingale model with a general
continuous filtration. We prove that C(B;\alpha) is (the first component of)
the unique solution of a backward stochastic differential equation with a
quadratic generator and obtain BMO estimates for the components of this
solution. This allows us to prove several new results about C_t(B;\alpha). We
obtain continuity in B and local Lipschitz-continuity in the risk aversion
\alpha, uniformly in t, and we extend earlier results on the asymptotic
behavior as \alpha\searrow0 or \alpha\nearrow\infty to our general setting.
Moreover, we also prove convergence of the corresponding hedging strategies.; Comment: Published at http://dx.doi.org/10.1214/105051605000000395 in the
Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute
of Mathematical Statistics (http://www.imstat.org)

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## Dynamic State Tameness

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 06/09/2005
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#Mathematics - Probability#Mathematics - Optimization and Control#Quantitative Finance - Pricing of Securities#91B28 (Primary) 60H10, 60G40 (Secondary)

An extension of the idea of state tameness is presented in a dynamic
framework. The proposed model for financial markets is rich enough to provide
analytical tools that are mostly obtained in models that arise as the solution
of SDEs with deterministic coefficients. In the presented model the
augmentation by a shadow stock of the price evolution has a Markovian
character. As in a previous paper, the results obtained on valuation of
European contingent claims and American contingent claims do not require the
full range of the volatility matrix. Under some additional continuity
conditions, the conceptual framework provided by the model makes it possible to
regard the valuation of financial instruments of the European type as a
particular case of valuation of instruments of American type. This provides a
unifying framework for the problem of valuation of financial instruments.; Comment: 19 pages

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## Properties of option prices in models with jumps

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

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We study convexity and monotonicity properties of option prices in a model
with jumps using the fact that these prices satisfy certain parabolic
integro-differential equations. Conditions are provided under which
preservation of convexity holds, i.e. under which the value, calculated under a
chosen martingale measure, of an option with a convex contract function is
convex as a function of the underlying stock price. The preservation of
convexity is then used to derive monotonicity properties of the option value
with respect to the different parameters of the model, such as the volatility,
the jump size and the jump intensity.; Comment: 14 pages

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## GKW representation theorem and linear BSDEs under restricted information. An application to risk-minimization

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

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In this paper we provide Galtchouk-Kunita-Watanabe representation results in
the case where there are restrictions on the available information. This allows
to prove existence and uniqueness for linear backward stochastic differential
equations driven by a general c\`adl\`ag martingale under partial information.
Furthermore, we discuss an application to risk-minimization where we extend the
results of F\"ollmer and Sondermann (1986) to the partial information framework
and we show how our result fits in the approach of Schweizer (1994).; Comment: 22 pages

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## On the Hedging of American Options in Discrete Time Markets with Proportional Transaction Costs

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 09/02/2005
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#Mathematics - Probability#Quantitative Finance - Pricing of Securities#MSC Classification (2000): 91B28, 60G40

In this note, we consider a general discrete time financial market with
proportional transaction costs as in Kabanov and Stricker (2001), Kabanov et
al. (2002), Kabanov et al. (2003) and Schachermayer (2004). We provide a dual
formulation for the set of initial endowments which allow to super-hedge some
American claim. We show that this extends the result of Chalasani and Jha
(2001) which was obtained in a model with constant transaction costs and risky
assets which evolve on a finite dimensional tree. We also provide fairly
general conditions under which the expected formulation in terms of stopping
times does not work.

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## Approximate Bermudan option pricing based on the r\'eduite or cubature: soundness and characterisation of perpetual prices as fixed points

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

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In this paper, it is shown that Bermudan option pricing based on either the
r\'eduite (in a one-dimensional setting: piecewise harmonic interpolation) or
cubature -- is sensible from an economic vantage point: Any sequence of
thus-computed prices for Bermudan options with increasing sets of exercise
times is increasing. Furthermore, under certain regularity assumptions on the
payoff function and provided the exercise times are equidistant of exercise
mesh size $h$, it has a supremum which coincides with the least fixed point of
the approximate pricing algorithm -- this algorithm being perceived as a map
that assigns to any real-valued function $f$ (on the basket of underlyings) the
approximate value of the European option of maturity $h$ and payoff function
$f$.; Comment: 16 pages; major revision, similar results for cubature-based pricing
included

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## A brief note on the soundness of Bermudan option pricing via cubature

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 12/03/2005
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The subject of this study is an iterative Bermudan option pricing algorithm
based on (high-dimensional) cubature. We show that the sequence of Bermudan
prices (as functions of the underlying assets' logarithmic start prices)
resulting from the iteration is bounded and increases monotonely to the
approximate perpetual Bermudan option price; the convergence is linear in the
supremum norm with the discount factor being the convergence factor.
Furthermore, we prove a characterisation of this approximated perpetual
Bermudan price as the smallest fixed point of the iteration procedure.; Comment: 5 pages

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## Utility Maximization with a Stochastic Clock and an Unbounded Random Endowment

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 24/03/2005
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#Mathematics - Probability#Quantitative Finance - Computational Finance#91B28 (Primary) 60G99, 60H99. (Secondary)

We introduce a linear space of finitely additive measures to treat the
problem of optimal expected utility from consumption under a stochastic clock
and an unbounded random endowment process. In this way we establish existence
and uniqueness for a large class of utility-maximization problems including the
classical ones of terminal wealth or consumption, as well as the problems that
depend on a random time horizon or multiple consumption instances. As an
example we explicitly treat the problem of maximizing the logarithmic utility
of a consumption stream, where the local time of an Ornstein-Uhlenbeck process
acts as a stochastic clock.; Comment: Published at http://dx.doi.org/10.1214/105051604000000738 in the
Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute
of Mathematical Statistics (http://www.imstat.org)

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## Sensitivity analysis of the early exercise boundary for American style of Asian options

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 16/01/2011
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#Quantitative Finance - Computational Finance#Quantitative Finance - Pricing of Securities#35K15, 35K55, 90A09, 91B28

In this paper we analyze American style of floating strike Asian call options
belonging to the class of financial derivatives whose payoff diagram depends
not only on the underlying asset price but also on the path average of
underlying asset prices over some predetermined time interval. The mathematical
model for the option price leads to a free boundary problem for a parabolic
partial differential equation. Applying fixed domain transformation and
transformation of variables we develop an efficient numerical algorithm based
on a solution to a non-local parabolic partial differential equation for the
transformed variable representing the synthesized portfolio. For various types
of averaging methods we investigate the dependence of the early exercise
boundary on model parameters.

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## High-order ADI scheme for option pricing in stochastic volatility models

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 08/12/2015
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#Quantitative Finance - Computational Finance#Mathematics - Numerical Analysis#65M06 \sep 91B28 65M06 \sep 91B28 65M06 \sep 91B28 65M06 \sep 91B28
65M06 \sep 91B28 65M06, 91B28

We propose a new high-order alternating direction implicit (ADI) finite
difference scheme for the solution of initial-boundary value problems of
convection-diffusion type with mixed derivatives and non-constant coefficients,
as they arise from stochastic volatility models in option pricing. Our approach
combines different high-order spatial discretisations with Hundsdorfer and
Verwer's ADI time-stepping method, to obtain an efficient method which is
fourth-order accurate in space and second-order accurate in time. Numerical
experiments for the European put option pricing problem using Heston's
stochastic volatility model confirm the high-order convergence.; Comment: 18 pages

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