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Strategic asset allocation under a fractional hidden markov model

Elliott, R.; Siu, T.
Fonte: Kluwer Academic Publishers Publicador: Kluwer Academic Publishers
Tipo: Artigo de Revista Científica
Publicado em //2014 Português
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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

Compatibility between pricing rules and risk measures: the CCVaR

Balbás, Alejandro; Balbás, Raquel
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 Português
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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...

Time Consistent Dynamic Limit Order Books Calibrated on Options

Bion-Nadal, Jocelyne
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 22/09/2008 Português
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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.

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

Roch, Alexandre F.
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 12/12/2008 Português
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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.

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

Bankowski, Damien; Klüppelberg, Claudia; Maller, Ross
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 05/01/2011 Português
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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.

Binomial Approximations for Barrier Options of Israeli Style

Dolinsky, Yan; Kifer, Yuri
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 23/07/2009 Português
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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.

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

Marinelli, Carlo
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Português
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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

Bubbles, convexity and the Black--Scholes equation

Ekström, Erik; Tysk, Johan
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 31/08/2009 Português
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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)

Optimal reinsurance/investment problems for general insurance models

Liu, Yuping; Ma, Jin
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 31/08/2009 Português
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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)

Utility maximization in incomplete markets

Hu, Ying; Imkeller, Peter; Muller, Matthias
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 24/08/2005 Português
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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)

Dynamic exponential utility indifference valuation

Mania, Michael; Schweizer, Martin
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 25/08/2005 Português
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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)

Dynamic State Tameness

Londoño, Jaime A.
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 06/09/2005 Português
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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

Properties of option prices in models with jumps

Ekström, Erik; Tysk, Johan
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Português
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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

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

Ceci, Claudia; Cretarola, Alessandra; Russo, Francesco
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Português
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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

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

Bouchard, Bruno; Temam, Emmanuel
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 09/02/2005 Português
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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.

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

Herzberg, Frederik S.
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Português
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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

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

Herzberg, Frederik S.
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 12/03/2005 Português
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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

Utility Maximization with a Stochastic Clock and an Unbounded Random Endowment

Zitkovic, Gordan
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 24/03/2005 Português
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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)

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

Sevcovic, Daniel; Takac, Martin
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 16/01/2011 Português
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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.

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

Düring, Bertram; Miles, James
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 08/12/2015 Português
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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