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Amogh Deshpande - One of the best experts on this subject based on the ideXlab platform.

  • On the role of Föllmer-Schweizer minimal Martingale Measure in risk-sensitive control asset management
    Journal of Applied Probability, 2020
    Co-Authors: Amogh Deshpande
    Abstract:

    Kuroda and Nagai (2002) stated that the factor process in risk-sensitive control asset management is stable under the Föllmer-Schweizer minimal Martingale Measure. Fleming and Sheu (2002) and, more recently, Föllmer and Schweizer (2010) observed that the role of the minimal Martingale Measure in this portfolio optimization is yet to be established. In this paper we aim to address this question by explicitly connecting the optimal wealth allocation to the minimal Martingale Measure. We achieve this by using a ‘trick’ of observing this problem in the context of model uncertainty via a two person zero sum stochastic differential game between the investor and an antagonistic market that provides a probability Measure. We obtain some startling insights. Firstly, if short selling is not permitted and the factor process evolves under the minimal Martingale Measure, then the investor's optimal strategy can only be to invest in the riskless asset (i.e. the no-regret strategy). Secondly, if the factor process and the stock price process have independent noise, then, even if the market allows short-selling, the optimal strategy for the investor must be the no-regret strategy while the factor process will evolve under the minimal Martingale Measure.

  • on the role of follmer schweizer minimal Martingale Measure in risk sensitive control asset management
    Journal of Applied Probability, 2015
    Co-Authors: Amogh Deshpande
    Abstract:

    Kuroda and Nagai \cite{KN} state that the factor process in the Risk Sensitive control Asset Management (RSCAM) is stable under the F\"ollmer-Schweizer minimal Martingale Measure . Fleming and Sheu \cite{FS} and more recently F\"ollmer and Schweizer \cite{FoS} have observed that the role of the minimal Martingale Measure in this portfolio optimization is yet to be established. In this article we aim to address this question by explicitly connecting the optimal wealth allocation to the minimal Martingale Measure. We achieve this by using a "trick" of observing this problem in the context of model uncertainty via a two person zero sum stochastic differential game between the investor and an antagonistic market that provides a probability Measure. We obtain some startling insights. Firstly, if short-selling is not permitted and if the factor process evolves under the minimal Martingale Measure then the investor's optimal strategy can only be to invest in the riskless asset (i.e. the no-regret strategy). Secondly, if the factor process and the stock price process have independent noise, then even if the market allows short selling, the optimal strategy for the investor must be the no-regret strategy while the factor process will evolve under the minimal Martingale Measure .

  • on the role of f ollmer schweizer minimal Martingale Measure in risk sensitive control asset management
    arXiv: Portfolio Management, 2014
    Co-Authors: Amogh Deshpande
    Abstract:

    Kuroda and Nagai \cite{KN} state that the factor process in the Risk Sensitive control Asset Management (RSCAM) is stable under the F\"ollmer-Schweizer minimal Martingale Measure . Fleming and Sheu \cite{FS} and more recently F\"ollmer and Schweizer \cite{FoS} have observed that the role of the minimal Martingale Measure in this portfolio optimization is yet to be established. In this article we aim to address this question by explicitly connecting the optimal wealth allocation to the minimal Martingale Measure. We achieve this by using a "trick" of observing this problem in the context of model uncertainty via a two person zero sum stochastic differential game between the investor and an antagonistic market that provides a probability Measure. We obtain some startling insights. Firstly, if short-selling is not permitted and if the factor process evolves under the minimal Martingale Measure then the investor's optimal strategy can only be to invest in the riskless asset (i.e. the no-regret strategy). Secondly, if the factor process and the stock price process have independent noise, then even if the market allows short selling, the optimal strategy for the investor must be the no-regret strategy while the factor process will evolve under the minimal Martingale Measure .

Martin Schweizer - One of the best experts on this subject based on the ideXlab platform.

  • minimal Martingale Measure
    Encyclopedia of Quantitative Finance, 2010
    Co-Authors: Hans Follmer, Martin Schweizer
    Abstract:

    Suppose discounted asset prices in a financial market are given by a P-semiMartingale of the form S = S0 + M + A. The minimal Martingale Measure for S is characterized by the properties that it turns S into a local Martingale and preserves the Martingale property for any local P-Martingale strongly P-orthogonal to M. It plays a key role in finding locally risk-minimizing strategies, and it comes up in various other contexts as well. Importantly, its density process can be written explicitly in terms of M and A, so that one can use it very generally and broadly. In some specific settings, it also has other optimality properties. Keywords: Martingale Measure; local risk-minimization; structure condition; Follmer–Schweizer decomposition; hedging; option pricing; quadratic hedging criteria

  • minimal entropy Martingale Measure
    Encyclopedia of Quantitative Finance, 2010
    Co-Authors: Martin Schweizer
    Abstract:

    Suppose that discounted asset prices in a financial market are given by a P-semiMartingale S. Among all probability Measures Q that turn S into a local Q-Martingale, the minimal entropy Martingale Measure is characterized by the property that it minimizes the relative entropy with respect to P. Via convex duality, it is intimately linked to the problem of maximizing expected exponential utility from terminal wealth. It also appears as a limit of p-optimal Martingale Measures as p decreases to 1. Like for most optimal Martingale Measures, finding its explicit form is easy if S is an exponential Levy process and quite difficult otherwise. Keywords: Martingale Measure; relative entropy; exponential utility maximization; duality; exponential Levy process; Esscher transform; utility indifference valuation; backward stochastic differential equations

  • Encyclopedia of Quantitative Finance - Minimal Martingale Measure
    Encyclopedia of Quantitative Finance, 2010
    Co-Authors: Hans Follmer, Martin Schweizer
    Abstract:

    Suppose discounted asset prices in a financial market are given by a P-semiMartingale of the form S = S0 + M + A. The minimal Martingale Measure for S is characterized by the properties that it turns S into a local Martingale and preserves the Martingale property for any local P-Martingale strongly P-orthogonal to M. It plays a key role in finding locally risk-minimizing strategies, and it comes up in various other contexts as well. Importantly, its density process can be written explicitly in terms of M and A, so that one can use it very generally and broadly. In some specific settings, it also has other optimality properties. Keywords: Martingale Measure; local risk-minimization; structure condition; Follmer–Schweizer decomposition; hedging; option pricing; quadratic hedging criteria

  • Encyclopedia of Quantitative Finance - Minimal Entropy Martingale Measure
    Encyclopedia of Quantitative Finance, 2010
    Co-Authors: Martin Schweizer
    Abstract:

    Suppose that discounted asset prices in a financial market are given by a P-semiMartingale S. Among all probability Measures Q that turn S into a local Q-Martingale, the minimal entropy Martingale Measure is characterized by the property that it minimizes the relative entropy with respect to P. Via convex duality, it is intimately linked to the problem of maximizing expected exponential utility from terminal wealth. It also appears as a limit of p-optimal Martingale Measures as p decreases to 1. Like for most optimal Martingale Measures, finding its explicit form is easy if S is an exponential Levy process and quite difficult otherwise. Keywords: Martingale Measure; relative entropy; exponential utility maximization; duality; exponential Levy process; Esscher transform; utility indifference valuation; backward stochastic differential equations

  • the minimal Martingale Measure
    2010
    Co-Authors: F Hans, Martin Schweizer, Unter Den Linden, Departement Mathematik
    Abstract:

    Suppose discounted asset prices in a flnancial market are given by a P -semimar- tingale of the form S = S0 +M +A. The minimal Martingale Measure for S is characterised by the properties that it turns S into a local Martingale and pre- serves the Martingale property for any localP -Martingale stronglyP -orthogonal to M. It plays a key role in flnding locally risk-minimising strategies, and it comes up in various other contexts as well. Importantly, its density process can be written explicitly in terms of M and A, so that one can use it very generally and broadly. In some speciflc settings, it also has other optimality properties.

Takuji Arai - One of the best experts on this subject based on the ideXlab platform.

Fred Espen Benth - One of the best experts on this subject based on the ideXlab platform.

  • The density process of the minimal entropy Martingale Measure in a stochastic volatility model with jumps
    Finance and Stochastics, 2020
    Co-Authors: Fred Espen Benth, Thilo Meyer-brandis
    Abstract:

    We derive the density process of the minimal entropy Martingale Measure in the stochastic volatility model proposed by Barndorff-Nielsen and Shephard [2]. The density is represented by the logarithm of the value function for an investor with exponential utility and no claim issued, and a Feynman-Kac representation of this function is provided. The dynamics of the processes determining the price and volatility are explicitly given under the minimal entropy Martingale Measure, and we derive a Black & Scholes equation with integral term for the price dynamics of derivatives. It turns out that the minimal entropy price of a derivative is given by the solution of a coupled system of two integro-partial differential equations. Copyright Springer-Verlag Berlin/Heidelberg 2005Stochastic volatility, Lévy processes, subordinators, minimal entropy Martingale Measure, density process, incomplete market, indifference pricing of derivatives, integro-partial differential equations,

  • the minimal entropy Martingale Measure and numerical option pricing for the barndorff nielsen shephard stochastic volatility model
    Stochastic Analysis and Applications, 2009
    Co-Authors: Fred Espen Benth, Martin Groth
    Abstract:

    Abstract We develop and apply a numerical scheme for pricing options in the stochastic volatility model proposed by Barndorff–Nielsen and Shephard. This non-Gaussian Ornstein–Uhlenbeck type of volatility model gives rise to an incomplete market, and we consider the option prices under the minimal entropy Martingale Measure. To numerically price options with respect to this risk neutral Measure, one needs to consider a Black and Scholes type of partial differential equation, with an integro-term arising from the volatility process. We suggest finite difference schemes to solve this parabolic integro-partial differential equation, and derive appropriate boundary conditions for the finite difference method. As an application of our algorithm, we consider price deviations from the Black and Scholes formula for call options, and the implications of the stochastic volatility on the shape of the volatility smile.

  • the density process of the minimal entropy Martingale Measure in a stochastic volatility model with jumps
    Finance and Stochastics, 2005
    Co-Authors: Fred Espen Benth, Thilo Meyerbrandis
    Abstract:

    We derive the density process of the minimal entropy Martingale Measure in the stochastic volatility model proposed by Barndorff-Nielsen and Shephard [2]. The density is represented by the logarithm of the value function for an investor with exponential utility and no claim issued, and a Feynman-Kac representation of this function is provided. The dynamics of the processes determining the price and volatility are explicitly given under the minimal entropy Martingale Measure, and we derive a Black & Scholes equation with integral term for the price dynamics of derivatives. It turns out that the minimal entropy price of a derivative is given by the solution of a coupled system of two integro-partial differential equations. Copyright Springer-Verlag Berlin/Heidelberg 2005

  • a pde representation of the density of the minimal entropy Martingale Measure in stochastic volatility markets
    Stochastics An International Journal of Probability and Stochastic Processes, 2005
    Co-Authors: Fred Espen Benth, Kenneth H Karlsen
    Abstract:

    Under general conditions stated in Rheinlander [An entropy approach to the stein/stein model with correlation. Preprint, 2003, ETH Zurich.], we prove that in a stochastic volatility market the Radon–Nikodym density of the minimal entropy Martingale Measure (MEMM) can be expressed in terms of the solution of a semilinear PDE. The semilinear PDE is suggested by the dynamic programming approach to the utility indifference pricing problem of contingent claims. One of our main results is the existence and uniqueness of a classical solution of the semilinear PDE in the case of a general stochastic volatility model with additive noise correlated with the asset price. Our results are applied to the Stein–Stein and Heston stochastic volatility models.

  • indifference pricing and the minimal entropy Martingale Measure in a stochastic volatility model with jumps
    2004
    Co-Authors: Fred Espen Benth, Thilo Meyerbrandis
    Abstract:

    We use the dynamic programming approach to derive an equation for the utility indierence price of Markovian claims in a stochastic volatility model proposed by Barndor-Nielsen and Shephard (3). The pricing equation is a Black & Scholes equation with a nonlinear integral term involving the risk preferences of the investor. Passing to the zero risk aversion limit, we present a Feynman-Kac representation of the minimal entropy price. The density of the minimal entropy Martingale Measure is found via the Girsanov transform of the Brownian motion and a subordinator process controlling the jumps in the volatility model. The density is represented by the logarithm of the value function for an investor with exponential utility and no claim issued, and a Feynman-Kac representation of this function is provided. We calculate the function explicitly in a special case, and show some properties in the general case.

Leunglung Chan - One of the best experts on this subject based on the ideXlab platform.

  • option pricing and esscher transform under regime switching
    Annals of Finance, 2005
    Co-Authors: Robert J Elliott, Leunglung Chan
    Abstract:

    We consider the option pricing problem when the risky underlying assets are driven by Markov-modulated Geometric Brownian Motion (GBM). That is, the market parameters, for instance, the market interest rate, the appreciation rate and the volatility of the underlying risky asset, depend on unobservable states of the economy which are modelled by a continuous-time Hidden Markov process. The market described by the Markov-modulated GBM model is incomplete in general and, hence, the Martingale Measure is not unique. We adopt a regime switching random Esscher transform to determine an equivalent Martingale pricing Measure. As in Miyahara [33], we can justify our pricing result by the minimal entropy Martingale Measure (MEMM).