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Martingale Measure (martingale + measure)

Distribution by Scientific Domains

Business, Economics, Finance and Accounting	100%

Kinds of Martingale Measure

equivalent martingale measure

Selected Abstracts

MORE ON MINIMAL ENTROPY,HELLINGER MARTINGALE MEASURE

MATHEMATICAL FINANCE, Issue 1 2006
Tahir Choulli
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. [source]

PROPERTIES OF OPTION PRICES IN MODELS WITH JUMPS

MATHEMATICAL FINANCE, Issue 3 2007
Erik Ekström
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. [source]

SOME REMARKS ON ARBITRAGE AND PREFERENCES IN SECURITIES MARKET MODELS

MATHEMATICAL FINANCE, Issue 3 2004
Marco Frittelli
We introduce the notion of a market-free-lunch that depends on the preferences of all agents participating in the market. In semimartingale models of securities markets, we characterize no arbitrage (NA) and no-free-lunch-with-vanishing-risk (NFLVR) in terms of the market-free-lunch and show that the difference between NA and NFLVR consists in the selection of the class of monotone, respectively monotone and continuous, utility functions that determines the absence of the market-free-lunch. We also provide a direct proof of the equivalence between the absence of a market-free-lunch, with respect to monotone concave preferences, and the existence of an equivalent (local/sigma) martingale measure. [source]

On the optimal portfolio for the exponential utility maximization: remarks to the six-author paper

MATHEMATICAL FINANCE, Issue 2 2002
Yuri M. Kabanov
This note contains ramifications of results of Delbaen et al. (2002). Assuming that the price process is locally bounded and admits an equivalent local martingale measure with finite entropy, we show, without further assumption, that in the case of exponential utility the optimal portfolio process is a martingale with respect to each local martingale measure with finite entropy. Moreover, the optimal value always can be attained on a sequence of uniformly bounded portfolios. [source]

Generic Existence and Robust Nonexistence of Numéraires in Finite Dimensional Securities Markets

MATHEMATICAL FINANCE, Issue 4 2000
Bruno Girotto
A numéraire is a portfolio that, if prices and dividends are denominated in its units, admits an equivalent martingale measure that transforms all gains processes into martingales. We first supply a necessary and sufficient condition for the generic existence of numéraires in a finite dimensional setting. We then characterize the arbitrage-free prices and dividends for which the absence of numéraires survives any small perturbation preserving no arbitrage. Finally, we identify the cases when any small, but otherwise arbitrary, perturbation of prices and dividends preserves either the existence of numéraires, or their nonexistence under no arbitrage. [source]

The Minimal Entropy Martingale Measure and the Valuation Problem in Incomplete Markets

MATHEMATICAL FINANCE, Issue 1 2000
Marco Frittelli
Let , be a family of stochastic processes on a given filtered probability space (,, F, (Ft)t,T, P) with T,R+. Under the assumption that the set Me of equivalent martingale measures for , is not empty, we give sufficient conditions for the existence of a unique equivalent martingale measure that minimizes the relative entropy, with respect to P, in the class of martingale measures. We then provide the characterization of the density of the minimal entropy martingale measure, which suggests the equivalence between the maximization of expected exponential utility and the minimization of the relative entropy. [source]

Alternative tilts for nonparametric option pricing

THE JOURNAL OF FUTURES MARKETS, Issue 10 2010
M. Ryan Haley
This study generalizes the nonparametric approach to option pricing of Stutzer, M. (1996) by demonstrating that the canonical valuation methodology introduced therein is one member of the Cressie,Read family of divergence measures. Alhough the limiting distribution of the alternative measures is identical to the canonical measure, the finite sample properties are quite different. We assess the ability of the alternative divergence measures to price European call options by approximating the risk-neutral, equivalent martingale measure from an empirical distribution of the underlying asset. A simulation study of the finite sample properties of the alternative measure changes reveals that the optimal divergence measure depends upon how accurately the empirical distribution of the underlying asset is estimated. In a simple Black,Scholes model, the optimal measure change is contingent upon the number of outliers observed, whereas the optimal measure change is a function of time to expiration in the stochastic volatility model of Heston, S. L. (1993). Our extension of Stutzer's technique preserves the clean analytic structure of imposing moment restrictions to price options, yet demonstrates that the nonparametric approach is even more general in pricing options than originally believed. © 2009 Wiley Periodicals, Inc. Jrl Fut Mark 30:983,1006, 2010 [source]

Equity swaps in a LIBOR market model

THE JOURNAL OF FUTURES MARKETS, Issue 9 2007
Ting-Pin Wu
This study extends the BGM (A. Brace, D. Gatarek, & M. Musiela, 1997) interest rate model (the London Interbank Offered Rate [LIBOR] market model) by incorporating the stock price dynamics under the martingale measure. As compared with traditional interest rate models, the extended BGM model is both appropriate for pricing equity swaps and easy to calibrate. The general framework for pricing equity swaps is proposed and applied to the pricing of floating-for-equity swaps with either constant or variable notional principals. The calibration procedure and the practical implementation are also discussed. © 2007 Wiley Periodicals, Inc. Jrl Fut Mark 27:893,920, 2007 [source]

COHERENT ACCEPTABILITY MEASURES IN MULTIPERIOD MODELS

MATHEMATICAL FINANCE, Issue 4 2005
Berend Roorda
The framework of coherent risk measures has been introduced by Artzner et al. (1999; Math. Finance 9, 203,228) in a single-period setting. Here, we investigate a similar framework in a multiperiod context. We add an axiom of dynamic consistency to the standard coherence axioms, and obtain a representation theorem in terms of collections of multiperiod probability measures that satisfy a certain product property. This theorem is similar to results obtained by Epstein and Schneider (2003; J. Econ. Theor. 113, 1,31) and Wang (2003; J. Econ. Theor. 108, 286,321) in a different axiomatic framework. We then apply our representation result to the pricing of derivatives in incomplete markets, extending results by Carr, Geman, and Madan (2001; J. Financial Econ. 32, 131,167) to the multiperiod case. We present recursive formulas for the computation of price bounds and corresponding optimal hedges. When no shortselling constraints are present, we obtain a recursive formula for price bounds in terms of martingale measures. [source]

ANALYTICAL COMPARISONS OF OPTION PRICES IN STOCHASTIC VOLATILITY MODELS

MATHEMATICAL FINANCE, Issue 1 2005
Vicky Henderson
This paper gives an ordering on option prices under various well-known martingale measures in an incomplete stochastic volatility model. Our central result is a comparison theorem that proves convex option prices are decreasing in the market price of volatility risk, the parameter governing the choice of pricing measure. The theorem is applied to order option prices under q -optimal pricing measures. In doing so, we correct orderings demonstrated numerically in Heath, Platen, and Schweizer (Mathematical Finance, 11(4), 2001) in the special case of the Heston model. [source]

No Arbitrage in Discrete Time Under Portfolio Constraints

MATHEMATICAL FINANCE, Issue 3 2001
Laurence Carassus
In frictionless securities markets, the characterization of the no-arbitrage condition by the existence of equivalent martingale measures in discrete time is known as the fundamental theorem of asset pricing. In the presence of convex constraints on the trading strategies, we extend this theorem under a closedness condition and a nondegeneracy assumption. We then provide connections with the superreplication problem solved in Föllmer and Kramkov (1997). [source]