Martingale Densities Research Articles

Robust estimation of superhedging prices

We consider statistical estimation of superhedging prices using historical stock returns in a frictionless market with $d$ traded assets. We introduce a plug-in estimator based on empirical measures and show it is consistent but lacks suitable robustness. To address this, we propose novel estimators which use a larger set of martingale measures defined through a tradeoff between the radius of Wasserstein balls around the empirical measure and the allowed norm of martingale densities. We then extend our study, in part, to estimation of risk measures, to the case of markets with traded options, to a multi-period setting and to settings with model uncertainty. We also study convergence rates of estimators and convergence of super-hedging strategies.

Description of Minimal Entropy Hellinger Sigma Martingale Density of Order One, Order <i>q</i> and Order Zero

Generally in this paper, we show how the new version of parameter in Jacod decomposition will change an expression of entropy-Hellinger process of order one, order q and order zero and consequently an equation of minimal entropy Hellinger sigma martingale density for all orders. This is because even the measurable function which is an important parameter of an equation of minimal martingale density changes. In order to get a required parameter , we introduce the function during our calculation for all orders. The result is different to order zero because we failed to get an equation of minimal entropy-Hellinger sigma martingale density of order zero.

Explicit Description of HARA Forward Utilities and Their Optimal Portfolios

This paper deals with forward performances of HARA type. Precisely, for a market model in which stock price processes are modeled by a locally bounded $d$-dimensional semimartingale, we elaborate a complete and explicit characterization for this type of forward utilities. In particular, under some mild technical assumptions of integrability, we prove that the risk aversion process of a power-type forward utility is constant. Furthermore, the optimal portfolios for all HARA forward utilities are explicitly described. Our approach is based on the minimal Hellinger martingale densities that are obtained from the important statistical concept of Hellinger process. These martingale densities were introduced recently and appeared herein tailor-made for these forward utilities. After outlining our parametrization method for the HARA forward, we provide illustrations on discrete-time market models.

Locally Ф-integrable σ-martingale densitiesfor general semimartingales

A --martingale density for a given stochastic process is a local -martingale starting at 1 such that the product is a --martingale. Existence of a --martingale density is equivalent to a classic absence-of-arbitrage property of , and it is invariant if we replace the reference measure with a locally equivalent measure . Now suppose that there exists a --martingale density for . Can we find another --martingale density for having some extra local integrability under ? We show that the answer is always positive for one part of that we identify, and we show that the complete answer depends in a precise quantitative way on the local integrability of the drift-to-jump ratio of the remaining ‘jumpy’ part of . Our proofs provide in addition new ideas and results in infinite-dimensional spaces.

On the Upper Hedging Price of Contingent Claims

We consider the problem of finding the upper hedging prices of a contingent claim in a semimartingale model of a financial market. In one case, an investor may use financial strategies from a given set, and in the other, the capital process of the investor should be nonnegative. In the first case, we prove that the price can be expressed as the supremum over the set of separating finitely additive measures and reduce the obtained formula to known results. In the second case, we obtain a representation of the corresponding price as the supremum over the sets of $\sigma$-martingale and supermartingale densities. Also we study the relation of the hedging prices introduced under different requirements on arbitrage.

Convex Comparison of Minimal Divergence Martingale Measures in Discrete Time Models

We refine some criteria for the convex comparison of martingale densities suggested in Franke et al. (1999) and Bellini and Sgarra (2012). We give sufficient conditions for comparison based on the classical notion of comparative convexity. We apply these conditions to the case of minimal f-divergence martingale measures, establishing an ordering result in the case of power divergences. We discuss the extension of the comparison to a multiperiod setting and provide several numerical examples.

SUPERHEDGING IN ILLIQUID MARKETS

We study superhedging of securities that give random payments possibly at multiple dates. Such securities are common in practice where, due to illiquidity, wealth cannot be transferred quite freely in time. We generalize some classical characterizations of superhedging to markets where trading costs may depend nonlinearly on traded amounts and portfolios may be subject to constraints. In addition to classical frictionless markets and markets with transaction costs or bid-ask spreads, our model covers markets with nonlinear illiquidity effects for large instantaneous trades. The characterizations are given in terms of stochastic term structures which generalize term structures of interest rates beyond fixed income markets as well as martingale densities beyond stochastic markets with a cash account. The characterizations are valid under a topological condition and a minimal consistency condition, both of which are implied by the no arbitrage condition in the case of classical perfectly liquid market models. We give alternative sufficient conditions that apply to market models with general convex cost functions and portfolio constraints.

No Arbitrage and the Growth Optimal Portfolio

Recently, several papers have expressed an interest in applying the Growth Optimal Portfolio (GOP) for pricing derivatives. We show that the existence of a GOP is equivalent to the existence of a strictly positive martingale density. Our approach circumvents two assumptions usually set forth in the literature: 1) infinite expected growth rates are permitted and 2) the market does not need to admit an equivalent martingale measure. In particular, our approach shows that models featuring credit constrained arbitrage may still allow a GOP to exist because this type of arbitrage can be removed by a change of numéraire. However, if the GOP exists the market admits an equivalent martingale measure under some numéraire and hence derivatives can be priced. The structure of martingale densities is used to provide a new characterization of the GOP which emphasizes the relation to other methods of pricing in incomplete markets. The case where GOP denominated asset prices are strict supermartingales is analyzed in the case of pure jump driven uncertainty.

MORE ON MINIMAL ENTROPY–HELLINGER MARTINGALE MEASURE

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.

On the law of one price

We consider the standard discrete-time model of a frictionless financial market and show that the law of one price holds if and only if there exists a martingale density process with strictly positive initial value. In contrast to the classical no-arbitrage criteria, this density process may change its sign. We also give an application to the CAPM.

Convergence in Incomplete Market Models

The problem of pricing and hedging of contingent claims in incomplete markets has led to the development of various valuation methodologies. This paper examines the mean-variance approach to risk-minimisation and shows that it is robust under the convergence from discrete- to continuous-time market models. This property yields new convergence results for option prices, trading strategies and value processes in incomplete market models. Techniques from nonstandard analysis are used to develop new results for the lifting property of the minimal martingale density and risk-minimising strategies. These are applied to a number of incomplete market models: It is shown that the convergence of the underlying models implies the convergence of strategies and value processes for multinomial models and approximations of the Black-Scholes model by direct discretisation of the price process. The concept of $D^2$-convergence is extended to these classes of models, including the construction of discretisation schemes. This yields new standard convergence results for these models. For ease of reference a summary of the main results from nonstandard analysis in the context of stochastic analysis is given as well as a brief introduction to mean-variance hedging and pricing.

Martingale densities for general asset prices

This paper discusses some properties of general asset prices in continuous time. We introduce the concept of a martingale density which is a generalization of an equivalent martingale measure, and we show that absence of arbitrage plus some technical conditions implies the existence of a martingale density. This is in turn already sufficient to derive a recent result of Back (1991) on local risk premia for asset returns. As an application, we obtain a simple condition, valid in arbitrary information structures, for the drift part of discounted security gains to be absolutely continuous with respect to the variance process of the martingale part.