Minimal-Variance Hedging in Large Financial Markets: Random Fields Approach

Abstract

We study a large financial market where the discounted asset prices are modeled by martingale random fields. This approach allows the treatment of both the cases of a market with a countable amount of assets and of a market with a continuum amount. We discuss conditions for these markets to be complete and we study the minimal variance hedging problem both in the case of full and partial information. An explicit representation of the minimal variance hedging portfolio is suggested. Techniques of stochastic differentiation are applied to achieve the main results. Examples of large market models with a countable number of assets are considered according to the literature and an example of market model with a continuum of assets is taken from the bond market.

A Maximum Principle for Mean-Field SDEs with Time Change

“…is then well-defined. See [8] (see also [2] for the specific case of martingale random fields with independently scattered values). The space of integrands denoted by…”

“…Notice that µ is a martingale random field with respect to G (and also F) in the sense of [8, Definition 2.1] and an Itô type non-anticipating integral I(φ) := T 0 R φ s (z)µ(ds, dz) is then well-defined. See [8] (see also [2] for the specific case of martingale random fields with independently scattered values). The space of integrands denoted by…”

“…, is a martingale (see [8], Remark 3.2)). Since F is right-continuous, then the martingale process of the integrals has a càdlàg version (see, e.g.…”

“…To give a uniform presentation we focus on martingale random fields generated by time changed Brownian and Poisson random fields. However we stress that the first part of the paper, dealing with the existence of solutions of a mean-field SDE, is valid for a general martingale random field with conditionally independent values as defined in [8], see also [5]. The reason for choosing these time changed driving noises comes from the balance between the relative easiness in generating noises in this way and the flexibility of this class of models from the point of view of applications.…”

Maximizing expected utility in the Arbitrage Pricing Model

“…Inspired by APM, an abstract framework for markets with "many" assets has been proposed in [14]. The theory of such "large financial markets" has been extensively investigated in [19,20,15,18,21,8,29,30,2,22,23,1,35,5,10]. There has also been a recent revival of interest in such models, see [25,6].…”