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.

Keywords:

• Bond market
• Large market
• Martingale random field
• Minimal variance hedging
• Random field
• Stochastic derivative

Mathematics Subject Classification:

• Primary 60H05, 60G57, 60G60
• Secondary 91B24, 91B28

Notes

Recall that a measure is tight if for every δ > 0 there exists a compact X _δ such that m(X∖X _δ) < δ.

From the point of view of applications, this is not a strong assumption in fact we recall that any σ-finite Borel measure on a complete separable metric space is tight.

The stated convergence is a.s. and in L ₁, but as the elements are in L ₂ convergence in this sense follows.