Making mean-variance hedging implementable in a partially observable market

Abstract

The mean-variance hedging (MVH) problem is studied in a partially observable market where the drift processes can only be inferred through the observation of asset or index processes. Although most of the literature treats the MVH problem by the duality method, here we study an equivalent system consisting of three BSDEs and try to provide more explicit expressions directly implementable by practitioners. Under the Bayesian and Kalman–Bucy frameworks, we find that a relevant BSDE can yield a semi-closed solution via a simple set of ODEs which allow quick numerical evaluation. This renders the remaining problems equivalent to solving European contingent claims under a new forward measure, and it is straightforward to obtain a forward looking non-sequential Monte Carlo simulation scheme. We also give a special example where the hedging position is available in a semi-closed form. For more generic set-ups, we provide explicit expressions of an approximate hedging portfolio by an asymptotic expansion. These analytic expressions not only allow the hedgers to update the hedging positions in real time but also make a direct analysis of the terminal distribution of the hedged portfolio feasible by standard Monte Carlo simulation.

Keywords:

• Mean-variance hedging
• BSDE
• Bayesian analysis
• Kalman–Bucy filter
• Asymptotic expansion
• Particle method

JEL Classification:

• G12
• C63

Acknowledgements

This research is partially supported by Center for Advanced Research in Finance (CARF).

Notes

All the contents expressed in this research are solely those of the authors and do not represent any views or opinions of any institutions. The authors are not responsible or liable in any manner for any losses and/or damages caused by the use of any contents in this research.

¹See also Pham and Quenez (2001) as an application of duality for utility maximization in a partially observable market.

²In fact, it is not difficult to add a stochastic interest rate. Especially, it is straightforward to include a set of interest rate futures, such as Euro-dollar Futures provided by CME group, in the current set-up. See Fujii and Takahashi (2013) for further details.

³Practically, one can still include bonds in his/her portfolio by modeling their dynamics directly just as equities or commodities.

⁴Put and use the corresponding for our first Bayesian model.

⁵The argument is omitted in for notational simplicity.

⁶Here, we put in (3.24). But the choice is free and what only matters for the dynamics of is .

⁷In theory, there is no need to expand by introducing since it already has a linear dynamics. However, if one treats exactly, the calculations associated with become hugely involved due to the presence of in its drift process most likely with only a minor improvement of accuracy.

⁸More precisely speaking, we need to consider the effect of time-integration together.