Mean-Variance Hedging with Limited Capital - a Decomposition Result

18 Pages Posted: 27 Mar 2002

See all articles by Nicole Branger

Nicole Branger

University of Muenster - Finance Center Muenster

Christian Schlag

Goethe University Frankfurt - Research Center SAFE

Angelika Esser

Sal. Oppenheim Jr. & Cie.

Yulia Bondarenko

Goethe University Frankfurt - Faculty of Economics and Business Administration

Date Written: March 19, 2002

Abstract

This paper deals with the problem of quadratic hedging with limited initial capital. We show (i) that the optimal amount of capital for the quadratic hedge of a portfolio of contingent claims is equal to the sum of optimal investments for the individual hedges of its components and (ii) that the optimal hedging strategies for individual claims add up to the optimal strategy for the total position. These results can be combined to derive the main result of the paper saying that the expected squared hedging error (ESHE) when using limited initial capital can be decomposed into two parts: (i) the ESHE for the claim using the optimal amount of initial capital and (ii) the ESHE for a zero payoff using a negative initial capital equal to the difference between available and optimal capital. It is further shown that the increase in ESHE arising from the capital restriction is entirely determined by the squared difference between available and optimal capital. Especially, it is independent of the original claim to be hedged. This result has the important implication for risk management that in the case of limited capital the quadratic hedge of a contingent claim can be decomposed into two problems: first, the claim is hedged as if the optimal amount of capital was available, and then an additional quadratic hedge is set up for a zero payoff where now the initial capital is given by the (negative) difference between available and optimal capital. Numerical examples show that the volatility of the underlying asset is crucial for the ESHE obtained for a zero payoff when starting with an initial capital of minus one.

Keywords: Mean-Variance hedging, risk management, limited capital, expected squared hedging error, optimal trading strategy, incomplete market, hedging numeraire

JEL Classification: G11, G13

Suggested Citation

Branger, Nicole and Schlag, Christian and Esser, Angelika and Bondarenko, Yulia, Mean-Variance Hedging with Limited Capital - a Decomposition Result (March 19, 2002). Available at SSRN: https://ssrn.com/abstract=304456 or http://dx.doi.org/10.2139/ssrn.304456

Nicole Branger

University of Muenster - Finance Center Muenster ( email )

Universitatsstr. 14-16
Muenster, 48143
Germany
+49 251 83 29779 (Phone)
+49 251 83 22867 (Fax)

HOME PAGE: http://www.wiwi.uni-muenster.de/fcm/fcm/das-finance-center/details.php?weobjectID=162

Christian Schlag (Contact Author)

Goethe University Frankfurt - Research Center SAFE ( email )

(http://www.safe-frankfurt.de)
Theodor-W.-Adorno-Platz 3
Frankfurt am Main, 60323
Germany
+49 69 798 33699 (Phone)

Angelika Esser

Sal. Oppenheim Jr. & Cie. ( email )

Untermainanlage 1
Frankfurt, 60329
Germany

Yulia Bondarenko

Goethe University Frankfurt - Faculty of Economics and Business Administration ( email )

Mertonstrasse 17-25
Uni-PF 77
Frankfurt am Main, D-60325
Germany

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