Mean-Variance Hedging with Limited Capital - a Decomposition Result

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