Hedging Large Portfolios of Options in Discrete Time*

*. We thank Bert Menkveld, Roman Kraeussl, and seminar participants at Tilburg University, University of Amsterdam, Vrije Universiteit Amsterdam, European Economic Association 2004 in Madrid, and Quantitative Risk Management Conference 2004 London for useful comments.

Abstract

The problem studied is that of hedging a portfolio of options in discrete time where underlying security prices are driven by a combination of idiosyncratic and systematic risk factors. It is shown that despite the market incompleteness introduced by the discrete time assumption, large portfolios of options have a unique price and can be hedged without risk. The nature of the hedge portfolio in the limit of large portfolio size is substantially different from its continuous time counterpart. Instead of linearly hedging the total risk of each option separately, the correct portfolio hedge in discrete time eliminates linear as well as second and higher order exposures to the systematic risk factors only. The idiosyncratic risks need not be hedged, but disappear through diversification. Hedging portfolios of options in discrete time thus entails a trade‐off between dynamic and cross‐sectional hedging errors. Some computations are provided on the outcome of this trade‐off in a discrete‐time Black–Scholes world.

*. We thank Bert Menkveld, Roman Kraeussl, and seminar participants at Tilburg University, University of Amsterdam, Vrije Universiteit Amsterdam, European Economic Association 2004 in Madrid, and Quantitative Risk Management Conference 2004 London for useful comments.

Keywords:

• Option hedging
• discrete time
• preference free valuation
• hedging errors
• cross‐sectional hedging
• static hedging
• JEL Codes: G13, G12

Notes

*. We thank Bert Menkveld, Roman Kraeussl, and seminar participants at Tilburg University, University of Amsterdam, Vrije Universiteit Amsterdam, European Economic Association 2004 in Madrid, and Quantitative Risk Management Conference 2004 London for useful comments.

1. Static hedging has also been analysed elsewhere in the literature, see, for example, Carr et al. (1998). The focus there, however, is on statically replicating complex derivatives with simpler derivatives. Here, we concentrate on hedging the (non‐linear) systematic risk exposure by holding a static portfolio of stocks and a money market account only.

2. See also Kabanov and Kramkov (1998) for a thorough discussion of market completeness in infinite economies.

3. If the Black–Scholes–Merton continuous time process (1) is replaced by a more complicated process, also the Black–Scholes–Merton price equation has to be replaced by an appropriate, more complicated counterpart.

4. This appears to be supported by the empirical results in, for example, Gilster (1990).

5. The covariance between securities S_i and S_j is equal to β _i β _j .

6. Not imposing this restriction and minimizing mean‐squared‐errors rather than variances yields the same results as presented in the current paper.

7. This involves the computation of 8th order derivatives of the Black–Scholes–Merton price equation.