Black Scholes for Portfolios of Options in Discrete Time

Tinbergen Institute Discussion Paper No. 2003-090/2

29 Pages Posted: 28 Nov 2003

See all articles by Bas Peeters

Bas Peeters

Vrije Universiteit Amsterdam, School of Business and Economics

Andre Lucas

Vrije Universiteit Amsterdam; Tinbergen Institute

Cees L. Dert

Vrije Universiteit Amsterdam, School of Business and Economics

Date Written: November 10, 2003

Abstract

Taking a portfolio perspective on option pricing and hedging, we show that within the standard Black-Scholes-Merton framework large portfolios of options can be hedged without risk in discrete time. The nature of the hedge portfolio in the limit of large portfolio size is substantially different from the standard continuous time delta-hedge. The underlying values of the options in our framework are driven by systematic and idiosyncratic risk factors. Instead of linearly (delta) hedging the total risk of each option separately, the correct hedge portfolio in discrete time eliminates linear (delta) as well as second (gamma) and higher order exposures to the systematic risk factor only. The idiosyncratic risk is not hedged, but diversified. Our result shows that preference free valuation of option portfolios using linear assets only is applicable in discrete time as well. The price paid for this result is that the number of securities in the portfolio has to grow indefinitely. This ties the literature on option pricing and hedging closer together with the APT literature in its focus on systematic risk factors. For portfolios of finite size, the optimal hedge strategy makes a trade-off between hedging linear idiosyncratic and higher order systematic risk.

Keywords: Option hedging, discrete time, portfolio approach, preference free valuation, hedging errors, Arbitrage Pricing Theory

JEL Classification: G13, G12

Suggested Citation

Peeters, Bas and Lucas, Andre and Dert, Cees L., Black Scholes for Portfolios of Options in Discrete Time (November 10, 2003). Tinbergen Institute Discussion Paper No. 2003-090/2, Available at SSRN: https://ssrn.com/abstract=469022 or http://dx.doi.org/10.2139/ssrn.469022

Bas Peeters (Contact Author)

Vrije Universiteit Amsterdam, School of Business and Economics ( email )

De Boelelaan 1105
Amsterdam, 1081HV
Netherlands

Andre Lucas

Vrije Universiteit Amsterdam ( email )

SBE/EDS, De Boelelaan 1105
Amsterdam, 1081 HV
Netherlands
+31 20 598 6039 (Phone)
+31 20 598 6020 (Fax)

HOME PAGE: http://personal.vu.nl/a.lucas

Tinbergen Institute

Roetersstraat 31
Amsterdam, 1018 WB
Netherlands

HOME PAGE: http://www.tinbergen.nl

Cees L. Dert

Vrije Universiteit Amsterdam, School of Business and Economics ( email )

De Boelelaan 1105
Amsterdam, 1081HV
Netherlands

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