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Theory of Rational Option Pricing: II (Revised: 1-96)

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  • Yaacov Z. Bergman
  • Bruce D. Grundy
  • Zvi Wiener

Abstract

This paper investigates the properties of contingent claim prices in a one dimensional diffusion world and establishes that (i) the delta of any claim is bounded above (below) by the sup (inf) of its delta at maturity, and (ii), if its payoff is convex (concave) then its current value is convex (concave) in the current value of the underlying. These properties are used as the foundation for a detailed study of the properties of option prices. Interestingly, although an upward shift in the term structure of interest rates will always increase a call’s value, a decline in the present value of the exercise price can be associated with a decline in the call price. We provide a new bound on the values of calls on dividend-paying assets. We establish that when the underlying’s instantaneous volatility is bounded above (below), the call price is bounded above (below) by its Black-Scholes value evaluated at the bounding volatility level. This leads to a new bound on a call’s delta. We also show that if changes in the value of the underlying follow a multidimensional diffusion (i.e., a stochastic volatility world), or are discontinuous or non-Markovian, then call option prices can exhibit properties very different from those of a Black-Scholes world: they can be decreasing, concave functions of the value of the underlying.

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Bibliographic Info

Paper provided by Wharton School Rodney L. White Center for Financial Research in its series Rodney L. White Center for Financial Research Working Papers with number 11-95.

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Handle: RePEc:fth:pennfi:11-95

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Cited by:
  1. Bernard Dumas & Jeff Fleming & Robert E. Whaley, 1996. "Implied Volatility Functions: Empirical Tests," NBER Working Papers 5500, National Bureau of Economic Research, Inc.
  2. Mark Broadie & Jérôme B. Detemple, 1996. "American Options on Dividend-Paying Assets," CIRANO Working Papers 96s-16, CIRANO.
  3. RØdiger Frey, 1998. "Perfect option hedging for a large trader," Finance and Stochastics, Springer, vol. 2(2), pages 115-141.
  4. Frey, Rüdiger, 1996. "The Pricing and Hedging of Options in Finitely Elastic Markets," Discussion Paper Serie B 372, University of Bonn, Germany.

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