Entropy 2001, 3[5], 300-324
Some Divergence Properties of Asset Price Models
Wolfgang Stummer
Department of Mathematical Stochastics, University of Karlsruhe,
Englerstrasse 2, D-76128 Karlsruhe, Germany.
E-mail: [email protected]
URL:
http://mspcdip.mathematik.uni-karlsruhe.de/personen/stummer/stummer.html
Received: 15 August 2001 / Accepted: 20 December 2001 / Published: 20 December 2001
Abstract:
We consider asset price processes Xt
which are weak solutions of one-dimensional stochastic differential equations of the form
dXt = b(t; Xt) dt + σt Xt dWt.
Such price models can be interpreted as non-lognormally-distributed
generalizations of the geometric Brownian motion. We study properties of
the Iα-divergence between the law of
the solution Xt and the corresponding drift-less measure
(the special case α=1 is the relative entropy).
This will be applied to some context in statistical information
theory as well as to arbitrage theory and contingent claim valuation. For
instance, the seminal option pricing theorems of Black-Scholes and Merton
appear as a special case.
Keywords:
Iα-divergence; relative entropy; statistical information;
equivalent martingale measure; option pricing; Black-Scholes-Merton.
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