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Arbitrage-Based Pricing When Volatility is Stochastic

Bossaerts, Peter and Ghysels, Eric and Gouriéroux, Christian (1996) Arbitrage-Based Pricing When Volatility is Stochastic. Social Science Working Paper, 977. California Institute of Technology , Pasadena, CA. (Unpublished) https://resolver.caltech.edu/CaltechAUTHORS:20170815-164236183

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Abstract

In one of the early attempts to model stochastic volatility, Clark [1973] conjectured that the size of asset price movements is tied to the rate at which transactions occur. To formally analyze the econometric implications, he distinguished between transaction time and calendar time. The present paper exploits Clark's strategy for a different purpose, namely, asset pricing. It studies arbitrage-based pricing in economies where: (i) trade takes place in transaction time, (ii) there is a single state variable whose transaction time price path is binomial, (iii) there are risk-free bonds with calendar-time maturities, and (iv) the relation between transaction time and calendar time is stochastic. The state variable could be interpreted in various ways. E.g., it could be the price of a share of stock, as in Black and Scholes [1973], or a factor that summarizes changes in the investment opportunity set, as in Cox, Ingersoll and Ross [1985] or one that drives changes in the term structure of interest rates (Ho and Lee [1986], Heath, Jarrow and Morton [1992]). Property (iv) generally introduces stochastic volatility in the process of the state variable when recorded in calendar time. The paper investigates the pricing of derivative securities with calendar-time maturities. The restrictions obtained in Merton [1973] using simple buy-and-hold arbitrage portfolio arguments do not necessarily obtain. Conditions are derived for all derivatives to be priced by dynamic arbitrage, i.e., for market completeness in the sense of Harrison and Pliska [1981]. A particular class of stationary economies where markets are indeed complete is characterized.


Item Type:Report or Paper (Working Paper)
ORCID:
AuthorORCID
Bossaerts, Peter0000-0003-2308-2603
Additional Information:The paper benefited from comments in the UCLA Theory Workshop and during the Workshop on the Mathematics of Finance in Montreal (April 30-May 3 1996). Helyette Geman's comments made us extend the scope of the theory beyond stock option pricing.
Group:Social Science Working Papers
Subject Keywords:Incomplete Markets, Transaction Time, Change of Time, Stochastic Volatility
Series Name:Social Science Working Paper
Issue or Number:977
Classification Code:JEL: D52, G13
Record Number:CaltechAUTHORS:20170815-164236183
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20170815-164236183
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:80452
Collection:CaltechAUTHORS
Deposited By: Jacquelyn Bussone
Deposited On:16 Aug 2017 16:37
Last Modified:03 Oct 2019 18:31

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