DYNAMIC SPANNING: ARE OPTIONS AN APPROPRIATE INSTRUMENT?¹

Abstract

Ross (1976) has shown, in a static framework, how options can complete financial markets. This paper examines the possible extensions of Ross's idea in a dynamic setup. Surprisingly enough, we find that the answer is very sensitive to the choice of the stochastic model for the underlying security returns. More specifically we obtain the following results: In a discrete-time model, classical European options typically become redundant with some probability (Proposition 2.1). Obnly path dependent ("exotic") options may generate dynamic spanning (Proposition 4.1). In a continuous-time model with stochastic volatility of the underlying security, and under reasonable assumptions, a European option is "always" a good instrument for completing markets (Proposition 5.2). Copyright 1996 Blackwell Publishers.

Partial Derivative Approach for Option Pricing in a Simple Stochastic Volatility Model

“…Obviously the same formula, just replacing C by D, is valid for the secondary option. This fact proves that the option D completes the market indeed [1,11]. Note that if χ(t, S, σ) = 0 we recover the classical Black-Scholes equation.…”

Perpetual American vanilla option pricing under single regime change risk: an exhaustive study

“…Since in the most usual situation markets do not trade such assets, we will consider the inclusion of a secondary option Q in the portfolio: a derivative of the same nature of P , but with a different set of contract specifications. In particular we will assume that Q has a striking price K ′ , different from K. This is a standard technique [32,33] intended to complete the market.…”

Nonreplication of Options

“…In Bajeux‐Besnainou and Rochet (1996) and also in a more general framework in Baptista (2005), the results of Ross (1976) are generalized in multiperiod markets. In Galvani (2009) the results of Ross are studied in L p spaces.…”