Contingent Claims and Market Completeness in a Stochastic Volatility Model

Abstract

In an incomplete market framework, contingent claims are of particular interest since they improve the market efficiency. This paper addresses the problem of market completeness when trading in contingent claims is allowed. We extend recent results by Bajeux and Rochet (1996) in a stochastic volatility model to the case where the asset price and its volatility variations are correlated. We also relate the ability of a given contingent claim to complete the market to the convexity of its price function in the current asset price. This allows us to state our results for general contingent claims by examining the convexity of their "admissible arbitrage prices." Copyright Blackwell Publishers Inc 1997.

The Econometrics of Option Pricing

“…In contrast to similar option pricing formulas derived by Romano and Touzi (1997) and Fouque, Papanicolaou and Sircar (2000), the expectation (2.29) is with respect to the historical distribution, while the risk-neutral probability distribution they consider corresponds to the particular case a(X t )´0. In particular, they have not addressed the di®erence between B ¤ (t; T ) and exp…”

“…Even though the GBS formula (2.11) is derived in a discrete time context, a similar pricing formula was derived by Romano and Touzi (1997) in the context of risk neutral continuous time models of stochastic volatility with leverage, as in Heston (1993), and by Fouque, Papanicolaou and Sircar (2000), where the variable » t;t+1 plays a similar role. However, the GBS formula (2.11) is even more convenient for empirical pricing since the expectation operator is considered with respect to the historical probability measure instead of the aforementioned formulas involving an equivalent martingale measure.…”

“…The resulting formulas will look very much the same whether we set the model in discrete time or in continuous time. We already mentioned that Romano and Touzi (1997) obtained at a formula which can be written as (2.11) in a Heston (1993) stochastic volatility di®usion model. The fundamental reason behind this similarity lies in the fact that the absence of arbitrage ensures the existence of a state-price density (and conversely) and hence of a stochastic discount factor.…”

“…White (1987) model) or of a fraction of it in case of leverage e®ect (see Romano and Touzi (1997) in the context of the Heston (1993) model). Then, by a simple application of the law of large numbers to time averages of the volatility process (assumed to be stationary and ergodic), one realizes that the e®ects of the randomness of the volatility should vanish when the time to maturity of the option increases and therefore the volatility smile should be erased.…”

Black-Scholes in a CEV Random Environment: A New Approach to Smile Modelling

“…In the case where V is a discrete random variable, this model reduces to the mixture of distributions, analysed, in the Gaussian case by Brigo and Mercurio [10,11]. In a stochastic volatility model where the instantaneous variance process (V t ) t≥0 is uncorrelated with the asset price process, the mixing result by Romano and Touzi [56] implies that the price of a European option with maturity τ is the same as the one evaluated from the SDE (2.1) with…”