On statistical indistinguishability of complete and incomplete market models

Purpose This paper aims to investigate possibility of statistical detection of market completeness for continuous time diffusion stock market models. Design/methodology/approach The paper uses theory of forecasting to find criteria of predictability of market parameters such as volatilities and the appreciation rates. Findings It is known that the market completeness is not a robust property: small random deviations of the coefficients convert a complete market model into an incomplete one. The paper shows that market incompleteness is also non-robust: for any incomplete market from a wide class of models, there exists a complete market model with arbitrarily close paths of the stock prices and the market parameters. Originality/value The paper results lead to a counterintuitive conclusion that the incomplete markets are indistinguishable in the terms of the market statistics.

Filtration shrinkage, the structure of deflators, and failure of market completeness

We analyse the structure of local martingale deflators projected on smaller filtrations. In a general continuous-path setting, we show that the local martingale parts in the multiplicative Doob–Meyer decomposition of projected local martingale deflators are themselves local martingale deflators in the smaller information market. Via use of a Bayesian filtering approach, we demonstrate the exact mechanism of how updates on the possible class of models under less information result in the strict supermartingale property of projections of such deflators. Finally, we demonstrate that these projections are unable to span all possible local martingale deflators in the smaller information market, by investigating a situation where market completeness is not retained under filtration shrinkage.

Dynamic effects of consumption tax reforms with durable consumption

This paper introduces durables into a dynamic general equilibrium overlapping generation model with idiosyncratic income shocks and endogenous borrowing constraints, which depend on durables. The aim of this paper is to evaluate the welfare effects of consumption tax reforms in a richer model that captures the difference between nondurable and durable consumption. When durables are considered, the standard results that a shift to consumption taxes is welfare improving are overturned. The mechanism of this opposing result is that consumption tax makes durable consumption more expensive without relaxing the borrowing constraint. The inability of borrowing to insure against income risk deviates the economy further away from market completeness and particularly hurts young and poor households. As a result, welfare decreases, coupled with negative redistribution.

The Martingale Approach to Optimal Investment

For the special case of optimal consumption/investment problems, there is an alternative to dynamic programming. The alternative is based on market completeness and martingale methods. This approach is sometimes much easier to apply than dynamic programming and we derive the necessary theory in some detail. The theory is then applied to several concrete problems which are solved in detail.

Completeness and Hedging

The concept of market completeness is discussed in some detail and we prove that the Black–Scholes model is complete. We also discuss how completeness and absence of arbitrage is related to the number of risky assets and the number of random sources in the model.

Integrability of exponential process and its application to backward stochastic differential equations

We consider the integrability problem of an exponential process with unbounded coefficients. The integrability is established under weaker conditions of Kazamaki type, which complements the results of Yong obtained under a Novikov type condition. As applications, we consider the solvability of linear backward stochastic differential equations (BSDEs) and market completeness, the solvability of a Riccati BSDE and optimal investment, all in the setting of unbounded coefficients.

FINANCIAL MARKETS WITH NO RISKLESS (SAFE) ASSET

We study markets with no riskless (safe) asset. We derive the corresponding Black–Scholes–Merton option pricing equations for markets where there are only risky assets which have the following price dynamics: (i) continuous diffusions; (ii) jump-diffusions; (iii) diffusions with stochastic volatilities, and; (iv) geometric fractional Brownian and Rosenblatt motions. No-arbitrage and market-completeness conditions are derived in all four cases.