Liquidity risk and arbitrage pricing theory

Abstract

Abstract. Classical theories of financial markets assume an infinitely liquid market and that all traders act as price takers. This theory is a good approximation for highly liquid stocks, although even there it does not apply well for large traders or for modelling transaction costs. We extend the classical approach by formulating a new model that takes into account illiquidities. Our approach hypothesizes a stochastic supply curve for a security's price as a function of trade size. This leads to a new definition of a self-financing trading strategy, additional restrictions on hedging strategies, and some interesting mathematical issues.

Pricing in an equilibrium based model for a large investor

“…We begin with economic primitives (such as agent's preferences and market equilibrium) and then derive the model. This is different from several papers in Mathematical Finance where the nature of illiquidity is postulated a priori, see for example [5,3,2,7].…”

“…Liquidity is a complex concept standing for the ease of trading of a security. (Il)liquidity can have different sources, such as inventory risk-Stoll [15], transaction costs-Cvitanić and Karatzas [4], uncertain holding horizons-Huang [13], asymmetry of information-Gârleanu and Pedersen [9], demand pressure-Gârleanu et al [10], search friction-Duffie et al [6], stochastic supply curve-Çetin et al [3] and demand for immediacy-Grossman and Miller [12], among many others (see Amihud et al [1] for a thorough literature overview).…”

Curve following in illiquid markets

“…The investor's market buy order is matched with prevailing limit orders and executed at prices higher than D(s); the more volume demanded, the higher the price paid per share. One may think of this as an increasing supply curve as in Çetin et al [13]. Similarly, market sell orders are executed at prices lower than D(s) and the price per share is decreasing in the volume sold.…”

Almost-sure hedging with permanent price impact

“…Some of them incorporate liquidity costs but no price impact, so that the price curve is not affected by the trading strategy. In the setting of [6], this does not affect the superhedging price because trading can essentially be done in a bounded variation manner at the marginal spot price at the origin of the curve. However, if additional restrictions are imposed on admissible strategies, this leads to a modified pricing equation, which exhibits a quadratic term in the second order derivative of the solution, and renders the pricing equation fully nonlinear, and even not unconditionally parabolic; see [7] and [20].…”