A Proposal on

The Cost of Illiquidity and its Effects on Hedging

By

Table of Contents

Background3

Purpose of the Study3

Objective of the Study4

Research Question4

Literature Review5

Structure of the Dissertation6

Anticipated Findings6

Proposed References8

The Cost of Illiquidity and its Effects on Hedging

Background

After credit risk, liquidity risk is probably the next most important risk faced by the finance industry; and yet the study of liquidity is far less advanced. This may be in part due to the fact that there is no agreed definition of what liquidity is, even in qualitative terms. Everyone would agree that the effect of illiquidity is to make it difficult or costly to trade large volumes of the underlying asset in small times, but there are different approaches to modelling this.

Purpose of the Study

The purpose of this study will be to find out the cost of illiquidity and its effect on hedging by using financial mathematics models. The illiquidity effect (i.e., the cost of trying to trade fast) is distinct from the price impact effect of a large trade, as we shall argue below in Chapter 2. Both can be considered as effects of supply and demand on price, but the illiquidity effect (or temporary price impact, if you prefer) arises because of the need to clear a market over a short time spell, whereas the permanent price impact comes from the clearing of the market over long periods. We can and shall consider the illiquidity costs ignoring the effects of permanent price impact, but it will be nonsensical to consider the second of these and ignore the first, as is pointed out by Schönbucher and Wilmott (2000). They consider the “free round trip” phenomenon, where the large agent rapidly sells and then buys back a large amount of stock, forcing the price instantaneously to drop, and if this round trip is not costly, then the large agent could make profits by selling down-and-out calls and subsequently knocking them out by a round trip. Another problem with permanent price impact models such as Frey (1988), Frey and Stremme (1997), Platen and Schweizer (1998), Papanicolaou and Sircar (1998), and Schönbucher and Wilmott (2000) is that they typically present the solution to a hedging problem in feedback form, exhibiting the hedge as a function of time, and current stock price—but if the initial portfolio is not at the exactly correct value, it will not be clear how it is to be moved to that value.

Objective of the Study

The objective here will be to steer the portfolio toward arandom endpoint, not known at earlier times, and this makes the problem a lot harder. This will be evidenced by the fact that earlier studies end up with portfolio rules which are deterministic functions of time, whereas the optimal rules which we arrive at have a genuinely interesting dependence on the underlying asset as well as time.

We will be working with a log-Brownian asset, and this requires solution of a partial differential equation (PDE) to arrive at the answer; many of the earlier studies cited above require only the ...