The last 30 years has seen the successful development of mathematical models for financial equity investing. Indeed the Black-Scholes-Merton theory, with its consequent formulas, has changed the way business worldwide handles questions of risk. Nevertheless, as currently used, the models still represent a crude approximation of what is actually going on, and improvements of the models can lead to an edge for an investor. This Learning Curve discusses the overlooked issue of liquidity risk. Economists classify risk into five rubrics:
* Market risk
* Credit risk
* Liquidity risk
* Model risk
* Operational risk
Market risk describes the mathematical models of equity prices incorporating market noise, together with the assumption that arbitrage opportunities do not exist. These of course are idealized models and they are usually fairly accurate if they are "almost true." Market risk models include the renowned Black-Scholes formula and paradigms. Credit risk refers to the mathematical estimation of the probability that an entity issuing a credit obligation will be able to meet its commitments fully or partially, and if the latter, then by how much. Mathematical analysis of credit risk is in its early stages of development. Model risk refers to the risk one takes by acting on a model of reality that is incorrect, being anywhere from close to far from reality. Operational risk refers to the risk one takes that incompetence, fraud, unethical, or criminal activity has given the investor incorrect or misleading information.
The first four risks lend themselves to mathematical analysis. Operational risk is the most disturbing, since even if one has perfect models, false data will lead to mistakes, perhaps grievous ones. In this essay we will ignore operational risk, in effect assuming it is zero, an assumption that appears to be a dangerous one in the current climate. Model risk has already been well studied, ranging from the inclusion of transaction costs in models to the inadequacy of the standard Black Scholes model; indeed, since the Black-Scholes model assumes the risky asset price follows a linear stochastic differential equation with constant coefficients, it is on the one hand easy to grasp with its accompanying formulas, but on the other hand a crude approximation of what is clearly a nonlinear reality. Nevertheless, just as tangent lines approximate curves well locally, the Black-Scholes model can approximate the true nonlinear system model well "locally", that is, for short time durations. If the volatility parameter is updated with sufficient frequency, Black-Scholes turns out to work rather well. There are other model risk issues, such as heavy tails observed from abnormally large deviations from the mean, and this has led to models with stochastic volatility and to models where the market noise is a Lévy process (a process with stationary and independent increments, but not normally distributed ones), which inevitably contain many small jumps. This article will not discuss model nor credit risk.
The standard mathematical models of equity prices assume unlimited liquidity as a given. That is, it is assumed that if an investor wants to place an order for a certain number of shares at the current price, there will be available to him or her a seller who is willing to sell the shares at the same price. For the NASDAQ, the depth of the quoted price is known, but for the New York Stock Exchange, only marginal trades of 100 or 200 shares can be expected to be executed at the listed price. A larger trade, such as 10,000 or more shares, the quantity can depend on the actual dollar range of the security price, is handled separately. It is conventional wisdom in the industry that a large or even moderate trade can move the price of an equity. There is both statistical and anecdotal evidence to support this belief. We give a three dimensional picture of one day of tick data for Cisco Systems common stock from 1998. The three axes are time of trade, selling price of the trade and trade volume is the vertical axis. The dots represent trade volume under 1,000 shares, the black lines are trade volume between 1,000 and 50,000 and the gray lines are trade volume over 50,000 shares.
Note the presence of large and huge trades, which tend to move the price.
Of course, for every trade there is both a purchaser and a seller, but the issue is in which direction the trade was initiated. For example, a buy order of 100,000 shares of IBM can be expected to result in a small run-up of the price as sell orders are collected to match it. IBM is considered to be one of the most liquid stocks. It belongs to the Standards & Poor's 100. As an extreme example, this is illustrated by the behavior of corporate raiders.
Umut Cetin, Robert Jarrow, and Philip Protter, in recent research at Cornell University in Ithaca, N.Y., postulate a supply curve to include in the usual equity models. Once the supply curve is known, one can develop a trading strategy that minimizes loss due to liquidity issues, while still approximating an ideal hedging strategy. Moreover, one can further analyze what percentage of the price of traded options is due to liquidity issues. Using a simplified model, in work with Mitch Warachka, the authors have determined the mean observed percentages for call options traded on the Chicago Board of Trade of five well known companies traded on the NYSE, during May 2000. The following table summarizes the average dollar and percentage value of the option's market price attributed to illiquidity.
From this table we see that liquidity costs can comprise a significant percentage of an option's value, and thus it is prudent to take them into account. To do this, one needs to estimate the supply curve from market data, and then use this approximation of the supply curve, together with the Cornell model which incorporates the supply curve in an arbitrage free capital asset pricing model. The supply curve will change with each stock, since different stocks have different liquidity. The extent of the variation of such supply curves is at present unknown. Indeed, the whole topic of liquidity influences in asset pricing has been overlooked in the past and is only now emerging as an important way to improve pricing models and hedging strategies.
This week's Learning Curve was written by Philip Protter, in the operations research group at Cornell University, Ithaca, N.Y. Company Sample Size Percentage Value Minimum Maximum
AMR 63 2.93% 0.82% 10.47%
MGM 9 4.36% 2.11% 8.66%
MO 150 2.53% 0.82% 8.89%
RBK 15 6.57% 2.55% 27.00%
TOY 43 5.00% 2.05% 12.73%
