Competence Center Corporate Finance & Risk Management

by Harald WeißStuttgart, Jan 27th 2006

Pricing Liquidity Risk

Agenda

1. Motivation

2. Liquidity Measures

3. Pricing Liquidity Risk with APT

4. Using OPT for Valuing the Time Dimension of Liquidity

5. Extending the Longstaff model

Practitioner´s view

majority of market participants consider the liquidity of assets in portfolio

decisions

Until recently

liquidity research was focussed on individual assets

asset value is reduced by lower liquidity

investor demands for a liquidity premium

Current research about liquidity

liquidity is a market-wide phenomena

Is liquidity risk a non-diversifiable risk?

1. Motivation

2. Liquidity Measures

Asset Liquidity

Time Dimension Price Dimension

Keynes (1930): „realisable at short notice without loss“

market liquidity is characterized by the trade-off between the selling-price and the time-till-sale for a given set of market conditions

2. Liquidity Measures

A closer to look to the Xetra-Orderbook

2. Liquidity Measures

Aitken and Comerton-Forde (Pacific-Basin Finance Journal, 2003 (1))

found 68 different measures used in the literature

Selected liquidity measures with summary statistics for NYSE (Roll, 2005)

3. Pricing Liquidity Risk with APT

Pastor and Stambaugh (2003) model

Pastor and Stambaugh (2003) model

3. Pricing Liquidity Risk with APT

4. Using OPT for Valuing the Time Dimension of Liquidity

Perfectly liquid asset:

t=0 t=1

S

t=0 t=1

S

Il liquid asset:

Restrictedfrom trading

more flexibility for investors holding the liquid asset standard no-arbitrage argument:

value of marketability is equal to the price difference between the two assets

(positive) value of marketability can be viewed as a trading option

tt

Idea: a put option provides a protection against illiquidity

a position hedged by a put self-liquidates as money is lost and markets become illiquid

put price can be viewed as the value of liquidity concept of portfolio insurance

In general: put option gives the holder the right to sell the underlying asset by a certain date T for a certain price X

Short Put

X

Payoff

ST (Terminal stock price)0

4. Using OPT for Valuing the Time Dimension of Liquidity

Longstaff, Journal of Finance (1995)

investor has perfect market timing ability, knows the optimal selling point in t=0

additional profit:

= payoff of a European lookback put option upper bound for the value of marketability

Lookback option

underlying stochastic process is normally a geometric Brownian motion

modification: Brownian bridge process

5. Extending the Longstaff model

5. Extending the Longstaff model

Definition of the underlying stochastic procees

Simulation of the underlying stochastic process70

8090

100

110

120

0 .2 .4 .6 .8 1t

S1 S2S3 S4S5

Lookback

5. Extending the Longstaff model

Pricing Lookback options using Monte Carlo Simulation

(STATA) Program:

(1) Select the starting parametershere: S0 , ST , σ , T , dt

(2) Simulate the underlying stochastic process from t=0 to t=T(3) Keep in mind the maximum stock price during t=0 and t=T(4) Calculate the option price as

(5) Repeat steps (1) to (4) (here: 1000 times)(6) Option price is calculated as the mean of (5)(7) Vary the starting parameters σ and T

and repeat the simulation steps (1) to (6)

5. Extending the Longstaff model

5. Extending the Longstaff model

Simulation results

Back-up: Further Research

Change the stochastic process of the underlying geometric Brownian motion Borwnian bridge process Ornstein-Uhlenbeck process Poisson process

Vary the option type American vs. European Lookback, Russian

Model the option parameters volatility model (GARCH) stochastic interest rate

-> add additional risk factors (stochastic interest rates)integrate multiple non-trading periods-> investigate time-dependence of liquidityasset remains illiquid-> no continuous trading (trading is only possible at certrain points in time)-> distribution of trading dates (not uniform?)-> dividend payments

Back-up: Further Research

Kempf

investor has no perfect foresight investor has an incentive to sell the asset when the reservations price Yt differs

from the market price value of marketability depends from investor preferences Xt follows a stochastic process with the restriction XT = 0

Koziol and Sauerbier, Working-Paper (2003) Lookback Option Stochastic interest rate, multiple non-trading periods, time varying liquidity of

bonds

Back-up: Option Based Models

Back-up: Limitations of the Model

Discussing the Black-Scholes assumptions (here: selection)

(1) underlying asset price follows a geometric Brownian Motion

check empirical return distributions

(2) assets are divisible

(3) trading of the underlying asset is continuously

same problem as for real options

see assumption: complete capital market is needed

Modelling Liquidity Risk

exercies of the option which corresponds to the sale of the property is stochastic

use Russian options which have no exipry date

No general equilibrium model

add further assumptions about investor preferences

Back-up: XETRA-Orderbook

Result

both approaches should deliver the same risk-return profile-> Compare the results!

=5%

=6%

we start here

Back-up: Motivation

3. Option Based Model for Valuing the Time Dimension

3. Option Based Model for Valuing the Time Dimension

Back-up

Back-up