ed hquidity drift. UsingdWt
= X[^^dt + dWt
dWt
=
X[^^dt-\-dWf
it holds as usual that A^ ^ = {p — r)/a^ while A^^ can be chosen arbitrarily,so that we obtain(3*it,p) = (3{t,p) - ^u{t,p)!^—^-
^/T^Xi'l
(7
Since liquidity is not traded, the market is incomplete. This leads to onedegree of freedom for the market price of liquidity risk, which cannot beeliminated in this setup. The market price of liquidity risk shows up in thedifference between dynamics under P and under P. To fix the drift of theliquidity process under P a unique price of at least one derivative dependingexplicitly on liquidity is needed. We provide a theoretical discussion on howthe market price of liquidity risk can be calculated from traded Europeanoptions in Subsection 1.7.4. This approach represents a perfect analogy to astochastic volatility model where the market price of volatility risk can onlybe computed from the prices of derivatives.Let us now come back to the stopping time r. By definition, the thresholdis hit at r—. This yields a jump in the stock price at r given by Sr- = Sand the reduced price at r:Sr = 5(1 - Pr-M
= S{1 - Pr4>0),
30
1 Modeling Feedback Effects with Stochastic Liquidity
assuming that p is a continuous function in r.
We propose a derivative
contract that compensates for the price difference between the reduced priceat r and the threshold S by paying Spr(j)oI{T < T) at r for one unit of theunderlying asset.To model a more reahstic scenario we additionally introduce a floor F > 0,which represents a deductible for the investor. This means that if the pricediscount due to illiquidity is less than or equal to F , the derivative paysnothing. Consequently, the contract would only compensate for a criticalliquidity discount. Thus, the payoff at r for one unit of the underlying assetisZr
= max[(5(/)oPr - F ) / ( r < T); 0]{S(t>oPr - F)I (T <T;Pr>
=
J^]
.
The price of the contract is given by the following proposition:Proposition 1.2 Let r := mi{t\St
< S}.
Then the price at t = 0 of a
derivative paying Z^ = max[(S'(/)oPr — F ) / ( r < T); 0] is given by
'o = E[e-^-•z,]=
S<f>oE
-
Scj>o fJo
r" ("'-!;)'(^-^^ "'"&).r
JF/S<J>O
e-^'(p-J-)g{t,p)dpdt,\
(1.18)
S4>o/
where E denotes the expectation under the risk-neutral measure and g represents the joint (risk-neutral) density of r and Pr^ The parameter p < oodenotes some upper bound for the liquidity process. In general, the priceof the derivative cannot be calculated explicitly, but for some special caseswe are able to derive semi-closed-form solutions of (1.18). One of these ispresented below,First of all, we need the distribution of the hitting time. From (1.16) we knowthat the process for 5 up to r under a risk-neutral measure P is a geometricBrownian motion such that the log of the process is an arithmetic Brownian
1.7 Application: Liquidity Derivative
31
motion. The distribution of the first hitting time of a Brownian motion withdrift is well-known (see, for example, Borodin and Salminen (1996)). In thisscenario we need to know the distribution of the first hitting time of In 5,starting at In SQ > InS under a risk-neutral measure. This is given by
The specification of the diffusion process for liquidity has remained arbitraryso far. In the following we consider some scenarios in which the price of theliquidity derivative can be calculated explicitly under certain assumptions onthe hquidity process.If liquidity is constant or a random variable uncorrelated with the underlyingasset, equation (1.18) simplifies toZo ==
500 ( E [p] - ^ )500 (E[p] - X ^ ^
E [e--^/(r < T)]e-'-7.n5o,in5(i)di.
If the Brownian motions exhibit zero correlation, i.e. 7 = 0, the hittingtime r and the process p are independent. In this case we can calculate theexpectation, taking the product of the corresponding risk-neutral densities,i.e. g(t,p)
= f{i)ht{p)^
where ht denotes the risk-neutral density of the
process p at time t, and we obtainZo = 50oE [ e - > . / ( r < T)]=
500 r
( r
(p-
J - ) ht{p)dp) e-^*/i„5o,in5(i)di- (1-19)
For F = 0 this simplifies toZo = S4>oJ
( E b t ] ) e-^*/in5o,insWdi.
(1.20)
Now, we are free to choose an appropriate process for p. To our knowledge thequestion of which specification of the liquidity process is empirically adequateis still unanswered in the literature.
32
1 Modeling Feedback Effects with Stochastic Liquidity
One may argue that a mean reversion process is a sensible choice, since thereexists a natural level of liquidity in the market. Thus, we again assume a CIRprocess for liquidity given by the following specification of the risk-neutralparameters in equation (1.17):
The asterisks denote risk-neutral parameters. For zero correlation the dynamics are given bydpt
= K{6 — pt)dt -f- (y/pldWt
dpt
=
under P and by
K'{e'-pt)dt-\-C,/p'tdWtundevP
with dWt = ^/ptXtdt + dWt. Here we set Xi = y/ptXt such that the hquidityprocess is again a CIR process after the change of measure. The parametersare then given by^* =
/^ + CA,
6' = —e.
(1.21)
In order to ensure a strictly positive liquidity path we have to impose theconstraint(9* > ; ^ .
(1.22)
The risk-neutral probability density of p at t with initial value po is given byv9/2
/i(t,p;Po)=ce-"-"(-jwherec
=
U
=
V
=
2KK*t
CpQt
cp
2K*eQ
=
e
1,(2 V^)
1.7 Application: Liquidity Derivative
33
Zz denotes the modified Bessel function of the first kind of order z, i.e.
My)-{^}
l . r ( . + n + i)n!-
Note that if the fioor F is equal to zero the expectation of hquidity in equation (1.20) can be computed explicitly:E [pt] = poe-^'' + r (1 - e-'^**) .
(1.23)
Hence, equation (1.20) simplifies to
JoIf one additionally assumes a zero market price of liquidity risk {9* = 9),then the price of the hquidity derivative does not depend on the volatilityparameter of the liquidity process C- However, as soon as these restrictiveassumptions are relaxed, for example by considering a non-zero market priceof hquidity risk (A^ 7^ 0) or a non-hnear payoff function ( F > 0), the volatilityof the liquidity process has an impact on the price of the liquidity derivative.This property is shown below in Figures 1.6 and 1.7 in Subsection 1.7.2.
1.7.2
Example
As an example we use the limit order book data for Medion already introduced in Section 1.3 to calibrate the stochastic processes for the best bidprice and liquidity. For numerical reasons we multiply the liquidity parameter by 5,000. Thus, it can be interpreted as the relative price differencebetween the best bid price and the average execution price if a hypotheticalmarket order of 5,000 shares is executed.Applying standard maximum likelihood estimation techniques and assuminga zero market price of liquidity risk, the following parameter estimates wereobtained:a =
0.48
K* = 248.12
9* =
0.020188
C = 3.1906347.
34
1 Modeling Feedback Effects with Stochastic Liquidity
0.15
35.0003.01.2002
28.03.2002
03.01.2002
28.03.2002
Fig. 1.5: Best bid prices and percentage liquidity discounts if 5,000 Medion sharesare sold in a single trade (daily data at 12.00 a.m. GET).The correlation between the best bid prices and the iUiquidity discounts inthe sample is —0.0354. Thus, assuming a correlation of zero seems to besensible.We assume that the derivative under consideration compensates for the pricedifference between the stop loss limit of €40.00 and the average executionprice if a stop loss order of 5,000 shares is executed within the next month.^Since the initial liquidity parameter po is scaled with 5,000 we have (J)Q — 1.0.Furthermore, we set ^o = 47.61, po = 0.05 and r = 0.05, and in the modelwith deterministic hquidity p = O"" = 0.020188. Now we are able to calculateprices for the liquidity derivative. In the model with constant hquidity oneobtains a price of 0.465449, whereas in the model with SL the price forone derivative is 0.489582. Thus the price difference for one contract (inour example 5,000 derivatives) between the stochastic and the deterministichquidity model would be € 120.67.
1.7.3
Sensitivity Analysis
How does variation in the parameters affect the results in the SL model?Recall that the parameters of the CIR process have to meet the constraint^* > C^/(2/^*) in order to ensure positivity of the hquidity process. When^For the sake of simplicity we assume that a stop loss order is executed if the best bidprice falls to the stop loss limit for the first time.
1.7 Application: Liquidity Derivative
35
0.01
Fig. 1.6: Price of the liquidity derivative ZQ as a function of the parameters (, 6*(left) and C, i^* (right). All other parameters are chosen in accordancewith the example; we assume especially that F = 0 and A^^^ = 0. Whilethe dark patterned surfaces represent ZQ in the SL framework, the lightsurfaces represent ZQ in the constant liquidity setup.
the boundary condition becomes binding, a reduction in /^* or ^* impliesa reduction in ( (for given ^* or /^*). Thus, the choice of parameters isrestricted.First we stay for a while in the slightly unrealistic world of the previousexample and consider a contract that is linear in p^, i.e. F = 0, and assumea zero market price of liquidity risk. Figure 1.6 illustrates how the priceof the derivative ZQ varies with either 9 and ( for fixed K (left graph) orwith K and ( for fixed 9 (right graph) such that inequality (1.22) is alwayssatisfied. The value of the derivative in the SL framework is representedby the dark patterned surfaces. The hght surfaces represent the derivativein the constant liquidity setup, which serves as a benchmark. Under thegiven assumptions F = 0 and A^^^ = 0, which imply 9* = 9, K"" = K, theworld is simple. Remember that in this case equation (1.19) simplifies to(1.20), where 'E[pt] is given by (1.23). In the SL setup the price of theliquidity derivative ZQ is a linear increasing function of the long-term mean 9and an exponentially decreasing function of the mean reversion parameter n.
36
1 Modeling Feedback Effects with Stochastic Liquidity
Fig. 1.7: Price of the Hquidity derivative ZQ as a function of the parameters C, F(left) and C, A^^^ (right). All other parameters are chosen in accordancewith the example.
The volatility parameter ( has no impact on the price of the derivative ZQ.Reconsidering the definition of E [pt] given by equation (1.23) one can statethat the stochastic and the deterministic liquidity setup give almost identicalresults if po and 9* are close to p^^^^*- and if the product hi*t is sufficientlylarge.If the assumptions of the previous examples are relaxed the volatility parameter ( has an impact on the price of the liquidity derivative ZQ. Figure 1.7shows the price of the derivative ZQ for combinations of (" and F (left graph)and of C and A^^^ (right graph) for fixed /^* and 6*.When the fioor F is increased, the derivative becomes less expensive. In thiscase the contract pays out in fewer states of the world. However, the pricereduction is less pronounced if the volatility parameter ^ is large. To put itdifferently: an increase in ( transfers weight to the tails of the distributionand raises the price of the derivative ZQ for a given positive F.Similar phenomena to those described above can be observed in a BS world.The price of a derivative that is linear in the asset price, for example aforward contract, is insensitive to the volatility of the underlying asset. Onthe other hand, volatility sensitivity of a plain vanilla European call option
1.7 Application: Liquidity Derivative
37
is a positive function of the strike price, which coincides with the fact thatthe greater the floor F of the hquidity derivative, the higher the sensitivityof the price with respect to changes in volatihty.The parameter A^^^ influences the drift of the hquidity process under therisk-neutral measure. Additionally, if A^^^ ^ 0, the volatility parameter ("appears in the risk-neutral drift, see equation (1.21). If A^^^ > 0, an increasein C causes an increase in /^* and a reduction in ^*. Both aspects reduce theprice of the derivative ZQ, The opposite eflFects can be observed for A^^^ < 0.
1.7.4
Market Price of Liquidity Risk
1.7.4.1 Overviev^The recent literature provides strong evidence that investors care about liquidity risk, which imphes that this type of risk is priced into asset returns.For example, Acharya and Pedersen (2003) analyze daily return and volumedata from 1962 until 1999 for all common shares listed on the NYSE and theAMEX. They corroborate the hypothesis that investors require a premiumfor a security that is illiquid when the market as a whole is illiquid and thatinvestors are willing to pay a premium for a security that is liquid whenstock returns are low. Moreover, they show that investors are willing to paya premium for a security with a high return when the market is ilhquid. Thisis empirically supported by Pastor and Stambaugh (2002) who use monthlydata for 34 years of common stocks traded on NYSE, AMEX, and NASDAQ.They find that stocks that are more sensitive to aggregated market liquidityhave higher expected returns. A comprehensive review of theoretical andempirical approaches can be found in Pritsker (2002).From this perspective and with the results of Subsection 1.7.2 in mind itseems important to provide a reasonable approach to determine the marketprice of liquidity risk from traded instruments. As an example of such atraded claim we consider a plain vanilla European put option that gives the
38
1 Modeling Feedback Effects with Stochastic Liquidity
holder the right to sell 0o units of the stock for the strike price X at maturityT. In the following we make four assumptions to simplify the explanation.Specifically, we suppose that:• the contract is settled via physical dehvery of the shares• the day of execution coincides with the day of delivery (note that inpractice there may be some exchange trading days in between)• the short seller of the put liquidates the (f)Q units of the stock immediately upon exercise of the option• neither the long nor the short party trades the underlying asset beforethe maturity date of the option.The first assumption is met at many exchanges for equity options.
The
second and the third assumption are not critical and can be relaxed if amore general framework is desired. The last assumption ensures that for7 = 0 the underlying price process and the hquidity process are independent.Easing this restriction, for example by assuming that the short seller hedgesthe option in the stock market, results in more complex dynamics for theunderlying asset as we have seen before when considering general tradingstrategies of a large investor.
1.7.4,2 IntuitionAssume that the holder of the put has to hquidate a long position in theunderlying asset at a certain time T. For this purpose the investor can eithersell the securities at the exchange or execute the put option and deliver thestocks to the short seller of the put. Since the market for the underlyingasset is illiquid, the terminal payoff of the put is given by VT — max[X —^ ^ ( l — pT^^o), 0]. This can be explained as follows.The execution of the put is optimal if it provides higher revenues than theliquidation of the underlying asset in the market. In the latter case one
1.7 Application: Liquidity Derivative
39
will receive 5^(1 — PTM due to the market illiquidity. Thus, it is optimalto exercise the put option if and only if X > 5 T ( 1 — PTM-
If the put is
exercised the short party faces a hquidity risk. Assume that the underlyingasset is liquidated immediately after the execution of the put. Then the shortseller has to bear a liquidity discount amounting to STPT4>Q'The story is similar to the liquidity derivative considered in the previoussection for a floor F = 0. If the stop loss order is executed at r < T, theholder of the liquidity derivative would sell the underlying asset at the marketfor 5(1 — Pr^o) and receive the payoff of the derivative Spr(t>o. In total heor she would obtain Scpo. On the other hand, if an investor is hedged by theEuropean put option, he or she has to deliver the shares to the short sellerat T and receives X(j)o.Although the payoff of the European put option and the liquidity derivativeseem closely related to each other for X = S, there are still important differences. The European put option matures at time T. Therefore, it may bea suitable instrument for an investor who has to liquidate stocks at a fixedpoint in time. In contrast to that, the liquidity derivative matures at thestopping time r < T^ when the stock price hits S for the first time. Considering this particular trading strategy a European put option would notprovide a perfect hedge. Imagine, for example a situation where S is hit attime r <T^ but then the stock price rises again and we observe ST > S, Inthis case the put would not be exercised at T, although the stop loss orderhad been executed. Only for r = T (which is a zero probability event) wouldthe investor be indifferent between the two claims.One may argue that an American put option better meets the needs of aninvestor with a stop loss strategy than a European put, since the Americanput option could be exercised at any stopping time r <T.
However, early ex-
ecution of an American put option following this particular strategy does nothave to be optimal. Hence, the American put option will be more expensivethan the hquidity derivative. Thus, a long position in the hquidity derivativewould be the cheapest way to hedge against the liquidity risk arising from astop loss trading strategy.
40
1 Modeling Feedback Effects with Stochastic Liquidity
1.7.4.3 Pricing a European Put on an Illiquid StockThis subsection proposes an innovative approach for the calculation of themarket price of hquidity risk. The price of a European put option can becomputed analytically in our framework. The put price is not unique anddepends on the market price of liquidity risk, since the market is incomplete.Therefore equating the theoretical prices to the observed market prices allowsus to extract the market price of hquidity risk.As before, the underlying asset follows a geometric Brownian motion withdrift r under the risk-neutral measure. For the liquidity process we again consider a CIR process. By fixing a risk-neutral measure P the pricing formulais given byVt
=
e-^(^-^)E[max[X-5T(l-PT0o),O]]
=
e-<^-'^t
=
e-'^^-'yXP{X
[{X - 5 T ( 1 - P T ^ O ) ) / ( ^ > ST{1
- e-'^^-'^E=
> ST{1 [ST{1
e-'^^-'^XP{X- e-'^^-'^E
[STI{X
PTM)
- PTMHX
> ST{1 -
- PT^O))]
> ST{1 -
PTM)]
PTM)
> ST{1 -
PTM)]
+ e-'^^-'^E [STPT<I>OI{X > ST{1 - PTM)] •
Let / and g denote the risk-neutral density of the asset price process andof the CIR process at T, respectively. The market price of liquidity riskis contained in the (risk-neutrally) adjusted parameters of the CIR density,compare Subsection 1.7.L In integral representation - assuming 7 = 0, i.e.independence of S and p - we obtain
Vt = e-^^'^-'^X f
fis)9{p)dsdp
Jx>s{l-p<t>o)
-e-'-(^-') /
sf{s)g{p)dsdp
Jx>sii-pM
-e-^^'^-'Uo /JX>S{1-P4>Q)
^-r(T-
spf{s)g{p)dsdp
1.8 Conclusion
41
using the substitution z = s{l — pct)^) in the last step. In order to determinethe market price of hquidity risk one has to find the root(s) of the equation
1.8
Conclusion
Criteria such as consistency with empirical phenomena, flexibility, and alsocomputational aspects should be considered in the development of a liquiditymodel. We have presented a framework that we feel meets these requirementsand has proved worthy of investigation.This chapter introduces a continuous-time model for an illiquid market,where the trading strategy of a large investor can move prices. The innovative features of our setup include on the one hand, a time-varying marketdepth and on the other hand, the modeling of liquidity feedback effects.We have analyzed two basic types of trading strategies. For positive feedbackstrategies and non-positively correlated Brownian motions, volatility generally is increased compared with BS. For contrarian feedback strategies onecan observe basically the opposite. The picture can change completely if theasset becomes highly illiquid and the asset price dynamics are dominated bythe trading activities of the large investor. These features are discussed bothanalytically and on the basis of simulation results.Furthermore, an application of the general framework is proposed. We derivea closed-form expression for the price of a liquidity option, the payoff ofwhich depends on the price difference between a stop loss hmit and theaverage execution price. Furthermore, a pragmatic way to determine themarket price of liquidity risk from traded European plain vanilla put optionsis presented.
42
1 Modeling Feedback Effects with Stochastic Liquidity
Further research may focus on derivative pricing under the effective priceprocesses or on the apphcation of hedging strategies from incomplete marketsto this setting.
An investigation of optimal liquidation strategies in the
proposed framework also seems promising.
1.9
Appendix
1.9.1
Proof of Proposition 1.1
Using Ito's lemma for (f){t^ 5, p) givesd(t) = Mt + 05d5 + 0pdp + (t)spd[S, p]t + - {(t)ssd[S]t + (j^ppiy^t) .Plugging this into equation (1.4) we getdSt
=
(a + p(t),iyj)StdWt + pcl^puSdWt + ptSt ({ptSt + (t>t)dt + (l)sdS+ ^pdp + v(j>spd[S, W]t + \ {(l>ssd[S]t + (t^ppv^dt) )
which in turn impliesdSt
=
^_ ^
((a + p(j>pU^)StdWt + St [pc^pW^ - 7^) dWti^^(l>t + P{put)^p + \(t>ppiy^]dt
-^pSi
+ ^cPssd[S]t + iycf>spd[S,W]tassuming ptSt4>s < 1- Using the trial solutiondSt = b{t, St, pt)dt + v{t, St, Pt)StdWt + v{t, St, pt)StdWtand comparing coefficients leads to
^(^'--) =
"t^
1.9 Appendix
43
b{t, S, p)dt
pS1 - pS<ps
- + 4>t + P{put)(t>p + \4>pp^'' ] At+ \<t>ssA[S]t + y4>spA[S,W]t
pS1 - pS(t>s
— + (j)t + li{put)4>p + -4>ppv''P^, 1^ ^^(j'^ + p'p''4>l + 2^p(ju(l>p+ 2^^^^
{l-pS4>s?1 - pb(j)s J
sinceA[S]t = S\v'
+ v^)di
^'('''t%T''^'^^and
6.[S,p]t = Sui'yv +
1.9.2
^/T^v)dt
Functional Form of 0(5, p)
In order to incorporate the idea that (/> is a function of both S and p wechoose a product approach by separating the strategy with respect to S andPA function that is able to model the features described in Section 1.6 is theincomplete gamma function, defined by
r(x,z) = j ^ e'H^-^dtWe read r{x, z) as a function of z for fixed x-values. It is monotonicallydecreasing to zero for 2: —> oo. The incomplete gamma function capturesthe dependency of the trading strategy with respect to p for fixed S. For
44
1 Modeling Feedback Effects with Stochastic Liquidity
the contrarian feedback strategy the scenario is similar with respect to Sfor fixed p. Since the image is reversed for the positive feedback strategythe functional form has to be adjusted. Therefore we multiply the gammafunction by —1 and then shift it to the positive quadrant. Thus, we choosethe following representation for the dependency on p:x{p) = r(6i,cip),and for the dependence on S:( r(62,C25)
(contrarian strategy),
[ r(62) — r(62,C2S')
(positive strategy).
Consequently, the contrarian feedback strategy is given byr(S,p) = ar{h,cip)rib2,C2S),
(1.24)
whereas the positive feedback strategy is given by0+(5, p) = ar(6i, cip) [r(62) - r(62, C2S)],where r(x) denotes the standard gamma function.
(1.25)
For the simulation in
Section 1.6 we use the following parameter constellation:61 =
2.1
62 =
6.0
ci =
5.5
C2 =
0.05
a =
-0.05.
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Stock-price manipulation,
The Review of
Financial Studies 5, 503-29.Almgren, R. and N. Chriss (2000): Optimal execution of portfolio transactions, Journal of Risk 3, 5-39.
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1 Modeling Feedback Effects with Stochastic Liquidity
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