Jump liquidity risk and its impacton CVaR

Harry Zheng and Yukun ShenDepartment of Mathematics, Imperial College, London, UK

Abstract

Purpose – The aim is to study jump liquidity risk and its impact on risk measures: value at risk(VaR) and conditional VaR (CVaR).

Design/methodology/approach – The liquidity discount factor is modelled with mean revisionjump diffusion processes and the liquidity risk is integrated in the framework of VaR and CVaR.

Findings – The standard VaR, CVaR, and the liquidity adjusted VaR can seriously underestimate thepotential loss over a short holding period for rare jump liquidity events. A better risk measure is theliquidity adjusted CVaR which gives a more realistic loss estimation in the presence of the liquidityrisk. An efficient Monte Carlo method is also suggested to find approximate VaR and CVaR of allpercentiles with one set of samples from the loss distribution, which applies to portfolios of securitiesas well as single securities.

Originality/value – The paper offers plausible stochastic processes to model liquidity risk.

Keywords Monte Carlo methods, Risk analysis, Liquidity

Paper type Research paper

1. IntroductionThe liquidity drying up was the prevailing trigger element of some high profile failuresin the recent past (e.g., the fall of long-term capital management). These highlyleveraged hedge funds had great difficulty in raising cash to meet margin calls byunwinding positions in markets where liquidity had almost disappeared. Dunbar(1998) explains that:

Portfolios are usually marked to market at the middle of the bid-offer spread, and many hedgefunds used models that incorporate this assumption. In late August, there was only onerealistic value for the portfolio: the bid price. Amid such massive sell-offs, only the first sellerobtains a reasonable price for its security; the rest lose a fortune by having to pay a liquiditypremium if they want a sale. [. . .] Models should be revised to include bid-offer behaviour.

The liquidity risk can be conceptually divided into an exogenous component and anendogenous component, the former depends on the general market condition and thelatter relates the specific position of a trader. (Lawrence and Robinson, 1996) assert thatignoring totally the liquidity risk can provoke an underestimate of the market risk upto 30 percent. Despite the wide recognition of the importance of the liquidity risk, thereis no universal agreement on the definition of liquidity. In the academic literature, theliquidity is usually defined in terms of the bid-ask spread and/or the transaction costwhereas in the practitioner literature the illiquidity is often viewed as the inability ofbuying and selling securities (at any price). The four major properties of the liquidityare the following Black (1971): the immediacy of the transaction, the tightness of the

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The authors thank the referee for several comments which have helped to improve the firstversion.

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The Journal of Risk FinanceVol. 9 No. 5, 2008

pp. 477-491q Emerald Group Publishing Limited

1526-5943DOI 10.1108/15265940810916139

spread, the resiliency of the market, and the depth of the market. The concept ofliquidity can be summarized as the ability for traders to execute large trades rapidly ata price close to current market price. The liquidity risk refers to the loss stemming fromcosts of liquidating a position.

To manage the liquidity risk a good risk measure is needed to account for the impactof the liquidity shock on tradable securities and portfolios. Value at risk (VaR) is the mostpopular market risk measure used in practice, which estimates the potential loss of afinancial instrument at a certain level of probability and for a given period of time.However, VaR ignores the liquidity component and can seriously underestimate thepotential loss if the loss distribution is fat-tailed. This is because VaR takes the value ofthe least loss among all possible losses if an event of a given probability does occur. Toovercome the underestimation of the potential loss VaR is often adjusted in an ad hocfashion either by lengthening the holding period or by magnifying the VaR calculatedwith the desired holding period. A different risk measure that addresses the shortcomingof VaR is the conditional VaR (CVaR). Unlike VaR in predicting the potential loss CVaRuses the average loss among all possible losses, which provides a more realistic lossestimation if an unexpected “bad” event occurs in a fat-tailed environment. CVaR is alsoa coherent risk measure whereas VaR is not (Artzner et al., 1999).

It is often difficult to compute directly the VaR and CVaR from their definitions asVaR requires to solve a nonlinear equation and CVaR to integrate over the taildistribution, especially when the closed-form expression of the loss distribution isunknown or is too complicated. Rockafellar and Uryasev (2002) suggest a viable methodfor the computation of VaR and CVaR by formulating a convex minimization problem inwhich the minimum value is the CVaR and the left end point of the minimum solution setgives the VaR. The resulting optimization problem can be easily solved in two steps byfirst generating samples of the loss distribution and then solving a large linearprogramming (LP) problem which gives the approximate VaR and CVaR. The sameformulation can also be used to find the minimum CVaR portfolio.

We focus in this paper on the loss of the realized value (bid-price) of a tradablesecurity, we define the bid-price as the product of the mid-price and the liquiditydiscount factor, both follow some stochastic processes. To highlight the key point andsimplify the discussion, we assume that the mid-price follows a geometric Brownianmotion (GBM) process, the liquidity discount factor follows a mean-reversion jumpdiffusion process, and two processes are independent of each other.

There are two main contributions in this paper. The first contribution is that itprovides an explicit solution to the LP problem of Rockafellar and Uryasev (2002) withthe following advantages:

(1) approximate VaR and CVaR can be computed by simply generating samples ofthe loss distribution and no optimization is needed;

(2) VaR and CVaR of any percentile can be computed with a given set of samples ofthe loss distribution;

(3) it works for both a single security and a portfolio of securities as long as thejoint distribution of security losses is known; and

(4) it opens up other optimization methods (e.g., the augmented Lagrangianmethod) to find the minimum CVaR portfolio in supplement to the nonsmoothoptimization method or the large-scaled LP method.

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The second contribution is that it defines liquidity-adjusted VaR (LVaR) andliquidity-adjusted CVaR (LCVaR) for market risk of tradable securities when there existsjump liquidity risk. It shows that the conventional VaR and CVaR for the mid-price of asecurity can seriously underestimate the potential loss, especially over a short periodsuch as one day, and can result in substantial loss if a “bad” rare event occurs. Thispartially explains the difficulty those hedge funds had to meet margin calls byunwinding the position when liquidity disappeared. The implication in riskmanagement is that financial institutions should reserve sufficient liquid assets,much larger than what the conventional VaR and CVaR for the mid-price would havesuggested, in their portfolios to withstand the potential large loss when a jump liquidityevent strikes.

The paper is organized as follows: Section 2 reviews the convex optimizationproblem of Rockafellar and Uryasev (2002) for VaR and CVaR, and solves the resultingLP problem by the dual method and the Kuhn-Tucker conditions. It also discusses theway of finding the minimum CVaR portfolio. Section 3 models the liquidity discountfactor with the mean reversion OU and CIR jump diffusion processes which seem tocharacterize well the general phenomenon of the liquidity risk: unpredictable suddenliquidity dry-up and gradual recovery afterwards. Section 4 compares LVaRs andLCVaRs of different models and parameters to see their effects for risk measures.Section 5 concludes and the appendix contains the proofs of theorems.

2. Computation of VaR and CVaRConsider a real-valued random variable L on a probability space ðV;F;P Þ thatrepresents the loss of an investment over a fixed time horizon. Let a [ (0, 1) be fixed.Then the VaR of L at level a is defined to be the smallest number x such that theprobability that the loss does not exceed is not less than a, i.e.:

VaRa ¼ min{x [ R : PðL # xÞ $ a}: ð1Þ

The CVaR at level a is defined to be the average loss given that the loss is at leastVaRa, i.e.:

CVaRa ¼ mean of the a2 tail distribution of L ð2Þ

where the a-tail distribution Fa(x) is defined by:

FaðxÞ ¼0 for x , VaRa

ðPðL # xÞ2 aÞ=ð1 2 aÞ for x $ VaRa:

(

Let St be the discounted asset price at time t, following a GBM process:

dSt ¼ sStdWt; ð3Þ

where s is a constant asset volatility and Wt a standard Brownian motion.Remark. In general asset price St is assumed to follow a GBM process with a driftm:

dSt ¼ mStdt þ sStdWt:

The discounted asset price ~St U e2rtSt satisfies SDE:

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d ~St ¼ ðm2 rÞ ~Stdt þ s ~StdWt:

where r is the risk-free interest rate. We can then apply the Girsanov theorem to changethe probability measure P to an equivalent probability measure P 0 such that:

d ~St ¼ s ~StdW0t

where W 0t is a standard Brownian motion under P 0 (Karatzas and Shreve, 1988). We

therefore assume, without loss of generality, that discounted asset price St has zero drift.The loss of the discounted asset price at time is given by:

L ¼ S0 2 ST ¼ S0ð1 2 e2ð1=2Þs 2TþsWT Þ:

(L , 0 represents a gain.) The distribution of is given by:

PðL # xÞ ¼ F2Inð1 2 ðx=S0ÞÞ2 ð1=2Þs 2T

sffiffiffiffiT

p

� �;

if x , S0 and P(L # x) ¼ 1 if x $ S0 where F(x) is the standard normal cumulativedistribution function. A simple calculation using (1) and (2) shows that:

VaRs ¼ S0ð1 2 e2ð1=2Þs 2T2sffiffiffiT

pF21ðaÞÞ;

CVaRa ¼ S0 21

1 2 aS0Fð2F21ðaÞ2 s

ffiffiffiffiT

pÞ;

In general, there are many factors which make the direct computation of VaR andCVaR with (1) and (2) difficult. For example, the closed form expression of the lossdistribution is unknown, or equation (1) is highly nonlinear, etc. (Rockafellar andUryasev, 2002) suggest a new way of computing VaR and CVaR by solving thefollowing convex optimization problem:

xminFaðxÞ U xþ

1

1 2 aEðL2 xÞþ; ð4Þ

where x þ ¼ max(x, 0). The VaR and CVaR are the optimal solution and optimal valueof problem (4), given by:

VaRa ¼ left endpoint ofx[R

argminFaðxÞ

and:

CVaRa ¼x[RminFaðxÞ ¼ FaðVaRaÞ:

Rockafellar and Uryasev (2002) suggest to solve (4) by first generating samples M ofrandom variable L to approximate (4) by:

x[Rmin xþ

1

1 2 a

1

M

XMi¼1

ðLi 2 xÞþ

and then solving an equivalent LP problem:

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min xþ1

1 2 a

1

M

XMi¼1

zi s:t: xþ zi $ Li and zi $ 0; i ¼ 1; . . . ;M : ð5Þ

The optimal solution and the optimal value to problem (5) are the approximate VaRand CVaR as we have replaced the expectation by the sample average. Theseapproximate VaR and CVaR tend to the exact VaR and CVaR as M ! 1. Weinvestigate the computation of these approximate VaR and CVaR and theirapplications in liquidity risk analysis. From now on, the VaR and CVaR in the paperrefer to these approximate VaR and CVaR, computed from (5).

We can solve problem (5) explicitly due to its special structure, and consequently wecan get VaRa and CVaRa explicitly by simply sorting the samples.

Theorem 1. Let the M samples of loss random variable L be arranged indecreasing order L1 $ · · · $ LM. Let a [ (0, 1) be a given percentile and N be theunique integer satisfying (1 2 a)M 2 1 , N # (1 2 a)M. Then the approximateVaR and CVaR are given by:

VaRa ¼ LNþ1 and CVaRa ¼ g ðL1 þ · · · þ LN Þ þ ð1 2 NgÞLNþ1

where g ¼ (1/(1 2 a))(1/M). Furthermore, as M ! 1 the approximate VaR and CVaRtend to the exact VaR and CVaR defined in (1) and (2).

Proof. See Appendix.Theorem 1 finds the approximate VaR and CVaR of all percentiles once a set. of

samples is generated and sorted, as the only difference with different a is to choosedifferent N. When a is close to 1 the number of samples should be sufficiently large toensure a stable result. For example, if 100 samples are generated, then CVaR is theaverage of the first ten sorted samples and VaR is the eleventh sorted sample whena ¼ 0.9 whereas CVaR is the first sorted sample and VaR is the second sorted samplewhen a ¼ 0.99, which is bound to be unstable.

Theorem 1 can be used to find the approximate VaR and CVaR of a portfolio ofsecurities, not necessarily a single security. Suppose there are n securities in aportfolio with Li representing the loss of security i. The loss of the portfolio isgiven by:

LðwÞ ¼Xni¼1

wiLi

where wi are weights of securities in the portfolio. For fixed w, we can find theVaR and CVaR of loss L(w) by first generating M samples of the joint distributionof (L1, . . . , Ln) say (L1k, . . . , Lnk) for k ¼ 1, . . . , M then sorting LkðwÞ ¼

Pni¼1wiLik

into a decreasing sequence, say L1(w) $ · · · $ LM (w), and finally applyingTheorem 1 to get VaRa(w) and CVaRa(w), i.e.:

VaRaðwÞ ¼Xni¼1

wiLi;Nþ1 and CVaRaðwÞ ¼Xni¼1

wiciðwÞ; ð6Þ

where ciðwÞ ¼ g ðLi1 þ · · · þ LiN Þ þ ð1 2 NgÞLi;Nþ1.

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As a byproduct we can also get the approximate marginal VaR and CVaR of L(w)with respect to the weights w under the mild condition that the sorted sequence Lk(w) isstrictly decreasing (as is typical). Then the perturbed loss sequence Lk(w þ e) keepsthe same order as that of Lk(w) if perturbation e ¼ (e1, . . . , en) has sufficiently smallmagnitude. We therefore have ciðwþ eÞ ¼ ciðwÞ for all and:

VaRaðwþ eÞ ¼Xni¼1

ðwi þ e iÞLi;Nþ1 and CVaRaðwþ eÞ ¼Xni¼1

ðwi þ e iÞciðwÞ:

This and (6) imply that the approximate gradients of VaR and CVaR are given by:

7wVaRaðwÞ ¼ ðL1;Nþ1; . . . ;Ln;Nþ1ÞT;

7wCVaRaðwÞ ¼ ðc1ðwÞ; . . . ; cnðwÞÞT:

To find the minimum CVaR portfolio, Rockafellar and Uryasev (2002) suggest to solvethe following convex optimization problem:

x[R;wminxþ

1

1 2 a

1

M

XMk¼1

Xni¼1

wiLik 2 x

!þ

¼w

minCVaRaðwÞ

subject to constraints w1 þ · · · þ wn ¼ 1 and w1, . . . , wn $ 0, and possibly some otherlinear constraints. The optimal value is the minimum CVaR of the portfolio, the optimalsolution w * is the optimal weight of the securities, and x * is the optimal VaR of theportfolio. Since the objective function is not differentiable, Rockafellar and Uryasev(2002) suggest to solve it either with the nonsmooth optimization method (n þ 1variables) or with the LP method (M þ n þ 1 variables). Since we know how tocompute CVaRa(w) and 7wCVaRa(w) explicitly for every w, we may also use otheroptimization method, such as the LANCELOT method of multipliers (Conn et al., 1992),to find the optimal weight and the minimum CVaR portfolio.

3. Jump liquidity risk processesLet (V, F, P) be a probability space and Ft be the filtration satisfying the usualconditions. Let St be the discounted mid-price of an asset following a GBM process (3).In Zheng (2006), it is suggested that the yield spread of corporate bonds is influencedby both credit risk and liquidity risk which is modelled with Poisson jump events. Thismotivates us to assume that the liquidity discount factor Xt of the asset price at time tfollows a jump-diffusion process:

dXt ¼ mðt;XtÞ dt þ s ðt;XtÞdBt þ Xt 2 dQt ð7Þ

where Bt is a standard Brownian motion, Qt ¼PNt

i¼1Yi is a marked point process, Nt isa Poisson process with intensity l, Yi s are independent and identically distributedrandom variables. Assume that Wt, Bt, Nt and Yis are independent to each other, andadapted to the filtration Ft. Assume also that m and s satisfy the conditions (e.g.,Lipschitz continuity and linear growth) that guarantee the existence of a unique strongsolution to (7).

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The discounted bid-price at time T is given by STXT and the discounted loss ofliquidating the asset at time T is given by:

L ¼ S0X0 2 STXT :

The LVaR and the LCVaR of L at level a are defined by (1) and (2), respectively. SinceST ¼ S0e2ð1=2Þs 2TþsWT we have by conditioning on WT that:

PðL # yÞ ¼

Z 1

21

P XT $ X0 2y

S0

� �eð1=2Þs 2T2s

ffiffiffiT

pz

� �dFðzÞ:

Let ti be the ith jump time. Conditional on j jumps over the interval [0, T ], i.e. NT ¼ jthe joint density function of t1, . . . , tj is given by:

f ðu1; . . . ; ujjNT ¼ j Þ ¼ j!T 2j1{0,u1,· · ·,uj,T}:

Let Xs; xt denote the strong solution to the SDE:

dXt ¼ mðt;XtÞdt þ s ðt;XtÞdBt ð8Þ

with the initial condition Xs ¼ x and Fs; xt ð yÞ ¼ PðXs;x

t # yÞ the correspondingdistribution function. If there is no jump in the interval [0, T ] then the conditionaldistribution of XT is simply given by PðXT # yjNT ¼ 0Þ ¼ F0;X0

T ð yÞ. We have byconditioning on the number of jumps NT that:

PðXT # yÞ ¼X1j¼0

PðNT ¼ j ÞPðXT # yjNT ¼ j Þ

and by conditioning on the jump times t1, . . . , tNTthat:

PðXT # yjNT ¼ jÞ ¼ j!T2j

Z T

0

Z T

u1

· · ·

Z T

uj21

PðXT # yjti ¼ uiÞduj. . .du1

and by conditioning on the jump sizes Y1, . . . , YNTthat:

PðXT # yjti ¼ uiÞ

¼ E{Yi}

j

i¼1

ZR

· · ·

ZR

Fuj;ð1þYjÞxjT ð yÞdFuj21;ð1þYj21Þxj21

ujðxjÞ. . .dF

0;X0u1

ðx1Þ

� �: ð9Þ

OU jump diffusion process. Assume that Xt follows a mean-revertingOrnstein-Uhlenbeck jump diffusion process (7) with m(t, x) ¼ k(u 2 x) and sðt; xÞ ¼ ~swhere k; u; ~s are constants. It is known that the solution to (8) is given by:

Xs; xt ¼ xe2kðt2sÞ þ uð1 2 e2kðt2sÞÞ þ ~se2kt

Z t

s

ekudBu; ð10Þ

and the distribution function of Xs;xt is given by:

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Fs; xt ð yÞ ¼ F

y2 xe2kðt2sÞ 2 u 1 2 e2kðt2sÞ� �

~sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1 2 e22kðt2sÞÞ=ð2kÞ

p !

: ð11Þ

We can express XT explicitly as follows.Theorem 2. Let the number of jumps in interval (0, T ) be NT and jump times be

0 , t1 , · · · , tNT, T with t0 ¼ 0 and tNT

þ 1 ¼ T. Then XT of the OU jumpdiffusion process is given by:

XT ¼ mNTþþ ~se2kT

XNTþ1

n¼1

Un;NT

Z tn

tn21

eksdBs; ð12Þ

where Un; j ¼Qj

i¼nð1 þ YiÞ for n $ 1 and Un, j ¼ 1 if n . j by convention, and

uj ¼ U 1; jX0e2kT þ ue2kTPjþ1

n¼1Un; jðektn 2 ektn21 Þ. The conditional probability of (9) is

equal to:

PðXT # yjti ¼ ui; i ¼ 1 . . . ;NTÞ ¼ E{Yi}

NTi¼1

Fy2 mNT

sNT

� �� �ð13Þ

where s 2j ¼ ð ~s 2=2kÞe22kT

Pjþ1n¼1U

2n; jðe

2kun 2 e2kun21 Þ.Proof. See Appendix. ACIR jump diffusion process. Assume that XT follows a mean-reverting

Cox-Ingersoll-Ross jump diffusion process (7) with m(t, x) ¼ ¼ k(u 2 x) and s ðt; xÞ ¼~sffiffiffix

pIt is known that there is no closed-form solution to (8) (Cox et al., 1985) and the

distribution function of Xs;xt has a noncentral x 2 distribution:

Fs; xt ð yÞ ¼ x 2 4ky

~s 2ð1 2 e2kðt2sÞÞ;4ku

~s 2;

4kx

~s 2ðekðt2sÞ 2 1Þ

� �; ð14Þ

where x 2(x; n, l) is the distribution function of a noncentral x 2 random variable withdegrees of freedom and noncentral parameter l.

Remark. The compensated GBM jump-diffusion process is given by (7) withmðt; xÞ ¼ 2blx and sðt; xÞ ¼ ~sx where b ¼ E(Yi). Although it is simple and has aclosed-form solution, it is not suitable for modelling the liquidity discount factor as Xt

increases (or decreases) at rate 2bl (ignoring the diffusion effect) between jump times,which is not in line with the empirical observation that the bid-ask spread is stable andrelatively flat in a normal market. To remove the obvious trend, one has to set the driftzero, but in doing so Xt is no longer a martingale and is unlikely to move back to itsoriginal level after jumps, which is again at odds with the empirical observation thatthe liquidity discount factor tends to recover to the normal market level after marketcrash. Owing to these reasons we do not use the GBM jump diffusion process to modelthe liquidity discount factor process Xt.

Figure 1 displays sample paths of GBM, OU, and CIR jump diffusion processes. It isobvious that sample paths for the GBM jump diffusion process either have a cleartrend between jumps if there is a compensator in the SDE or have no mean reversionafter jumps if there is no compensator. Sample paths for OU and CIR jump diffusionprocesses are similar and both display the mean reversion property as expected.

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Remark. In between jumps OU process Xt is driven by Brownian motion and there isa positive probability that Xt can be greater than 1 or less than 0, this is due to thenature of the Brownian motion, a well-known phenomenon for Gaussian interest ratemodels (Hull, 2000). To keep the liquidity discount factor process Xt within the range of0 and 1, we may use the reflected stochastic process. For example, the reflected OUprocess is modelled by:

dXt ¼ kðu2 XtÞdt þ ~sdWt þ dLt 2 dUt;

where both L and U are continuous nondecreasing processes with L0 ¼ U0 ¼ 0 andand increase only on the sets {t [ Rþ : Xt ¼ 0} and {t [ Rþ : Zt ¼ 1}. The reflectedprocess Xt is guaranteed to stay in between 0 and 1. The other possibility is to definethe liquidity discount factor process as an exponential process Xt ¼ X0exp(2Yt) whereYt is a basic affine process:

dYt ¼ �k ð �y2 YtÞdt þ �sffiffiffiffiffiffiYt

pdBt þ j dNt

andNt is a Poisson process with intensityl and j is an exponential random variable withmean g. All parameters �k; �y; �s; l; g are constants (Duffie and Singleton, 2003). SinceYt . 0 a.s. the liquidity discount factor process Xt takes values in range of 0 and 1.If there is a jump of size j at time t then the liquidity discount factor jumps downwardsfrom Xt2 to Xt ¼ e 2 jXt2 .

4. Numerical testsTo find numerical values of LVaR and LCVaR one may apply the results of Rockafellarand Uryasev (2002) to solve a convex optimization problem (4) with the Monte Carlomethod. In fact, if S i

T and XiT ; i ¼ 1; . . . ;M are samples of random variables ST and

XT set Li ¼ S0X0 2 SiTX

iT sort Li in decreasing order, and apply Theorem 1 to find

LVaRa and LCVaRa.

Figure 1.Sample paths of GBM, OU,

and CIR jump diffusionprocesses. Data: X0 ¼ 1,k ¼ 2, u ¼ 0.98, ~s ¼ 0:1,

l ¼ 2, Yi , U [20.5,20.2]0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

0 0.2 0.4 0.6 0.8 1

GBM with compensatorGBM without compensator

OUCIR

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Since ST follows a GBM process, it is easy to generate ST. To generate XT, we needto know the distribution of XT which is known for the mean reversion OU and CIRprocesses. In each simulation run, we first generate jump times ti and jump sizes Yi,i ¼ 1, . . . , NT, in the interval [0, T ]. If Xt follows an OU jump diffusion process, then wegenerate further NT þ 1-independent standard normal random variables Zn, n ¼ 1, . . . ,NT þ 1, and compute XT by (12) with the Ito integral:

Z tn

tn21

eksdBs ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðe2ktn 2 e2ktn21 Þ

ð2kÞ

sZn:

If Xt follows a CIR jump diffusion process, then we generate recursively further NT þ 1noncentral x 2 variables Xtiþ 1 2

from the noncentral x 2 distribution function (14) withs ¼ ti, x ¼ Xti2

(1 þ Yi) and t ¼ tiþ 1 for i ¼ 0, . . . , NT and finally setXT ¼ XtNT

þ 1 2 . (Here, we denote Xt02(1 þ Y0) ¼ X0 and t0 ¼ 0 and

tNTþ 1 ¼ T.) Noncentral x 2 random variables can be generated with the algorithm

discussed in Glasserman (2003).Table I lists the values of LVaR and LCVaR with the OU and CIR jump diffusion

processes. The parameters used represent a market in which the liquidity premium issmall and stable (mean reversion level close to 1 and volatility close to 0), the liquiditydry up event is rare (once every five years on average), potential liquidity loss is severe(20-50 percent of asset value), the holding period is two weeks. The number ofsimulations is ¼ 100,000.

Table I clearly shows the following outcomes:. The OU and CIR jump diffusion processes produce very similar values for LVaR

and LCVaR, which implies that one can essentially use either of these two modelsto compute liquidity adjusted risk measures.

. LCVaR is much larger than LVaR at 0.99 percentile, which implies that LVaR canstill seriously underestimate the potential loss even after the jump liquidity riskis included, LCVaR is a more realistic potential loss indicator.

. LVaR and LCVaR are close at 0.999 percentile, which implies these two riskmeasures produce similar results at the extreme tail part of the loss distribution.

Table II shows that jump intensity l affects greatly the values of LVaR and LCVaR.When there are no jumps (l ¼ 0) LVaR and LCVaR are close to VaR and CVaR, thedifference is mainly due to the CIR mean reversion diffusion process for the liquidity

a Model VaR LVaR CVaR LCVaR

0.99 CIR 8.96 11.05 10.18 29.90OU 11.02 29.66

0.999 CIR 11.70 45.63 12.66 48.30OU 45.45 48.13

Notes: Data: S0 ¼ 100, s ¼ 0.2, X0 ¼ 1, k ¼ 1, u ¼ 0.98, ~s ¼ 0:02, l ¼ 0.2, T ¼ 0.04, Yi , U [20.5,20.2]

Table I.LVaR and LCVaR for OUand CIR jump diffusionprocesses

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discount factor. When l increases, both LVaR and LCVaR increase at different speed.For example, at 0.99 percentile, when l ¼ 0.2, LVaR is increased by 23 percent over thestandard VaR, while LCVaR is increased by 194 percent over the standard CVaR, andthe ratio of LCVaR to LVaR is about 2.7. This implies that the traditional VaR andCVaR are inappropriate risk measures in the presence of jump liquidity risk, and thatone should take cautious views on the loss suggested by the LVaR as it may seriouslyunderestimate the potential average loss for rare jump liquidity events.

Table III shows that as the holding period T increases both LVaR and LCVaRincrease and LVaR gives a good indication of the average loss. When the holdingperiod T is very short (e.g., one day) the LVaR, VaR, and CVaR all suggest a small loss.However, LCVaR points out a much larger loss. At 0.999 percentile, the ratio of LCVaRto LVaR is 6.0, which implies that if one manages the risk with the liquid assetsuggested by VaR/CVaR/LVaR, one is possibly unable to withstand the potentialsevere loss. This sheds some light to the cause of the fall of the LTCM which had greatdifficulty in raising sufficient cash in a short spell of time to meet margin calls byliquidating the asset in a market where the liquidity essentially disappeared.

We have also tested cases for different mean-reversion rate k, mean-reversion levelu, and volatility ~s. We find that LVaR and LCVaR are not very sensitive to changes ofthese parameters. This is because over a short period (two weeks) the change causedby diffusion part of the liquidity discount factor process is small, but if there is a jumpliquidity event, then there is little time to recover and the loss is likely to be substantial.On the other hand, LVaR and LCVaR are sensitive to the magnitude of the jump size.

5. ConclusionWe have suggested in this paper some plausible stochastic processes to model liquidityrisk and discussed their impact on VaR and CVaR. We have shown that VaR, CVaR,

a T VaR LVaR CVaR LCVaR

0.99 0.0028 2.44 2.48 2.79 4.590.04 8.96 11.03 10.18 29.891 38.45 51.55 42.37 57.00

0.999 0.0028 3.22 3.48 3.50 20.910.04 11.70 45.62 12.66 48.291 47.17 63.66 49.94 67.38

Notes: Data: S0 ¼ 100, s ¼ 0.2, X0 ¼ 1, k ¼ 1, u ¼ 0.98, ~s ¼ 0:02, l ¼ 0.2, Yi , U [20.5, 20.2]

Table III.LVaR and LCVaR for theCIR jump diffusion model

with varying T

a l VaR LVaR CVaR LCVaR

0.99 0 8.96 9.07 10.18 10.290.2 11.03 29.891 42.09 46.83

0.999 0 11.70 11.82 12.66 12.800.2 45.62 48.291 50.63 56.86

Notes: Data: S0 ¼ 100, s ¼ 0.2, X0 ¼ 1, k ¼ 1, u ¼ 0.98, ~s ¼ 0:02, T ¼ 0.04, Yi , U [20.5, 20.2]

Table II.LVaR and LCVaR for theCIR jump diffusion model

with varying l

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and LVaR can seriously underestimate the potential loss over a short holding periodfor rare jump liquidity events. This has significant implication for short-term riskmanagement, i.e. one should keep a much larger liquid asset reserve than the suggestedVaR value to withstand the potential severe loss if a rare “bad” event does happen. TheLTCM’s fall is a recent example to illustrate such a need. A better risk measure is theLCVaR which gives a more realistic loss estimation in the presence of the liquidity risk.

We have also suggested a simple and fast Monte Carlo method to computeapproximate VaR and CVaR without having to solve nonlinear equations and tointegrate tail expectations. The only work involved is to generate and sort samples ofthe loss distribution, which is sufficient to find VaR and CVaR of all percentiles, andtheir marginal values for a portfolio of securities.

Many questions remain to be answered, especially in the area of calibration,empirical studies, and correlation modelling. For example, how to calibrate the jumpliquidity risk process to the market data? how good are these models in explainingmarket liquidity-crunch and crash behaviour? What is the correlation of the liquidityrisk with other market risks such as credit risk? We will focus on these questions in ourfuture work.

References

Artzner, P., Delbaen, F., Eber, J.M. and Heath, D. (1999), “Coherent measures of risk”,Mathematical Finance, Vol. 9, pp. 203-28.

Black, F. (1971), “Towards a fully automated exchange: Part 1”, Financial Analyst Journal,Vol. 27, pp. 29-34.

Conn, A.R., Gould, N.I.M. and Toint, P.L. (1992), LANCELOT: A FORTRAN Package forLarge-scale Nonlinear Optimization, No. 17 in Springer Series in ComputationalMathematics, Springer, New York, NY.

Cox, J.C., Ingersoll, J.E. Jr. and Ross, S.A. (1985), “A theory of the term structure of interest rates”,Econometrica, Vol. 53, pp. 385-408.

Duffie, D. and Singleton, S.K. (2003), Credit Risk: Pricing, Measurement, and Management,Princeton University Press, Princeton, NJ.

Dunbar, N. (1998), “Meriwether’s meltdown”, Risk, October, pp. 32-6.

Glasserman, P. (2003), Monte Carlo Methods in Financial Engineering, Springer, New York, NY.

Hull, J. (2000), Options, Futures, and Other Derivatives, Prentice-Hall, London.

Karatzas, I. and Shreve, S.E. (1988), Brownian Motion and Stochastic Calculus, Springer, NewYork, NY.

Lawrence, C. and Robinson, G. (1996), “Liquidity, dynamic hedging and VAR risk”, Managementfor Financial Institutionals, pp. 63-72.

Rockafellar, R.T. and Uryasev, S. (2002), “Conditional value-at-risk for general lossdistributions”, Journal of Banking & Finance, Vol. 26, pp. 1443-71.

Zheng, H. (2006), “Interaction of credit and liquidity risks: modelling and valuation”, Journal ofBanking & Finance, Vol. 30, pp. 391-407.

Further reading

Nocedal, J. and Wright, S.J. (1999), Numerical Optimization, Springer, New York, NY.

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Appendix: ProofsProof of Theorem 1. The dual problem for (5) is:

max L1 y1 þ · · · þ LMyM s:t: y1 þ y2 þ · · · þ yM ¼ 1; 0 # yi # g; i ¼ 1; . . . ;M : ðA1Þ

The choice of integer ensures that Ng # 1 and (N þ 1)g . 1. Since L1 $ · · · $ LM weknow the optimal solution to (A1) is:

y*1 ¼ y*2 ¼ · · · ¼ y*N ¼ g; y*Nþ1 ¼ 1 2 Ng; y*Nþ2 ¼ y*Nþ3 ¼ · · · ¼ y*M ¼ 0: ðA2Þ

The Lagrangian function for (A1) is given by:

L ¼ 2XMi¼1

L iyi þ xXMi¼1

yi 2 1

!þXMi¼1

zið yi 2 gÞ2XMi¼1

mi yi;

where x, zi, mi, i ¼ 1, . . . , M are Lagrange multipliers. The optimal solution to the dual problem(A1) is characterized by the following Kuhn-Tucker conditions:

2Li þ xþ zi 2 mi ¼ 0; ðA3Þ

zið yi 2 gÞ ¼ 0; ðA4Þ

mi yi ¼ 0; ðA5Þ

zi $ 0; ðA6Þ

mi $ 0; ðA7Þ

for i ¼ 1, . . . , M. Since the optimal solution to the dual problem is (A2) we can find the optimalLagrange multipliers x, zi and mi, i ¼ 1, . . . , M from the Kuhn-Tucker conditions as follows:

. For i ¼ 1, . . . , N

y*i ¼ gðA5Þ)mi ¼ 0

ðA3Þ)2 Li þ xþ zi ¼ 0:

. For i ¼ N þ 2, . . . , M

y*i ¼ 0ðA4Þ)zi ¼ 0:

. For i ¼ N þ 1

y*Nþ1 ¼ 1 2 Ng,g

ðA4Þ)zNþ1 ¼ 0:

. If Ng , 1 then

y*Nþ1 . 0ðA5Þ)mNþ1 ¼ 0

ðA3Þ)2 LNþ1 þ xþ zNþ1 ¼ 0

ðcÞ)x ¼ LNþ1:

and zi ¼ Li 2 LNþ1, i ¼ 1, . . . , N, from (a). The optimal solution to the primal problem (5)is unique: x* ¼ LNþ1; z

*i ¼ Li 2 LNþ1, and z*i ¼ 0; i ¼ N þ 1; . . . ;M :

. If 1 2 Ng ¼ 0, then

y*Nþ1 ¼ 0ðA3Þ)xþ zNþ1 ¼ LNþ1 þ mNþ1

ðA7Þ)xþ zNþ1 $ LNþ1

ðcÞ)x $ LNþ1

Since (a) and (A6) imply zi ¼ Li 2 x $ 0, i ¼ 1, . . . , N, must also satisfy x # LN as {Li} isa non-increasing sequence. The optimal solution to the primal problem (5) is not unique:x * [ [LNþ1, LN], z*i ¼ Li 2 x*, i ¼ 1, . . . ,N and z*i ¼ 0; i ¼ N þ 1; . . . ;M .

Jump liquidityrisk

489

Rockafellar and Uryasev (2002) show that VaR is equal to the left endpoint of the optimalsolution set, which implies VaRa ¼ LNþ1 whether the primal problem has a unique solution ornot, and CVaRa ¼ ð1 2 NgÞLNþ1 þ gðL1 þ · · · þ LN Þ is the corresponding optimal value. A

Proof of Theorem 2. We first use the induction method to prove (12). In fact, when NT ¼ 0,i.e. there is no jump in interval [0, T ], then (12) is the same as (10). Now assume that (12) is correctfor NT ¼ j 2 1( j $ 1) and we only need to show that (12) is correct for NT ¼ j too. Since there isno jump between the jth jump time tj and the terminal time the solution XT is given by:

XT ¼ X tje2kðT2tjÞ þ u 1 2 e2kðT2tjÞ

� �þ ~se2kT

Z T

tj

eksdBs: ðA8Þ

On the other hand, since tj is a jump time and there are only j 2 1 jumps in interval [0, tj) theinduction assumption implies:

X tj ¼ ð1 þ YjÞXtj2 ¼ ð1 þ YjÞ mj21 þ ~se2ktjXjn¼1

Un:j21

Z tn

tn21

eksdBs

!

¼ U 1;jX0e2ktj þ ue2ktjXjn¼1

Un; jðektn 2 ektn21 Þ þ ~se2ktj

Xjn¼1

Un; j

Z tn

tn21

eksdBs:

Substituting Xtjinto (A8) we see that (12) holds true for NT ¼ j.

Given jump sizes Y1, . . . , Yj XT defined in (12) is a normal variable with mean mj and variances2j . Here, we have used the independent increment property of a Brownian motion and the Ito

isometry property. The conditional probability (9) is therefore given by (13).We can also prove (13) by substituting (11) directly into (9). First, note that ifZ 1 , Nðm1;s

21Þ and

Z 2 , Nð0;s22Þ are two independent normal variables, then cZ1 þ Z2 (c is a constant) is a normal

variable with mean cm1 and variance c 2s21 þ s2

2, and:

PðcZ 1 þ Z 2 # yÞ ¼ Fy2 cm1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffic 2s2

1 þ s22

q0B@

1CA:

On the other hand, by conditioning on Z1 we have:

PðcZ 1 þ Z 2 # yÞ ¼

Z 1

21

Fy2 cz

s2

� �dF

z2 m1

s1

� �:

Therefore, the following relation holds:

Z 1

21

Fy2 cz

s2

� �dF

z2 m1

s1

� �¼ F

y2 cm1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffic 2s2

1 þ s22

q0B@

1CA: ðA9Þ

With the expression (11) for Fs; xt ð yÞ and the relation (A9) we can get:Z

R

Fuj;ð1þYjÞxjT ð yÞdFuj21 ;ð1þYj21Þxj21

ujðxjÞ

¼ Fy2 e2kðT2uj21ÞUj21;jxj21 2 u e2kT

Pjþ1n¼jUn:jðe

kun 2 ekun21 Þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�s 2e22kT

Pjþ1n¼jU

2n; jðe

2kun 2 e2kun21 Þ=ð2kÞq

0B@

1CA

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Repeating the same argument, also noting u0 ¼ 0, x0 ¼ X0, and Y0 ¼ 0, we get:ZR

· · ·

ZR

Fuj;ð1þYjÞxjT ð yÞdFuj21;ð1þYj21Þxj21

ujðxjÞ. . .dF

0;X0u1

ðx1Þ ¼ Fy2 mj

sj

� �:

A

Corresponding authorHarry Zheng can be contacted at: h.zheng@imperial.ac.uk

Jump liquidityrisk

491

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