Liquidity Alexandre Roch_6

8/4/2019 Liquidity Alexandre Roch_6

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Hedging under Liquidity Risk and Price Impacts

Mathematical Finance and Probability Seminar - Rutgers University

Alexandre F. Roch

Center for Applied MathematicsCornell University

November 4th, 2008
http://find/http://goback/


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Outline

1 The Cetin-Jarrow-Protter Model for Liquidity Risk

2 Liquidity Model with Trade Impacts and Resiliency

3 No Arbitrage and Self-Financing Strategies

4 The Replication Problem and Quadratic BSDEsOption replication

Properties of Prices

Replication Error

5 Solving with PDEs and Viscosity Solutions

6 References


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The Cetin-Jarrow-Protter Model for Liquidity Risk

S(t, x) is price per share to buy (x> 0) or sell (x< 0) at time t.Total price to pay for x shares is then xS(t, x). In practice,S(t, x) = St + Mtx. (See Marcel Blaiss thesis)


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Mtx is the liquidity premium for a transaction of size x.Mt changes in time : liquidity risk.


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Mtx is the liquidity premium for a transaction of size x.Mt changes in time : liquidity risk.

Self-financing strategies (X,Y) satisfy

YT = Y0 +

T0

XudSu

T0

Mud[X]u.

Xt denotes the number of shares held at time t and Yt the money inthe bank account (0 interest).


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Figure: Typical order book density for linear model.


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We want to buy Xt shares at time t.

With one block trade, we pay XtSt + Mt(Xt)2


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Or we can break itdown into n smaller trades.

n

i=11

nXt(St + Mt

1

nXt)

= XtSt +1n

Mt(Xt)2

When n is large, the cost converges to XtSt.


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Or we can break itdown into n smaller trades.

n

i=11

nXt(St + Mt

1

nXt)

= XtSt + 1n

Mt(Xt)2

When n is large, the cost converges to XtSt.

CJP show that any reasonable strategy can be approximated usingcontinuous FV strategies.

YT = Y0 +

T0

XudSu

T0

Mud[X]u.


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Liquidity Model with Trade Impacts and Resiliency

Figure: Typical order book density for linear model.


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Figure: Price Impact at time t+ .


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We take the point of view of an impatient hedger. All the tradesare made at the market price S0(t, x). We can take r = 0 withoutloss of generality.


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S0t denotes the actual observed marginal price at time t. It dependson the process X up to time t. S0t+ = S

0t + 2MtXt.


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0t + 2MtXt.

We denote by St the marginal price (or unaffected price) that would

have been observed if not trades had been executed by the hedgeruntil time t (Xs = 0, 0 s t).It is a fictitious price and cannot be observed. It includes everyother traders activity except the hedgers.


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0t + 2MtXt.

We denote by St the marginal price (or unaffected price) that would

have been observed if not trades had been executed by the hedgeruntil time t (Xs = 0, 0 s t).It is a fictitious price and cannot be observed. It includes everyother traders activity except the hedgers.

We define the price after the hedgers impact by

S0t+ = St + 2

t0

MudXu + 2

t0

d[M,X]u.

for any t T. ( = 0 is the original CJP Model)


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dvt = (vt + a)dt+ b(vt, t)dB1,t,

dMt = (Mt + )dt+ 1(Mt, t)dB1,t + 2(Mt, t)dB2,t

dSt = fundamental + other market orders impact

= fundamental +

i 2MtdX

it + 2d[M,X

i]t

=

3j=1

jvtStdBj,t

fundamental

+ 23

j=1

jMtStdBj,t

market orders

We work directly under the measure Q that makes S a (local)martingale.

dS0t =3

j=1j(vt + Mt)StdBj,t + 2MtdXt + 2d[M,X]t


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dSt =3

j=1 j(vt + Mt)StdBj,t

Figure: Volatility vs Liquidity, Coefficient of determination= 0.3745

S f S


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No Arbitrage and Self-Financing Strategies

Let n : 0 = n0

n1 . . .

nkn

= t be a sequence of randompartitions tending to the identity and nkX = Xnk X

nk1

. A pair

(Xt,Yt)t0 is a self-financing trading strategy (s.f.t.s) if X is acadlag process and Y is an optional process satisfying

Yt = Y0 limn

knk=1

nkXS0(nk,

nkX).

We will always define trading strategies with X0 = Y0 = 0.

N A bi d S lf Fi i S i


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Let n : 0 = n0

n1 . . .

nkn

= t be a sequence of randompartitions tending to the identity and nkX = Xnk X

nk1

. A pair

(Xt,Yt)t0 is a self-financing trading strategy (s.f.t.s) if X is acadlag process and Y is an optional process satisfying

Yt = Y0 limn

knk=1

nkXS0(nk,

nkX).

We will always define trading strategies with X0 = Y0 = 0.

Theorem 1

Let X be a cadlag process and Y an optional process. If(Xt,Yt)t0 is aself-financing trading strategy then

YT = Y0 +

T0

XudSu

T0

X2udMu (1)

(1 )M0X2

0

T

0

(1 )Mud[X,X]u.


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N A bit d S lf Fi i St t i


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ZT = T

0 XudSu T

0 X

2

udMu T

0 (1 )Mud[X,X]u.

An arbitrage opportunity is an admissible s.f.t.s. whose payoff ZTsatisfies P{ZT 0} = 1 and P{ZT > 0} > 0.

Hypothesis (1) : There exists a measure Q equivalent to P such that Sis a Q-local martingale and M is a Q-submartingale.

Theorem 2

Under Hypothesis (1) there are no arbitrage opportunities.

N A bit d S lf Fi i St t i


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ZT = T

0 XudSu T

0 X

2

udMu T

0 (1 )Mud[X,X]u.

An arbitrage opportunity is an admissible s.f.t.s. whose payoff ZTsatisfies P{ZT 0} = 1 and P{ZT > 0} > 0.

Hypothesis (1) : There exists a measure Q equivalent to P such that Sis a Q-local martingale and M is a Q-submartingale.

Theorem 2

Under Hypothesis (1) there are no arbitrage opportunities.

Since S is a Q-local martingale by construction, it suffices to take, 0 to rule out arbitrage opportunities in our setting.

dMt = (Mt + )dt+ 1(Mt, t)dB1,t + 2(Mt, t)dB2,t

O tli


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Outline

1 The Cetin-Jarrow-Protter Model for Liquidity Risk

2 Liquidity Model with Trade Impacts and Resiliency

3 No Arbitrage and Self-Financing Strategies

4

The Replication Problem and Quadratic BSDEsOption replication


Replication Error

5 Solving with PDEs and Viscosity Solutions

6 References

The Volatility Swaps


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We add two volatility swaps, denoted Gi,t for i = 1, 2. To ensure noarbitrage, we assume the existence of an equivalent probability

measure Q such that S is martingale, M is submartingale and

Gi,t = EQ

vTi + MTi

Ft Kifor i = 1, 2.



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Gi,t = EQ

vTi + MTi

Ft Kifor i = 1, 2.

i,t will represent the number of shares invested in the swap Gi attime t. We assume that these swaps have liquidity constraintssimilar to the asset S except that their liquidity is constant.



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Gi,t = EQ

vTi + MTi

Ft Kifor i = 1, 2.

i,t will represent the number of shares invested in the swap Gi attime t. We assume that these swaps have liquidity constraintssimilar to the asset S except that their liquidity is constant.

S.f.t.s now satisfy

YT = Y0 +T0

XudSuT0

X2udMuT0

(1 )Mud[X]u

+i

T0

i,udGi,u i

T0

(1 i)Nid[i]u.

Here Ni and Ki refer to the liquidity constants of Gi.

Approximate Completeness and S f t s


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Approximate Completeness and S.f.t.s.

Recall the definition of self-financing:

YT = Y0 +T0

XudSu T0

X2udMu T0

(1 i)Mud[X]u

+i

T0

i,udGi,u i

T0

(1 i)Nid[i]u.

Lemma 1

Let U be a semimartingale and X be predictable and integrable withrespect to U. There exists a sequence {Xn}n of bounded continuous

processes with finite variation such that Xn

0 = Xn

T = 0 and Xn

convergesto X in H2. In particular, XndU XdU in H2.

Approximate Completeness and S f t s


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Approximate Completeness and S.f.t.s.

Recall the definition of self-financing:

YT = Y0 +T0

XudSu T0

X2udMu T0

(1 i)Mud[X]u

+i

T0

i,udGi,u i

T0

(1 i)Nid[i]u.

Lemma 1

Let U be a semimartingale and X be predictable and integrable withrespect to U. There exists a sequence {Xn}n of bounded continuous

processes with finite variation such that Xn

0 = Xn

T = 0 and Xn

convergesto X in H2. In particular, XndU XdU in H2.Definition H L1 can be approximately replicated if there exists asequence (Xn, n,Yn)n1 of s.f.t.s. such that Y

nT H in L

1.

Option replication


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Option replication

We want to replicate options with payoff function h. However, the payoff

depends on the observed value of the stock :S0T = ST+2

T0 MsdXs + 2[M,X]T= ST2

T0 XsdMs,

h(S0T) = Yt + T

t

XsdSs T

t

(Xs)2dMs +i

T

t

i,sdGi,s.

Option replication


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Option replication

We want to replicate options with payoff function h. However, the payoff

depends on the observed value of the stock :S0T = ST+2

T0 MsdXs + 2[M,X]T= ST2

T0 XsdMs,

h(S0T) = Yt + T

t

XsdSs T

t

(Xs)2dMs +i

T

t

i,sdGi,s.

Instead we consider:

h(ST 2

T

0 XsdMs) = Yt

T

t

XsdSs +

T

t

(Xs)2dMs

i T

t

i,sdGi,s.

in which X is the solution of the replication problem in a simpler case.Jarrow (1994) calls it the market perception of the delta.

Option replication


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Option replication

X is the solution of the replication problem in the case of no tradeimpacts:

h(St) = Yt + TtXudSu +

i=1,2

Tti,udGi,u. (2)

Its a linear BSDE and it has a unique solution.



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The claim h has liquidity constraints associated to it : thereplicating cost of units is not times the replicating cost of 1 unit.



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p


Let (X

,

,Y

) be the solution of BSDE with terminal conditionh(ST) in which ST = ST T0 XtdMt.



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p


Let (X

,

,Y

) be the solution of BSDE with terminal conditionh(ST) in which ST = ST T0 XtdMt.Let Ht() be the replicating price per share of shares of the claim

h starting at time t, i.e. Y

t

.

Ht(0) := lim0 Ht().



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p


Let (X

,

,Y

) be the solution of BSDE with terminal conditionh(ST) in which ST = ST T0 XtdMt.Let Ht() be the replicating price per share of shares of the claim

h starting at time t, i.e. Y

t

.

Ht(0) := lim0 Ht().

Theorem 3

Let h : R+ R be Lipschitz continuous. Then xh(SxT) can beapproximately replicated for all x R, i.e. the BSDE

xh(SxT) = Yt Tt

XsdSs + Tt

(Xs)2dMs

i

Tt

i,sdGi,s.

has a solution.



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p

Theorem 4

Let h : R+ R be Lipschitz continuous. We have that

Ht(0) = Yt = EQ(h(ST)|Ft) and 1 X X in L2(dQ dt) as 0.



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Theorem 4

Let h : R+ R be Lipschitz continuous. We have that

Ht(0) = Yt = EQ(h(ST)|Ft) and 1 X X in L2(dQ dt) as 0.

h(St) = Yt + Tt XudSu + i=1,2T

t i,udGi,uis the same as

H = Yt + 3i=1

TtZi,udBi,u

with H = h(ST),

Zi,u = iuSu

Xu +

j=1,2 i,j,t

j,u for i = 1, 2,

Z3,u = 3uSuXu.



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Also,

h(ST) = Y

t +T

t

XsdSs T

t

(Xs)2dMs +i

T

t

i,sdGi,s

is the same as

H = Yt

Tt

(Z3,u)2Cudu+

3

i=1Tt

Zi,udBi,u

with H = h(ST), Cu =(Mu+)

23

2uS

2u

and a similar change of variables.



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Also,

h(ST) = Y

t +T

t

XsdSs T

t

(Xs)2dMs +i

T

t

i,sdGi,s

is the same as

H = Yt

Tt

(Z3,u)2Cudu+

3

i=1Tt

Zi,udBi,u


23

2uS

2u


2(H)2 = (Yt )2 2

Tt

Cu(Z

3,u)2Yu

1

2|Zu|

2

du+ 2

Tt

YuZ

udBu

Tt

12

|Zu|2du+ 2 T

t

YuZ

udBu.

for small .



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Also,

h(ST) = Y

t +T

t

XsdSs T

t

(Xs)2dMs +i

T

t

i,sdGi,s

is the same as

H = Yt

Tt

(Z3,u)2Cudu+

3

i=1Tt

Zi,udBi,u


23

2uS

2u


2(H)2 = (Yt )2 2

Tt

Cu(Z

3,u)2Yu

1

2|Zu|

2

du+ 2

Tt

YuZ

udBu

Tt

12

|Zu|2du+ 2 T

t

YuZ

udBu.

for small .We have

EQ(T

t

|Zu|2du|Ft) 2EQ(

2(H)2|Ft)

2


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Yt = EQ

H +

t

Cu(Z

3,u)2du

Ft ,Yt = EQ HFt .

2


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Yt = EQ

H +

t

Cu(Z

3,u)2du

Ft ,Yt = EQ HFt .

|1

Yt

Yt| |

1

Yt EQ(H

|Ft)| + |EQ(H|Ft) EQ(H|Ft)|

C1

EQ(Tt

|Zu|2du|Ft) + EQ(|H H||Ft)

2CEQ((H)2|Ft) + EQ(|H

H||Ft).

2


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Yt = EQ

H +

t

Cu(Z

3,u)2du

Ft ,Yt = EQ HFt .

|1

Yt

Yt| |

1

Yt EQ(H

|Ft)| + |EQ(H|Ft) EQ(H|Ft)|

C1

EQ(Tt

|Zu|2du|Ft) + EQ(|H H||Ft)

2CEQ((H)2|Ft) + EQ(|H

H||Ft).

EQ T

t0| 1

Zu Zu|2du

= EQ|H H|2

Y0 1

Y0

2+ 2EQ

T0

Cu1

2(Z3,u)

2(Yu Yu)d

EQ|H H|2 +

C

EQ

T

0|Zu|

2du



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Theorem 5

If h is bounded and differentiable everywhere except at a finite number of

points, then H0(x) is a.s. differentiable at x = 0 and

H0(0) = EQ

T0

(Ms + )

X2sds

F0

2EQh(ST)(T0XsdMs)F0 .



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Theorem 5

If h is bounded and differentiable everywhere except at a finite number of

points, then H0(x) is a.s. differentiable at x = 0 and

H0(0) = EQ

T0

(Ms + )

X2sds

F0

2EQh(ST)(T0 XsdMs)F0 .H0(0) is the level of liquidity or liquidity premium of the option. Theanalogy of M for the underlying asset.

Ht(x) = EQ(h(ST)|Ft) + xHt(0) + O(x

2)St(x) = St + xMt


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1

1 Yt Yt= EQ

t

Cu1

2(Z3,u)

2du+1

(H H)

Ft

EQ

t CuZ23,uduFt+ EQ1 h(ST 2 T

0 XsdMs) h(ST) EQ

t

CuZ23,uduFt

+ EQ

2

T0XsdMs

h(ST)

Ft

Replication Error


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We replicated: h(ST 2T0 XsdMs) using the strategy X. But thetrue payoff is h(ST 2T0 XsdMs).How far off are we?

Replication Error


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We replicated: h(ST 2T0 XsdMs) using the strategy X. But thetrue payoff is h(ST 2T0 XsdMs).How far off are we?

Because,

X is close to X for small epsilon, we have:

Theorem 6

If h is Lipschitz continuous then

EQ h(S0T) h(S

T)2

= O(3)

as 0.

Solving with PDEs and Viscosity Solutions


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Define Lt = L0 + 2 t

0 XsdMs and let (Xt,S,L,M,Yt,S,L,M) be the

solution of the BSDE starting at (St, Lt,Mt) = (S, L,M) at timet T with terminal condition H = h(ST LT). Define

y(t,S, L,M) = Yt,S,L,Mt .



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Define Lt = L0 + 2 t

0 XsdMs and let (Xt,S,L,M,Yt,S,L,M) be the

solution of the BSDE starting at (St, Lt,Mt) = (S, L,M) at timet T with terminal condition H = h(ST LT). Define

y(t,S, L,M) = Yt,S,L,Mt .

The Markov properties of S and M carry on to y. And we find a

PDE for y :Theorem 7

y is a viscosity solution of

y

t Ly (M + )(

y

S)2

= 0 (3)

in which ...



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Ly = (M+ ) yM

+ X(S,M)yL+ 1

22M2S2 2y

S2

+1

2(i

2i )

2y

M2+ X(S,M)2

2y

L2

+(11 + 22)MS 2ySM

+ X(S,M) 2y

SL

+(21 + 22 )X(S,M)

2y

ML

with boundary conditions

y(t, S, L,M) = h(S L) if t = T or S = 0 or M = 0. (4)


D fi


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Definition 2

We say that a continuous function y is a viscosity supersolution (resp.

subsolution) of (3)-(4) if y satisfies (4) and if for any functions C1,2(U) and any (t0,0) U such that

y(t0,0) = (t0,0) and

y(t,) (t,) (resp. y(t,) (t,))

for any (t,) U, then

y

t(t0,0) Ly(t0,0)

(M0 + )(yS

(t0,0))2 0,

(resp. 0). The function y is a viscosity solution if it is both asupersolution and subsolution.

References


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Blais, M. (2006). Liquidity and Modelling the Stochastic SupplyCurve for a Stock Price. PhD Thesis, Cornell University.

Cetin, U., Jarrow, R. and Protter, P. (2004). Liquidity Riskand Arbitrage Pricing Theory, Finance and Stochastics 8 311341.

Kobylanski, M. (1997). Resultats dexistence et dunicite pour desequations differentielles stochastiques retrogrades avec des generateursa croissance quadratique, C. R. Acad. Sci. Paris Ser. I Math. 324(1),81-86.

Weber, P. and Rosenow, B. (2005). Order book approach toprice impact. Quantitative Finance, 5(4) 357364.

Documents

Liquidity Alexandre Roch_6