liquidity alexandre roch_6
TRANSCRIPT
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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
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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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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)
Mtx is the liquidity premium for a transaction of size x.Mt changes in time : liquidity risk.
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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)
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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The Cetin-Jarrow-Protter Model for Liquidity Risk
Figure: Typical order book density for linear model.
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The Cetin-Jarrow-Protter Model for Liquidity Risk
We want to buy Xt shares at time t.
With one block trade, we pay XtSt + Mt(Xt)2
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The Cetin-Jarrow-Protter Model for Liquidity Risk
We want to buy Xt shares at time t.
With one block trade, we pay XtSt + Mt(Xt)2
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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The Cetin-Jarrow-Protter Model for Liquidity Risk
We want to buy Xt shares at time t.
With one block trade, we pay XtSt + Mt(Xt)2
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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Liquidity Model with Trade Impacts and Resiliency
Figure: Price Impact at time t+ .
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Liquidity Model with Trade Impacts and Resiliency
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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Liquidity Model with Trade Impacts and Resiliency
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.
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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Liquidity Model with Trade Impacts and Resiliency
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.
S0t denotes the actual observed marginal price at time t. It dependson the process X up to time t. S0t+ = S
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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Liquidity Model with Trade Impacts and Resiliency
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.
S0t denotes the actual observed marginal price at time t. It dependson the process X up to time t. S0t+ = S
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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Liquidity Model with Trade Impacts and Resiliency
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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Liquidity Model with Trade Impacts and Resiliency
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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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.
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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No Arbitrage and Self-Financing Strategies
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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No Arbitrage and Self-Financing Strategies
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
Properties of Prices
Replication Error
5 Solving with PDEs and Viscosity Solutions
6 References
The Volatility Swaps
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The Volatility Swaps
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.
The Volatility Swaps
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The Volatility Swaps
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.
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.
The Volatility Swaps
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The Volatility Swaps
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.
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.
Properties of Prices
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Properties of Prices
The claim h has liquidity constraints associated to it : thereplicating cost of units is not times the replicating cost of 1 unit.
Properties of Prices
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p
The claim h has liquidity constraints associated to it : thereplicating cost of units is not times the replicating cost of 1 unit.
Let (X
,
,Y
) be the solution of BSDE with terminal conditionh(ST) in which ST = ST T0 XtdMt.
Properties of Prices
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p
The claim h has liquidity constraints associated to it : thereplicating cost of units is not times the replicating cost of 1 unit.
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().
Properties of Prices
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p
The claim h has liquidity constraints associated to it : thereplicating cost of units is not times the replicating cost of 1 unit.
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.
Properties of Prices
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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.
Properties of Prices
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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.
Properties of Prices
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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.
Properties of Prices
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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.
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 .
Properties of Prices
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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.
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
Properties of Prices
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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 .
Properties of Prices
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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 .
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 .
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)
Solving with PDEs and Viscosity Solutions
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.
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