Download Free boundary regularity close to initial state and

Free boundary
regularity close to
initial state and
applications to
finance
Free boundary regularity close to initial state
and applications to finance
Teitur Arnarson
KTH, Stockholm
American options
History and
background
The blow-up
technique
Teitur Arnarson
KTH, Stockholm
PDE Methods in Finance 2007
October 15-16, University of Marne-la-Vallée
Application to
indifference pricing
Outline
Free boundary
regularity close to
initial state and
applications to
finance
Teitur Arnarson
KTH, Stockholm
American options
?
?
?
?
Problem and setup
History
The blow-up technique
Application to indifference pricing
History and
background
The blow-up
technique
Application to
indifference pricing
American option
Free boundary
regularity close to
initial state and
applications to
finance
Teitur Arnarson
KTH, Stockholm
American options
History and
background
I
A contract on one or several underlying assets that can
be exercised during some predetermined period [t, T ].
The blow-up
technique
Application to
indifference pricing
American option
Free boundary
regularity close to
initial state and
applications to
finance
Teitur Arnarson
KTH, Stockholm
American options
History and
background
I
I
A contract on one or several underlying assets that can
be exercised during some predetermined period [t, T ].
Payoff g :
Rn
→ R at exercise τ ∈ T[t,T ] .
The blow-up
technique
Application to
indifference pricing
Example: American put option
Gives you the right, but not the obligation, to sell the
underlying stock Xs for a predetermined price K any time
s ∈ [t, T ].
Free boundary
regularity close to
initial state and
applications to
finance
Teitur Arnarson
KTH, Stockholm
American options
History and
background
The blow-up
technique
Application to
indifference pricing
Free boundary
regularity close to
initial state and
applications to
finance
Example: American put option
Gives you the right, but not the obligation, to sell the
underlying stock Xs for a predetermined price K any time
s ∈ [t, T ].
Teitur Arnarson
KTH, Stockholm
American options
History and
background
The blow-up
technique
At exercise τ the payoff is g (Xτ ) = max(K − Xτ , 0).
g (x)
K
x
Application to
indifference pricing
Free boundary
regularity close to
initial state and
applications to
finance
Complete markets
Teitur Arnarson
KTH, Stockholm
The market consists of
American options
I
non-risky asset
History and
background
dBs
Bt
I
= ρBs ds
= B.
traded asset
dXs
Xt
= µXs ds + σXs dWs
= x
Ws is Brownian motion.
The blow-up
technique
Application to
indifference pricing
Free boundary
regularity close to
initial state and
applications to
finance
Option price
Teitur Arnarson
KTH, Stockholm
American options
The price h of an American option with payoff g is given by
Theorem (Risk-neutral valuation formula)
History and
background
The blow-up
technique
Application to
indifference pricing
h(x, t) = sup E (e −ρ(τ −t) g (Xτ )|Xt = x).
τ ∈[t,T ]
Free boundary
regularity close to
initial state and
applications to
finance
Variational inequality
h solves the following linear variational inequality
1
min − ht − σ 2 x 2 hxx − ρxhx + ρh,
2
h(x, t) − g (x)
= 0
Teitur Arnarson
KTH, Stockholm
American options
in R × [0, T )
h(x, T ) = g (x)
in [0, T )
History and
background
The blow-up
technique
Application to
indifference pricing
Free boundary
regularity close to
initial state and
applications to
finance
Variational inequality
h solves the following linear variational inequality
1
min − ht − σ 2 x 2 hxx − ρxhx + ρh,
2
h(x, t) − g (x)
= 0
Teitur Arnarson
KTH, Stockholm
American options
in R × [0, T )
h(x, T ) = g (x)
in [0, T )
A free boundary Γ separates the sets
C
E
1
= {−ht − σ 2 x 2 hxx − ρxhx + ρh = 0}
2
= {h − g = 0}.
History and
background
The blow-up
technique
Application to
indifference pricing
History
Free boundary
regularity close to
initial state and
applications to
finance
Teitur Arnarson
KTH, Stockholm
American options
Results for the American put.
History and
background
The blow-up
technique
I
Kuske & Keller (1998)
I
Bunch & Johnsson (2000)
I
Stamicar, Sevcovic & Chadam (1999)
Application to
indifference pricing
Free boundary
regularity close to
initial state and
applications to
finance
Chen, Chadam: Reformulation
Teitur Arnarson
KTH, Stockholm
American options
In dimensionless variables the price function h̃(x, t) solves
h̃t − h̃xx − (k − 1)h̃x + k h̃ = 0
for x > β̃(t)
h̃ = 1 − e x
for x < β̃(t)
h̃(0, x) = (1 − e x )+ ,
where x = β̃(t) is a parameterization of the free boundary Γ.
History and
background
The blow-up
technique
Application to
indifference pricing
Free boundary
regularity close to
initial state and
applications to
finance
Fundamental solution
Teitur Arnarson
KTH, Stockholm
Find the fundamental solution for the PDE
1
(x + (k − 1)t)2
√ exp −
Φ(x, t) =
4t
2 πt
and get the following integral representation
Z 0
(1 − e y )Φ(x − y , t)dy
h̃(x, t) =
−∞
Z tZ
β̃(t−θ)
Φ(x − y , θ)dydθ.
+k
0
−∞
American options
History and
background
The blow-up
technique
Application to
indifference pricing
ODE for the free boundary
Free boundary
regularity close to
initial state and
applications to
finance
Teitur Arnarson
KTH, Stockholm
Derive an ODE for the free boundary
2Φx (β̃(t), t)
−2
β̃˙ = −
k
Z
0
t
˙ − θ)dθ.
Φx (β̃(t) − β̃(t − θ), θ)β̃(t
American options
History and
background
The blow-up
technique
Application to
indifference pricing
ODE for the free boundary
Free boundary
regularity close to
initial state and
applications to
finance
Teitur Arnarson
KTH, Stockholm
Derive an ODE for the free boundary
2Φx (β̃(t), t)
−2
β̃˙ = −
k
Z
t
˙ − θ)dθ.
Φx (β̃(t) − β̃(t − θ), θ)β̃(t
0
Asymptotic expansion
β̃ 2
1
1
17
= −ξ −
+
+
+ ...
4t
2ξ 8ξ 2 24ξ 3
√
where ξ = 4πk 2 t.
American options
History and
background
The blow-up
technique
Application to
indifference pricing
Summary of the expansion method
Free boundary
regularity close to
initial state and
applications to
finance
Teitur Arnarson
KTH, Stockholm
American options
Advantage
I
Good precision
History and
background
The blow-up
technique
Application to
indifference pricing
Drawback
I
One-dimensional, linear setting.
Free boundary
regularity close to
initial state and
applications to
finance
Teitur Arnarson
KTH, Stockholm
American options
The blow-up technique
(only 4 slides)
History and
background
The blow-up
technique
Application to
indifference pricing
Free boundary
regularity close to
initial state and
applications to
finance
A general obstacle problem
Teitur Arnarson
KTH, Stockholm
American options
Obstacle problem with a non-linear, n + 1-dimensional,
parabolic operator
min(Dt u − F (D 2 u, Du, u, x, t), u − g ) = 0
in B1 × (0, 1)
u(x, 0) = g (x)
where B1 is the unit ball in Rn .
in B1
History and
background
The blow-up
technique
Application to
indifference pricing
Scaling in the point (0, 0)
Free boundary
regularity close to
initial state and
applications to
finance
Teitur Arnarson
KTH, Stockholm
For simplicity assume: u(0, 0) = g (0) = 0.
American options
Scaled function
History and
background
u(r x, r 2 t)
ur (x, t) =
αr
Scaled operator
Fr (D 2 u, Du, u, x, t) = F (D 2 u, r Du, r 2 u, r x, r 2 t).
The blow-up
technique
Application to
indifference pricing
Scaling in the point (0, 0)
Free boundary
regularity close to
initial state and
applications to
finance
Teitur Arnarson
KTH, Stockholm
For simplicity assume: u(0, 0) = g (0) = 0.
American options
Scaled function
History and
background
u(r x, r 2 t)
ur (x, t) =
αr
Scaled operator
Fr (D 2 u, Du, u, x, t) = F (D 2 u, r Du, r 2 u, r x, r 2 t).
Choose αr so that 0 < limr →0 ur < ∞ .
The blow-up
technique
Application to
indifference pricing
Scaled obstacle problem
Free boundary
regularity close to
initial state and
applications to
finance
Teitur Arnarson
KTH, Stockholm
American options
Under standard assumptions on F the scaled function ur
solves
min(Dt ur − Fr (D 2 ur , Dur , ur , x, t),
1
ur − gr ) = 0 in B1/r × (0, 2 )
r
ur (x, 0) = gr (x) in B1/r .
History and
background
The blow-up
technique
Application to
indifference pricing
Free boundary
regularity close to
initial state and
applications to
finance
Blow-up limit
Teitur Arnarson
KTH, Stockholm
American options
Take the so called blow-up limit by letting r → 0.
History and
background
If we have the right growth and continuity of u the limit
function u0 = limr →0 ur will solve
Application to
indifference pricing
2
min(Dt u0 − F0 (D u0 , Du0 , u0 , x, t),
u0 − g0 ) = 0
The blow-up
technique
in R × R+
u0 (x, 0) = g0 (x)
in R.
Free boundary
regularity close to
initial state and
applications to
finance
Blow-up limit
Teitur Arnarson
KTH, Stockholm
American options
Take the so called blow-up limit by letting r → 0.
History and
background
If we have the right growth and continuity of u the limit
function u0 = limr →0 ur will solve
Application to
indifference pricing
2
min(Dt u0 − F (D u0 , 0, 0, 0, 0),
u0 − g0 ) = 0
The blow-up
technique
in R × R+
u0 (x, 0) = g0 (x)
in R.
Free boundary
regularity close to
initial state and
applications to
finance
Free boundary regularity
Teitur Arnarson
KTH, Stockholm
Assume we have a free boundary.
American options
t
u=g
History and
background
Dt u − Fu = 0
The blow-up
technique
Γ
Application to
indifference pricing
x
Free boundary
regularity close to
initial state and
applications to
finance
Free boundary regularity
Teitur Arnarson
KTH, Stockholm
Assume that the free boundary stays above t = cx 2 .
American options
t = cx 2
t
u=g
History and
background
Dt u − Fu = 0
The blow-up
technique
Γ
Application to
indifference pricing
x
Free boundary
regularity close to
initial state and
applications to
finance
Free boundary regularity
Pick a sequence X1 , X2 . . . ∈ {t = cx 2 },where Xj = (xj , tj ).
Teitur Arnarson
KTH, Stockholm
American options
t = cx 2
t
u=g
X1
History and
background
Dt u − Fu = 0
The blow-up
technique
Γ
Application to
indifference pricing
X2
X3
Xi
x
Free boundary
regularity close to
initial state and
applications to
finance
Free boundary regularity
Teitur Arnarson
KTH, Stockholm
Set rj = |Xj |. . .
American options
t = cx 2
t
u=g
History and
background
Dt u − Fu = 0
The blow-up
technique
Γ
Application to
indifference pricing
Xi
Brj × (0, rj2 )
x
Free boundary
regularity close to
initial state and
applications to
finance
Free boundary regularity
. . . and scale the problem by rj . X̃j = (xj /rj , tj /rj2 ).
Teitur Arnarson
KTH, Stockholm
American options
t = cx 2
History and
background
t
urj = grj
X̃i
Dt urj − Frj urj = 0
Γ
Application to
indifference pricing
B1 × (0, 1)
Xi
The blow-up
technique
Brj × (0, rj2 )
x
Free boundary
regularity close to
initial state and
applications to
finance
Free boundary regularity
Teitur Arnarson
KTH, Stockholm
Take the limit as j → ∞. Note |X̃∞ | = 1.
American options
t = cx 2
t
u0 = g0
X̃∞
History and
background
Dt u0 − F0 u0 = 0
Γ
The blow-up
technique
Application to
indifference pricing
B1 × (0, 1)
x
The blow-up limit problem
I
I
For the limit problem no lower order terms occur in the
PDE.
The limit obstacle g0 is possibly simpler than the
original g .
Free boundary
regularity close to
initial state and
applications to
finance
Teitur Arnarson
KTH, Stockholm
American options
History and
background
The blow-up
technique
Application to
indifference pricing
The blow-up limit problem
I
I
For the limit problem no lower order terms occur in the
PDE.
The limit obstacle g0 is possibly simpler than the
original g .
Free boundary
regularity close to
initial state and
applications to
finance
Teitur Arnarson
KTH, Stockholm
American options
History and
background
The blow-up
technique
Application to
indifference pricing
⇓
When analysing the limit problem we might arrive at
different conclusios:
I
We might find an analytic solution.
I
We might be able to contradict the picture above.
The obstacle is a strict subsolution
g0 is a strict subsolution if
2
−F (D g0 , 0, 0, 0, 0) < 0 in B1 × (0, 1).
Free boundary
regularity close to
initial state and
applications to
finance
Teitur Arnarson
KTH, Stockholm
American options
History and
background
The blow-up
technique
Application to
indifference pricing
The obstacle is a strict subsolution
g0 is a strict subsolution if
2
−F (D g0 , 0, 0, 0, 0) < 0 in B1 × (0, 1).
Free boundary
regularity close to
initial state and
applications to
finance
Teitur Arnarson
KTH, Stockholm
American options
Dt u0 − F (u0 , 0, 0, 0, 0) ≥ 0 in B1 × (0, 1) and the maximum
principle
⇓
u0 > g0 in B1 × (0, 1).
History and
background
The blow-up
technique
Application to
indifference pricing
The obstacle is a strict subsolution
g0 is a strict subsolution if
2
−F (D g0 , 0, 0, 0, 0) < 0 in B1 × (0, 1).
Free boundary
regularity close to
initial state and
applications to
finance
Teitur Arnarson
KTH, Stockholm
American options
Dt u0 − F (u0 , 0, 0, 0, 0) ≥ 0 in B1 × (0, 1) and the maximum
principle
⇓
u0 > g0 in B1 × (0, 1).
⇓
No free boundary exists for the limit problem, i.e.
Γ ∈ {t < x 2 · σ(x)}
for some modulus of continuity σ(x).
History and
background
The blow-up
technique
Application to
indifference pricing
Incomplete markets: Market components
The market consists of
Free boundary
regularity close to
initial state and
applications to
finance
Teitur Arnarson
KTH, Stockholm
I
non-risky asset (zero interest rate for simplicity)
American options
Bs
I
The blow-up
technique
traded asset
dXs
Xt
I
= B.
= µXs ds + σXs dWs
= x
non-traded asset
dYs
Yt
History and
background
= b(Ys , s)ds + a(Ys , s)dWs0
= y
Ws and Ws0 are correlated with correlation ρ ∈ (−1, 1).
Application to
indifference pricing
Aim
Free boundary
regularity close to
initial state and
applications to
finance
Teitur Arnarson
KTH, Stockholm
American options
History and
background
Define the indifference price h of a call option written on the
non-traded asset Ys .
The blow-up
technique
Application to
indifference pricing
Investment alternatives
Alternative 1: Invest in stock Xs and bond Bs
Free boundary
regularity close to
initial state and
applications to
finance
Teitur Arnarson
KTH, Stockholm
American options
I
Allocation in traded stock Xs : πs
Allocation in bond: πs0
History and
background
I
Wealth: Zs = πs0 + πs .
The blow-up
technique
dZs
Zt
= πs µds + πs σdWs
= z.
Application to
indifference pricing
Investment alternatives
Alternative 1: Invest in stock Xs and bond Bs
Free boundary
regularity close to
initial state and
applications to
finance
Teitur Arnarson
KTH, Stockholm
American options
I
Allocation in traded stock Xs : πs
Allocation in bond: πs0
History and
background
I
Wealth: Zs = πs0 + πs .
The blow-up
technique
dZs
Zt
= πs µds + πs σdWs
= z.
Alternative 2: Invest in stock Xs , bond Bs and buy a call
option on non-traded asset Ys at time t for price h
I
American call payoff: g (y ) = (y − K )+ .
Application to
indifference pricing
Indifference pricing
I
Alternative 1 (Stock and bond only)
Initial wealth:
z
Terminal wealth: ZT
Value function:
V1 (z, t) = sup E (U(ZT )|Zt = z).
π
where U(z) =
−e −γz .
Free boundary
regularity close to
initial state and
applications to
finance
Teitur Arnarson
KTH, Stockholm
American options
History and
background
The blow-up
technique
Application to
indifference pricing
Indifference pricing
I
Alternative 1 (Stock and bond only)
Initial wealth:
z
Terminal wealth: ZT
Value function:
V1 (z, t) = sup E (U(ZT )|Zt = z).
π
−e −γz .
I
where U(z) =
Alternative 2 (Stock, bond and call option)
Initial wealth:
z −h
Wealth at exercise time τ : Zτ + g (Yτ )
Value function:
V2 (z, y , t) = sup E (V1 (Zτ + g (Yτ ), τ )|Zτ = z, Yτ = y )
π,τ
Free boundary
regularity close to
initial state and
applications to
finance
Teitur Arnarson
KTH, Stockholm
American options
History and
background
The blow-up
technique
Application to
indifference pricing
Indifference pricing
I
Alternative 1 (Stock and bond only)
Initial wealth:
z
Terminal wealth: ZT
Value function:
V1 (z, t) = sup E (U(ZT )|Zt = z).
π
−e −γz .
I
where U(z) =
Alternative 2 (Stock, bond and call option)
Initial wealth:
z −h
Wealth at exercise time τ : Zτ + g (Yτ )
Value function:
V2 (z, y , t) = sup E (V1 (Zτ + g (Yτ ), τ )|Zτ = z, Yτ = y )
π,τ
I
Definition: The indifference price h satisfies
V1 (z, t) = V2 (z − h, y , t)
Free boundary
regularity close to
initial state and
applications to
finance
Teitur Arnarson
KTH, Stockholm
American options
History and
background
The blow-up
technique
Application to
indifference pricing
Free boundary
regularity close to
initial state and
applications to
finance
Variational inequality
Teitur Arnarson
KTH, Stockholm
American options
min(Hh, h − g ) = 0
h(y , T ) = g (y )
in R × [0, T )
in R
where
µ
1
Hu = Dt u − a2 (y , t)Dy2 u − b(y , t) − ρ a(y , t) Dy u
2
σ
1
+ γ(1 − ρ2 )a2 (y , t)(Dy u)2 .
2
History and
background
The blow-up
technique
Application to
indifference pricing
Free boundary at expiry
I
I
I
Parameterization of free boundary: Γ = (β(t), t)
Location at expiry: β0 = limt→0 β(t)
call
A(y , t) = −Hg = b −
ρ σµ a
−
1
2 γ(1
−
ρ2 )a2
Free boundary
regularity close to
initial state and
applications to
finance
Teitur Arnarson
KTH, Stockholm
American options
History and
background
The blow-up
technique
Application to
indifference pricing
Free boundary
regularity close to
initial state and
applications to
finance
Free boundary at expiry
I
I
I
Parameterization of free boundary: Γ = (β(t), t)
Location at expiry: β0 = limt→0 β(t)
call
A(y , t) = −Hg = b −
ρ σµ a
−
Lemma 1 If A(y0 , 0) = 0 and
A(y0 + δ, 0)A(y0 − δ, 0) < 0 for all
small δ then either no free boundary exists or
1
2 γ(1
−
Teitur Arnarson
KTH, Stockholm
ρ2 )a2
American options
t
History and
background
The blow-up
technique
Application to
indifference pricing
β0 = y0 .
K
A>0
y0
A<0
x
Free boundary
regularity close to
initial state and
applications to
finance
Free boundary at expiry
I
I
I
Parameterization of free boundary: Γ = (β(t), t)
Location at expiry: β0 = limt→0 β(t)
call
A(y , t) = −Hg = b −
ρ σµ a
−
Lemma 1 If A(y0 , 0) = 0 and
A(y0 + δ, 0)A(y0 − δ, 0) < 0 for all
small δ then either no free boundary exists or
1
2 γ(1
−
Teitur Arnarson
KTH, Stockholm
ρ2 )a2
American options
t
History and
background
The blow-up
technique
Application to
indifference pricing
β0 = y0 .
K
A>0
y0
A<0
x
t
Lemma 2 If A(y , 0) < −ε for
some ε > 0 and all y ∈ {g > 0}
then
β0 = K .
K
A<0
x
Free boundary regularity: β0 6= K
Free boundary
regularity close to
initial state and
applications to
finance
Teitur Arnarson
KTH, Stockholm
Theorem 1 There exists ξ0 and r > 0 such that for
ξ1 < ξ0−2 < ξ2 and t < r
(β(t), t) ∈ {(y , t) : ξ1 (y − β0 )2 ≤ t ≤ ξ2 (y − β0 )2 }.
American options
History and
background
The blow-up
technique
Application to
indifference pricing
Free boundary
regularity close to
initial state and
applications to
finance
Free boundary regularity: β0 6= K
Teitur Arnarson
KTH, Stockholm
Theorem 1 There exists ξ0 and r > 0 such that for
ξ1 < ξ0−2 < ξ2 and t < r
(β(t), t) ∈ {(y , t) : ξ1 (y − β0 )2 ≤ t ≤ ξ2 (y − β0 )2 }.
ξ0 solve u(ξ0 ) −
ξ0 u 0 (ξ0 )
u(ξ) = ξ(6a2 (β0 , 0) + ξ 2 )
ξ
−∞
The blow-up
technique
Application to
indifference pricing
= 0 where
Z
American options
History and
background
exp
(6a2 (β0
−x 2
0 ,0)
dx.
, 0) + x 2 )2 x 2
4a2 (β
Proof
? Rewrite equation
Ĥu = A(y , t)χ{u>0}
where Ĥ = H + γ(1 − ρ2 )a2 gy Dy .
Free boundary
regularity close to
initial state and
applications to
finance
Teitur Arnarson
KTH, Stockholm
American options
History and
background
The blow-up
technique
Application to
indifference pricing
Proof
? Rewrite equation
Ĥu = A(y , t)χ{u>0}
where Ĥ = H + γ(1 − ρ2 )a2 gy Dy .
? Scale by r 3
u(ry + β0 , r 2 t)
ur (y , t) =
r3
and take the limit r → 0
1
Dt u0 − a02 Dy2 u0 = A0 y χ{u0 >0} .
2
Free boundary
regularity close to
initial state and
applications to
finance
Teitur Arnarson
KTH, Stockholm
American options
History and
background
The blow-up
technique
Application to
indifference pricing
Free boundary
regularity close to
initial state and
applications to
finance
Proof
? Rewrite equation
Teitur Arnarson
KTH, Stockholm
Ĥu = A(y , t)χ{u>0}
where Ĥ = H + γ(1 − ρ2 )a2 gy Dy .
? Scale by r 3
u(ry + β0 , r 2 t)
ur (y , t) =
r3
and take the limit r → 0
1
Dt u0 − a02 Dy2 u0 = A0 y χ{u0 >0} .
2
√
? Self-similar solution in the variable ξ = −y / t.
ũ(ξ) = u(y , t).
−ũ 00 −
1
3
ξũ 0 + 2 = −A0 ξ
2
2a0
2a0
in {ũ > 0}
American options
History and
background
The blow-up
technique
Application to
indifference pricing
Free boundary regularity: β0 = K
Free boundary
regularity close to
initial state and
applications to
finance
Teitur Arnarson
KTH, Stockholm
American options
History and
background
Theorem 2 There exists a modulus of continuity σ(r ) such
that
2
(β(t), t) ∈ {(y , t) : t < (y − K ) σ(y − K )}.
The blow-up
technique
Application to
indifference pricing
Free boundary
regularity close to
initial state and
applications to
finance
Proof
? Scale by r
hr (y , t) =
h(ry + K , r 2 t)
r
and take limit r → 0
1
min(Dt h0 − a02 Dy2 h0 , h0 − g0 ) = 0
2
h0 (y , 0) = g0 (y )
Teitur Arnarson
KTH, Stockholm
American options
History and
background
The blow-up
technique
Application to
indifference pricing
Free boundary
regularity close to
initial state and
applications to
finance
Proof
? Scale by r
hr (y , t) =
h(ry + K , r 2 t)
r
Teitur Arnarson
KTH, Stockholm
American options
and take limit r → 0
1
min(Dt h0 − a02 Dy2 h0 , h0 − g0 ) = 0
2
h0 (y , 0) = g0 (y )
? g0 = y + is a strict subsolution to the limit PDE.
⇓
The limit problem does not have a free boundary.
⇓
(β(t), t) ∈ {t < (y − K )2 σ(y − K )}.
History and
background
The blow-up
technique
Application to
indifference pricing