We prove existence, regularity and a Feynman-Ka\v{c} representation formula
of the strong solution to the free boundary problem arising in the financial
problem of the pricing of the American Asian option with arithmetic average

Monti, Laura

Pascucci, Andrea

English

arXiv

We prove existence, regularity and a Feynman-Kac representation formula of the strong solution to the free boundary problem arising in the financial problem of the pricing of the American Asian option with arithmetic average

AMS Acta - Alm@DL - Università di Bologna

ar
X
iv
:0
91
0.
42
57
v1
  [
ma
th.
AP
]  
22
 O
ct 
20
09
Mathematical Economics
Obstacle problem for Arithmetic Asian options
Laura Monti a, Andrea Pascucci a
aDipartimento di Matematica, Universita` di Bologna, Piazza di Porta S. Donato 5, 40126 Bologna (Italy)
Received *****; accepted after revision +++++
Presented by
Abstract
We prove existence, regularity and a Feynman-Kacˇ representation formula of the strong solution to the free
boundary problem arising in the financial problem of the pricing of the American Asian option with arithmetic
average.
Re´sume´
Proble`me de l’obstacle pour l’option ame´ricain asiatique a` moyenne arithme´tique. On de´montre
l’existence, la re´gularite´ et une formule de repre´sentation de Feynman-Kacˇ de la solution forte d’un proble`me
avec frontie`re libre. Ce type de proble`me on le retrouve en finance pour e´valuer le prix d’une option asiatique a`
moyenne arithme´tique de style ame´ricain.
1. Introduction
According to the classical financial theory (see, for instance, [12]) the study of Asian options of American
style leads to free boundary problems for degenerate parabolic PDEs. More precisely, let us assume that,
in the standard setting of local volatility models, the dynamics of the underlying asset is driven by the
SDE
dSt = µ(t, St)Stdt+ σ(t, St)StdWt, (1)
and consider the process dAt = f(St)dt, where f(S) = S and f(S) = logS occur respectively in the study
of the Arithmetic average and Geometric average Asian options. Then the price of the related Amerasian
option with payoff function ϕ is the solution of the obstacle problem with final condition{
max{Lu, ϕ− u} = 0, ]0, T [×R2+,
u(T, s, a) = ϕ(T, s, a), s, a > 0,
(2)
Email addresses: monti@unibo.it (Laura Monti), pascucci@dm.unibo.it (Andrea Pascucci).
Preprint submitted to the Acade´mie des sciences October 22, 2009
where
Lu =
σ2s2
2
∂ssu+ rs∂su+ f(s)∂au+ ∂tu− ru, s, a > 0, (3)
is the Kolmogorov operator of (St, At) and r is the risk free rate. Recently this problem has also been
considered in the study of pension plans in [7] and stock loans in [2].
Typical Arithmetic average payoffs are of the form
ϕ(t, s, a) =
(a
t
−K
)+
(fixed strike),
ϕ(t, s, a) =
(a
t
− s
)+
(floating strike).
(4)
A direct computation shows that in these cases a super-solution1 to (2) with f(s) = s is given by
u¯(t, s, a) =
α
t
(
1 + e−βt
√
s2 + a2
)
(5)
for α, β are positive constants, with β suitably large. On the other hand it is well-known that generally
(2) does not admit a smooth solution in the classical sense.
Recently the Geometric Asian option has been studied under the following hypotheses:
(H1) σ is bounded, locally Ho¨lder continuous and such that σ ≥ σ0 for some positive constant σ0;
(H2) ϕ is locally Lipschitz continuous on ]0, T [×R2+ and the distributional derivative ∂ssϕ is locally
lower bounded (to fix ideas, this includes ϕ(s) = (s−K)+ and excludes ϕ(s) = −(s−K)+).
In [4], [11] it is proved that problem (2), with f(s) = log s, has a strong solution u in the Sobolev space
Sploc = {u ∈ Lp | ∂su, ∂ssu, (f(s)∂a + ∂t)u ∈ Lploc}, p ≥ 1. (6)
Moreover uniqueness has been proved via Feynman-Kacˇ representation. However, as we shall see below,
the Geometric and Arithmetic cases are structurally quite different.
The aim of this note is to give an outlined proof of the following
Theorem 1.1 Consider problem (2) with f(s) = s, under the assumptions (H1) and (H2). Then we
have:
i) if there exists a super-solution u¯ then there also exists a strong solution u ∈ Sploc ∩ C
(
]0, T ]× R2+
)
for any p ≥ 1, such that u ≤ u¯;
ii) if u is a strong solution to (2) such that
|u(t, s, a)| ≤ C
t
(1 + sq + aq), s, a > 0, t ∈]0, T ], (7)
for some positive constants C, q, then
u(t, s, a) = sup
τ∈Tt,T
E
[
e−rτϕ(τ, St,s,aτ , A
t,s,a
τ )
]
, t, s, a > 0, (8)
where Tt,T = {τ ∈ T | τ ∈ [t, T ] a.s.} and T is the set of all stopping times with respect to the Brownian
filtration. In particular there exists at most one strong solution of (2) verifying the growth condition (7).
For simplicity here we only consider the case of constant σ. Then by a transformation (cf. formula (4.4)
in [1]), operator L in (3) with f(s) = s can be reduced in the canonical form
LA = x
2
1∂x1x1 + x1∂x2 + ∂t, x = (x1, x2) ∈ R2+. (9)
Before proceeding with the proof, we make some preliminary comments.
1 u¯ is a super-solution of (2) if {
max{Lu¯, ϕ− u¯} ≤ 0, ]0, T [×R2+,
u¯(T, s, a) ≥ ϕ(T, s, a), s, a > 0.
2
Remark 1. In the Geometric case f(s) = log s, by a logarithmical change of variables the pricing operator
L takes the form
LG = ∂x1x1 + x1∂x2 + ∂t, x = (x1, x2) ∈ R2. (10)
It is known (cf. [9]) that LG has remarkable invariance properties with respect to a homogeneous Lie
group structure: precisely, LG is invariant with respect to the left translation in the law (t
′, x′) ◦ (t, x) =
(t′ + t, x′1 + x1, x
′
2 + x2 + tx
′
1) and homogeneous of degree two with respect to the dilations δl(t, x) =
(l2t, lx1, l
3x2) in the sense that, setting z = (t, x),
LG(u(z
′ ◦ z)) = (LGu)(z′ ◦ z), LG(u(δl(z))) = l2(LGu)(δl(z)), z, z′ ∈ R3, l > 0.
Moreover LG has a fundamental solution ΓG(t, x;T,X) (here (t, x) and (T,X) represent respectively the
starting and ending points of the underlying stochastic process) of Gaussian type whose explicit expression
is known explicitly:
ΓG(t, x;T,X) = ΓG
(
(T,X)−1 ◦ (t, x); 0, 0)
where (T,X)−1 = (−T,−X1,−X2 + TX1) and
ΓG (t, x; 0, 0) =
√
3
2πt2
exp
(
x21
t
+
3x1(x2 − tx1)
t2
+
3(x2 − tx1)2
t3
)
, t < 0, x1, x2 ∈ R.
On the contrary, the Arithmetic operator LA does not admit a homogeneous structure: nevertheless in
this case we are able to find an interesting invariance property with respect to the “translation” operator
ℓ(t′,x′)(t, x) = (t
′ + t, x′1x1, x
′
2 + x
′
1x2) , t, t
′ ∈ R, x, x′ ∈ R2+; (11)
precisely we have
LA(u
(
ℓ(t′,x′)
)
) = (LAu)
(
ℓ(t′,x′)
)
. (12)
We remark that the fundamental ΓA of LA is not known explicitly and one of the key point in the
proof consists in showing suitable summability properties of ΓA near the pole. Note also that, due to the
invariance property (12), the fundamental solution ΓA verifies
ΓA(t, x;T,X) =
1
X21
ΓA
(
t− T, x1
X1
,
x2 −X2
X1
; 0, 1, 0
)
, x1, x2, X1, X2 > 0, x2 > X2, t < T.
Remark 2. Given a domain Ø and α ∈]0, 1[, we denote by CαG(Ø) and C1,αG (Ø) the intrinsic Ho¨lder
spaces defined by the norms
‖u‖Cα
G
(Ø) = sup
Ø
|u|+ sup
z,z0∈Ø
z 6=z0
|u(z)− u(z0)|
‖z−10 ◦ z‖αG
, ‖u‖C1,α
G
(Ø) = ‖u‖CαG(Ø) + ‖∂x1u‖CαG(Ø)
where ‖·‖G is a δl-homogeneous norm in R3. Due to the embedding Theorem 2.1 in [4], the strong solution
u of Theorem 1.1 is locally in C1,αG for any α ∈]0, 1[. In particular this ensures the validity of the standard
“smooth pasting” condition in the x1 variable (corresponding to the asset price). We also recall that the
(optimal) S∞loc-regularity of the solution has been recently proved in [5].
Remark 3. In the standard Black & Scholes setting and for particular homogeneous payoffs, it is possible
to reduce the spatial dimension of the pricing problem (from two to one). In the case of European Asian
options, this was first suggested in [8] and [13] respectively for the floating and fixed strike payoffs. In
the American case, the dimensional reduction is only possible for the floating strike payoff while the fixed
strike Amerasian option necessarily involves a 2-dimensional degenerate PDE.
Proof of Theorem 1.1. It is not restrictive to assume that ϕ is continuous on [0, T ]×R2+ or equivalently
we may study the problem on [ε, T ] × R2+ for a fixed, but arbitrary, positive ε. We divide the proof in
some steps.
3
Step 1. Let Dr(x) denote the Euclidean ball centered at x ∈ R2, with radius r. We construct a sequence
of “lentil shaped” domains On = Dn
(
n+ 1
n
, 0
)∩Dn (0, n+ 1n) covering R2+. For any n ∈ N, the cylinder
Hn =]0, T [×On is a LA-regular domain in the sense that there exists a barrier function at any point of
the parabolic boundary ∂PHn := ∂Hn\({0}×On) and therefore the obstacle problem on Hn has a strong
solution (cf. Theorem 3.1 in [4]). Indeed on any compact subset H of R× R2+, we have LA ≡ LH where
LH = aH(t, x)∂x1x1 + x1∂x2 + ∂t (13)
and aH is a some smooth function such that 0 < aH ≤ aH ≤ a¯H on ]0, T [×R2+, with aH , a¯H suitable
positive constants. Note that LH is a perturbation of the Geometric operator LG: as such, by Theorem
1.4 in [3], it has a fundamental solution ΓH that is bounded from above and below by Gaussian functions.
Then by Theorem 3.1 in [4] we have: for any n ∈ N and g ∈ C(Hn ∪ ∂PHn), g ≥ ϕ, there exists a strong
solution 2 un ∈ Sploc (Hn) ∩ C (Hn ∪ ∂PHn) to problem{
max{LAu, ϕ− u} = 0 in Hn,
u|∂PHn = g.
(14)
Moreover, for every p ≥ 1 and H ⊂⊂ Hn there exists a positive constant C, only dependent on
H,Hn, p, ‖ϕ‖L∞(Hn), ‖g‖L∞(Hn) such that ‖un‖Sp(H) ≤ C.
Step 2. To prove part i), we consider a sequence of cut-off functions χn ∈ C∞0 (R2+) such that χn = 1 on
On−1, χn = 0 on R
2
+ \On and 0 ≤ χn ≤ 1. We set gn(t, x, y) = χn(x, y)ϕ(t, x, y) + (1−χn(x, y))u¯(t, x, y)
and denote by un the strong solution to (14) with g = gn. By the comparison principle we have ϕ ≤
un ≤ un+1 ≤ u¯ and therefore, by the a priori estimate in Sp, for every p ≥ 1 and H ⊂⊂ Hn we have
‖un‖Sp(H) ≤ C for some constant C dependent on H but not on n. Then we can pass to the limit as
n → ∞, on compacts of ]0, T [×R2+, to get a solution of max{Lu, ϕ − u} = 0. A standard argument
based on barrier functions shows that u(t, x) is continuous up to t = T and attains the final datum. This
concludes the proof of part i).
Step 3. To prove part ii), we first construct the fundamental solution ΓA of LA as the limit of an
increasing sequence of Green functions for LA on the cylinders Hn, n ∈ N: to this end we combine some
classical PDE technique (cf. Chapter 15 in [6]) with the recent interior and boundary Schauder estimates
for degenerate Kolmogorov operators proved in [10]. We also show that ΓA is the transition density of
the underlying stochastic process.
Next we observe that, for fixed (t, x) ∈]0, T [×R2+ and n ∈ N suitably large so that x ∈ On, by the
maximum principle we have
ΓA(t, x; ·, ·) ≤ ΓHn(t, x; ·, ·) + max
[t,T ]×∂On
ΓA(t, x; ·, ·) in ]t, T ]×On.
Consequently we infer (cf. formula (4.8) in [11]) that ΓA(t, x; ·, ·) ∈ Lp(Hn) for some p > 1. This local
summability property of ΓA can be combined with the standard maximal estimate
E
[
sup
t≤τ≤T
|Xt,xτ |q
]
<∞, q ≥ 1,
valid for the solution X of a SDE whose coefficients have at most linear growth: then, adapting the
arguments used in the proof of Theorem 4.3 in [11], we can show formula (8). This concludes the proof
of part ii). 2
2 Here Sp
loc
is the Sobolev space defined in (6) with f(s) = s and s = x1, a = x2.
4
References
[1] E. Barucci, S. Polidoro, and V. Vespri, Some results on partial differential equations and Asian options, Math.
Models Methods Appl. Sci., 11 (2001), pp. 475–497.
[2] M. Dai and Z. Q. Xu, Optimal redeeming strategy of stock loans, arXiv:0906.0702v1, (2009).
[3] M. Di Francesco and A. Pascucci, On a class of degenerate parabolic equations of Kolmogorov type, AMRX Appl.
Math. Res. Express, (2005), pp. 77–116.
[4] M. Di Francesco, A. Pascucci, and S. Polidoro, The obstacle problem for a class of hypoelliptic ultraparabolic
equations, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 464 (2008), pp. 155–176.
[5] M. Frentz, K. Nystro¨m, A. Pascucci, and S. Polidoro, Optimal regularity in the obstacle problem for Kolmogorov
operators related to American Asian options, to appear in Math. Ann., (2009).
[6] A. Friedman, Stochastic differential equations and applications. Vol. 2., Probability and Mathematical Statistics. Vol.
28. Academic Press, 1976.
[7] A. Friedman and W. Shen, A variational inequality approach to financial valuation of retirement benefits based on
salary, Finance Stoch., 6 (2002), pp. 273–302.
[8] J. E. Ingersoll, Theory of Financial Decision Making, Blackwell, Oxford, 1987.
[9] E. Lanconelli and S. Polidoro, On a class of hypoelliptic evolution operators, Rend. Sem. Mat. Univ. Politec. Torino,
52 (1994), pp. 29–63. Partial differential equations, II (Turin, 1993).
[10] K. Nystro¨m, A. Pascucci, and S. Polidoro, Regularity near the initial state in the obstacle problem for a class of
hypoelliptic ultraparabolic operators, submitted, (2009).
[11] A. Pascucci, Free boundary and optimal stopping problems for American Asian options, Finance Stoch., 12 (2008),
pp. 21–41.
[12] G. Peskir and A. Shiryaev, Optimal stopping and free-boundary problems, Lectures in Mathematics ETH Zu¨rich,
Birkha¨user Verlag, Basel, 2006.
[13] L. C. G. Rogers and Z. Shi, The value of an Asian option, J. Appl. Probab., 32 (1995), pp. 1077–1088.
5


Obstacle problem for Arithmetic Asian options

PASCUCCI, ANDREA

MONTI, LAURA

Archivio istituzionale della ricerca - Alma Mater Studiorum Università di Bologna

Numérisation de Documents Anciens Mathématiques

C. R. Acad. Sci. Paris, Ser. I 347 (2009) 1443–1446Mathematical Economics/Partial Differential EquationsObstacle problem for Arithmetic Asian optionsLaura Monti, Andrea PascucciDipartimento di Matematica, Università di Bologna, Piazza di Porta S. Donato 5, 40126 Bologna, ItalyReceived 22 October 2009; accepted 23 October 2009Available online 6 November 2009Presented by Olivier PironneauAbstractWe prove existence, regularity and a Feynman–Kač representation formula of the strong solution to the free boundary problemarising in the financial problem of the pricing of the American Asian option with arithmetic average. To cite this article: L. Monti,A. Pascucci, C. R. Acad. Sci. Paris, Ser. I 347 (2009).© 2009 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.RésuméProblème de l’obstacle pour l’option américain asiatique à moyenne arithmétique. On démontre l’existence, la régularitéet une formule de représentation de Feynman–Kač de la solution forte d’un problème avec frontière libre. Ce type de problèmeon le retrouve en finance pour évaluer le prix d’une option asiatique à moyenne arithmétique de style américain. Pour citer cetarticle : L. Monti, A. Pascucci, C. R. Acad. Sci. Paris, Ser. I 347 (2009).© 2009 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.According to the classical financial theory (see, for instance, [12]) the study of Asian options of American styleleads to free boundary problems for degenerate parabolic PDEs. More precisely, let us assume that, in the standardsetting of local volatility models, the dynamics of the underlying asset is driven by the SDEdSt = μ(t, St )St dt + σ(t, St )St dWt, (1)and consider the process dAt = f (St )dt , where f (S) = S and f (S) = logS occur respectively in the study of theArithmetic average and Geometric average Asian options. Then the price of the related Amerasian option with payofffunction ϕ is the solution of the obstacle problem with final condition{max{Lu,ϕ − u} = 0, ]0, T [×R2+,u(T , s, a) = ϕ(T , s, a), s, a > 0, (2)whereLu = σ2s22∂ssu + rs∂su + f (s)∂au + ∂tu − ru, s, a > 0, (3)E-mail addresses: monti@unibo.it (L. Monti), pascucci@dm.unibo.it (A. Pascucci).1631-073X/$ – see front matter © 2009 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.doi:10.1016/j.crma.2009.10.0191444 L. Monti, A. Pascucci / C. R. Acad. Sci. Paris, Ser. I 347 (2009) 1443–1446is the Kolmogorov operator of (St ,At ) and r is the risk free rate. Recently this problem has also been considered inthe study of pension plans in [7] and stock loans in [2].Typical Arithmetic average payoffs are of the formϕ(t, s, a) =(at− K)+(fixed strike),ϕ(t, s, a) =(at− s)+(floating strike). (4)A direct computation shows that in these cases a super-solution1 to (2) with f (s) = s is given byū(t, s, a) = αt(1 + e−βt√s2 + a2 ) (5)where α, β are positive constants, with β suitably large. On the other hand it is well known that generally (2) does notadmit a smooth solution in the classical sense.Recently the Geometric Asian option has been studied under the following hypotheses:(H1) σ is bounded, locally Hölder continuous and such that σ  σ0 for some positive constant σ0;(H2) ϕ is locally Lipschitz continuous on ]0, T ] × R2+ and the distributional derivative ∂ssϕ is locally lower bounded(to fix ideas, this includes ϕ(s) = (s − K)+ and excludes ϕ(s) = −(s − K)+).In [4,11] it is proved that problem (2), with f (s) = log s, has a strong solution u in the Sobolev spaceS ploc ={u ∈ Lp ∣∣ ∂su, ∂ssu, (f (s)∂a + ∂t)u ∈ Lploc}, p  1. (6)Moreover uniqueness has been proved via Feynman–Kač representation. However, as we shall see below, the Geo-metric and Arithmetic cases are structurally quite different.The aim of this Note is to give an outlined proof of the following:Theorem 1. Consider problem (2) with f (s) = s, under the assumptions (H1) and (H2). Then we have:i) if there exists a super-solution ū then there also exists a strong solution u ∈ S ploc ∩ C(]0, T ] × R2+) for any p  1,such that u  ū;ii) if u is a strong solution to (2) such that∣∣u(t, s, a)∣∣  Ct(1 + sq + aq), s, a > 0, t ∈]0, T ], (7)for some positive constants C, q , thenu(t, s, a) = supτ∈Tt,TE[e−rτ ϕ(τ, St,s,aτ ,At,s,aτ)], t, s, a > 0, (8)where Tt,T = {τ ∈ T | τ ∈ [t, T ] a.s.} and T is the set of all stopping times with respect to the Brownian filtration.In particular there exists at most one strong solution of (2) verifying the growth condition (7).For simplicity here we only consider the case of constant σ . Then by a transformation (cf. formula (4.4) in [1]),operator L in (3) with f (s) = s can be reduced in the canonical formLA = x21∂x1x1 + x1∂x2 + ∂t , x = (x1, x2) ∈ R2+. (9)Before proceeding with the proof, we make some preliminary comments.1 ū is a super-solution of (2) if{max{Lū,ϕ − ū}  0, ]0, T [×R2+,ū(T , s, a)  ϕ(T , s, a), s, a > 0.L. Monti, A. Pascucci / C. R. Acad. Sci. Paris, Ser. I 347 (2009) 1443–1446 1445Remark 1. In the Geometric case f (s) = log s, by a logarithmical change of variables the pricing operator L takesthe formLG = ∂x1x1 + x1∂x2 + ∂t , x = (x1, x2) ∈ R2. (10)It is known (cf. [9]) that LG has remarkable invariance properties with respect to a homogeneous Lie group structure:precisely, LG is invariant with respect to the left translation in the law (t ′, x′) ◦ (t, x) = (t ′ + t, x′1 + x1, x′2 + x2 + tx′1)and homogeneous of degree two with respect to the dilations δλ(t, x) = (λ2t, λx1, λ3x2) in the sense that, set-ting z = (t, x), LG(u(z′ ◦ z)) = (LGu)(z′ ◦ z), LG(u(δλ(z))) = λ2(LGu)(δλ(z)), z, z′ ∈ R3, λ > 0. Moreover LGhas a fundamental solution ΓG(t, x;T ,X) (here (t, x) and (T ,X) represent respectively the starting and end-ing points of the underlying stochastic process) of Gaussian type whose explicit expression is known explic-itly: ΓG(t, x;T ,X) = ΓG((T ,X)−1 ◦ (t, x);0,0), where (T ,X)−1 = (−T ,−X1,−X2 + T X1) and ΓG(t, x;0,0) =√32πt2exp(x21t+ 3x1(x2−tx1)t2+ 3(x2−tx1)2t3), t < 0, x1, x2 ∈ R. On the contrary, the Arithmetic operator LA does not admita homogeneous structure: nevertheless in this case we are able to find an interesting invariance property with respectto the “translation” operator(t ′,x′)(t, x) =(t ′ + t, x′1x1, x′2 + x′1x2), t, t ′ ∈ R, x, x′ ∈ R2+; (11)precisely we haveLA(u((t ′,x′))) = (LAu)((t ′,x′)). (12)We remark that the fundamental ΓA of LA is not known explicitly and one of the key point in the proof consistsin showing suitable summability properties of ΓA near the pole. Note also that, due to the invariance property (12),the fundamental solution ΓA verifies ΓA(t, x;T ,X) = 1X21ΓA(t − T , x1X1 ,x2−X2X1;0,1,0), x1, x2,X1,X2 > 0, x2 > X2,t < T .Remark 2. Given a domain Ω and α ∈]0,1[, we denote by CαG(Ω) and C1,αG (Ω) the intrinsic Hölder spaces definedby the norms ‖u‖CαG(Ω) = supΩ |u| + sup z,z0∈Ωz =z0|u(z)−u(z0)|‖z−10 ◦z‖αG, ‖u‖C1,αG (Ω)= ‖u‖CαG(Ω) +‖∂x1u‖CαG(Ω), where ‖ · ‖G is aδλ-homogeneous norm in R3. Due to the embedding Theorem 2.1 in [4], the strong solution u of Theorem 1 is locallyin C1,αG for any α ∈]0,1[. In particular this ensures the validity of the standard “smooth pasting” condition in the x1variable (corresponding to the asset price). We also recall that the (optimal) S ∞loc-regularity of the solution has beenrecently proved in [5].Remark 3. In the standard Black & Scholes setting and for particular homogeneous payoffs, it is possible to reducethe spatial dimension of the pricing problem (from two to one). In the case of European Asian options, this was firstsuggested in [8] and [13] respectively for the floating and fixed strike payoffs. In the American case, the dimensionalreduction is only possible for the floating strike payoff while the fixed strike Amerasian option necessarily involves a2-dimensional degenerate PDE.Proof of Theorem 1. It is not restrictive to assume that ϕ is continuous on [0, T ]× R2+ or equivalently we may studythe problem on [ε,T ] × R2+ for a fixed, but arbitrary, positive ε. We divide the proof in some steps.Step 1. Let Dr(x) denote the Euclidean ball centered at x ∈ R2, with radius r . We construct a sequence of “lentilshaped” domains On = Dn(n + 1n ,0) ∩ Dn(0, n + 1n ) covering R2+. For any n ∈ N, the cylinder Hn =]0, T [×On isa LA-regular domain in the sense that there exists a barrier function at any point of the parabolic boundary ∂P Hn :=∂Hn \ ({0} × On) and therefore the obstacle problem on Hn has a strong solution (cf. Theorem 3.1 in [4]). Indeed onany compact subset H of R × R2+, we have LA ≡ LH whereLH = aH (t, x)∂x1x1 + x1∂x2 + ∂t (13)and aH is a smooth function such that 0 < a H  aH  āH on ]0, T [×R2+, with a H , āH suitable positive constants.Note that LH is a perturbation of the Geometric operator LG: as such, by Theorem 1.4 in [3], it has a fundamental1446 L. Monti, A. Pascucci / C. R. Acad. Sci. Paris, Ser. I 347 (2009) 1443–1446solution ΓH that is bounded from above and below by Gaussian functions. Then by Theorem 3.1 in [4] we have: forany n ∈ N and g ∈ C(Hn ∪ ∂P Hn), g  ϕ, there exists a strong solution2 un ∈ S ploc(Hn) ∩ C(Hn ∪ ∂P Hn) to problem{max{LAu,ϕ − u} = 0 in Hn,u|∂P Hn = g. (14)Moreover, for every p  1 and H  Hn there exists a positive constant C, only dependent on H , Hn, p, ‖ϕ‖L∞(Hn),‖g‖L∞(Hn) such that ‖un‖S p(H)  C.Step 2. To prove part i), we consider a sequence of cut-off functions χn ∈ C∞0 (R2+) such that χn = 1 on On−1, χn = 0on R2+ \ On and 0  χn  1. We set gn(t, x, y) = χn(x, y)ϕ(t, x, y) + (1 − χn(x, y))ū(t, x, y) and denote by unthe strong solution to (14) with g = gn. By the comparison principle we have ϕ  un  un+1  ū and therefore, bythe a priori estimate in S p , for every p  1 and H  Hn we have ‖un‖S p(H)  C for some constant C dependenton H but not on n. Then we can pass to the limit as n → ∞, on compacts of ]0, T [×R2+, to get a solution ofmax{Lu,ϕ −u} = 0. A standard argument based on barrier functions shows that u(t, x) is continuous up to t = T andattains the final datum. This concludes the proof of part i).Step 3. To prove part ii), we first construct the fundamental solution ΓA of LA as the limit of an increasing sequenceof Green functions for LA on the cylinders Hn, n ∈ N: to this end we combine some classical PDE technique (cf.Chapter 15 in [6]) with the recent interior and boundary Schauder estimates for degenerate Kolmogorov operatorsproved in [10]. We also show that ΓA is the transition density of the underlying stochastic process.Next we observe that, for fixed (t, x)∈]0, T [×R2+ and n ∈ N suitably large so that x ∈ On, by the maximumprinciple we have ΓA(t, x; ·,·)  ΓHn(t, x; ·,·) + max[t,T ]×∂On ΓA(t, x; ·,·) in ]t, T ] × On. Consequently we infer (cf.formula (4.8) in [11]) that ΓA(t, x; ·,·) ∈ Lp(Hn) for some p > 1. This local summability property of ΓA can becombined with the standard maximal estimate E[suptτT |Xt,xτ |q ] < ∞, q  1, valid for the solution X of a SDEwhose coefficients have at most linear growth: then, adapting the arguments used in the proof of Theorem 4.3 in [11],we can show formula (8). This concludes the proof of part ii). References[1] E. Barucci, S. Polidoro, V. Vespri, Some results on partial differential equations and Asian options, Math. Models Methods Appl. Sci. 11(2001) 475–497.[2] M. Dai, Z.Q. Xu, Optimal redeeming strategy of stock loans, arXiv:0906.0702v1, 2009.[3] M. Di Francesco, A. Pascucci, On a class of degenerate parabolic equations of Kolmogorov type, AMRX Appl. Math. Res. Express (2005)77–116.[4] M. Di Francesco, A. Pascucci, S. Polidoro, The obstacle problem for a class of hypoelliptic ultraparabolic equations, Proc. R. Soc. Lond. Ser.A Math. Phys. Eng. Sci. 464 (2008) 155–176.[5] M. Frentz, K. Nyström, A. Pascucci, S. Polidoro, Optimal regularity in the obstacle problem for Kolmogorov operators related to AmericanAsian options, Math. Ann. (2009), in press.[6] A. Friedman, Stochastic Differential Equations and Applications, vol. 2., Probability and Mathematical Statistics, vol. 28, Academic Press,1976.[7] A. Friedman, W. Shen, A variational inequality approach to financial valuation of retirement benefits based on salary, Finance Stoch. 6 (2002)273–302.[8] J.E. Ingersoll, Theory of Financial Decision Making, Blackwell, Oxford, 1987.[9] E. Lanconelli, S. Polidoro, On a class of hypoelliptic evolution operators, Rend. Sem. Mat. Univ. Politec. Torino 52 (1994) 29–63. Partialdifferential equations, II (Turin, 1993).[10] K. Nyström, A. Pascucci, S. Polidoro, Regularity near the initial state in the obstacle problem for a class of hypoelliptic ultraparabolicoperators, 2009, submitted for publication.[11] A. Pascucci, Free boundary and optimal stopping problems for American Asian options, Finance Stoch. 12 (2008) 21–41.[12] G. Peskir, A. Shiryaev, Optimal Stopping and Free-Boundary Problems, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel,2006.[13] L.C.G. Rogers, Z. Shi, The value of an Asian option, J. Appl. Probab. 32 (1995) 1077–1088.2 Here S ploc is the Sobolev space defined in (6) with f (s) = s and s = x1, a = x2.

Laura Monti

Andrea Pascucci

Comptes Rendus Mathématique

AMS Acta

arXiv:0910.4257v1  [math.AP]  22 Oct 2009Mathematical EconomicsObstacle problem for Arithmetic Asian optionsLaura Monti a, Andrea Pascucci aaDipartimento di Matematica, Universita` di Bologna, Piazza di Porta S. Donato 5, 40126 Bologna (Italy)Received *****; accepted after revision +++++Presented byAbstractWe prove existence, regularity and a Feynman-Kacˇ representation formula of the strong solution to the freeboundary problem arising in the financial problem of the pricing of the American Asian option with arithmeticaverage.Re´sume´Proble`me de l’obstacle pour l’option ame´ricain asiatique a` moyenne arithme´tique. On de´montrel’existence, la re´gularite´ et une formule de repre´sentation de Feynman-Kacˇ de la solution forte d’un proble`meavec frontie`re libre. Ce type de proble`me on le retrouve en finance pour e´valuer le prix d’une option asiatique a`moyenne arithme´tique de style ame´ricain.1. IntroductionAccording to the classical financial theory (see, for instance, [12]) the study of Asian options of Americanstyle leads to free boundary problems for degenerate parabolic PDEs. More precisely, let us assume that,in the standard setting of local volatility models, the dynamics of the underlying asset is driven by theSDEdSt = µ(t, St)Stdt+ σ(t, St)StdWt, (1)and consider the process dAt = f(St)dt, where f(S) = S and f(S) = logS occur respectively in the studyof the Arithmetic average and Geometric average Asian options. Then the price of the related Amerasianoption with payoff function ϕ is the solution of the obstacle problem with final condition{max{Lu, ϕ− u} = 0, ]0, T [×R2+,u(T, s, a) = ϕ(T, s, a), s, a > 0,(2)Email addresses: monti@unibo.it (Laura Monti), pascucci@dm.unibo.it (Andrea Pascucci).Preprint submitted to the Acade´mie des sciences October 22, 2009whereLu =σ2s22∂ssu+ rs∂su+ f(s)∂au+ ∂tu− ru, s, a > 0, (3)is the Kolmogorov operator of (St, At) and r is the risk free rate. Recently this problem has also beenconsidered in the study of pension plans in [7] and stock loans in [2].Typical Arithmetic average payoffs are of the formϕ(t, s, a) =(at−K)+(fixed strike),ϕ(t, s, a) =(at− s)+(floating strike).(4)A direct computation shows that in these cases a super-solution1 to (2) with f(s) = s is given byu¯(t, s, a) =αt(1 + e−βt√s2 + a2)(5)for α, β are positive constants, with β suitably large. On the other hand it is well-known that generally(2) does not admit a smooth solution in the classical sense.Recently the Geometric Asian option has been studied under the following hypotheses:(H1) σ is bounded, locally Ho¨lder continuous and such that σ ≥ σ0 for some positive constant σ0;(H2) ϕ is locally Lipschitz continuous on ]0, T [×R2+ and the distributional derivative ∂ssϕ is locallylower bounded (to fix ideas, this includes ϕ(s) = (s−K)+ and excludes ϕ(s) = −(s−K)+).In [4], [11] it is proved that problem (2), with f(s) = log s, has a strong solution u in the Sobolev spaceSploc = {u ∈ Lp | ∂su, ∂ssu, (f(s)∂a + ∂t)u ∈ Lploc}, p ≥ 1. (6)Moreover uniqueness has been proved via Feynman-Kacˇ representation. However, as we shall see below,the Geometric and Arithmetic cases are structurally quite different.The aim of this note is to give an outlined proof of the followingTheorem 1.1 Consider problem (2) with f(s) = s, under the assumptions (H1) and (H2). Then wehave:i) if there exists a super-solution u¯ then there also exists a strong solution u ∈ Sploc ∩ C(]0, T ]× R2+)for any p ≥ 1, such that u ≤ u¯;ii) if u is a strong solution to (2) such that|u(t, s, a)| ≤ Ct(1 + sq + aq), s, a > 0, t ∈]0, T ], (7)for some positive constants C, q, thenu(t, s, a) = supτ∈Tt,TE[e−rτϕ(τ, St,s,aτ , At,s,aτ )], t, s, a > 0, (8)where Tt,T = {τ ∈ T | τ ∈ [t, T ] a.s.} and T is the set of all stopping times with respect to the Brownianfiltration. In particular there exists at most one strong solution of (2) verifying the growth condition (7).For simplicity here we only consider the case of constant σ. Then by a transformation (cf. formula (4.4)in [1]), operator L in (3) with f(s) = s can be reduced in the canonical formLA = x21∂x1x1 + x1∂x2 + ∂t, x = (x1, x2) ∈ R2+. (9)Before proceeding with the proof, we make some preliminary comments.1 u¯ is a super-solution of (2) if {max{Lu¯, ϕ− u¯} ≤ 0, ]0, T [×R2+,u¯(T, s, a) ≥ ϕ(T, s, a), s, a > 0.2Remark 1. In the Geometric case f(s) = log s, by a logarithmical change of variables the pricing operatorL takes the formLG = ∂x1x1 + x1∂x2 + ∂t, x = (x1, x2) ∈ R2. (10)It is known (cf. [9]) that LG has remarkable invariance properties with respect to a homogeneous Liegroup structure: precisely, LG is invariant with respect to the left translation in the law (t′, x′) ◦ (t, x) =(t′ + t, x′1 + x1, x′2 + x2 + tx′1) and homogeneous of degree two with respect to the dilations δl(t, x) =(l2t, lx1, l3x2) in the sense that, setting z = (t, x),LG(u(z′ ◦ z)) = (LGu)(z′ ◦ z), LG(u(δl(z))) = l2(LGu)(δl(z)), z, z′ ∈ R3, l > 0.Moreover LG has a fundamental solution ΓG(t, x;T,X) (here (t, x) and (T,X) represent respectively thestarting and ending points of the underlying stochastic process) of Gaussian type whose explicit expressionis known explicitly:ΓG(t, x;T,X) = ΓG((T,X)−1 ◦ (t, x); 0, 0)where (T,X)−1 = (−T,−X1,−X2 + TX1) andΓG (t, x; 0, 0) =√32πt2exp(x21t+3x1(x2 − tx1)t2+3(x2 − tx1)2t3), t < 0, x1, x2 ∈ R.On the contrary, the Arithmetic operator LA does not admit a homogeneous structure: nevertheless inthis case we are able to find an interesting invariance property with respect to the “translation” operatorℓ(t′,x′)(t, x) = (t′ + t, x′1x1, x′2 + x′1x2) , t, t′ ∈ R, x, x′ ∈ R2+; (11)precisely we haveLA(u(ℓ(t′,x′))) = (LAu)(ℓ(t′,x′)). (12)We remark that the fundamental ΓA of LA is not known explicitly and one of the key point in theproof consists in showing suitable summability properties of ΓA near the pole. Note also that, due to theinvariance property (12), the fundamental solution ΓA verifiesΓA(t, x;T,X) =1X21ΓA(t− T, x1X1,x2 −X2X1; 0, 1, 0), x1, x2, X1, X2 > 0, x2 > X2, t < T.Remark 2. Given a domain Ø and α ∈]0, 1[, we denote by CαG(Ø) and C1,αG (Ø) the intrinsic Ho¨lderspaces defined by the norms‖u‖CαG(Ø) = supØ|u|+ supz,z0∈Øz 6=z0|u(z)− u(z0)|‖z−10 ◦ z‖αG, ‖u‖C1,αG(Ø) = ‖u‖CαG(Ø) + ‖∂x1u‖CαG(Ø)where ‖·‖G is a δl-homogeneous norm in R3. Due to the embedding Theorem 2.1 in [4], the strong solutionu of Theorem 1.1 is locally in C1,αG for any α ∈]0, 1[. In particular this ensures the validity of the standard“smooth pasting” condition in the x1 variable (corresponding to the asset price). We also recall that the(optimal) S∞loc-regularity of the solution has been recently proved in [5].Remark 3. In the standard Black & Scholes setting and for particular homogeneous payoffs, it is possibleto reduce the spatial dimension of the pricing problem (from two to one). In the case of European Asianoptions, this was first suggested in [8] and [13] respectively for the floating and fixed strike payoffs. Inthe American case, the dimensional reduction is only possible for the floating strike payoff while the fixedstrike Amerasian option necessarily involves a 2-dimensional degenerate PDE.Proof of Theorem 1.1. It is not restrictive to assume that ϕ is continuous on [0, T ]×R2+ or equivalentlywe may study the problem on [ε, T ] × R2+ for a fixed, but arbitrary, positive ε. We divide the proof insome steps.3Step 1. Let Dr(x) denote the Euclidean ball centered at x ∈ R2, with radius r. We construct a sequenceof “lentil shaped” domains On = Dn(n+ 1n, 0)∩Dn (0, n+ 1n) covering R2+. For any n ∈ N, the cylinderHn =]0, T [×On is a LA-regular domain in the sense that there exists a barrier function at any point ofthe parabolic boundary ∂PHn := ∂Hn\({0}×On) and therefore the obstacle problem on Hn has a strongsolution (cf. Theorem 3.1 in [4]). Indeed on any compact subset H of R× R2+, we have LA ≡ LH whereLH = aH(t, x)∂x1x1 + x1∂x2 + ∂t (13)and aH is a some smooth function such that 0 < aH ≤ aH ≤ a¯H on ]0, T [×R2+, with aH , a¯H suitablepositive constants. Note that LH is a perturbation of the Geometric operator LG: as such, by Theorem1.4 in [3], it has a fundamental solution ΓH that is bounded from above and below by Gaussian functions.Then by Theorem 3.1 in [4] we have: for any n ∈ N and g ∈ C(Hn ∪ ∂PHn), g ≥ ϕ, there exists a strongsolution 2 un ∈ Sploc (Hn) ∩ C (Hn ∪ ∂PHn) to problem{max{LAu, ϕ− u} = 0 in Hn,u|∂PHn = g.(14)Moreover, for every p ≥ 1 and H ⊂⊂ Hn there exists a positive constant C, only dependent onH,Hn, p, ‖ϕ‖L∞(Hn), ‖g‖L∞(Hn) such that ‖un‖Sp(H) ≤ C.Step 2. To prove part i), we consider a sequence of cut-off functions χn ∈ C∞0 (R2+) such that χn = 1 onOn−1, χn = 0 on R2+ \On and 0 ≤ χn ≤ 1. We set gn(t, x, y) = χn(x, y)ϕ(t, x, y) + (1−χn(x, y))u¯(t, x, y)and denote by un the strong solution to (14) with g = gn. By the comparison principle we have ϕ ≤un ≤ un+1 ≤ u¯ and therefore, by the a priori estimate in Sp, for every p ≥ 1 and H ⊂⊂ Hn we have‖un‖Sp(H) ≤ C for some constant C dependent on H but not on n. Then we can pass to the limit asn → ∞, on compacts of ]0, T [×R2+, to get a solution of max{Lu, ϕ − u} = 0. A standard argumentbased on barrier functions shows that u(t, x) is continuous up to t = T and attains the final datum. Thisconcludes the proof of part i).Step 3. To prove part ii), we first construct the fundamental solution ΓA of LA as the limit of anincreasing sequence of Green functions for LA on the cylinders Hn, n ∈ N: to this end we combine someclassical PDE technique (cf. Chapter 15 in [6]) with the recent interior and boundary Schauder estimatesfor degenerate Kolmogorov operators proved in [10]. We also show that ΓA is the transition density ofthe underlying stochastic process.Next we observe that, for fixed (t, x) ∈]0, T [×R2+ and n ∈ N suitably large so that x ∈ On, by themaximum principle we haveΓA(t, x; ·, ·) ≤ ΓHn(t, x; ·, ·) + max[t,T ]×∂OnΓA(t, x; ·, ·) in ]t, T ]×On.Consequently we infer (cf. formula (4.8) in [11]) that ΓA(t, x; ·, ·) ∈ Lp(Hn) for some p > 1. This localsummability property of ΓA can be combined with the standard maximal estimateE[supt≤τ≤T|Xt,xτ |q]<∞, q ≥ 1,valid for the solution X of a SDE whose coefficients have at most linear growth: then, adapting thearguments used in the proof of Theorem 4.3 in [11], we can show formula (8). This concludes the proofof part ii). 22 Here Splocis the Sobolev space defined in (6) with f(s) = s and s = x1, a = x2.4References[1] E. Barucci, S. Polidoro, and V. Vespri, Some results on partial differential equations and Asian options, Math.Models Methods Appl. Sci., 11 (2001), pp. 475–497.[2] M. Dai and Z. Q. Xu, Optimal redeeming strategy of stock loans, arXiv:0906.0702v1, (2009).[3] M. Di Francesco and A. Pascucci, On a class of degenerate parabolic equations of Kolmogorov type, AMRX Appl.Math. Res. Express, (2005), pp. 77–116.[4] M. Di Francesco, A. Pascucci, and S. Polidoro, The obstacle problem for a class of hypoelliptic ultraparabolicequations, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 464 (2008), pp. 155–176.[5] M. Frentz, K. Nystro¨m, A. Pascucci, and S. Polidoro, Optimal regularity in the obstacle problem for Kolmogorovoperators related to American Asian options, to appear in Math. Ann., (2009).[6] A. Friedman, Stochastic differential equations and applications. Vol. 2., Probability and Mathematical Statistics. Vol.28. Academic Press, 1976.[7] A. Friedman and W. Shen, A variational inequality approach to financial valuation of retirement benefits based onsalary, Finance Stoch., 6 (2002), pp. 273–302.[8] J. E. Ingersoll, Theory of Financial Decision Making, Blackwell, Oxford, 1987.[9] E. Lanconelli and S. Polidoro, On a class of hypoelliptic evolution operators, Rend. Sem. Mat. Univ. Politec. Torino,52 (1994), pp. 29–63. Partial differential equations, II (Turin, 1993).[10] K. Nystro¨m, A. Pascucci, and S. Polidoro, Regularity near the initial state in the obstacle problem for a class ofhypoelliptic ultraparabolic operators, submitted, (2009).[11] A. Pascucci, Free boundary and optimal stopping problems for American Asian options, Finance Stoch., 12 (2008),pp. 21–41.[12] G. Peskir and A. Shiryaev, Optimal stopping and free-boundary problems, Lectures in Mathematics ETH Zu¨rich,Birkha¨user Verlag, Basel, 2006.[13] L. C. G. Rogers and Z. Shi, The value of an Asian option, J. Appl. Probab., 32 (1995), pp. 1077–1088.5

https://comptes-rendus.academie-sciences.fr/mathematique/item/10.1016/j.crma.2009.10.019.pdf

Obstacle problem for Arithmetic Asian options

Abstract

Similar works

Full text

Available Versions

AMS Acta - Alm@DL - Università di Bologna

Archivio istituzionale della ricerca - Alma Mater Studiorum Università di Bologna

Numérisation de Documents Anciens Mathématiques

Comptes Rendus Mathématique

AMS Acta