Obstacle problem for Arithmetic Asian options

We prove existence, regularity and a Feynman–Kač 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. 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. Résumé Problème de l’obstacle pour l’option américain asiatique à moyenne arithmétique. On démontre l’existence, la régularité et une formule de représentation de Feynman–Kač de la solution forte d’un problème avec frontière libre. Ce type de problème on le retrouve en finance pour évaluer le prix d’une option asiatique à moyenne arithmétique de style américain. Pour citer cet 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. 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 )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 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 [×R+, u(T , s, a)= φ(T , s, a), s, a > 0, (2) where Lu= σ 2s2 2 ∂ssu+ rs∂su+ f (s)∂au+ ∂tu− ru, s, a > 0, (3) E-mail addresses: [email protected] (L. Monti), [email protected] (A. Pascucci). 1631-073X/$ – see front matter © 2009 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved. doi:10.1016/j.crma.2009.10.019 1444 L. Monti, A. Pascucci / C. R. Acad. Sci. Paris, Ser. I 347 (2009) 1443–1446 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 ū(t, s, a)= α t ( 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 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 Hölder continuous and such that σ σ0 for some positive constant σ0; (H2) φ is locally Lipschitz continuous on ]0, T ] ×R+ 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 S loc = { u ∈ L ∣∣ ∂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 Geometric 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 loc ∩C(]0, T ] ×R+) for any p 1, such that u ū; ii) if u is a strong solution to (2) such that ∣∣u(t, s, a)∣∣ C t ( 1 + s + 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−rτ φ ( τ, S τ ,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 = x2 1∂x1x1 + x1∂x2 + ∂t , x = (x1, x2) ∈R+. (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 [×R+, 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 1445 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 δλ(t, x) = (λ2t, λx1, λx2) in the sense that, setting z = (t, x), LG(u(z′ ◦ z)) = (LGu)(z′ ◦ z), LG(u(δλ(z))) = λ(LGu)(δλ(z)), z, z′ ∈ R3, λ > 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( x2 1 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 (t ′,x′)(t, x)= ( t ′ + t, x′ 1x1, x′ 2 + x′ 1x2 ) , t, t ′ ∈R, x, x′ ∈R+; (11)