The early exercise premium representation for American options on multiply assets

In the paper we consider the problem of valuation of American options written on dividend-paying assets whose price dynamics follow the classical multidimensional Black and Scholes model. We provide a general early exercise premium representation formula for options with payoff functions which are convex or satisfy mild regularity assumptions. Examples include index options, spread options, call on max options, put on min options, multiply strike options and power-product options. In the proof of the formula we exploit close connections between the optimal stopping problems associated with valuation of American options, obstacle problems and reflected backward stochastic differential equations.


Introduction
In the paper we study American options written on dividend-paying assets. We assume that the underlying assets dynamics follow the classical multidimensional Black and Scholes model. It is now well known that the arbitrage-free value of American options can be expressed in terms of the optimal stopping problem (Bensoussan [3], Karatzas [16]; see also [17] for nice exposition and additional references), in terms of variational inequalities (Jaillet, Lamberton and Lapeyre [15]) and in terms of solutions of reflected BSDEs (El Karoui and Quenez [10]). Although these approaches provide complete characterization of the option value (see Section 2 for a short review), the paper by Broadie and Detemple [5] shows that it is of interest to provide alternative representation, which expresses the value of an American option as the value of the corresponding European option plus the gain from early exercise. The main reason is that the representation, called the early exercise premium formula, gives useful information on the determinants of the option value. The formula was proved first by Kim [18] in the case of standard American put option on a single asset. Another important contributions in the case of single asset include El Karoui and Karatzas [9] and Jacka [14] (see also Section 2.7 in [17] and the references therein). The case of options on multiply dividend-paying assets is more difficult and has received rather little attention in the literature. In the important paper [5] and next in Detemple, Feng and Tian [7] the early exercise premium formula was established for concrete classes of options on multiply assets.
In the present paper we provide a unified way of treating a wide variety of seemingly disparate examples. It allows us to prove a general exercise premium formula for options with convex payoff functions satisfying the polynomial growth condition or payoff function satisfying quite general condition considered in Laurence and Salsa [21]. Verifying the last condition requires knowledge of the payoff function and the structure of the exercise set. Therefore it is a complicated task in general. Fortunately, in most interesting cases one can easily check convexity of the payoff function or check some simpler condition implying the general condition from [21]. The class of options covered by our formula includes index options, spread options, call on max options, put on min options, multiply strike options, power-product options and others.
In the proof of the exercise premium formula we rely on some results on reflected BSDEs and their links with optimal stopping problems (see [10]) and with parabolic variational inequalities established in Bally, Caballero, Fernandez and El Karoui [2]. We also use classical results on regularity of the solution of the Cauchy problem for parabolic operator with constant coefficients. Perhaps it is worth mentioning that we do not use any regularity results on the free boundary problem for an American option. The basic idea of the proof comes from our earlier paper [20] devoted to standard American call and put options on single asset.

Preliminaries
We will assume that under the risk-neutral probability measure the underlying assets prices X s,x,1 , . . . , X s,x,n evolve on the time interval [s, T ] according to stochastic differential equation of the form Here r ≥ 0 denotes the rate of interest, d i ≥ 0 is the dividend rate of the asset i, σ = {σ ij } is the n-dimensional volatility matrix and W = (W 1 , . . . , W n ) is a standard n-dimensional Wiener process. As for the payoff function ψ we will assume that it satisfies the assumptions: (A1) ψ is a nonnegative continuous function on R n with polynomial growth, or (A3) ψ is a nonnegative convex function on R n with polynomial growth.
Note that convex functions are locally Lipschitz, so assumption (A3) implies (A1). Assumption (A2) is considered in [21]. It is satisfied for instance if (A2 ′ ) The region where ψ is strictly positive is the union of several connected components in which ψ is smooth.
Following [21] let us also note that unlike (A2 ′ ) or (A3), condition (A2) cannot be verified by appealing to the structure of the payoff alone. Verifying (A2) requires additional knowledge of the structure of the exercise set {u = ψ}.
Let Ω = C([0, T ]; R n ) and let X be the canonical process on Ω. For (s, x) ∈ Q T let P s,x denote the law of the process X s,x = (X s,x,1 , . . . , X s,x,n ) defined by (1) and let {F s t } denote the completion of σ(X θ ; θ ∈ [s, t]) with respect to the family {P s,µ ; µ a finite measure on B(R n )}, where P s,µ (·) = R n P s,x (·) µ(dx). Then for each s ∈ [0, T ), X = (Ω, (F s t ) t∈[s,T ] , X, P s,x ) is a Markov process on [0, T ]. In Bensoussan [3] and Karatzas [16] (see also Section 2.5 in [17]) it is shown that under (A1) the arbitrage-free value V of an American option with payoff function ψ and expiration time T is given by the solution of the stopping problem where the supremum is taken over the set T s of all {F s t }-stopping times τ with values in [s, T ].
From the results proved in [8] it follows that under (A1) for every (s, x) there exists a unique solution (Y s,x , Z s,x , K s,x ), on the space (Ω, F s T , P s,x ), to the reflected BSDE with terminal condition ψ(X T ), coefficient f : R d → R defined as f (y) = −ry, y ∈ R n , and barrier ψ(X) (RBSDE s,x (ψ, −ry, ψ) for short). This means that the processes Y s,x , Z s,x , K s,x are {F s t }-progressively measurable, satisfy some integrability conditions and P s,x -a.s., where B s,· is some standard Wiener process on [s, T ] under P s,x . In [8] it is also proved that for every (s, where u is a viscosity solution to the obstacle problem with From [8,10] we know that V defined by (2) is equal to Y s,x s . Hence In the next section we analyze V via (6) but as a matter of fact instead of viscosity solutions of (5) we consider variational solutions which provide more information on the value function V .

Obstacle problem for the Black and Scholes equation
Assume that ψ : R n → R + is continuous and satisfies the polynomial growth condition. [2,20] we adopt the following definition.
Here ·, · ̺,T stands for the duality pairing between L 2 (0, T ; H 1 ̺ ) and L 2 (0, T ; H −1 ̺ ), ·, · 2,̺,T is the scalar product in L 2 (0, T ; H 1 ̺ ) and ·, · 2,T = ·, · 2,̺,T with ̺ ≡ 1. (b) If µ in the above definition admits a density (with respect to the Lebesgue measure) of the form Φ u (t, x) = Φ(t, x, u(t, x)) for some measurable Φ :Q T × R → R + , then we say that u is a variational solution to the semilinear problem In our main theorems below we show that if ψ satisfies (A1) and (A2) or (A3) then the measure µ is absolutely continuous with respect to the Lebesgue measure and its density has the form Φ u (t, and Ψ is determined by ψ and the parameters r, d, a. In the next section we compute Ψ for some concrete options.
Remark. One can check that if u is a solution to (7) then v defined as Furthermore, a simple calculation shows that if η is a smooth function on R n with compact support and U ⊂ R n is a bounded open set such that supp[η] ⊂ U thenṽ = vη is a solution of the Cauchy-Dirichlet problemṽ whereψ =ψη,L is some uniformly elliptic operator with smooth coefficients not depending on t and f ∈ L 2 (0, T ; L 2 (U )). By classical regularity results (see, e.g., Theo- . From this and the construction ofṽ we infer that the regularity properties ofṽ are retained by u. It follows in particular that Theorem 1. Assume (A1), (A2).
Moreover, from [2, Theorem 3] it follows that u ∈ L 2 (0, T ; H 1 ̺ ) ∩ C(Q T ) and u is a variational solution of the Cauchy problem By the above and (8), On the other hand, by the definition of Ψ, Thus Ψ = −αΨ − a.e., which implies that αΨ = Ψ a.e. on {u = ψ} and hence that Accordingly (9) is satisfied. By Itô's formula, Therefore if s ∈ [0, T ) and x ∈ D ι for some ι ∈ I then P s,x (X t ∈ D ι , t ≥ s) = 1 and for each t ∈ (s, T ] the random variable X t has strictly positive density on D ι under P s,x . From this and (13) it follows that for every (s, x) ∈ P T . In [19] it is proved that the function 1 {u=ψ} α is a weak limit in L 2 (Q T ) of some sequence {α n } of nonnegative functions bounded by 1 and such that α n (t, X t ) → α s,x t weakly in L 2 ([0, T ] × Ω; dt ⊗ P s,x ) for every (s, x) ∈ Q T . Therefore using once again the fact that for every (s, x) ∈ P T the process X has a strictly positive transition density under P s,x we conclude that (12) holds for every (s, x) ∈ P T , which when combined with (14) implies (10). What is left is to prove that for every (s, x) ∈ P T , From the results proved in [8, Section 6] it follows that for every (s, x) ∈ Q T , where (Y s,x,n , Z s,x,n ) is a solution of the BSDE It is known (see [23]) that Y s,x,n t = u n (t, X t ), t ∈ [s, T ], P s,x -a.s., where u n is a viscosity solution of the Cauchy problem (u n ) t + L BS u n = −ru n + n(u n − ψ) − ), u n (T, ·) = ψ.
We know that P s,x (X t ∈ D ι , t ≥ s) = 1 if x ∈ D ι . Moreover, by classical regularity results (see, e.g., [13, Theorem 1.5.9] and Remark preceding Theorem 1), u n ∈ C 1,2 (P T ). From this, the fact that X s,x,n is a solution o SDE with smooth coefficients and Proposition 1.2.3 and Theorem 2.2.1 in [22] it follows that Y s,x,n t ∈ D 1,2 for every (s, x) ∈ P T , where D 1,2 is the domain of the derivative operator D t in L 2 (Ω) (see [22,Section 1.2] for the definition). Therefore as in the proof of [20, Theorem 2.3] one can show that for every t ∈ (s, T ] and x ∈ P , D t Y s,x,n t = Z s,x,n t , P s,x -a.s., and on the other hand, that D t Y s,x,n t = σ(X t )(u n ) x (t, X t ), P s,x -a.s. Hence, by (17), for every (s, x) ∈ P T . By (16) and (17), u n → u pointwise in Q T . Moreover, from (17), (18) and standard estimates for solutions of BSDEs (see, e.g., [8,Section 6]) it follows that there is C > 0 such that for any (s, x) ∈ P T , while from (16), (18) it follows that as n, m → ∞. From (19) one can deduce that u n ∈ L 2 (0, T ; H ̺ ) and then, by using (20), that u n → u in L 2 (0, T ; H ̺ ) (see the arguments following (2.12) in the proof of [20,Theorem 2.3]). From the last convergence and (16), (18) it may be concluded that for (s, x) ∈ P T , which implies (15). ✷

Convex payoffs
Assume that ψ : R n → R is convex. Let m denote the Lebesgue measure on R n , ∇ i ψ denote the usual partial derivative with respect to x i , i = 1 . . . , n, and let E be set of all x ∈ R n for which the gradient exists. Since ψ is locally Lipschitz function, m(E c ) = 0 and ∇ψ = (ψ x 1 , . . . , ψ xn ) a.e. Moreover, for a.e.
x ∈ E there exists matrix i.e. ∇ 2 ij ψ are defined as limits through the set where ∇ i ψ exists (see, e.g., [1, Section 7.9]). By Alexandrov's theorem (see, e.g., [1,Theorem 7.10], if x ∈ E is a point where (21) holds then ψ has second order differential at x and The second order derivative of ψ in the distribution sense D 2 ψ = {ψ x i x j } is a matrix of real-valued Radon measures {µ ij } on R n such that µ ij = µ ji and for each Borel set B, {µ ij (B)} is a nonnegative definite matrix (see, e.g., [12, Section 6.3]). Let µ ij = µ a ij +µ s ij be the Lebesgue decomposition of µ ij into the absolutely continuous and singular parts with respect to m. By Theorem 1 in Section 6.4 in [12], For R > 0 set D R = P ∩ {x ∈ R n : |x| < R} and τ R = inf{t ≥ s : X t / ∈ D R }. LetL BS denote the operator formally adjoint to L BS . By [24, Theorem 4.2.5] for a sufficiently large α > 0 there exist the Green's functions G α R ,G α R for α − L BS and α −L BS on D R . Let A be a continuous additive functional of X and let ν denote the Revuz measure of A (see, e.g., [25]). By the theorem proved in Section V.5 of [25], for every nonnegative f ∈ C 0 (R d ), Since G α R (x, y) =G α R (y, x) by [24,Corollary 4.2.6], it follows that for any nonnegative g ∈ C 0 (D R ), where E s,g·m denotes the expectation with respect to the measure P s,g·m (·) = P s,x (·)g(x) dx and Note that if g is not identically equal to zero thenG α R g is strictly positive (see [24,Theorem 4.2.5]). Set Theorem 2. Assume (A3). Then assertions (i), (ii) of Theorem 1 hold true with L BS replaced by L BS .
Proof. We use the notation of Theorem 1. Fix s ∈ [0, T ). Since ψ is a continuous convex function, from Itô's formula proved in [3] it follows that there exists a continuous increasing process A such that for x ∈ R n , ψ(X t ) = ψ(X s ) + A t + t s ∇ψ(X θ ) dX θ , t ∈ [s, T ], P s,x -a.s.
From (24) it follows that A is a positive continuous additive functional (PCAF for short) of X. Let ν denote the Revuz measure of A. We are going to show that 1 P · ν = 1 P · µ where µ is the measure on R n defined as To this end, let us set where {ρ ε } ε>0 is some family of mollifiers. Fix a nonnegative g ∈ C 0 (D R ) such that g(x) > 0 for some x ∈ D R and denote by A ε the PCAF of X in Revuz correspondence with µ ε . Then for a sufficiently large α > 0, for all nonnegative f ∈ C 0 (R d ). By [6,Theorem 2], E s,x sup t≥s |A ε t∧τ R − A t∧τ R | → 0 as ε ↓ 0 for every x ∈ R d . Hence and sup ε>0 sup |x|≤R |∇ψ ε (x)| ≤ C(R) < ∞ by Lemma in [6], it follows that for every compact subset K ⊂ R n , sup x∈K sup ε>0 E s,x |A ε t∧τ R | 2 < ∞. Therefore as ε ↓ 0. On the other hand, since µ ε ij → µ ij weakly * for i, j = 1, . . . , n and, by [24,Theorem 4.2.5] Combining this with (23), (25), (26) we see that for every f ∈ C 0 (R n ), SinceG α R g is strictly positive on D R , we conclude from the above that µ = ν on D R for each R > 0. Consequently, µ = ν on P . For x ∈ P , P s,x (X t ∈ R n \ P ) = 0 for t ≥ s. Hence for x ∈ P . Let µ a denote the absolutely continuous part in the Lebesgue decomposition of 1 P · µ. By (22), µ a (dx) = n i,j=1 for x ∈ P . From (24), (27), (28) and [8,Remark 4.3] it follows that Let u be a viscosity solution of (5). From the above and the results proved in [2] (see the reasoning following (11)) we conclude that u ∈ L 2 (0, T ; H 1 ̺ ) ∩ C(Q T ) and there is a function α on Q T such that 0 ≤ α ≤ 1 a.e., (12) is satisfied and u is a variational solution of the Cauchy problem with Ψ = −rψ + L BS ψ on {u = ψ}.
By Remark preceding Theorem 1, u(t, ·) ∈ H 2 loc (R n ). Therefore by Remark (ii) following Theorem 4 in Section 6.1 in [12] the distributional derivatives u x i , u x i x j are a.e. equal to the approximate derivatives ∇ ap i u, (∇ ap ) 2 ij u. Let L ap BS denote the operator defined as L BS but with ∇ i , ∇ ij replaced by ∇ ap i , (∇ ap ) 2 ij . Then u is a variational solution of (29) with L BS replaced by L ap BS and (8) holds with L BS replaced by L ap BS . Hence u t + L ap BS u = ru − α1 {u=ψ} Ψ − a.e. on Q T . Consequently, Since ψ is convex, L BS ψ = L ap BS ψ a.e. on R n by Remark (i) following Theorem 4 in Section 6.1 in [12]. Therefore combining (30) with (31) we see that Ψ = −αΨ − a.e. on {u = ψ} from which as in the proof of Theorem 1 we get (14). To complete the proof it suffices now to repeat step by step the arguments following (14) in the proof of Theorem 1. ✷

The early exercise premium representation
Let ξ denote the payoff process for an American option with payoff function ψ, i.e.
Corollary 3. For every (s, x) ∈ Q T the Snell envelope admits the representation Taking t = s in (32) and using (4) we get the early exercise premium representation for the value function. is the value of the European option with payoff function ψ and expiration time T .
In closing this section we show by examples that for many options Ψ − can be explicitly computed. Using results of §4 and §5 in [26] one can check that the payoff functions ψ in examples 1-4 below satisfy (A3). It is also easy to see that the payoff function ψ in example 5 satisfies (A2 ′ ). Note that the payoff function in example 1 also satisfies (A2 ′ ) and, by [5,21], the payoff functions in examples 2-4 satisfy (A2). We would like to stress that the last assertion is by no means evident. On the other hand, the convexity of ψ in examples 2-4 is readily checked.

Index options and spread options
(Here w i ∈ R for i = 1, . . . , n).