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 [ [20]; see also [21] for nice exposition and additional references), in terms of variational inequalities (Jaillet et al. [19]) and in terms of solutions of reflected BSDEs (El Karoui and Quenez [14]). Although these approaches provide complete characterization of the option value (see Sect. 2 for a short review), the paper by Broadie and Detemple [7] 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 [22] in the case of standard American put option on a single asset. Another important contributions in the case of single asset include Broadie and Detemple [6], El Karoui and Karatzas [13] and Jacka [18] (see also [10,21] and the references therein). The case of options on multiply dividendpaying assets is more difficult and has received rather little attention in the literature. In the important paper [7] and next in Detemple et al. [11] (see also [10]) the early exercise premium formula was established for concrete classes of options on multiply assets. Note that in the last paper call on min option, i.e. option with nonconvex payoff function is investigated. A subclass of call on min options consisting of capped options is studied in [6][7][8] (see also [10]).
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 [26]. 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 [26]. 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 [14]) and with parabolic variational inequalities established in Bally et al. [2]. We also use classical results on regularity of the solution of the Cauchy problem for parabolic operator with constant coefficients, and in case of convex payoffs, some fine properties of convex functions. 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 [25] 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 W = (W 1 , . . . , W n ) is a standard n-dimensional Wiener process, r ≥ 0 is the rate of interest, d i ≥ 0 is the dividend rate of the asset i and σ = {σ i j } is the ndimensional volatility matrix. We assume that a = σ · σ * is positive definite. Since the distributions of the processes X s,x,i depend σ only through a, we may and will assume that σ is a symmetric square root of a. 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, × R n and u is the value of an option with payoff ψ; see (5) and (9) below) 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 [26]. 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 [26] 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 = C([0, T ]; R n ) and let X be the canonical process on . 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).
Therefore if s ∈ [0, T ) and x ∈ D ι for some ι ∈ I then P s,x (X t ∈ D ι , t ≥ s) = 1. From this and the fact that a is positive definite it follows that if x ∈ P T then det σ (X t ) > 0, P s,x -a.s.
is a continuous process. Therefore, if x ∈ P T then by Lévy's theorem the process B s,· defined as B s, i.e.
The above forms of the assets price dynamics will be more convenient for us than (1) or (2). Note that from the definition of the process B s,· and (4) it follows that is the completion of the Brownian filtration. In Bensoussan [4] and Karatzas [20] (see also Sect. 2.5 in [21]) 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 From the results proved in [12] it follows that under (A1) for every (s, [12] it is also proved that for every (s, where u is a viscosity solution to the obstacle problem with From [12,14] we know that V defined by (5) is equal to Y s,x s . Hence In the next section we analyze V via (9) but as a matter of fact instead of viscosity solutions of (8) 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. Let Following [2,25] we adopt the following definition.
(b) If μ in the above definition admits a density (with respect to the Lebesgue measure) of the form u (t, x) = (t, x, u(t, x)) for some measurable :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 u (t, and is determined by ψ and the parameters r, d, a. In the next section we compute for some concrete options.
(i) u defined by (9) is a variational solution of the semilinear Cauchy problem with where for x ∈ R n such that (t, x) ∈ {u = ψ}, Then for every (s, x) ∈ P T the triple (u(·, X ), σ (X )u x (·, X ), K s,· ) is a unique solution of RBSDE s,x (ψ, −r y, ψ).
Since the coefficients of the stochastic differential equation (3) By the above and (11), so by Lemma A.4 in Chapter II in [23], On the other hand, by the definition of , Thus = −α − a.e. on {u = ψ}, which implies that α = a.e. on {u = ψ}, and hence that Accordingly (12) is satisfied. From (2) it is clear that 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 every t ∈ (s, T ] the random variable X t has strictly positive density on D ι under P s,x . From this and (16) it follows that for every (s, x) ∈ P T . In [24] 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 ] × ; 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 (15) holds for every (s, x) ∈ P T , which when combined with (17) implies (13). What is left is to prove that for every (s, x) ∈ P T , From the results proved in [12,Sect. 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 [27]) that where u n is a viscosity solution of the Cauchy problem 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., [17,Theorem 1.5.9] and Remark preceding Theorem 1), u n ∈ C 1,2 (P T ). Therefore applying Itô's formula shows that (20) holds true with Z s,x,n θ replaced by σ (X θ )(u n ) x (θ, X θ ). Since (20) has a unique solution (see [12,Corollary 3.7]), it follows that for every (s, x) ∈ P T . By (19) and (21), u n → u pointwise in Q T . Moreover, from (21), (22) and standard estimates for solutions of BSDEs (see, e.g., [12,Sect. 6]) it follows that there is C > 0 such that for any (s, x) ∈ P T , while from (19), (22) it follows that as n, m → ∞. From (23) one can deduce that u n ∈ L 2 (0, T ; H ) and then, by using (24), that u n → u in L 2 (0, T ; H ) (see the arguments following (2.12) in the proof of [25,Theorem 2.3]). From the last convergence and (19), (22) it may be concluded that for (s, x) ∈ P T , which implies (18).

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 x n ) a.e. (recall that ψ x i stands for the partial derivative in the distribution sense). Moreover, for a.e. x ∈ E there exists an n-dimensional symmetric matrix  (25) holds then ψ has second order differential at x and H (x) is the hessian matrix of ψ at x, i.e. H (x) = {∇ 2 i j ψ(x)}. 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 i j } on R n such that μ i j = μ ji and for each Borel set B, {μ i j (B)} is a nonnegative definite matrix (see, e.g., [16,Sect. 6.3]). Let μ i j = μ a i j + μ s i j be the Lebesgue decomposition of μ i j into the absolutely continuous and singular parts with respect to m. By Theorem 1 in Sect. 6.4 in [16], 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 [28, 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., [29]). By the theorem proved in Sect. V.5 of [29], for every nonnegative f ∈ C 0 (R d ), 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 [28,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 [5] it follows that there exists a continuous increasing process A such that for x ∈ R n , From (28) 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 [9, and sup ε>0 sup |x|≤R |∇ψ ε (x)| ≤ C(R) < ∞ by Lemma in [9], 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 i j → μ i j weakly * for i, j = 1, . . . , n and, by [28,Theorem 4.2.5] Combining this with (27), (29), (30) 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 (26), for x ∈ P. From (28), (31), (32) and [12,Remark 4.3] it follows that Let u be a viscosity solution of (8). From the above and the results proved in [2] (see the reasoning following (14)) we conclude that u ∈ W ∩ C(Q T ) and there is a function α on Q T such that 0 ≤ α ≤ 1 a.e., (15) is satisfied and u is a variational solution of the Cauchy problem By Remark preceding Theorem 1, u(t, ·) ∈ H 2 loc (R n ). Therefore by Remark (ii) following Theorem 4 in Sect. 6 On the other hand, since ψ is convex, ψ ∈ BV loc (R n ) as a locally Lipschitz continuous function and, by Theorem 3 in Sect. 6.3 in [16], ψ x i ∈ BV loc (R n ), i = 1, . . . , n. Therefore ψ is twice approximately differentiable a.e. by Theorem 4 in Sect. 6.1 in [16]. It follows now from Theorem 3 in Sect. 6.1 in [16] that L ap u = L ap ψ a.e. on {u = ψ}. Consequently, Moreover, since ψ is convex, L BS ψ = L ap BS ψ a.e. on R n by Remark (i) following Theorem 4 in Sect. 6.1 in [16]. Therefore combining (34) with (35) we see that = −α − a.e. on {u = ψ} from which as in the proof of Theorem 1 we get (17). To complete the proof it suffices now to repeat step by step the arguments following (17) 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 (36) and using (7) we get the early exercise premium representation for the value function. In closing this section we show by examples that for many options − can be explicitly computed. Using results of Sects. 4 and 5 in [30] 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 [7,26],

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