The viscosity solutions approach to swing options pricing under a regime-switching mean-reverting model

In this paper, we study the valuation of swing options on electricity markets with local volume and refraction time constraints, under the setting that the dynamic of the underlying spot price is a 2-state regime-switching mean-reverting process. We derive the corresponding optimal multiple stopping problem, reduce it to a sequence of optimal single stopping problems, and further find that those value functions satisfy HJB variational inequalities subject to suitable conditions. Then after a prior estimation for the value functions, the viscosity solutions approach is adopted to get existence and uniqueness results in viscosity sense.


Introduction
In real electricity markets, swing options provide the holder with rights to exercise repeatedly under certain constraints, such as limits on the local volume, the global volume, and refraction time etc. However, those trading rules add complexity to the associated valuation, which can be handled as stochastic optimal control problems. Besides the specification of option contracts, the other one of fundamental settings of the whole valuation procedure is the dynamic process characterizing the behavior of electricity prices, which remain high for a period of time after a jump, according to the empirical researches like [1][2][3][4] and [5]. Among various kinds of models, the regime-switching type is widely adopted. Wahab et al. [6] investigate the valuation problem under a three-state regime-switching model, where one regime is the Ornstein-Uhlenbeck type, and another two are Gaussian processes with constant coefficients. Wahab et al. [7] deal with a n-state regime-switching model where processes in all regime are geometric Brownian motion. Chiarella et al. [8] handle the valuation problem under a regime-switching forward curve model, and Bauerle et al. [9] choose a 2-state regime-switching model.
In all of [6][7][8][9], the lattice method is applied to price swing options, though with different choices of underlying price models. The lattice method concentrates on how to find the value function (a specific type of conditional expectation) by directly computing expectation iteratively with lattice. More specifically, in the light of stochastic control theory, the dynamic programming principle is satisfied by the value function leads us to the Bellman equation. Then lattices in different form can be applied to dealing with the conditional expectations arising in the corresponding Bellman's equations. Different from the lattice method, another approach to tackling this problem is to seek the PDE characterization of the value function, which leads to the HJB variational inequalities. In this direction, Dahlgren et al. in [10] and [11] derive the variational inequalities and solve them by the finite difference method. In addition, basing on [12], Wilhelm and Winter [13] solve the related variational inequalities by finite element method. Besides, according to [14] and [15], the viscosity solutions approach is a powerful tool to deal with stochastic optimal control problems, and it is well known that the value function is the unique viscosity solution to the HJB-type equation or variational inequality subject to certain conditions, and that this approach can also help studying properties (e.g. convergence and stability) of numerical schemes. For pricing swings, the viscosity solution approach has been used in [16] and [17].
In this paper, the electricity price model we use is of a regime-switching mean-reverting type with two states, where the mean-reverting characteristic is added to the model adopted in [6], and this modification is motivated by empirical researches in [1] and [18], which report that electricity spot price is equipped with the mean-reversion feature. As for the swing option contract, we assume that there exist: (i) the local volume constraint, that is, the holder can either exercise one volume of spot or none each time she uses one right, and (ii) the refraction time constraint, which requires a fixed time break between two successive exercises. Under those settings, we formulate the pricing of the swing option as an optimal multiple stopping problem. Basing on the work of Carmona and Touzi [12], we reduce this problem. After the reduction, continuity and boundedness of value functions is obtained and then we try to characterize the value functions as the unique viscosity solution of the associated HJB variational inequalities. First, existence results can be received. In addition, we prove the comparison principle fitting this problem which ensure the uniqueness of the solutions in viscosity sense.
The structure of this paper is as follows. Section 2 introduces the model for the underlying electricity price and derives the variational inequalities that the value functions formally satisfy. Section 3 treats the properties of the value functions. Section 4 characterizes the value functions as viscosity solutions of the associated variational inequalities. Section 5 gives the comparison principle and the uniqueness result.

Formulation of the problem 2.1 Mean-reverting regime-switching model and swing options
For a fixed maturity T, we work on a probability space (Ω, F, Let {α(t)} 0≤t≤T be an independent time-homogeneous Markov chain with values in a finite state space M := {1, 2}, and the related generator matrix P = (q ij ) 2×2 , satisfying q ij ≥ 0 and 2 j=1 q ij = 0 for each i, j ∈ M and i = j. We model the electricity price by the stochastic process {S(t)} t∈[0,T] , which is governed by the following: where S 0 and i are initial data; β(i), ξ (i) and σ (i) for each i ∈ M are constants representing the speed of adjustment, the long-term mean level, and the volatility. Moreover, we notice that the usual Lipschitz and linear growth conditions are satisfied to ensure the existence and uniqueness of the strong solution to the SDE(1). On the settings of swings, we follow [12]. Consider a swing option with n ∈ N + exercise rights during the time period [0, T]. The strike price is K ∈ R + , thus the holder gets the payoff φ(x) = (Kx) + . All of the above parameters are fixed. And there are two rules the holder must obey: 1 Each time the holder use the right, she can exercise either one volume or none.
2 If one right is used at time t, the holder cannot use another right until after the time t + δ, where δ > 0 is the refraction period. We call the first item above the local volume constraint and the second the refraction time constraint.

Variational inequalities of value functions
In this subsection, we start from introducing the value function of this problem. For simplicity, we sometimes use f i (t, s) to stand for any function f (t, s, i) For fixed integer n ∈ N, we define the value function where T (n) t,T is the collection of τ := (τ 1 , τ 2 , . . . , τ n ) having the following properties: s., ∀i = 1, 2, . . . , n, • τ i := T+ on {τ i > T} a.s., ∀i = 1, 2, . . . , n; moreover, T+ > T is a cemetery time and we set φ(S T+ ) ≡ 0.
Here, the notation V (n) i (t, s) denotes the value of swing option at time t with state value (S(t), α(t)) = (s, i), and the superscript n is used to emphasize that the holder still has n rights before the expiry.
For the above value functions, we state in the following proposition an iterative equation showing the relationship between V (n) and V (n-1) . It is this relationship that transforms the optimal multiple stopping problem into a sequence of optimal single stopping problems.
(ii) For n ≥ 1, where T t,T is the same as in the above definition, and where δ is the refraction time.
Proof Notice that the reward (or payoff ) process φ(S t ) = (K -S t ) + has continuous paths and the strong Markov property, which indicates that it satisfies conditions (2.1), (2.2) and (2.8) in [12]. In addition, according to [19] we find that the filtration {F (t)} t∈[0,T] satisfies the usual conditions and additional requirements (2.10) in [12]. In this way, we can apply Theorem 1 in [12] to rewrite the problem (2) into the iteration form (4) with (5).

Remark 1 The specific form of G (n)
i of (5) in Proposition 1 can be further written as follows. (i) For n = 1, (ii) For n ≥ 2, G (n) i involves a European option price, i.e., in which To see how (6) comes about, we define Theorem 1 For n ∈ N + , the pair (V (n) 1 (t, s), V (n) 2 (t, s)) formally satisfy the coupled variational inequalities where L i V (n) i and G (n) i are the same as in Proposition 1.
Let τ be a stopping time in T t,T . Then we have from the Dynkin formula that where M is a martingale under measure Q, and Taking the expectation on both sides, we obtain Then under the assumption that we have and by taking the supreme on both sides, it follows that Hence, we assume that (ζ 1 (t, s), ζ 2 (t, s)) is the solution of On the other hand, define Above all, if V (n) is as smooth as ζ , then it is the solution of the problem (7).

Viscosity solutions
In this subsection, after introducing the notion of viscosity solution, we state that the value functions in this paper is the unique viscosity solution of the problem (7).
Then we say f (n) i (t, s) is a continuous viscosity supersolution of the variational inequalities in (7) on [0, T) × R + .
i (t, s) is a continuous viscosity solution of (7) on [0, T] × R + if it is both a continuous supersolution and a continuous subsolution of the variational inequalities in (7). In Sects. 4 and 5, we will show the proofs of the existence and uniqueness results, respectively.

Properties of the value function
In this part, we derive properties including continuity and local boundedness of the value function to demonstrate that it satisfies the first two conditions in Definition 2 (or in Definition 3, notice that they are the same).
Proof For each i ∈ M, given s 1 , s 2 and t ∈ [0, T], let S 1 and S 2 be two solutions of the SDE (1) with S 1 (t) = s 1 and S 2 (t) = s 2 , respectively. We adopt mathematical induction.
(i) For n = 1, we have Here we need to obtain the upper bound of (1), by using the Cauchy-Schwarz inequality and Tonelli's theorem we find that Notice that C is a generic positive constant, whose value varies in different cases. Next we adopt Gronwall's inequality to deal with (10), and we obtain Then, by using the Cauchy-Schwarz inequality again, we get an upper bound of E Q [|S 1 (u)-S 2 (u)|], that is, With the inequality (11), we can continue to deal with the inequality (9) and finally get Hence V (1) (t, s, i) is (Lipschitz) continuous respect to s.
(ii) For n = k, k ≥ 2, we assume that under which we can compute for n = k, Here we need to investigate the property of |G (k) (τ , S 1 (τ ), α(τ )) -G (k) (τ , S 2 (τ ), α(τ ))| as follows: In this way, we get which completes the proof of continuity of V (n) (t, s, i) with respect to s.
Proof Firstly, we define the objective function Next, for any t and t , we set 0 < t < t < τ < τ := τ + tt < T, our discussion is based on mathematical induction. (i) For n = 1, we have and Then we investigate the property of h (1)
(ii) For n = k, k ≥ 2, we assume that Then we start from defining and Next we need to investigate the property of h (k) (t, S(t), α(t)) (t ∈ [0, T]). For this purpose, let t 1 , t 2 > 0, S(t 1 ) = s 1 , S(t 2 ) = s 2 , and we estimate here we need to deal with I 1 and I 2 . Note that := C(I 11 + I 12 ), where and in this way we get For I 2 , we get Hence we can conclude that Now we come back to Eq. (15), and we estimate With the fact that we find that Similarly, we can restart from the equality (16) to get In this way, we finally arrive at the estimation that and this indicates the continuity of V (k) with respect to t. Therefore we complete the proof of the continuity of V (n) with respect to t. Proof For each i ∈ M, given any t ∈ [0, T], assume that S(t) = s and α(t) = i. Then, by mathematical induction, we have the following.
(i) For n = 1, (ii) For n = k, k ≥ 2, assuming that then using the results in Lemma 1 and Lemma 2, we get

Existence of the viscosity solution
This section shows the existence result. We break the whole proof into two parts, where the first part corresponds to the supersolution property of the value function, and in the second part we deal with the subsolution property.
Proof We only need to show that, for any i ∈ M, V (n) (t, s, i) is a viscosity supersolution to where L i V and G (n) i are the same as in Propersition 1.
Given (t,s) ∈ [0, T) × R + , assume that α(t) = i ∈ M, and let φ ∈ C 1,2 ([0, T) × R + ) such that Let Since φ ∈ C 1,2 , we have Next, let θ be the time such that α(t) ≡ i in [t, θ ]. In addition, we denote by S m (t) := S t m ,s m ,i (t) the associated state process with initial data S m (t m ) = s m . For m sufficiently large, we have Then we define the stopping time where α > 0 is some given constant. By the dynamic programming principle we have and, together with (18) and (19), we have Similar to [20], we set Therefore by Dynkin's formula, we continue the inequality (20) as follows: Next we divide both sides by γ mt m , then the mean value theorem leads us to Note that, for m sufficiently large, γ m = t m + h m , therefore Finally, we send m to infinity, and the inequality (21) becomes In addition, Theorem 1 states and together with Lemma 1, we find that V (n) (t, s, i) is the continuous viscosity supersolution of (17).
Proof We only need to show that, for any i ∈ M, V (n) (t, s, i) is a viscosity subsolution to where L i V and G (n) i are the same as in Lemma 1.
We argue by contradiction. Given (t,s) ∈ [0, T) × R + , assume α(t) = i ∈ M and let φ 1,2 ([0, T) × R + ) such that We assume to the contrary that Next, let θ be the time such that α(t) ≡ i in [t, θ ]. We try to work towards a contradiction of the weak dynamic programming principle, i.e., for all (t,s) ∈ [0, T] × R + and τ ∈ Tt ,θ such that S¯t ,s (τ ) is L ∞ -bounded. Firstly by the properties that G (n) ∈ C([0, T] × R + × M) and the first equality of (23), we know that there exist δ > 0 and 0 < h ≤ θ such that the following inequalities hold: and Secondly, by the second equality of (23), we get Thirdly, by the continuity of Next we define the stopping time, and denote Then, for arbitrary stopping time τ ∈ T t m ,T , And we define Therefore, Noting that the process (t, S t m ,s m ,i (t)) is in D, then, by (25), (26) and (27) we find that since η m → 0 as m → ∞, and τ is arbitrary. This provides the contradiction of (24).

Uniqueness of the viscosity solution
The main task of this section is to prove the comparison principle of the HJB variational inequality problem (7), which ensures the uniqueness of the solution.
Before that, we try to prove the following comparison principle of the HJB equation problem: Since the problem (P) is of the weakly coupled type, we only need to consider the problem where V is as above. To begin with, we introduce Definitions 5, 6,7 and Ishii's lemma, which will play important roles in the succeeding proof. For more details as regards the above definitions and Ishii's lemma, we refer to the classical literature [21] and [22]. Definition 7 An u.s.c. (resp. l.s.c.) function w i on [0, T) × R + is a viscosity subsolution (resp. supersolution) of (P) on [0, T) × R + if and only if for all (t, s) ∈ [0, T) × R + and all (q, p, M) ∈ P 2,+ w(t, s) (resp. P 2,w i (t, s)), F t, s, w i (t, s), q, p, M ≤ 0 (resp. ≥ 0).
Now we proceed to the statement of the comparison principle for the problem (P ), and the proof of it.
Theorem 3 Let U i be a u.s.c. viscosity subsolution of problem (P ) with polynomial growth condition, and let V i be l.s.c. viscosity supersolution of problem (P ) with polynomial growth condition. In addition, let U i (T, s) = V i (T, s) = (Ks) + for s ∈ R + . If U i (T, ·) ≤ V i (T, ·), then U i ≤ V i in [0, T) × R + .
Proof We argue by contradiction.
2 Or F i (t , x , W , q, p, M) ≤ 0, which can be combined with the supersolution part F i (τ , y , W , q, p, M) ≥ 0 exactly as in the proof of Theorem 3, and this leads to the same contradiction as above.