Valuation of American Continuous-Installment Options Under the Constant Elasticity of Variance Model

Theorem 1

Let the asset price S satisfies(1)–(2), then the value function C^a(t,S;q) at time t of the American CI call option has the following integral representation

Moreover, the optimal stopping and exercise boundaries, A_t and B_t, are solved by the following nonlinear integral equation systems:

Proof

For u ∈ [t,T], let Z(u,S) = e^−r(u−t)C^a(u,S;q) be the discounted option value function defined on 𝓓. Then, the function Z(u,S) is convex in S for all u and belongs to the class C^1,2(𝓓^o). Applying the Itô’s Lemma to Z(u,S), we have

which yields

Since Et(∫tT|e−r(u−t)σSθ/2∂Ca∂S|2du)<∞, then ∫tTe−r(u−t)σSθ/2∂Cua∂SdWu is a 𝓕_t-martingale and Brownian motion. Substituting C^a(T,S_T;q) = (S_T − K)⁺ and C^a = 1_{(Au < Su < Bu)}C^a + 1_{(0 ≤ Su ≤ Au)} × 0 + 1_{(Su ≥ Bu)} × (S_u − K) into (20), and taking the conditional expectation with respect to 𝓕_t on both sides of (20) under the risk-neutral measure Q, we obtain

By Appendix A, one has

and

Substituting the above equations (22) and (23) into (21), this leads to (16). Applying the boundary condition (14), and inserting instead of S_t the value A_t and B_t of the optimal stopping and exercise boundary at time t in (16), respectively. we obtain the nonlinear integral equation system (17) and (18) for the optimal stopping and exercise boundaries A_t and B_t.

The integral representation (16) express the initial premium of the American CI call option as the sum of the corresponding European call value, the early exercise premium, and the expected present value of installment payments along the optimal stopping boundary.

Propositoin 1

The optimal stopping and exercise boundaries, A_t and B_t, have the following properties:

1. A_t ∈ ℂ[0,T] and is nondecreasing and with B_t ∈ ℂ[0,T] and is non-increasing;

2. A_T = K andBT=max{K,rK−qδ}.

Proof

1) Here only gives the proof of non-decreasing for the function A_t, for the non-increasing of the function B_t can beproved in the same way. Note that, Since t| → C^a(t,S_t;q) is non-increasing, then for any t ≤ s, s, t ∈[0,T], we have C^a(t,S_t;q) ≥ C^a(s,S_s;q). Suppose thatA_t is strictly decreasing function for t ∈[0,T], that is, there exists some t ≤ s, s, t ∈[0,T] such that A_t > A_s. But if (t,S) ∈𝓢₁ and there is a S_t ≥ A_s, we then have (s,S_t)∈𝓢₁. Since (t,S_t) ∈𝓢₁, we deduce that C^a(t,S_t;q) = 0 and (s,S_t)∈𝓢₁, thus C^a(s,S_t;q) ≥ (S_t − K)⁺ > 0 = C^a(t,S_t;q). This is in contradiction with the non-increasing function of time t ∈[0,T].

Next we prove the continuity of the function A_t on [0,T]. We first show the left-continuous. Recall that A_t is non-decreasing, so that A_{t −} ≤ A_t. Therefore, we only prove that A_{t −} ≥ A_t, which implies A_t is left-continuous. Suppose that v < t and let (v,A_v) ∈𝓢₁. Therefore, on the closed set 𝓢₁, we have lim_v↑t(v,A_v) = (t,A_{t −}) ∈𝓢₁. Furthermore, using the definition of the optimal stopping boundary, we get A_t = sup{S: S ∈𝓢₁(t)}, that is, A_{t −} ≥ A_t. Hence, A_{t −} = A_t which leads to A_t being left-continuous.

We now prove that A_t is right-continuous. Since the function A_t is non-decreasing, the inequality A_v ≥ A_t holds for any v > t. Suppose that (v,A_v) ∈ 𝓢₁, we then have lim_{v↓ t}(v,A_v) = (t,A_{t +}) ∈𝓢₁, so that C^a(t,A_{t +};q) = 0. On the other hand, we have C^a(t,A_{t +};q) ≥ C^a(t,A_t;q) = 0. Therefore, C^a(t,A_{t +};q) > 0 in 𝓓, which is obviously false. Hence, A_{t +} ≤ A_t, that is A_{t +} = A_t, and right-continuous is proved.

In the following, we prove 2). Firstly, we prove A_T = K. It is easy to see that C^a(t,S_t;q) = 0 for (t, S_t) ∈𝓢₁, which implies that A_T ≤ K. If we suppose that A_T ≠ K, then there exists a domain 𝓓_ε = {T − ε ≤ t ≤ T, A_t < S < K} ⊂ 𝓒₁(ε is small enough) such that

So, at t = T in the domain {A_T < S_T < K}, C^a(t, S; q) satisfies

Noting that C^a(T,S_T;q) = 0, Hence, we have C^a(t,S_t;q) < C^a(T,S_T;q) = 0 in the domain 𝓓_ε, which contradicts with the definition of C^a(t,S;q), and we obtain that A_T = K.

Now, we prove BT=max{K,rK−qδ}. we consider in the following two cases, respectively.

(i) If (r − δ)K < q, we then deduce from (9) that B_T ≥ K. If B_T ≠ K, we then get B_T > K, and moreover, there exists a domain D~ε = {T − ε ≤ t ≤ T, K < S < B_t} (ε is small enough), lying in 𝓒₁ such that

So, at t = T in the domain {K < S_T < B_T}, C^a(t,S;q) satisfies

Noting that C^a(T,S_T;q) = S_T − K. Hence, we have C^a(t,S_t;q) < C^a(T,S_T;q) = S_T − K in the domain D~ε, which contradicts with C^a(t,S_t;q) > (S − K)⁺, and we conclude that B_T = K.

(ii) If (r − δ)K ≥ q, we prove that BT=rK−qδ. If it does not hold, we assume that BT<rK−qδ. Since B_T ≥ K, then there exists a domain Eε={T−ε≤t≤T,BT<S<rK−qδ} involving in 𝓔₁ such that C^a(t,S_t;q) = S_t − K, that is, if (t,S) ∈ 𝓔_ε, then

However, ∂Ca∂t+12σ2Sθ∂2Ca∂S2+(r−δ)S∂Ca∂S−rCa−q<0 in 𝓔₁, which contradicts (24). On the other hand, if we assume that BT>rK−qδ. Since rK−qδ>K, then there exists a domain Cε={T−ε≤t≤T,rK−qδ<S<BT} involving in the continuous region 𝓒₁ such that

In particular, at t = T we have

That means, C^a(t,S_t;q) < C^a(T,S_T;q) = S_T − K, which contradicts with the result of C^a(t,S_t;q) ≥ (S − K)⁺. Combining (i) and (ii), thus BT=max{K,rK−qδ}.

Theorem 2

Let the asset price S satisfies(1)–(2), then the value function P^a(t,S;q) at time t of the American CI put option with maturity date T and the strike price K, has the integral representation

Moreover, the early exercise and the early stopping boundaries F_t and G_t are determined by the following nonlinear integral equation system

Remark 2

Similar to the call case above, it is proved that the two optimal boundaries, F_t and G_t, are continuous functions in [0,T], and F_t is nondecreasing and G_t is non-increasing.

Corollary 1

The Δ value of an American CI call or put option is respectively given by

Remark 3

Under the CEV model (1)–(2), the delta values of both the European vanilla call and put option, ΔcE and ΔpE, are respectively given by Larguinho et al.[27].