VALUATION OF AMERICAN STRANGLE OPTION: VARIATIONAL INEQUALITY APPROACH

. In this paper, we investigate a parabolic variational inequality problem associated with the American strangle option pricing. We obtain the existence and uniqueness of W 2 , 1 p, loc solution to the problem. Also, we analyze the smoothness and monotonicity of two free boundaries. Finally, numerical results of the model based on this problem are described and used to show the boundary properties and the price behavior.


1.
Introduction. Various strategies have been studied to reduce the risk of options since the financial crisis. According to Chaput and Ederington [1], strangle and straddle account for more than 80% of options strategies. A strangle option is a strategy that holds a position at the same time in both a call and a put with different strike prices but with the same expiry. If we are expecting large movements in underlying assets, but are not sure which direction the movement will be, we can buy or sell them to reduce the risk exposed by a European call or put option. In particular, a straddle option is one of strangle options where the strike price of the call portion is the same as the strike price of the put portion.
Here we focus on American strangle options. The definition of an American type option is an option contract that allows option holders to exercise their rights at any time before expiry. Since option holders in the American option can exercise their rights at any time before expiry, the pricing of such options is often categorized as the optimal stopping problem or the free boundary problem. In this paper, we study the parabolic variational inequality associated with the model of American strangle options pricing. In other words, we will investigate V (t, s) satisfying if V > (s − K 2 ) + + (K 1 − s) + , (t, s) ∈ (0, T ] × (0, +∞), where r, σ, K 1 , K 2 are positive constants with K 1 < K 2 and q is a constant with q ≥ 0. In Appendix A, we present the formulation and the financial background of the problem (1.1).
There are various studies on the American strangle option. Chiarella and Ziogas derived the integral equation satisfying the American strangle option in [3] by using the incomplete Fourier transform method. Qiu [10] gave an alternative method to derive the EEP representation of the American strangle option value and analyzed the properties of the option value and the early exercise boundary. Ma et al. [9] construct tight lower and upper bounds for the price of an American strangle. In addition, there are a variety of studies on parabolic variational inequality arising in option pricing. Yang et al. [12], [15] considered parabolic variational inequalities associated with European-style installment call or put option pricing and obtained the existence and uniqueness of W 2,1 p,loc solution to the problem and the monotonicity, smoothness and boundedness properties of free boundaries. Also, Chen et al. [2] proved existence and uniqueness of weak solution in variational inequality in the case of American lookback option with fixed strike price. However, the parabolic variational inequalities in the above researches, have only one free boundary.
Of course, Yang and Yi [13] already considered a parabolic variational inequality problem associated with the American-style continuous-installment options with two free boundaries, the lower obstacle of the variational inequality is a monotone function in spatial variables. In the present paper, variational inequality with two free boundaries does not have monotonicity condition on the lower obstacle function in (1.1). The novelty of this paper is that we analyze more general case of this problem.
The rest of this paper is organized as follows. In section 2, we prove the existence and uniqueness of W 2,1 p,loc solution to problem (1.1). In section 3, we show that the monotonicity and C ∞ -regularity of two free boundaries based on the results in section 2. Moreover, we will prove the starting points of the free boundaries. In section 4, we conduct comparative static analysis of variational inequality (1.1). In section 5, we solve the stationary problem arising from American strangle option and use it to show that two free boundaries are bounded. In section 6, we describe the numerical result applying the finite difference scheme. Appendix A is the formulation of the model. Appendix B shows that the unique solution to the problem (1.1) coincides with the expected value of the American strangle option.
2. Existence and uniqueness of a solution. We first transform the degenerate backward parabolic problem (1.1) into a familiar forward non-degenerate parabolic problem. Setting We now consider the problem in the bounded domain [0, T ) × (−n, n): where n ∈ N with n > ln 2 κ .
The lower obstacle and terminal condition function of variational inequality (2.1) are not monotonic for spatial variable x. Thus, we appropriately transform the value function to have monotonicity. The following lemma provides the existence, uniqueness and properties of a solution to the above problem.
Lemma 2.1. For each fixed n ∈ N with n > ln 2 κ , there exists a unique solution Proof. We first define a penalty function β ε ∈ C ∞ (R) (0 < ε < 1) satisfying Since the functions (e x − 1) + and (κ − e x ) + are not smooth enough, we also define a function ϕ ε ∈ C ∞ (R) satisfying (2.7) We then consider the following approximation of the problem (2.3): (2.8) By Schauder's fixed point theorem, the above problem (2.8) has a unique W 2,1 p solution, see [14]. We next claim that for sufficiently large n. Observe from (2.7) that (2.10) Then we deduce from (2.7) and (2.10) that Furthermore, we see from the boundary conditions in (2.8) that if 0 < ε < κ 2 , By the comparison principle, we get On the other hand, it follows from (2.6) and (2.7) that if 0 < ε < κ 2 < 1 2 , Also, from the boundary conditions in (2.8), we get By the comparison principle, we obtain Therefore, the claim (2.9) follows from (2.11) and (2.12). We see from (2.9 [8,Theorem 6.33]), we get for some constant c > 0 which is independent of ε. We then apply the method in [5] to deduce that as ε → 0+, where Y n is the solution to the problem (2.3). Hence, (2.4) follows immediately from (2.9). Now let us prove (2.5). For any small δ > 0, we see from (2.3) and (2.4) that the function Y n (τ + δ, x) satisfies Applying the monotonicity of solution of variational inequality with respect to initial value (see [5]), we obtain which yields the first inequality in (2.5). For the second inequalities in (2.5), we differentiate (2.8) with respect to x, then we have where Z := ∂ x Y n,ε and β ε (· · · ) = β ε (Y n,ε − ϕ ε (e x − 1) − ϕ ε (κ − e x )). Letting Then we conclude from the maximum principle that Similarly, we set Z 2 := Z − e x and then deduce that It follows from the maximum principle that Z 2 ≤ 0, and hence that ∂ x Y n,ε ≤ e x . Thus we obtain (2.5).
To prove the uniqueness, suppose that Y n and Y n are two solutions to the problem (2.3) and that the set where ∂ p N is the parabolic boundary of the domain N . Then it follows from the ABP maximum principle (see [11]) that U ≤ 0 in N , which contradicts the definition of the set N . Hence N = ∅ and Y n ≥ Y n . Similarly, we conclude that Y n ≤ Y n , and finally that Y n = Y n .
Theorem 2.2. There exists a unique solution to the problem (2.1) for all R > ln 2 κ , ρ > 0 and 1 < p < ∞. Moreover, we have Here, χ A denotes the characteristic function of the set A. Then we see that where the constant c(R) depends on R, but is independent of n. Therefore, it follows from the W 2,1 p estimates (see [8]) and (2.4) that for n > R > ln 2 κ , for some constant c(R) which is independent of n. Letting n → ∞, we deduce that, up to a subsequence, ) . In addition, we obtain from the Sobolev embedding theorem that Furthermore, the C α estimate yields Y ∈ C([0, T ] × R) and the inequalities (2.14) and (2.15) follow from (2.4) and (2.5), respectively. The proof of the uniqueness is the same as that of Lemma 2.1.
3. Analysis of the free boundaries. In this section we analyze the free boundaries of variational inequality (2.1). If q = 0, we will show that the variation inequality (2.1) has two free boundaries. Since the two free boundaries interact with each other, it is not easy to prove the monotonicity and smoothness of the free boundaries.
Let D = [0, T ] × R be the whole region. We denote Hence we can define the free boundaries We note that the free boundary A(τ ) separates E A from C and that the free bound- Then, the S A (τ ) and S B (τ ) are the free boundaries of the variational inequality (1.1). When the underlying asset hits below S A (τ ) or above S B (τ ), the option holder can benefit by exercising his/her rights. Also, the region surrounded by free boundaries S A (τ ) and S B (τ ) is the continuation region C of American strangle options.
We now prove the regularity and the monotone property of the free boundaries. In addition, we describe the limiting behavior of the free boundaries as time to maturity goes to zero.
(1) If q = 0, then E B is the empty set, and hence B(τ ) does not exist.
(2) If q > 0, then B(τ ) is smooth and strictly increasing in (0, T ]. Moreover, When q = 0, it follows that (∂ τ − L)Y = −r < 0 in E B . On the other hand, (2.1) leads to (∂ τ − L)Y ≥ 0 in E B . This is a contradiction. Therefore, we conclude that E B = ∅, and hence that B(τ ) does not exist.
a contradiction. Therefore, we conclude that B(0) = 0 in the case q ≥ r.
We now turn to the case q < r. Suppose that B(0) = x 0 > ln r q . In the same manner we can see that a contradiction. Therefore, B(0) ≤ ln r q . To show that B(0) = ln r q , we now suppose that B(0) = x 0 < ln r q . Let x 1 := 1 2 x 0 + ln r q . We deduce that there exists τ 0 > 0 such that B(τ 0 ) ≤ x 1 , and hence Then it is clear that On the other hand, we discover from (2.1) that We have arrived at a contradiction which proves B(0) = ln r q in the case q < r. Thus (3.1) is proved.
We now consider the case q > r. Suppose that A(0) = x 0 < ln rκ q . We obtain similarly that a contradiction. Therefore, A(0) ≥ ln rκ q . We now suppose that A(0) = x 0 > ln rκ q . As in the proof of the previous theorem, there exists τ 0 > 0 such that On the other hand, we deduce from (2.1) that which contradicts (3.7). This yields A(0) ≤ ln rκ q , and hence A(0) = ln rκ q . Analysis similar to that in the proof of Theorem 3.3 shows that A(τ ) is strictly decreasing in (0, T ]. Moreover, the smoothness of A(τ ) follows by the same method as Theorem 3.3.
Remark 3.2. Contrary to Remark 3.1, Theorem 3.4 means that when the underlying asset decrease sufficiently, if it is to exercise the short-term American strangle, the long-term American strangle already has been exercised. Remark 3.3. If q = 0, Theorem 3.3 and Theorem 3.4 imply that the free boundaries S A (τ ) and S B (τ ) are well-defined. If q = 0, (1) in Theorem 3.3 implies that the free boundary B(τ ) or S B (τ ) does not exist. That is, for the American strangle option written on an underlying asset without dividends, although the underlying asset increases enough, the option holder does not exercise his/her right. This phenomena is consistent with American call option with non-dividend. Since the price of the American call option on a non-dividend-paying stock always exceeds its intrinsic value prior to expiration, the early exercise is never optimal. 4. Comparative static analysis. In this section we conduct comparative static analysis with respect to important model parameters. First, we analyze the effect of value σ, the volatility of the American strangle option.
Proof. We denote where D = (0, T ] × R. In the continuation region C, it follows from (2.1) that On the other hand, we observe from Lemma 3.2 that Hence, we see that in the exercise region E, We thus get in the exercise region E. Combining (4.2) and (4.3) gives For δ > 0, we now set Y δ (τ, x) := Y (τ, x − δ). We also denote Then we obtain similarly that (4.5) In the region E ∩ {x < ln κ}, we know that Y (τ, x) = κ − e x . Then we deduce from Section 3 that Y δ (τ, x) = Y (τ, x − δ) = κ − e x−δ , and so In addition, we notice that We now combine (4.4)-(4.7) to discover that Then we conclude from (4.8) and (4.9) that It follows that for all (τ, x) ∈ D and δ > 0. This yields the desired inequality (4.1).
Using the above Lemma, we prove the following theorem, the behavior of the free boundaries according to σ. (1) A(τ ) is decreasing with respect to σ.
By the comparison principle, we have Y 1 ≥ Y 2 , From this we deduce that for j = 1, 2. Therefore, we obtain A 1 (τ ) ≤ A 2 (τ ) and B 1 (τ ) ≥ B 2 (τ ), where A j (τ ) and B j (τ ) are the free boundaries of Y j for j = 1, 2. This completes the proof.
For fixed r, q, σ and maturity T , let us define F p (τ ) and F c (τ ) as the free boundaries of the degenerate backward parabolic problem arising from the American put option with strike price K 1 and the American call option with strike price K 2 , respectively. Then, by the following theorem, we can compare A(τ ) and F p (τ ), B(τ ) and F c (τ ), respectively. Proof. We first prove that B(τ ) ≥ F c (τ ). Let Y c be the solution to the problem which is the forward non-degenerate parabolic problem arising from the model of American call option. In view of the monotonicity of solution of variational inequality with respect to initial value Now let Y c be the solution to the forward non-degenerate parabolic problem arising from the model of American put option: Since we have Y ≥ Y c in [0, T ] × R, and hence A(τ ) ≤ F p (τ ) for all τ ∈ (0, T ].

5.
Stationary problem for American strangle option. In this section, we consider a stationary problem arising in the American strangle option. According to Theorem 3.3, the problem (2.1) is divided into two cases according to the range of q: (1) There exist two free boundaries as q > 0, (2) There is only one free boundary as q = 0. Therefore, we also need to consider about the stationary problems corresponding to problem (2.1) separately.
(1) For q > 0, the stationary problem is where L is defined in (2.2).
Theorem 5.1. The variational inequality (5.1) has a unique W 2 p,loc solution, which is x < s p , and n 1 , n 2 are the positive and negative roots of the algebraic equation The constants s c and s p are defined as e sp = n 2 n 2 − 1 · κy n1 + 1 y n1 + y , e sc = e sp · y.
Next, we show that V defined as (5.2) is the unique W 2 p,loc solution to the variational inequality (5.1). By a method similar to Section 2, we can get the uniqueness of the solution of the problem (5.1). Also, it is not hard to check that V (x) ∈ W 2 p,loc . Therefore, we only need to show that V satisfies the variational inequality (5.1).
(2) As q = 0, the subregion E B of exercise region E does not exist and for the uniqueness of the solution to the corresponding stationary problem, we need to impose an additional condition, which comes from the properties of strangle options. The problem is Theorem 5.2. The variational inequality (5.15) has a unique W 2 p,loc solution, which is x < s p , The general solution form of (5.17) is given by It is easily seen that then (5.16) follows if we extend V by V (x) = κ − e x for any x < s p . We note that e sp < κ and c 2 > 0, and so V (x) ≥ 0. Next, we prove that V possessing the form (5.16) is the unique W 2 p,loc solution to the variational inequality (5.15). Similarly in case (1), we only need to prove that V satisfies (5.15).
First, for x < s p , Secondly, from (5.16), we see that for any x > s p , there holds Moreover, for x ≥ s p , Since V (s p ) = κ − e sp , we have V (x) ≥ κ − e x , x ∈ [s p , +∞). We also have Thus, V ≥ (e x − 1) + + (κ − e x ) and V is the W 2 p,loc solution to variational inequality (5.15).
Applying the comparison principle with respect to initial value of variational inequality, we can obtain the following theorem.
In fact, the constants s p and s c are defined in Theorem 5.1 and Theorem 5.2.
6. Numerical methods and results. In this section, we obtain the numerical solution of the value function and free boundaries of the American strangle option by applying the finite difference scheme.
Starting from (2.1), we have (6.1) Given mesh size ∆τ , ∆x > 0, Y n j = Y (n∆τ, j∆x) represents the value of numerical approximation at (n∆τ, j∆x), then the PDE is converted to the following difference equation: Applying the Taylor expansion, we see that as ∆τ → 0 + , there holds Ignoring a higher order of √ ∆τ , we get (6.5)

JUNKEE JEON AND JEHAN OH
Consider the point (τ, x) = (n∆τ, j∆x), then    Figure 1 shows the value function of the American strangle option using the numerical method described above. Since the American strangle option has both the properties of the call option and the put option, the value of the option increases as the underlying asset increases or decreases sufficiently. That is, the value function V does not hold monotonicity with respect to s. This intuition is consistent with the numerical result in Figure 1.   In Figure 2, it can be seen that as the volatility σ of the underlying asset increases, A(τ ) decreases and B(τ ) increases. As mentioned in the introduction, the American strangle option can obtain more profits as the underlying asset fluctuates more. That is, as the volatility σ of the underlying assets increases, the price of the option increases, so it should be exercised when the underlying asset price is higher or lower than when the volatility is lower. This is consistent with the result of Theorem 4.2. In addition, the area enclosed by the two free boundaries A(τ ) and B(τ ) is the continuation region C of Problem (2.1) and the complement of this region is the exercise region E of Problem (2.1). Figure 3 plots the behavior of free boundaries B(τ ) and F c (τ ) according to τ . Observe that B(0) = F c (0) ,B(τ ) ≥ F c (τ ), B(τ ) and F c (τ ) are increasing. Likewise, Figure 4 shows the behavior of free boundaries A(τ ) and F p (τ ) according to τ. Observe that A(0) = F p (0), A(τ ) ≤ F p (τ ), A(τ ) and F p (τ ) are decreasing. Therefore, Figure 4 and Figure 5 illustrate the result of Theorem 4.3 numerically. The American strangle option is more expensive than the American call option and the American put option because it can obtain benefits both cases in which the underlying asset increases or decreases. Therefore, the American strangle option should be exercised at a higher price than the corresponding American call option and at a lower price than the corresponding American put option. This intuition is also consistent with the numerical results in Figure 3-4. Figure 5 shows the upper and lower bounds of free boundaries A(τ ) and B(τ ), respectively. In other words, s p ≤ A(τ ) ≤ ln(κ max(1, r q )) and ln(max(1, r q )) ≤ B(τ ) ≤ s c . The numerical results coincide with Theorem 5.3.
Appendix A. Formulation of the model. An American strangle option whose underlying asset is the stock which has the price S t given by dS t = (r − q)S t dt + σS t dW t , (A.1) where t is calender time, r represents the risk-free interest rate, q(≥ 0) is the continuous dividend rate, and σ > 0 is the constant volatility of S t . Also, (W t ) t≥0 is a standard Brownian motion on a filtered probability space (Ω, (F t ) t≥0 , P) with a risk-neutral measure P, where (F t ) t≥0 is the natural filtration generated by (W t ) t≥0 . Let T t,T be the set of all stopping time in [t, T ], and the value of American strangle option price at time t is defined by where K 1 and K 2 denote the strike price of put and call options, respectively, and we assume K 1 < K 2 . It is obvious that By the definition of (A.2) and the law of iterated expectations, Appendix B. Verification. In this section, we show that the unique solution V (t, s) to the problem (1.1) coincides with the expected value of the American strangle option price, i.e., (A.2) holds. By Theorem 2.2, we know v(·, ·) ∈ W 2,1 p,loc ((0, T ) × (0, +∞), so V (·, ·) and ∂ s V (·, ·) are continuous in (0, T ) × (0, +∞) by embedding theorem [7]. Since ∂ t V ≤ 0 and (S − K 2 ) + + (K 1 − s) + is the lower obstacle, we can obtain from a standard method as explained in Friedman [4] that ∂ t V (·, ·) is continuous across s = S A (t) and s = S B (t), where S A (t), S B (t) are the free boundaries of problem (1.1). Therefore ∂ t V (·, ·) is continuous in (0, T ) × (0, +∞). Moreover, ∂ t V (·, ·) and ∂ ss V (·, ·) are locally bounded in (0, T ) × (0, +∞).