DISCRETE AND CONTINUOUS doi:10.3934/dcdsb.2018206 DYNAMICAL SYSTEMS SERIES B Volume 24, Number 2, February 2019 pp. 755{781 VALUATION OF AMERICAN STRANGLE OPTION: VARIATIONAL INEQUALITY APPROACH Junkee Jeon Department of Mathematical Sciences Seoul National University Seoul 08826, Republic of Korea Jehan Oh∗ Fakult¨atf¨urMathematik Universit¨atBielefeld Postfach 100131, D-33501 Bielefeld, Germany (Communicated by Bei Hu) Abstract. In this paper, we investigate a parabolic variational inequality problem associated with the American strangle option pricing. We obtain the 2;1 existence and uniqueness of Wp;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 2010 Mathematics Subject Classification. Primary: 35R35; Secondary: 91G80. Key words and phrases. American strangle options, free boundary problem, option pricing, variational inequality. The first author gratefully acknowledges the support of the National Research Foundation of Korea grant funded by the Korea government (Grant No. NRF-2017R1C1B1001811), BK21 PLUS SNU Mathematical Sciences Division and the POSCO Science Fellowship of POSCO TJ Park Foundation. ∗ Corresponding author: Jehan Oh. 755 756 JUNKEE JEON AND JEHAN OH study the parabolic variational inequality associated with the model of American strangle options pricing. In other words, we will investigate V (t; s) satisfying 8 2 σ 2 > @tV + s @ssV + (r − q)s@sV − rV = 0; > 2 > + + > if V > (s − K2) + (K1 − s) ; (t; s) 2 (0;T ] × (0; +1); > < σ2 @ V + s2@ V + (r − q)s@ V − rV ≤ 0; (1.1) > t 2 ss s > + + > if V = (s − K2) + (K1 − s) ; (t; s) 2 (0;T ] × (0; +1); > > + + :> V (T; s) = (s − K2) + (K1 − s) ; s 2 [0; +1); where r; σ; K1;K2 are positive constants with K1 < K2 and q is a constant with q ≥ 0. In AppendixA, 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 2;1 the existence and uniqueness of Wp;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 contributions of the paper are threefold: (i) we prove the existence and 2;1 uniqueness of Wp;loc ((0;T ] × (0; +1)) \ C([0;T ] × (0; +1)) solution to the par- abolic variational inequality (1.1). (ii) We prove that two free boundaries (1.1) are monotone and C1-regular. (iii) We prove the existence and uniqueness of 2 Wp;loc ((0; +1)) solution to the stationary problem of (1.1) and utilize it to show that the free boundaries are bounded. The rest of this paper is organized as follows. In section2, we prove the exis- 2;1 tence and uniqueness of Wp;loc solution to problem (1.1). In section3, we show that the monotonicity and C1-regularity of two free boundaries based on the results in section2. Moreover, we will prove the starting points of the free boundaries. In section4, we conduct comparative static analysis of variational inequality (1.1). In section5, we solve the stationary problem arising from American strangle op- tion and use it to show that two free boundaries are bounded. In section6, we describe the numerical result applying the finite difference scheme. AppendixA is AMERICAN STRANGLE OPTION 757 the formulation of the model. AppendixB 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 s V (t; s) τ = T − t; x = ln ;Y (τ; x) = ; K2 K2 we have 8 @ Y − LY = 0; if Y > (ex − 1)+ + (κ − ex)+; (τ; x) 2 [0;T ) × ; > τ R <> x + x + @τ Y − LY ≥ 0; if Y = (e − 1) + (κ − e ) ; (τ; x) 2 [0;T ) × R; (2.1) > x + x + :> Y (0; x) = (e − 1) + (κ − e ) ; x 2 R; where K κ := 1 2 (0; 1) K2 and σ2 σ2 LY := @ Y + r − q − @ Y − rY: (2.2) 2 xx 2 x We now consider the problem in the bounded domain [0;T ) × (−n; n): x + x + 8 @τ Yn − LYn = 0; if Yn > (e − 1) + (κ − e ) ; (τ; x) 2 [0;T ) × (−n; n); > > x + x + <> @τ Yn − LYn ≥ 0; if Yn = (e − 1) + (κ − e ) ; (τ; x) 2 [0;T ) × (−n; n); −n n (2.3) @xYn(τ; −n) = −e ;@xYn(τ; n) = e ; τ 2 [0;T ]; > > x + x + :> Yn(0; x) = (e − 1) + (κ − e ) ; x 2 [−n; n]; 2 where n 2 with n > ln . N 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. 2 Lemma 2.1. For each fixed n 2 with n > ln , there exists a unique solution N κ 2;1 Yn 2 C ([0;T ] × [−n; n]) \ Wp (([0;T ) × (−n; n)) n (Bρ(0; 0) [ Bρ(0; ln κ))) to the 2 2 problem (2.3), where 1 < p < 1, ρ > 0 and Bρ(0; x0) = (τ; x): τ + (x − x0) ≤ 2 ρ . Furthermore, if n 2 N is large enough, then we have x + x + x (e − 1) + (κ − e ) ≤ Yn ≤ e + κ, (2.4) x x @τ Yn ≥ 0; −e ≤ @xYn ≤ e : (2.5) 1 Proof. We first define a penalty function β" 2 C (R) (0 < " < 1) satisfying 8 0 00 β"(t) ≤ 0; β"(t) ≥ 0; β" (t) ≤ 0; 8t 2 R; > <> n β"(t) = 0 if t ≥ "; β"(0) = −C0 for C0 = 3(r + q)e + 3r; (2.6) > lim β"(t) = 0 if t > 0; lim β"(t) = −∞ if t < 0: :> "!0 "!0 758 JUNKEE JEON AND JEHAN OH Since the functions (ex − 1)+ and (κ − ex)+ are not smooth enough, we also define 1 a function '" 2 C (R) satisfying 8 0 00 '"(t) ≥ 0; 0 ≤ '"(t) ≤ 1;'" (t) ≥ 0; 8t 2 R; > <> '"(t) = t if t ≥ "; '"(t) = 0 if t ≤ −"; (2.7) > + > lim '"(t) = t ; 8t 2 R: :> "!0 We then consider the following approximation of the problem (2.3): x x 8 @τ Yn;" − LYn;" + β" (Yn;" − '"(e − 1) − '"(κ − e )) = 0; > > <> (τ; x) 2 [0;T ) × (−n; n); −n n (2.8) @xYn;"(τ; −n) = −e ;@xYn;"(τ; n) = e ; τ 2 [0;T ]; > > x x :> Yn;"(0; x) = '"(e − 1) + '"(κ − e ); x 2 [−n; n]: 2;1 By Schauder's fixed point theorem, the above problem (2.8) has a unique Wp solution, see [14]. We next claim that x x x '"(e − 1) + '"(κ − e ) ≤ Yn;" ≤ e + κ (2.9) for sufficiently large n. Observe from (2.7) that + + t ≤ '"(t) ≤ (t + ") ; 8t 2 R: (2.10) Then we deduce from (2.7) and (2.10) that x x x x @τ ['"(e − 1) + '"(κ − e )] − L ['"(e − 1) + '"(κ − e )] x x x x + β" ('"(e − 1) + '"(κ − e ) − '"(e − 1) − '"(κ − e )) x x = −L ['"(e − 1) + '"(κ − e )] + β"(0) σ2 = − ['00(ex − 1) + '00(κ − ex)] e2x + (q − r)['0 (ex − 1) − '0 (κ − ex)] ex 2 " " " " x x + r ['"(e − 1) + '"(κ − e )] − C0 x x + x + ≤ 2(q + r)e + r (e − 1 + ") + (κ − e + ") − C0 x x ≤ 2(q + r)e + r [(e + ") + (κ + ")] − C0 x n ≤ (2q + 3r)e + 3r − C0 ≤ 3(q + r)e + 3r − C0 = 0: κ Furthermore, we see from the boundary conditions in (2.8) that if 0 < " < , 2 8 ' (ex − 1) + ' (κ − ex) = Y (τ; x); τ = 0; > " " n;" <> x x −n @x ['"(e − 1) + '"(κ − e )] = −e = @xYn;"(τ; x); x = −n; > x x n :> @x ['"(e − 1) + '"(κ − e )] = e = @xYn;"(τ; x); x = n: By the comparison principle, we get x x '"(e − 1) + '"(κ − e ) ≤ Yn;": (2.11) AMERICAN STRANGLE OPTION 759 κ 1 On the other hand, it follows from (2.6) and (2.7) that if 0 < " < < , 2 2 x x x x x @τ [e + κ] − L [e + κ] + β" (e + κ − '"(e − 1) − '"(κ − e )) x x x x = −L [e + κ] + β" (e + κ − '"(e − 1) − '"(κ − e )) x x x x = qe + rκ + β" (e − '"(e − 1) + κ − '"(κ − e )) x x ≥ qe + rκ + β"(1 − ") = qe + rκ ≥ 0: Also, from the boundary conditions in (2.8), we get 8 ex + κ ≥ ' (ex − 1) + ' (κ − ex) = Y (τ; x); τ = 0; > " " n;" <> x −n −n @x [e + κ] = e ≥ −e = @xYn;"(τ; x); x = −n; > x n :> @x [e + κ] = e = @xYn;"(τ; x); x = n: By the comparison principle, we obtain x e + κ ≥ Yn;": (2.12) Therefore, the claim (2.9) follows from (2.11) and (2.12).