Computation for Pricing Shout Option

Computation for Pricing Shout Option Irma Palupi Graduate School of Natural Sciences and Technology Kanazawa University, Japan Faculty of Mathematics and Natural Sciences Institut Teknologi Bandung, Indonesia [email protected] [email protected] Abstract Shout option is a modified contract of European type option. This contract can only be exercised at the expiration date, and holder has a right to reset the exercise price into the current level of underlying asset in along time to expiry date. This work try to price the option by solving a linear complementary problem, and consider to use Discrete Morse Flow (DMF) to approximate the solution. DMF is a minimizing scheme to solve differential equation depend on time, and concerning space variable. It solves differential equation in divergence form, and it can ensure gradient continuity of solution. Keywords : Shout option, Black-Scholes, Linear Complementary Problem, DMF 1 INTRODUCTION In the financial industry, option can be used as security for reducing the risk of changing underlying asset price, and also for speculating and arbitrage. The purpose for Speculating and arbitrage usually dominate in the market, and this purpose make market become more dynamic. There are many types of option in financial market with their unique features. For example, European option, American option, Bermudan option, Shout option, Asian option, etc. The names are usually coming from their unique features, for example the reason why people name European option and American option. All of options type are also distinguished into two kind of option, based on contents of the contract. It is called Call option if contains the right to buy asset, and Put option if contains the right to sell asset. In this work, we will formulate Call option type to valued. It will not take so much difference for Put option type. Since option price is depend on underlying asset, the price will be influenced by some parameters that also influence the underlying asset. These parameters are interest rate, dividend rate, and volatility. Actually, they can never be constant in the real market, because they always move for every times. However, this work will assume those parameters are constant for simpler model. Shout option is one of modified European style option, which is the right to buy or to sell some underlying asset for certain price at the specified time. Shout option contains special feature such that make the value is higher than normal European option. Holder is able to make such an interruption in along time to expiration date by resetting the exercise price into the current level of underlying asset. Resetting an exercise price aim to lock the return of option at expiry date. Thus, although the new exercise price implies ’out-of-the-money’ for option, holder is safe with the minimal amount of return. 1 This work applies a numerical PDE approach for pricing shout option. Discrete Morse Flow (DMF) method is used to approach the solution price of option. This method ensure the gradient continuity of solution against to asset variable, which is important to obtain the smooth solution in pricing. DMF method for solving the pricing problem involve minimization problem, and in this work, minimization problem will be solved numerically by using Steepest Descent combined with Bisection method. DMF method for solving the pricing problem involve minimization problem which is explained on [2] and [3] for pricing American option. The method used in the computation need to use a finite computational domain, although the actual pricing involve the infinite domain. For the boundary condition issue, it will be used artificial boundary condition related with truncation of solution domain. The purpose of this thesis is to arrange robust method for pricing shout option, although it is still under the assumption of constant parameters. Robust method is required for pricing such this option, because numerical schemes are using in financial industry still do not work well. Explicit method which work well for American option can not be used for this option, because this method could not give the smooth solution, even though free boundary problem appears as with on American option. And even the Monte Carlo simulation that commonly used for pricing many kind of options, could not handle well this option in the optimization frame. 2 FORMULATION FOR PRICING SHOUT CALL OPTION Shout Option is European option type which holder is given a right to ’shout’ at anytime during the options life time. ’Shout’ means holder reset the exercise price of the option into the current level of asset, when the option is in-the-money 1 . There is an amount of money that must be paid when holder is resetting the exercise price. It is usually defined as the difference between the old and the new exercise price, anyhow these amount can be defined as another value according to the consideration of options rule. Shouting has purpose to lock the gain of option in certain level. For example shout call Option, if holder is given one chance to shout during time to maturity, and he reset the exercise price at tf , then after time (T − tf ) holder will get payoff as follow; { S(tf ) − K + max{S(T ) − Stf , 0} if shouting occur at tf (1) P (S, T ) = max{S(T ) − K, 0} if no shouting occur. It is obvious that the value of Shout option must be higher than Eroupean option, because if shouting occur during time to maturity, then holder will get a nonzero minimal return at the terminal time, eventhough the new exercise price implies ’outthe-money’. Figure (1) describes how shout option lock the return at terminal time, it is clear that the Shout option value can not be less than European option. 1 In the money is the condition when the payoff against to current underlying asset value give positive value. At the money is the condition when the payoff against to current underlying asset value give zero. Out the money is the condition when the payoff against to current underlying asset value give negative value. 2 Graph of Payoff function Shout Call and European Call option 10 european option shout call option 8 option value 6 4 2 0 0 5 10 S(t) 15 20 Figure 1: Graph of shout call Option and European call option in the terminal time (payoff) with K=10. On the graph, shout call option is reset by new exercise price K=12. So that, after shouting holder locked the options payoff at 2. In pricing, the Black-Scholes PDE will be used as the consideration of shout option is included into the European type, even though with additional condition that must be satisfied. Free boundary problem accurs as with American Options. The free boundary curve will devide the domain of asset into two areas, The area where holder is not allowed to reset the exercise price, because he would not get the optimal return, and the area where holder can reset the option’s exercise price optimally. Sf (t) is the boundary where the exercise price can be reset optimally. This value should be greater than current exercise price. If the current underlying asset have not touched the resetting boundary Sf (t), the exercise price is not reset optimally, even though the current underlying asset S(t) is greater than exercise price. 2.1 Mathematical Model To value the Shout call option, consider the Black-Scholes Formula for European Call option value at time t. CBS (S, t) = Se−q(T −t) N (d1 ) − Ke−r(T −t) N (d2 ) where, S ln K + (r − q + 12 σ 2 )(T − t) √ σ T −t √ d2 = d1 − σ T − t ∫ x ξ2 1 e− 2 dξ N (x) = √ 2π −∞ d1 = (2) (3) (4) (5) Equation (2) can be represented as solution of Black-Scholes PDE which is European Call option value. Here, CBS (S, t) is the European Call Option price follows the BlackScholes’s formula, S is stock price, the positive constants σ, q, r, K, and T respectively 3 denote volatility, dividend rate, interest rate, exercise price, and time to expiration date. Consider the option price which following the Black-Scholes equation, when the underlying asset is in the same level with exercise price (S = K). The call option value is determined as; CBS (S, t) = S[e−q(T −t) N (d1 ) − e−r(T −t) N (d2 )] (6) Equation (6) can be defined as a function of Call option value after holder reset the exercise price into the current asset value, but we need a modification for pricing the shout option. Consider the shout call option which has one chance to reset its exercise price during time to maturity. If holder decides to shout the option, it is clear that holder must reset the exercise price optimally, and the contract should be ’in-themoney’ at that time. Resetting the exercise price intend to lock the option’s return on the exipiry date, therefore there must be a simultaneous payment to increase the price, hence we define as the difference between the old and the new exercise price. Let denote Csh (S, t) as shout call option value, and PˆC (S, t) as function of shout option’s value after shouting. PˆC (S, t) = S[e−q(T −t) N (d1 ) − e−r(T −t) N (d2 )] + R(S, K) (7) In pricing, writer must consider about the possibility if holder prefer to reset the exercise price rather than keep it. If writer issue the option such a shout option, the option’s value can not be less than the value of option after shouting. Therefore, with Csh (S, t) ≥ PˆC (S, t) = S[e−q(T −t) N (dˆ1 ) − e−r(T −t) N (dˆ2 )] + (S − K) (8) ⇒ Csh (S, t) − PˆC (S, t) ≥ 0 (9) (r − q + 12 σ 2 )(T − t) √ dˆ1 = σ T −t (10) (r − q − 21 σ 2 )(T − t) √ . dˆ2 = σ T −t (11) The constraint above must be satisfied for pricing shout call Option Csh (S, t) in the domain (0, ∞) × [0, T ) with gradient continuity. Here, R(S, K) = S − K is a simultaneous payment that must be paid when shouting. 2.2 Shouting Curve and Free Boundary Problem Shout option can never has a value that is smaller than the function of option after shouting occur. If Csh (S, t) < PˆC (S, t) is happened, the Shout option’s pricing is becoming worthless, because holder does not want to keep it. Free boundary problem appears in Shout option’s pricing as an American Option. The free boundary that is unknown explicitly divides the area where option is optimally can be reset and the area where keeping the option is better than shouting. Shout Option has at least the value as written on (8), which is a lower bound that move depend on time. For all time 4 t ∈ (0, T ], there should be a contact point Sf that devide domain S. Let’s define Sf (t) as follow; { Csh (S, t) > PˆC (S, t) for S < Sf (t) option can not be reset (12) Csh (S, t) = PˆC (S, t) for S ≥ Sf (t) option can be reset. Therefore, Sf (t) is a curve that separates the area of S into two areas. The solution Csh (S, t) must hold a gradient continuity, and the contact point Sf varies in time. The contact point Sf is always greater than exercise price K, and there is a possibility for value S, such that K < S < Sf . If the underlying asset price implies K < S < Sf , then holder still can not reset the exercise price optimally, even tough S > K. It will make the option price down rather than if holder keep the old exercise price. Holder can reset exercise price of shout call option, only if the underlying asset is satisfied S ≥ Sf for all t ∈ [0, T ). According to the Black-Scholes PDE, when S > Sf the shout call option Csh (S, t) equals to the solution of PDE Black-Scholes, and in this situation Csh (S, t) would be greater than function P̂C (S, t). Meanwhile in the other side when S ≤ Sf , the shout call option Csh (S, t) equals to the function P̂C (S, t). Under the assumption r > q and the fact that σ is a positive parameter, the inequality of Black-Scholes PDE appears as follow (see: [4]); ∂Csh σ 2 2 ∂ 2 Csh ∂Csh − S − (r − q)S + rCsh ≥ 0 (13) 2 ∂t 2 ∂S ∂S The inequalities (12) and (13) must be hold for all (S, t) ∈ [0, ∞) × [0, T ). If the strict inequality ”>” is hold by Csh (S, t) according to equation (12), then the equality must be hold for (13), and vice versa. The shouting curve which is the boundary that divides a definition area of Csh (S, t) into two areas as explained before, each of them appropriate the convention as follow: { Csh (S, t) = PˆC (S, t) for S ≥ Sf (shouting) Csh (S, t)is a solution of Black-Scholes PDE for S < Sf (keeping the old exercise price). (14) According to (12) and the location of Sf that is unknown priory, then Csh (S, t) is called a solution of free boundary problem. In order to find an unknown contact point sh ˆ Sf , we should consider about the slope ∂C ∂S when the solution Csh (S, t) touch PC (S, t) − ˆ (S,t) (S,t) (Sf , t) = PC∂S (Sf , t), which at contact point Sf . It should be satisfied that ∂Csh ∂S is a gradient continuity. Therefore, through the linear complementary problem we can define our problem into the simpler as following;  ∂Csh ∂Csh σ 2 2 ∂ 2 Csh ˆ  min{− ∂t − 2 S ∂S 2 − (r − q)S ∂S + rCsh , Csh (S, t) − PC (S, t)} = 0 in (0, T ) × (0, ∞) Csh (S, T ) = max{S − K, 0} for S ∈ [0, +∞)   Csh (0, t) = 0 for t ∈ (0, T ) (15) 3 APPROXIMATION SCHEME AND NUMERICAL METHOD We have a problem as written at (15). Reformulate the problem into the parabolic type of partial differential equation to reduce complexity in computation. By using the 5 transformation as following;    τ =T −t S x = log K   Csh (S, t) = Keαx u(x, τ ) (16) and by choosing α = 12 + q−r , and β = − 12 σ 2 α2 − αr + αq + 21 σ 2 α + r, the problem σ2 turns into the parabolic type as follow;  σ2 ∂ 2 u ∂u  min{ ∂τ − 2 ∂x2 + βu, u − ϕ} = 0 in (0, T ) × R (17) u(x, 0) = e−αx max{ex − 1, 0} for x ∈ R   limx→−∞ u(x, τ ) = 0 for τ ∈ (0, T ) −αx where, ϕ(x, τ ) = e K PˆC (S, t). Then, we can make restriction for domain of x on (17). This restriction is reasonable, because for each τ , limx→−∞ u(x, τ ) = 0. By taking a large M0 > 0 such that S (t) free boundary {xf = log fK }0<t<T ⊂ (0, M0 ), we consider the following problem;  ∂u σ2 ∂ 2 u  min{ ∂τ − 2 ∂x2 + βu, u − ϕ} = 0 u(x, 0) = e−αx max{ex − 1, 0}   u(−M0 , τ ) = 0, and u(M0 , τ ) = ϕ(M0 , τ ) in (0, T ) × Ω := (−M0 , M0 ) for x ∈ Ω̄ (18) On the selection value of M0 , we should be sure that {xf (τ )}0<t<T ⊂ (−M0 , M0 ). To solve the problem (18), defined the energy function which is initiated from the discrete Morse flow. Let h > 0 be a time step. The energy function is defined as; { ∫ 2 2 Jm (u) := 21 Ω { |u−uhm−1 | + σ2 (ux )2 + βu2 } dx for u ∈ κ, m = 1, 2, ... (19) κ := {v ∈ H 1 (Ω), v ≥ ϕ(τ ) in Ω}. We will obtain the sequence {um }∞ m=1 in κ, by putting u0 = u(x, 0) as a initial data in κ, and then define um as a minimizer of Jm (u) in κ. In this work, minimization problem of functional Jm (u) through DMF formulation is solved by using Steepest Descent method combined with Golden Search method. These simple minimization method still work well for this option’s pricing, because the problem involve only a simple domain. Firstly, choose the big number M correlated T becomes small enough. The algorithm to solve the numerical with T , such that h = M problem is described as following; 1. Define u0 ∈ κ as an initial value 2. For m = 1, 2, ..., M , determine um using the following prosedures (a). v 1 = um (b). For k = 1, 2, ..., Nm (i). Finding gradient of functional Jm (v k ), ∇Jm (v k ) as a direction to search a minimizer. (ii). By combining Stepest descent and Golden Search method, find a minimizer ṽ k along the direction −∇Jm (v k ). 6 (iii). Projecting a minimizer from step (ii) onto κ :vk+1 = P roj(ṽ k ) (c). um+1 = v Nm +1 By Using step (a) until (c), we can get approximation minimizer of functional Jm , i.e um for m = 1, 2, ..., M . 4 NUMERICAL RESULTS AND DISCUSSION Numerical scheme via DMF method which is based on Black-Scholes model can be alternative for pricing Shout option. Since this computation works through PDE model, the result is quite robust, even though it is still under the assumption of constant parameter. Robust method is required for pricing this option, because numerical schemes are using in financial industry still do not work well. Graph of Shout Option Value 5 terminal condition t=0.75 t=0.5 t=0.25 t=0.0 r=0.03 ; q=0.02 ; volatility=0.2 ; K=10 ; T=1. Shout Call Price 4 3 2 1 0 0 2 4 6 8 10 12 14 S(t) Figure 2: Graph of shout call Option Value, by using parameters: r=0.03, q=0.02, σ =0.2, K=10 Figure (2) shows the value of Shout call option Csh (S, t) with time to expiry T = 1 (unit time). Those figure shows the option’s value for t = 0.75, t = 0.5, t = 0.25 and t = 0 (when option is issued). We can see that shout call option’s pricing works well when holder is optimally allowed to reset the exercise price which is lock the return in the certain level. Locking the return is important to avoid uncertainty of underlying asset’s price at the expiry date. If holder does not have any chance to reset the option, then shout option’s value will be a normal European call option. Through the figure (3) which is a free boundary curve of shout call option relate with (2), holder obtain a reference along the option’s life time. Holder can estimate if the option needs to shout, cause it is in the optimal shouting area, otherwise holder needs to keep the exercise price. Since Shout Call option is a modified European option which is added a feature to lock its gain, then the price must be greater than European option. The calculation through this work consider about that, and it is showed in figure (4). Along the time to expiry, the price of Shout option never be less than European option. 7 free boundary of shout call option 11.8 "free.dat" u 1:2 11.6 r=0.03 ; q=0.02 ; volatility=0.2 ; K=10 ; T=1. 11.4 11.2 S(t) 11 10.8 10.6 10.4 10.2 10 0 0.2 0.4 0.6 0.8 1 1.2 t Figure 3: Free boundary shout call Option, by using parameters: r=0.03, q=0.02, σ =0.2, K=10 t=0.0 t=0.25 7 7 european shout european shout 6 5 Option value Option value 6 4 3 2 1 5 4 3 2 1 0 0 0 2 4 6 8 S(t) 10 12 14 0 2 4 6 t=0.50 10 12 14 12 14 t=0.75 6 6 european shout european shout 5 Option value 5 Option value 8 S(t) 4 3 2 1 4 3 2 1 0 0 0 2 4 6 8 10 12 14 0 S(t) 2 4 6 8 10 S(t) Figure 4: Comparation shout call and Eropean Call at t=2.5, by using parameters: r=0.03, q=0.02, σ =0.2, K=10 Figure (5) and (6) show a value of Shout call option with various initial exercise price K, and its curves of free boundary respectively. The fact that for the call option type that has a smaller value of exercise price must be more expensive than the greater one shown in the figure. And its free boundary curve is interesting to be analyzed further as in the American option’s phenomena. Figure (7) show the value of Shout call option at the issuing date with various time to expiry T . The shout option which has a longer life time is more expensive than a shorter life times. And figure(8) is a free boundary curve with various life time T , related with the shout option in figure (7). 8 t=0.0 t=0.25 35 35 K=10 K=15 K=20 K=30 25 K=10 K=15 K=20 K=30 30 Option value Option value 30 20 15 10 5 25 20 15 10 5 0 0 0 5 10 15 20 S(t) 25 30 35 40 0 5 10 15 t=0.50 25 30 35 40 35 40 t=0.75 35 35 K=10 K=15 K=20 K=30 25 K=10 K=15 K=20 K=30 30 Option value 30 Option value 20 S(t) 20 15 10 5 25 20 15 10 5 0 0 0 5 10 15 20 25 30 35 40 0 5 10 15 S(t) 20 25 30 S(t) Figure 5: Graph of shout call Option with some exercise price K, by using parameters: r=0.03, q=0.02, σ =0.2, K=10 K=15 17.5 free boundary curve free boundary curve 17 S(t) S(t) K=10 11.8 11.6 11.4 11.2 11 10.8 10.6 10.4 10.2 10 9.8 16.5 16 15.5 15 0 0.2 0.4 0.6 t 0.8 1 1.2 0 0.2 0.4 K=20 free boundary curve 22 S(t) S(t) 22.5 21.5 21 20.5 20 0 0.2 0.4 0.6 0.8 1 1.2 1 1.2 K=30 23.5 23 0.6 t 0.8 1 1.2 35 34.5 34 33.5 33 32.5 32 31.5 31 30.5 30 free boundary curve 0 t 0.2 0.4 0.6 0.8 t Figure 6: Graph of Free Boundary with some exercise price K, by using parameters: r=0.03, q=0.02, σ =0.2, K=10 9 Graph Shout Call option value at t=0.0 with various T 8 T=1 T=2 T=4 T=5 7 6 Option value 5 4 3 2 1 0 0 2 4 6 8 10 12 14 S(t) Figure 7: Graph of shout call Option with some time to expiration T, by using parameters: r=0.03, q=0.02, σ =0.2, K=10 T=2 10.6 "freeT-1.dat" u 1:2 "freeT-2.dat" u 1:2 10.5 10.4 S(t) S(t) T=1 11.8 11.6 11.4 11.2 11 10.8 10.6 10.4 10.2 10 10.3 10.2 10.1 10 9.9 0 0.2 0.4 0.6 t 0.8 1 1.2 0 0.5 1 2 T=5 12.5 "freeT-4.dat" u 1:2 "freeT-5.dat" u 1:2 12 11.5 S(t) S(t) T=4 11 10.9 10.8 10.7 10.6 10.5 10.4 10.3 10.2 10.1 10 9.9 1.5 t 11 10.5 10 9.5 0 0.5 1 1.5 2 2.5 3 3.5 4 0 t 1 2 3 4 5 t Figure 8: Graph of Free Boundary with some time to expiration T, by using parameters: r=0.03, q=0.02, σ =0.2, K=10 10 5 CONCLUSION In this work we have applied Black-Scholes model which is usually used for pricing European option to calculate Shout option’s value. This model can be alternative to make a robust calculation for Shout option. Furthermore, the calculation can also be modified by considering non-constant related parameter to make the more realistic situation as in market. Shout option is more expensive than normal European option because of the feature to reset the exercise price. Free boundary problem appears related with a contact point which is divide an optimal shouting area, and the area where shouting is not allowed. And the obstacle which is function PˆC (S, t) can be modified into another form that more realistic, especially for the simultaneous payment, but however the differences between old and new expiration price quite nice and simple to use. DMF method to solve the linear complementary problem that occurs to handle the model for Shout option is solved numerically by using minimization strategy which apply simple minimization method, that is Steepest descent method combined with Golden Search method. Those simple minimization method can work good enough for this problem, because the computation involve only the simple domain, under the assumption that the convexity is satisfied. The gradient continuity for free boundary is very required, because the fact that the decision for shouting or not is depend on the underlying asset’s condition. The free boundary is also required to be analyzed further, it is interesting to consider a phenomena such as the American option. The computation scheme on this work is also enable to modified for manage the computation scheme for Multi Shout option. For Put option type the calculation will not being so much different with Call option type, it requires only to consider the related characteristic. Acknowledgements The author would like to gratefully acknowledge the advices, guidances, and supervision of Prof. Seiro Omata during this work, and also thank to the members of Omata Lab for their suggestions, guidances, and supports. Besides, The author would also like to convey thanks to JASSO for financial support. Reference [1] Min Dai and Yue Kuen KMin Dai, Yue Kuen Kwokwok, Optimal Multiple Stopping Models Of Reload Options And Shout Options, Journal of Economic Dynamics and Control. Vol.32, No.7. pp.2269 - 2290, 2008. [2] S. Omata, A numerical method based on the discrete morse flow related to parabolic and hyperbolic equation, Nonlinear Analysis, Theory, Methods and Applications. Vol.30, No.4. pp.2181-2187, 1997. [3] S.Omata, H.Iwasaki, K.Nakane, Xiaohua Xiong, and M.Sakuma, A Numerical Computation to the American Option Pricing via the Discrete Morse Flow, Kanazawa University and Osaka Institute of Technology, 2003. [4] R.U. Seydel., 2009, Tools of Computational Finance Fourth Edition, Springer, German, pp. 164-166. 11