Minhyun Yoo, Darae Jeong, Seungsuk Seo, and Junseok Kim

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1 Hoam Mathematical J. 37 (15), No. 4, pp A COMPARISON STUDY OF EXPLICIT AND IMPLICIT NUMERICAL METHODS FOR THE EQUITY-LINKED SECURITIES Mihyu Yoo, Darae Jeog, Seugsuk Seo, ad Juseok Kim Abstract. I this paper, we perform a compariso study of explicit ad implicit umerical methods for the equity-liked securities (ELS). The optio prices of the two-asset ELS are typically computed usig a implicit fiite differece method because a explicit fiite differece scheme has a restrictio for time steps. Nowadays, the three-asset ELS is gettig popularity i the real world fiacial market. I practical applicatios of the fiite differece methods i computatioal fiace, we typically use relatively large space steps ad small time steps. Therefore, we ca use a accurate ad efficiet explicit fiite differece method because the implemetatio is simple ad the computatio is fast. The computatioal results demostrate that if we use a large space step, the the explicit scheme is better tha the implicit oe. O the other had, if the space step size is small, the the implicit scheme is more efficiet tha the explicit oe. 1. Itroductio Equity-liked securities (ELS) are auto-callable optios whose retur o ivestmet is depedet upo the path of the uderlyig equities liked to the securities. ELS ca be made from a few umber of stocks or stock idexes such as the KOSPI i Korea. ELS is a derivative product i the market. ELS guaratees a debt ad is similar to a barrier optio. ELS comprises a large portio of exchage volume i Korea fiacial market. A distiguishig feature of ELS is the automatic earlyredemptio coditio before its maturity. Geerally, i order to get price Received April 15, 15. Accepted October 6, Mathematics Subject Classificatio. 91G6, 65M6. Key words ad phrases. Black Scholes partial differetial equatio, log trasformatio, explicit fiite differece method, equity-liked securities, o-uiform grid. *Correspodig author. Tel.: ; fax:

2 44 Mihyu Yoo, Darae Jeog, Seugsuk Seo, ad Juseok Kim of ELS, Mote Carlo simulatio (MCS) ad fiite differece method (FDM) are used. Typically, we use the implicit scheme with operator split method (OSM) or alteratig directio implicit (ADI) because they are stable. Although the implicit scheme with OSM has a advatage i stability, it is costly to solve the tridiagoal matrix implicitly. I particular, we eed to solve the system three times i each time step for the three-asset problems. I practice, we calculate oe time step with oe day, which is about 1/365 ad discretize the asset by oe uit. Therefore, the time step is small eough ad space step is large eough. Furthermore, if we wat to calculate the Greeks of the optio price, especially theta which is the rate of chage of the optio value with respect to chages i the time to maturity, the we have to use much smaller time step. For these cosideratios, it is better to use a explicit scheme with smaller time step, which gives much accurate solutios. The program implemetatio is simple ad fast. The mai purpose of this paper is to develop a explicit scheme o a o-uiform grid to solve value of ELS. The paper is orgaized as follows. We itroduce Black Scholes model i Sectio. I Sectio 3, we preset three-asset step-dow ELS. Numerical methods are preseted i Sectio 4. I Sectio 5, we perform umerical experimets. I Sectio 6, we take coclusio for this paper.. Black Scholes model To evaluate value of the ELS optio, we cosider the stadard Black Scholes model [1], which ca be writte as u(x, t) t = 1 d i,j=1 for (x, t) R d + [, T ]. ρ ij σ i σ j x i x j u(x, t) x i x j r d i=1 x i u(x, t) x i + ru(x, t), Here, u(x, t) is the value of the optio, where x = (x 1, x,..., x d ), d is the total umber of uderlyig assets, x i is the value of the i-th uderlyig assets, ad t is the time. Also, r represets the riskless iterest rate, σ i is the volatility of i-th the uderlyig assets, ρ ij is the correlatio coefficiet betwee x i ad x j, ad T is the maturity time of the optio. Switchig to the ew coordiate X = log x [4] ad usig the

3 A compariso study of umerical method for the equity-liked securities 443 trasformatio τ = T t, the stadard BS equatio ca be rewritte as (1) U(X, τ) τ = d i=1 (r σ i ru(x, τ), ) U(X, τ) X i + 1 d i,j=1 ρ ij σ i σ j U(X, τ) X i X j where U(X, τ) is the value of the optio ad X = (X 1, X,..., X d ). 3. Iitial coditio I this sectio, we briefly describe the cocept of step-dow ELS optio ad we itroduce a three-asset step-dow ELS optio as a example Step-dow ELS The payoff of ELS is determied by early redemptio or fial maturity redemptio. I oe-asset step-dow ELS, if a uderlyig asset is i the predetermied exercise price at certai maturity dates accordig to cotract, the ELS gives desigated retur ad is extermiated. However, if the uderlyig asset is ot i a certai price, the cotract is ot extermiated ad will be cotiued util ext maturity. If the cotract cotiues at fial maturity ad the uderlyig asset is ot i fial exercise barrier, the payoff is determied whether the cotract hit kock i barrier. If the uderlyig asset did ot hit the kock i barrier, the ELS gives predetermied retur, dummy. Otherwise, the ELS will make a loss i face value. The step-dow meas that the desigated strike price decreases. I this paper, we cosider a three-asset step-dow ELS which is similar to a oe asset step-dow ELS, as we explaied above. The differece with a oe asset ELS is that the base price is referred from the miimum of three uderlyig assets at certai maturity dates. The payoff structure of three-asset step-dow ELS is as follows [3]: Early redemptio occurs, ad the cotract is extermiated with predetermied retur if the value of the worst performer, which meas the miimum value of uderlyig assets is greater tha or equal to a give exercise price o a give date. If the early redemptio do ot occur util the fial maturity, the retur depeds upo whether Kock-I occurs or ot.

4 444 Mihyu Yoo, Darae Jeog, Seugsuk Seo, ad Juseok Kim 3.. Example for three-asset step-dow ELS To help readers uderstadig of the step-dow ELS optio, we iclude the example for three-asset step-dow ELS. We cosider the parameters as the referece price E = 1, the riskless iterest rate r =.3, the volatilities of the uderlyig assets σ 1 = σ = σ 3 =.3, the correlatios of uderlyig assets ρ 1 = ρ 13 = ρ 3 =.5, the face value F = 1, the Kock-I barrier level KIb =.65E, the dummy rate d =.3, ad the fial maturity time T = 1. At the early redemptio, observatio date (τ i ), exercise price (K i ), ad retur rate (c i ) are described i Table 1. Observatio date (τ i ) Exercise price (K i ) Retur Rate (c i ) τ 6 = T K 6 =.85E c 6 =.3 τ 5 = 5T/6 K 5 =.85E c 5 =.5 τ 4 = 4T/6 K 4 =.9E c 4 =. τ 3 = 3T/6 K 3 =.9E c 3 =.15 τ = T/6 K =.95E c =.1 τ 1 = T/6 K 1 =.95E c 1 =.5 Table 1. Observatio date (τ i ), exercise price (K i ), ad retur rate (c i ) used i example for three-asset step-dow ELS. Figure 1 illustrates the payoff of early obligatory redemptio before fial maturity. Payoff at early redemptio U(X 1,X,X 3,τ i ) V(X 1,X,X 3,τ i ) (1+c 5 )F (1+c 4 )F (1+c 3 )F (1+c )F (1+c 1 )F Observatio date at τ = τ 5 at τ = τ 4 at τ = τ 3 at τ = τ at τ = τ 1.85E.9E.95E S = mi(x 1,X,X 3 ) Figure 1. Payoff at early redemptio before fial maturity.

5 A compariso study of umerical method for the equity-liked securities 445 Alog with the characteristics of the step-dow ELS, we must take two cosideratios of the payoffs at maturity. Accordig to whether or ot the value of ELS hits the Kock-I barrier (KIb) durig the cotract, we defie two values V (X, τ) ad U(X, τ). The, the iitial coditios of U ad V are set to () ad U(X 1, X, X 3, ) = (1 + c 6 )F if S K 6 (1 + d)f if KIb < S < K 6 S F/E otherwise, (3) { (1 + c6 )F if S V (X 1, X, X 3, ) = K 6 S F/E otherwise, where S = mi(x 1, X, X 3 ) which is the value of the worst performer. I Fig., we ca see the correspodig payoffs of U ad V whe c 6 = d. U(X 1,X,X 3,) (1+d)F V(X 1,X,X 3,) (1+c 6 )F Payoff at maturity.65f Payoff at maturity.85f.65f KIb (a).85e S KIb (b).85e S Figure. Payoffs of (a) U ad (b) V at fial maturity. 4. Numerical solutio I this sectio, we describe the umerical discretizatio of Eq. (1) usig explicit scheme o the computatioal domai Ω = [1, S max ] 3, which is log-trasformed by the three-dimesioal fiite domai Ω = [e, e S max ]. Also, to prevet the spurious oscillatory solutio by explicit scheme, we derive the coditio for time step size.

6 446 Mihyu Yoo, Darae Jeog, Seugsuk Seo, ad Juseok Kim 4.1. Discretizatio of log-trasformed BS equatio Let X, Y, ad Z deote the log-trasform of x-, y-, ad z-variables, respectively. Now, we discretize the log-trasformed computatioal domai Ω = [1, S max ] 3 with o-uiform spatial step size h i 1 = X i X i 1 = Y i Y i 1 = Z i Z i 1 (see Fig. 3) ad temporal step size τ = T/N τ. Here, X = 1, X Nx = S max, where N x ad N τ are the umbers of grid poits i the X- ad τ-directios, respectively. 1 h i 1 h i X X 1 X i 1 X i X i+1 S max X Nx X Figure 3. Nouiform mesh o log-trasformed grid X. We deote the umerical solutio by U U(X i, Y j, Z k, τ ) for i = 1,,, N x, j = 1,,, N y, k = 1,,, N z, ad =, 1,, N τ. Applyig the explicit fiite differece scheme to Eq. (1) gives U +1 U ( = σ 1 ) ( U τ X + σ U (4) Y ( ) ( ) + r σ 1 U ( ) ( ) + r σ U X Y ( +ρ 1 σ1σ ) ( U + ρ 13 σ X Y 1σ3 U X Z ( +ρ 3 σσ 3 ) U ru Y Z. ) + σ 3 ) + ( r σ 3 ( U Z ) ) ( U Z Here, spatial differeces o the o-uiform grid are defied by ( ) U h i = X h i 1 (h i 1 + h i ) U i 1,jk + h i h i 1 U h i 1 h i ( ) U X = h i 1 + h i (h i 1 + h i ) U i+1,jk, h i 1 (h i 1 + h i ) U i 1,jk h i 1 h i U + h i (h i 1 + h i ) U i+1,jk, )

7 A compariso study of umerical method for the equity-liked securities 447 ( ) U = U i+1,j+1,k + U i 1,j 1,k U i+1,j 1,k U i 1,j 1,k. X Y h i h j + h i h j 1 + h i 1 h j + h i 1 h j 1 Other spatial differeces ca be similarly defied. Equatio (4) is rewritte as follows: [ U +1 = U + τ σ1 h ic 1 (5) h i 1 (h i 1 + h i ) U i 1,jk + σ 1 + (h i h i 1 )C 1 h i 1 h i U + σ 1 + h i 1C 1 h i (h i 1 + h i ) U i+1,jk + σ h jc h j 1 (h j 1 + h j ) U ij 1,k + σ + (h j h j 1 )C h j 1 h j + σ + h j 1C h j (h j 1 + h j ) U ij+1,k + σ 3 h kc 3 h k 1 (h k 1 + h k ) U 1 U + σ 3 + (h k h k 1 )C 3 U h k 1 h + σ 3 + h k 1 C 3 k h k (h k 1 + h k ) U +1 ru ( U +ρ 1 σ1σ i+1,j+1,k + Ui 1,j 1,k U i+1,j 1,k U i 1,j+1,k ) h i h j + h i h j 1 + h i 1 h j + h i 1 h j 1 ( U +ρ 13 σ1σ 3 i+1,jk+1 + Ui 1,jk 1 U i+1,jk 1 U i 1,jk+1 ) h i h k + h i h k 1 + h i 1 h k + h i 1 h k 1 ( U +ρ 3 σσ 3 ij+1,k+1 + Uij 1,k 1 U ij+1,k 1 U ij 1,k+1 ) ], h j h k + h j h k 1 + h j 1 h k + h j 1 h k 1 where C 1 = r.5σ 1, C = r.5σ, ad C 3 = r.5σ Coditio for the o-oscillatory solutio Now, we derive the coditios uder which the explicit scheme for Eq. (4) will ot make spurious oscillatios by usig the idea i referece []. The, we rewrite Eq. (4) as (6) U +1 ( σ = τ 1 + (h i h i 1 )C 1 + σ + (h j h j 1 )C h i h i 1 h j h j 1 ) U + σ 3 + (h k h k 1 )C 3 h k h k 1 r + 1 τ + τ(σ 1 h ic 1 ) h i 1 (h i + h i 1 ) U i 1,jk + τ(σ 1 + h i 1C 1 ) h i (h i + h i 1 ) + τ(σ h jc ) h j 1 (h j + h j 1 ) U ij 1,k + τ(σ + h j 1C ) h j (h j + h j 1 ) U i+1,jk U ij+1,k

8 448 Mihyu Yoo, Darae Jeog, Seugsuk Seo, ad Juseok Kim + τ(σ 3 h kc 3 ) h k 1 (h k + h k 1 ) U 1 + τ(σ 3 + h k 1C 3 ) U+1 h k (h k + h k 1 ) ( U i+1,j+1k + Ui 1,j 1,k + τρ 1 σ 1 σ U i+1,j 1,k U ) i 1,j+1,k h i h j + h i h j 1 + h i 1 h j + h i 1 h j 1 ( U i+1,jk+1 + Ui 1,jk 1 + τρ 13 σ 1 σ U i+1,jk 1 U ) i 1,jk+1 3 h i h k + h i h k 1 + h i 1 h k + h i 1 h k 1 + τρ 3 σ σ 3 ( U ij+1,k+1 + U ij 1,k 1 U ij+1,k 1 U ij 1,k+1 h j h k + h j h k 1 + h j 1 h k + h j 1 h k 1 Next, we substitute U +1 = β +1 (1 r τ) ito Eq. (6), where the superscript for (1 r τ) represets a expoet. The, we obtai (7) β +1 = τ [ σ 1 + (h i h i 1 )C 1 + σ + (h j h j 1 )C (1 r τ)h i h i 1 (1 r τ)h j h j 1 + σ 3 + (h k h k 1 )C 3 (1 r τ)h k h k 1 r (1 r τ) + 1 τ(1 r τ) ] β τ(σ1 + h ic 1 ) (1 r τ)h i 1 (h i + h i 1 ) β i 1,jk + τ(σ1 + h i 1C 1 ) (1 r τ)h i (h i + h i 1 ) β i+1,jk τ(σ + h jc ) (1 r τ)h j 1 (h j + h j 1 ) β ij 1,k + τ(σ + h j 1C ) (1 r τ)h j (h j + h j 1 ) β ij+1,k τ(σ3 + h kc 3 ) (1 r τ)h k 1 (h k + h k 1 ) β 1 + τ(σ3 + h k 1C 3 ) (1 r τ)h k (h k + h k 1 β +1 + τρ ( 1σ 1 σ β i+1,j+1k + βi 1,j 1,k β i+1,j 1,k ) β i 1,j+1,k (1 r τ) h i h j + h i h j 1 + h i 1 h j + h i 1 h j 1 + τρ ( 13σ 1 σ 3 β i+1,jk+1 + βi 1,jk 1 β i+1,jk 1 ) β i 1,jk+1 (1 r τ) h i h k + h i h k 1 + h i 1 h k + h i 1 h k 1 + τρ ( 3σ σ 3 β ij+1,k+1 + βij 1,k 1 β ij+1,k 1 ) β ij 1,k+1. (1 r τ) h j h k + h j h k 1 + h j 1 h k + h j 1 h k 1 Sice all coefficiets of β i Eq. (7) are positive, the followig coditios should be satisfied. σ1 + (h i h i 1 )C 1 + σ + (h j h j 1 )C (8) (1 r τ)h i h i 1 (1 r τ)h j h j 1 + σ 3 + (h k h k 1 )C 3 r (1 r τ)h k h k 1 (1 r τ) + 1 τ(1 r τ) >. Also, whe we solve the log-trasformed BS equatio o the adaptive grid, we will use o-decreasig spatial step size, that is, h i 1 h i for ).

9 A compariso study of umerical method for the equity-liked securities 449 all i. By settig the lower boud of spatial step size as h mi satisfyig h mi h 1, coditio (8) gives (9) r τh mi + h mi τσ 1 τσ τσ 3 >. Therefore, we have the followig restrictio coditio for time step size (1) τ < h mi rh mi + σ 1 + σ +. σ 3 5. Numerical experimets I this sectio, we implemet umerical tests with our umerical method. For umerical tests, we cosider three-asset step-dow ELS optio as the example described i sectio 3.. We compare o-oscillatory explicit FDM ad implicit FDM with respect to computatioal costs ad errors Numerical treatmet for three-asset step-dow ELS optio pricig Before evaluatig the optio value of the test problem which is stated i sectio. 3., we first have to cosider two cases accordig to the iitial payoff. Let U ad V be the umerical solutios with payoffs which kock-i evet does ot happe ad happe, respectively. With the iitial payoffs (3) ad () which are described i Fig., we solve Eq. (5). After solvig Eq. (5) oce, we replace the values of less tha KIb i U with the values of V [3]. Ad the, we update the value of U by usig the FDM scheme. These processes are repeated from to T every time step. Also, at the early redemptio before τ = T, U ad V follow the coditios which are described i Table 1 ad Fig Numerical test We perform umerical experimets for pricig three-asset step-dow ELS. The parameters we have used are listed i sectio 3.. Also, the computatio domai is used as Ω = [1, ] [1, ] [1, ]. I Figs. 4-5, we ca see the iitial payoff fuctio ad umerical result at τ = 1 of U ad V, respectively. Now, we compare the results from MCS ad the FDM, the implicit scheme with OSM ad o-oscillatory explicit scheme. We focus o the value of U(1, 1, 1). I this test, MCS is performed 1 6 samples with τ = 1/144 usig atithetic variates of variace reductio, ad results from MCS are used as a referece value [7]. I order to

10 45 Mihyu Yoo, Darae Jeog, Seugsuk Seo, ad Juseok Kim U(X1, X, 1) 13 5 V(X1,X,1) 13 5 X 1 1 X1 X 1 1 X1 Figure 4. Iitial payoff fuctios of U ad V. U(X1, X, 1) 13 5 U(X1, X, 1) 13 5 X 1 1 X1 (a) X 1 1 X1 (b) Figure 5. Fial solutio of U at X 3 = 1 ad τ = 1 usig the (a) explicit ad (b) implicit scheme. reduce errors of simulatio, we calculate the average of the 1 MCS cases. The optio value obtaied from MCS is as a referece value, ad the MCS takes 145 secods at a time. For FDM tests, a various of o-uiform mesh for each directio is used. We fix τ for 1/36 i the implicit scheme ad adjust τ to prevet from spurious oscillatio of explicit scheme satisfyig the umber of early redemptios. [ a : h : b ] i Tables 6 meas that computatioal mesh is [ a, a+h, a+h,, b h, b, ]. All tests were performed o Itel(R) Core(TM) Duo E84 CPU@3.GHZ with 3.46GB of RAM loaded MATLAB 14a [6]. The optio value of U(1, 1, 1) ad absolute relative percet error with the value of MCS ad each FDM are show i Tables 6, where h is spatial step size for o-uiform mesh. Table shows that o-oscillatory explicit scheme is more superior tha implicit scheme with OSM i terms of computatioal cost ad the error with MCS i every spatial step. Note that the more space steps are take, the faster computatioal time is take i our scheme. O the other had, Tables

11 A compariso study of umerical method for the equity-liked securities preset that explicit scheme has similar error with implicit scheme. Overall, the explicit scheme is better tha the implicit if we use a small umber of mesh poits. However, if the spatial step size is small, the the implicit scheme is more efficiet tha the explicit scheme. h N x N τ U(1, 1, 1) Error CPU time (s) Table. Error ad the value of U(1, 1, 1) with o-uiform mesh [1 6:h: ] ad various values of h. h N x N τ U(1, 1, 1) Error CPU time (s) Table 3. Error ad the value of U(1, 1, 1) with o-uiform mesh [1 6:h: ] ad various values of h. h N x N τ U(1, 1, 1) Error CPU time (s) Table 4. Error ad the value of U(1, 1, 1) with o-uiform mesh [1 6:h: ] ad various values of h. Next, we similarly examie the optio value U(1, 1, 1) with respect to chagig volatility ad riskless iterest rate. We use the solutio of MCS with variace reductio. Tables 9 ad 1 show the optio price ad error with chagig the volatility ad riskfree iterest

12 45 Mihyu Yoo, Darae Jeog, Seugsuk Seo, ad Juseok Kim h N x N τ U(1, 1, 1) Error CPU time (s) Table 5. Error ad the value of U(1, 1, 1) with o-uiform mesh [1 6:h: ] ad various values of h. h N x N τ U(1, 1, 1) Error CPU time (s) Table 6. Error ad the value of U(1, 1, 1) with o-uiform mesh [1 6:h: ] ad various values of h. rate. Chages of volatility ad iterest rate require the differet umber of time steps o o-oscillatory explicit scheme. O the other had, chages of iterest rate have less of a effect o the umber of time step tha the chages of volatility. The referece values whe σ =. ad σ =.4 are ad , respectively. Also, ad are referece values for r =.1 ad r =.5. h N x N τ U(1, 1, 1) Error CPU time (s) Table 7. Error ad the value of U(1, 1, 1) with o-uiform mesh [1 6:h: ] ad various values of h (σ 1 = σ = σ 3 =.).

13 A compariso study of umerical method for the equity-liked securities 453 h N x N τ U(1, 1, 1) Error CPU time (s) Table 8. Error ad the value of U(1, 1, 1) with o-uiform mesh [1 6:h: ] ad various values of h (σ 1 = σ = σ 3 =.4). h N x N τ U(1, 1, 1) Error CPU time (s) Table 9. Error ad the value of U(1, 1, 1) with o-uiform mesh [1 6:h: ] ad various values of h (r =.1). h N x N τ U(1, 1, 1) Error CPU time (s) Table 1. Error ad the value of U(1, 1, 1) with o-uiform mesh [1 6:h: ] ad various values of h (r =.5). 6. Coclusios I geeral, the implicit scheme is used with Thomas algorithm. However, there are several drawbacks for practical calculatio. First, the implicit scheme has more time complexity i terms of computatios for solvig optio values. Next, it is hard to use the methodology for multidimesioal problem sice it is ecessary to apply the OSM or ADI. Hece we compared the explicit scheme satisfyig o-oscillatory coditio ad the implicit method for multi-dimesioal fiacial optios.

14 454 Mihyu Yoo, Darae Jeog, Seugsuk Seo, ad Juseok Kim Throughout the paper, the o-oscillatory explicit scheme is simple to implemet ad superior if we use a small umber of mesh poits. O the other had, the implicit scheme is more efficiet tha the explicit scheme if the spatial step size is small. Refereces [1] F. Black, ad M. Scholes, The pricig of optios ad corporate liabilities, J. Polit. Eco., 81 (1973), [] R. Zva, P. A. Forsyth, ad K. R. Vetzal, Robust umerical methods for PDE models of Asia optios, J. Comput. Fiac., 1 (1998), [3] D. Jeog, I. Wee, ad J. Kim. A operator splittig method for pricig the ELS optio, J. Korea Soc. Id. Appl. Math., 14(3), (1), [4] L. Trigeorgis, A log-trasformed biomial umerical aalysis method for valuig complex multi-optio ivestmets, J. Fiac. Quat. Aal., 6, (1991), [5] C. Reisiger, ad G. Wittum, O multigrid for aisotropic equatios ad variatioal iequalities Pricig multi-dimesioal Europea ad America optios. Comput. Visual. Sci., 7, (4), [6] MathWorks, Ic., MATLAB: the laguage of techical computig, The MathWorks, Natick, MA., [7] R. Seydel, Tools for computatioal fiace, Spriger Sciece & Busiess Media, 1. Mihyu Yoo Departmet of Fiacial Egieerig, Korea Uiversity, Seoul , Korea. ymh1989@korea.ac.kr Darae Jeog Departmet of Mathematics, Korea Uiversity, Seoul , Korea. tiayoyo@korea.ac.kr Seugsuk Seo Garam Aalytics, Yosei Uiversity, Seoul 1-749, Korea. sseo@gaalytics.co.kr Juseok Kim Departmet of Mathematics, Korea Uiversity, Seoul , Korea.

15 A compariso study of umerical method for the equity-liked securities Homepage: cfdkim

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