In. J. Conemp. Mah. Sciences, Vol. 8, 2013, no. 20, 987-991 HIKARI Ld, www.m-hikari.com hp://dx.doi.org/10.12988/ijcms.2013.311123 Pricing FX Targe Redempion Forward under Regime Swiching Model Ho-Seok Lee FX & Derivaives Trading Div., Korea Exchange Bank 181, Euljiro 2-ga, Jung-gu, Seoul 100-793, Korea kais.hoseoklee@gmail.com Copyrigh 2013 Ho-Seok Lee. This is an open access aricle disribued under he Creaive Commons Aribuion License, which permis unresriced use, disribuion, and reproducion in any medium, provided he original work is properly cied. Absrac In his paper, we employ regime swiching model in valuing an exoic FX derivaive, called arge redempion forward. Marke condiions are assumed o be governed by a hidden Markov process and he coupled parial differenial equaions can be solved numerically. Keywords: Regime Swiching, Hidden Markov Process, Finie Difference Mehod 1 Inroducion The mos simple way o hedge fuure cash flows of a foreign currency is o ener ino an FX forward conrac hrough OTC(Over The Couner) marke. FX forward is a conrac o buy or sell a foreign currency a he prespecified exchange rae(srike price). Bu one can sell(buy) a a higher(lower) srike price hrough exoic FX derivaives. In his paper, we inroduce an exoic derivaive called FX arge redempion forward and use regime swiching model in valuing he produc numerically. FX arge redempion forward is composed of serial FX forwards, each of which has heir own payoff and if he accumulaed profi exceeds he prespecified knock ou level a he nh fixing ime before expiry, he conrac erminaes. For example, le he conrac noional amoun and srike price N, K repecively and
988 Ho-Seok Lee here are n fixing imes, ha is he conrac consiss of n serial underlying FX forwards. Formally, le P = N ( K S ), i = 1, L,n. S i is he underlying asse price a ih fixing ime. i k = 1 i C = max(0, P ) is he accumulaed profi for fixing ime. Then, P i is he payoff for he clien, ha is he clien receives P i if P i is posiive bu if P i is negaive, he clien pays he absolue value of P i a each fixing ime i. If is he firs fixing ime such ha C M for n, M - C will be finally received and remaining payoffs P i( < i) are cancelled. This produc conains an embedded opion ha knocks ou fuure cash flows if he accumulaed profi reaches he prespecified knock ou level. This opion canno be decomposed vanilla opions which have analyic pricing formula and should be solved numerically. Marke pracice in FX opion is o assume he reurn of a foreign currency follows a log-normal disribuion wih deerminisic volailiy. Bu he volailiy is no deerminisic bu sochasic. A simplified approach o incorporae sochasic behavior of volailiy is o adop a regime swiching model. [4] proposed an analyic pricing formula for European opion under regime-swiching model and [1] developed a laice mehod for opion valuaion using regime-swiching model. [2] invesigaed numerical algorihm for valuing European syle exoic opion wih regime-swiching marke condiions which are modulaed by a hidden Markov process In his paper, we use regime-swiching model for valuing FX arge redempion forward. I is assumed ha marke condiions are governed by a hidden Markov process and he relaed coupled parial differenial equaions will be inroduced and are solved numerically. The res of his paper is organized as follows. Secion 2 describes he model and derive he governing parial differenials. A numerical example is presened in Secion 3 and Secion 4 concludes. k 2 Model 2.1 Two saes regime swiching model Consider a probabiliy riple ( Ω, F,P) wih he filraion { F }, which is generaed by a sandard Brownian moion B() and a hidden Markov process y(). I is assumed ha B() and y() are independen. Under he risk-neural probabiliy measure Q, he price of one uni of foreign currency S() is given by ds()/ S() = ( r () r ()) d + σ () db(), d where r d () and r f () are domesic and foreign risk-free ineres rae, respecively f
Pricing FX arge redempion forward 989 and σ() is he volailiy. We assume ha he hidden Markov process y () can ake wo values 0, 1. r d (), r f () and σ() are assumed o ake corresponding wo sae values (r d,0,r d,1 ), (r f,0,r f,1 ) and (σ 0, σ 1 ) respecively. As in [2], y() is generaed by a ransiion marix a0,0 a0,1 λ0 λ0 =. a1,0 a 1,1 λ1 λ1 2.2 Governing parial differenial equaions In his subsecion, we consruc he parial differenial equaions for valuing FX arge redempion forward which is inroduced in he Secion 1. The sae variables ha deermine he value of FX arge redempion forward are he price of foreign currency S() and he accumulaed benefi C. Formally, if we denoe V (i) he price of FX arge redempion forward a ime for regime i(i=0, 1), we can wrie V (i) V(,S(),C,i). Le Σ = {,,, } 1 2 L n be he se of fixing imes. n = T is he conrac mauriy. Then, we have he following coupled parial differenial equaions for he price V(,S(),C,i). Theorem 2.1 V(,S(),C,i) saisfies he following coupled parial differenial equaions VSCi (,,, ) VSCi (,,, ) 1 VSCi (,,, ) + ( r r ) S + S r V(, S, Ci, ) + a V(, S, C, k) = 0 S S 2 1 2 2 di, f, i σi 2 di, ik, 2 k= 0 beween fixing imes. The jump condiions a fixing imes are given by wih he final condiions M < C; V(, S, C, i) = 0, C M C + max( P,0); V(, S, C, i) = M C, + C + max( P,0) < M; V(, S, C, i) = V(, S, C + max( P,0), i) + P, M < C; V( T, S, C, i) = 0 C M C + max( PT,0); V( T, S, C, i) = M C C + max( PT,0) < M; V( T, S, C, i) = PT. Proof. The accumulaed benefi C are consan beween fixing imes. Therefore, we may consider C as a consan beween fixing imes. Following [2], we can obain he coupled parial differenials beween fixing imes. The jump condiions
990 Ho-Seok Lee and final condiions are easily obained and are similar o hose of clique opion, which is inroduced in [3]. 3 Resuls In his secion, we apply he numerical pricing procedure explained in he previous secion o a USD/KRW arge redempion forward. 3.1 Two saes regime in he marke In general, when USD/KRW increases, USD/KRW opion volailiy σ() also increases and he domesic risk-free ineres rae r d () plummes. So, we can assume a wo saes regime economy, ha is regime0 wih a lower opion volailiy wih a higher domesic risk-free ineres rae and regime1 wih a higher opion volailiy wih a lower domesic risk-free ineres rae. We assume ha he foreign risk-free ineres rae is consan, ha is r f,0 = r f,1. 3.2 Numerical resuls In his secion, we assume ha λ 0 = λ 1 = λ and solve he coupled parial differenial equaions numerically. We use he finie difference mehod wih he ieraive algorihm inroduced in [2] and he parameers are given as follows: N=$1,000,000, K=1,180, M =400, r d,0 =0.025, r d,1 =0.010, r f,1 = r f,1 =0.005, σ 0 = 0.10, σ 1 = 0.25. In Table1, we give he prices for differen values of ransiion inensiy λ. Firs, he effec of regime swiching is larger for small FX raes(usd/krw). Second, as he ransiion inensiy increases, he difference beween prices in wo regimes decreases and he effec of regime swiching decreases. In Table2, we consider he prices for differen values of he conrac mauriy T. We can see ha he effec of regime swiching is more subsanial for large mauriies. Table1 Prices for differen values of λ. T = 1. (Hundred million won) λ Regime0 Regime0 Regime0 FX=1,160 FX=1,160 FX=1,180 FX=1,180 0.00-1.12-5.44-2.98-7.06-4.65-8.44 0.25-1.42-5.24-3.26-6.86-4.91-8.25 0.50-1.67-5.08-3.50-6.70-5.12-8.09 0.75-1.88-4.94-3.69-6.57-5.30-7.97 1.00-2.05-4.83-3.86-6.46-5.46-7.86 1.25-2.20-4.73-4.00-6.36-5.58-7.77 1.50-2.33-4.65-4.12-6.28-5.69-7.69 1.75-2.45-4.58-4.22-6.21-5.79-7.62 2.00-2.54-4.51-4.31-6.15-5.87-7.56 10.0-3.42-3.96-5.12-5.62-6.59-7.05 20.0-3.59-3.86-5.27-5.52-6.72-6.96
Pricing FX arge redempion forward 991 Table2 Prices for differen mauriy T. λ = 1. (Hundred million won) T Regime0 FX=1,160 FX=1,160 Regime0 FX=1,180 FX=1,180 Regime0 0.25 0.45 0.37-0.12-0.15-0.75-0.75 0.50 0.37-0.67-0.65-1.52-1.74-2.43 0.75-0.59-2.57-1.93-3.74-3.41-5.00 1.00-2.05-4.83-3.86-6.46-5.46-7.86 1.25-4.02-7.47 5.90-9.19-8.47-11.50 1.50-6.22-10.28-8.60-12.47-11.18-14.79 1.75-8.25-12.88-11.17-15.57-14.31-18.42 2.00-10.71-15.86-14.26-19.10-18.05-22.54 4 Conclusion In his paper, we derive he coupled parial differenial equaion for valuing FX arge redempion forward under regime swiching marke model. We can solve he problem by using he ieraive mehod inroduced in [2]. The effec of regime swiching is large for small level of FX, small ransiion inensiy and large mauriy. References [1] N. P. B. Bollen, Valuing opions in regime-swiching models, Journal of Derivaives, 6 (1998), 38-49 [2] P. Boyle and T. Draviam, Pricing exoic opions under regime swiching, Insurance: Mahemaics and Economics, 40 (2007), 267-282. [3] P. Wilmo, Clique Opions and Volailiy models, Wilmo Magazine, December (2002), 78-83 [4] V. Naik, Opion Valuaion and Hedging Sraegies wih Jump in he Volailiy of Asse Reurns, Journal of Finance, 5 (1993), 1969-1984 Received: Ocober 1, 2013