 Journal of Financial Risk Management 2013. Vol.2, No.4, 67-70 Published Online December 2013 in SciRes (http://www.scirp.org/journal/jfrm) http://dx.doi.org/10.4236/jfrm.2013.24011 Open Access 67 Pricing Double Barrier Parisian Option Using Finite Difference Xuemei Gao South Western University of Finance and Econom i cs, Chengdu, China Email: gaoxuemei2000@sina.com Received September 15th, 2013; revised October 15th, 2013; accepted October 23rd, 2013 Copyright © 2013 Xuemei Gao. This is an open access article distributed under the Creative Commons Attribu-tion License, which permits unrestricted use, distri b ut io n, and reproduction in any medium, provided the original work is properly cited. In this paper, we price the valuation of double barrier Parisian options, under the Black-Scholes frame-work. The approach is based on fundamental partial differential equations. We reduce the dimension of partial differential equations，then using finite difference scheme to solve the partial differential equations. Keywords: Black-Scholes Model; Double Barrier; Parisian Options; Finite Difference Scheme Introduction It is well known that valuation of financial derivatives, such as options, is one of the major topics in quantitative finance research. A Parisian option is a special kind of barrier options for which the knock-in or knock-out feature is only activated if the underlying price remains continually in breach of the barrier for a pre-specified time period. The valuation of Parisian op-tions can be done by using several different methods: Monte Carlo simulations (Baldi, Caramellino, & Iovino, 2000), lattices (Avellaneda & Wu, 1999), Laplace transforms (Zhu & Chen, 2013) or partial differential equations. An approach based on partial differential equations has been developed by (Wilmott, 1998; Haber, Schönbucher, & Wilmott, 1999). The options we study in this paper are called double barrier Parisian options. The paper (Chesney, Jeanblanc-Picqué, & Yor, 1997) intro-duced the standard Parisian options with two barriers. Double barrier Parisian options are options where the conditions im-posed on the assets involve the time spent out of the range de-fined by two barriers. Double barrier Parisian options have already been priced by (Baldi, Caramellino, & Iovino, 2000) using Monte Carlo simulations corrected by the means of sharp large deviation estimates, by (Labart & Lelong, 2009) using Laplace transforms. We use partial differential equations to price double barrier Parisian options. There are two different ways of measuring the time outside the barrier range. One ac-cumulates the time spent in a row and resets the counting whenever the stock price crosses the barrier(s). This type is referred to as continuous double barrier Parisian options. The other adds the time spent in the relevant excursions without resuming the counting from 0 whenever the stock price cros- ses the barrier(s). These options are named as cumulative double barrier Parisian options. In practice, these two ways of counting time raise different questions about the paths of Brownian motion. In this work, we only focus on continuous knock-out double barrier Parisian call options. We establish the partial differential equation systems for the prices of dou-ble barrier Parisian options, and reduce the dimension of par-tial differential equations, then using finite difference scheme to solve the equations. The State Space and Boundary Conditions Unavoidable The pricing of double barrier Parisian options requires the value of a state variable (clock) J, which dictates the time underlying price outside the barrier range (Zhu & Chen, 2013). When the underlying price is outside the barrier range, the state variable SJ starts to accumulate values at the same rate as the passing time , and when the underlying is inside the barrier range, tJ is reset to zero ，and remains zero: 1212120, d0,dd, 0< JJLSLJtS LorS LLL where 12LL is a preset lower(up) barrier of the underlying. According to (Zhu & Chen, 2013), pricing domain can be defined as: 21:0 ,0,0:, ,0:0 ,,0IStTJJIILS JtJTJJJIIISLJtJTJJJ   J is the barrier time triggering parameter. When the variable J reaches J the option becomes worthless. is the expiration time. For simplicity we suppose that does not jump from 1TSL to 2L and does not jump from 2L to 1L. The value of a double barrier Parisian option depends on the underlying price , the current time t and the barrier time SJ, the volatility, risk-free interest rate and the expiry time etc.. Under the Black-Scholes framework, the volatility  is a positive constant, r denotes the risk-free interest rate，the parameter  is the dividend rate if the underlying is a stock or the foreign interest rate in case of a currency. is given by SddttS rSt SW dtt where is a standard Brownian motion. Let tW1,VSt, 2VS,,tJ and 3,,VStJ denote the option prices in the X. M. GAO region , and respectively. By applying the Feyn-man-Kac theorem (Simon, 2000), III III1,VSt should satisfy the classical BS (Black-Scholes) equation 211 112102VV VSrSrVtS S22 . In region where the underlying price rises above the bar-rier 2IIL, in region where the underlying price moves below the barrier III1L, the barrier time J starts to accumulate. As a result, 2, ,,VStJ3,,VStJ are governed by a modi-fied Black-Scholes Equation (Haber, Schönbucher & Wilmott, 1999) respectively, 222210, 2,32i iiVV VVSrSrViS S   iitJ . We show below how the solutions are linked in these three regions. At barrier we impose pathwise continuity of option price, which means the option price does not jump at a barrier. The continuity of the price across the barrier 2L demands 222limm 0SL LV1liSt,VS , ,St1. The continuity of the option price across the barrier L demands 1113lim,lim, ,0SL SLVSt VSt. Appropriate boundary conditions are also needed. In most general form, the option is specified as follows: If the knock out option has not been triggered by expiration , then the option has the price contingent payoff which might also depend on TJ at expiration; if the knock out option has been triggered during the lifetime of the option，the option pays off the option value at point ,,JSt . The terminal condition in pricing domain can be given by the payoff function of a European call of ma-turity T and exercise price , IK ,TSK1VS . A knock out double barrier Parisian call option is lost if un-derlying price made an excursion outside the barrier range older than SJ before , T2limJJ, ,0VStJ, 3limJJ, ,0VStJ. That it would take infinite amount of time for an infinitely large underlying price to fall back to the barrier 2L, the option must be worth nothing when becomes very large gives S2lim, 0StJ ,VS . A call option becomes worthless when the underlying price approaches zero, gives 3lim, ,0SVStJ. The boundary condition at barrier is specified by the so called “reset condition”, 2221li, ,lim,SLtJVStmSLVSmSLVS,  1131li, ,lim,SLtJVSt. PDE Systems for Pricing Double Barrier Parisian Options Under the Black-Scholes framework, the PDE (partial dif-ferential equation) systems for the prices of double barrier Pari-sian options with above boundary conditions have already been established in (Haber, Schönbucher, & Wilmott, 1999):  112222 211 1121131202,,lim,lim, ,0lim,lim,,0BSSL SLSL SLVSV VrS rVtS SVSTJVSJVSt VStVSt VSt     2222 222 2222222102,, 0lim, ,0lim, ,lim,SSL SLVV SVVrS rVtJ SSVStJVStJVStJ VSt    1122 233 33323303102,, 0lim, ,0lim, ,lim,SSL SLVV SVVrS rVtJ SSVStJVStJVStJVSt   The above PDE systems are in 3-D and can implified to 2-D PDE systems. 1 is already in 2-D. We need to deal with the system governing 2, 3. To reduce dimensionality of a PDE system usually requires the application of some sorts of trans-formation techniques, such as the Fourier transform, the Laplace transform, and so on. Without applying any transfor-mation methods，the pricing domain is a parallelepipedon, and can be decomposed into infinite many cross-sections (which will be referred to as “slides” hereafter)，all of which are of 45˚ to both of the plane, t = 0 and J = 0 (Zhu & Chen, 2013). In the pricing domain , the positions of the regions are reversed. It is clear that the option value 2, 3 at any given point VV VIIIIIV V,,StJ can be uniquely determined as long as enough information along the every slide passing through that static point is known. In other words，the original 3-D problem can be decomposed into a set of 2-D problems defined on each slide，if viewed from a 45˚ rotated coordinate system. Mathematically, to obtain the PDE governing , , in the rotated coordinate 2V3Vsystem. We can use the directional derivative 222Vl , 332Vl which represents the instantaneous rate of change of the function 2V, 3 at the point V,tJ, in the direction of (22,22 ), to replace the sum of the two partial deriva- tives 22VVtJ, 3VVtJ 3, respectively. Furthermore, let 222ll, 332ll. As a result, the governing equation in the new coordinate system can be written as 2222102,; , 2,3ii iiiiiVVVVSrSrVlS SSl ti   which is the BS equation. In the new coordinate system, 22 222,;, ,VSlt VStll, , serves as a parameter. The boundary conditions sets for 33 333,;, ,VSlt VStllt22,;VSlt can be extracted from the corresponding boundary Open Access 68 X. M. GAO conditions that 2,,VStJ needs to satisfy (Zhu & Chen, 2013):  22222222212lim, ;0lim, ;0lim, ;lim,SlJSLSLVSltVSltVSlt VStl The boundary conditions set for 33,;VSlt ditions can be extracted fr thom the corresponding boundary conat 3,,VStJ needs to satisfy: 31133033331 30lim, ;0lim, ;lim,SlJSL SLVSltVSltVStl Therefore, the2-D PDE systems that govern the price of double lim, ;VSltbarrier Parisian options can be now summarized as:   1221VV V 2211 1121111202,,lim ,lim ,BSSLSLSrSrVtS SVSTJVSJVSt tVStt  (1) 222222 2222222222 2102,; 0lim, ;0lim, ;SSLVV VSrSrVlS SVSJtVSltVSlt tl   (2) 122233 33233330331 3102,; 0lim, ;0lim, ;SSLVV VSrSrVlS SVSJtVSltVSlt tl  (3) for 0,tTJ, 20,lJ, 30,lJ. Algorithm The numerical solutio(1)-(3) is implemented usare discretized asn to Equations ing a finite difference scheme in 2-D. Although explicit finite difference schemes are similar to the binomial numerical me- thod in spirit, they are more general and thus more flexible. The method is time-efficient because it is extremely easy to pro-gram，and the programs run very quickly. It is suitable for many types of contracts including most common path-dependent de- rivatives and is trivially—with one extra line of code—exten- ded to American-style early exercise. In Equation (1), the price S and t S, bet, respectively. For stabili of the scheme, t has to sen small enough. We denote ,ijV by the erical ap-proximation to the option value at iS, tjt. We call the discrete barrier cho ty numSi (i.e., 1iS L), i (i.e., 2S Li) and j (i.e., jtTJ  ). 22,1,,1, 1,,222ij 1,jiji jij ijijiLVVVir VV rV i,1 ,,,ij ijijVVtLiii . In Equation (2), the price and are diretized asS2lsc S, 2l respectively. 2l has to be chosen small enough, eter pa-ram 0, ,2,tt Jt,t . We denote ,hjV by the nu-merical e option value at ShSationapproxim to th, 22ljl. We call the discrete barrier h (i.e., 2LhS), and j (i.e., 2jlJ). 22,1,,1, 1,,222hjh jhjh jhj hjhjhLVVVhr VV rV 1,In Equation (3), the price and are discretized as,1 ,,2hj hjhjVVlL S1l S, 1l, respectively. 1l has be chsen small enough, eter to opa-ram 0, ,2,tt Jt,t . We denote ,kjV by the nu-merical e option value at SkSationapproxim to th, 11ljl. We call the discre te ba rrier k (i.e., 2L)kS , andj (i.e., 1jlJ, 22,1,,1, 1,,222kjk jkjk jkj kjkjkLVVVkr VV rV 1, The price is discretized using equ steps in and no e prices of a double barrier Parisi call w,1 ,,3kj kjkjVVlL.S are thal lnS an outt in S. We pcomith 0100SK , 190L, 2110L, 0.095r, 0 and 1T obtained with our methodo method with 0 samples. The programs of finite difference schemes run very quickly. Comparison corrected Monte Carlo and finite difference: and Monte Carl1000MC Price FD Price 0.645 1.142 2.747 2.214 3.065 0.632 1.121 2.732 2.352 3.031 REFERENCES Baldi, P. Caramellino, 000). Pricing complex AL., & Iovino, M. G. (2barrier options with general features using sharp large deviation es-timates. In Monte Carlo and quasi-Monte Carlo methods in scientific computing (3rd ed., pp. 149-162). Claremont, CA: Springer. vellaneda, M., & Wu, L. (1999). Pricing parisian-style options with a lattice method. International Journal of Theoretical and Applied Fi-nance, 2, 1-16. http://dx.doi.org/10.1142/S0219024999000029 hu, S. P., & Chen, W. T. (2013). Pricing Parisian and Parasian optionsZ analytically. Journal of Economic Dynamics and Control, 37, 875- 896. http://dx.doi.org/10.1016/j.jedc.2012.12.005 ilmott, P. (1998). 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