An efficient algorithm for Bermudan barrier option pricing

Transcription

1 Appl. Math. J. Chinese Univ. 2012, 27(1): An efficient algorithm for Bermudan barrier option pricing DING Deng 1 HUANG Ning-ying 2, ZHAO Jing-ya 1 Abstract. An efficient option pricing method based on Fourier-cosine expansions was presented by Fang and Oosterlee for European options in 2008, and later, this method was also used by them to price early-exercise options and barrier options respectively, in In this paper, this method is applied to price discretely American barrier options in which the monitored dates are many times more than the exercise dates. The corresponding algorithm is presented to practical option pricing. Numerical experiments show that this algorithm works very well and efficiently for different exponential Lévy asset models. 1 Introduction Financial market is developing explosively, although it is struck by the financial tsunami recently. Many new financial derivatives, including options, warrants and swaps are popping out. They are widely used as risk management tool by investors, stock brokers and bankers. But still, options are the most popular derivative products as hedging tools in constructing a portfolio. Since Bachelier, a French mathematician, first tried to give a mathematical definition for Brownian motion and used it to model the dynamics of stock process in 1900, financial mathematics has developed in leaps and bounds. The Black-Scholes model [4], one of the major breakthroughs of modern finance, commend easy ways to compute option prices. Many good ideas have been proposed to model the stock pricing processes since then. For example, Merton s model [18], Kou s model [15], Variance Gamma (VG) model [17], Inverse Gaussian (IG) model [8], Normal Inverse Gaussian (NIG) model [2], Heston s model [14], Bates model [3], CGMY model [5], etc. Meanwhile, many efficient numerical methods for option pricing have been proposed. These methods can be classified into three major groups: numerical solutions Received: MR Subject Classification: 42A10, 62P05, 65T40. Keywords: American barrier option, Bermudan option, Fourier transform, Fourier-cosine expansion. Digital Object Identifier(DOI): /s The work was partially supported by the research grants (UL020/08-Y4/MAT/JXQ01/FST and MYRG136(Y1-L2)-FST11-DD) from University of Macau. Corresponding author: HuangNY@gmail.com.

2 50 Appl. Math. J. Chinese Univ. Vol. 27, No. 1 to partial integro-differential equations (PIDEs), Monte Carlo simulation techniques, and numerical integration methods. Each of them has its advantages and disadvantages for different financial models and specific applications. The conditional densities of many stock price processes are usually unknown. Meanwhile, the Fourier transforms of these densities, i.e., the characteristic functions, are often available. Hence, the Fourier transform methods have been considered naturally by many authors (see [6] and references there in) in the numerical integration methods. In recent years, some new numerical integration methods are proposed. The QUAD method was introduced by Andricopoulos et al [1]. The CONV method was presented by Lord et al [16]. A fast Hilbert transform approach was considered by Feng and Linetsky in [12]. Meanwhile, a novel numerical method based on Fourier-cosine series expansion, called the COS method, was proposed by Fang and Oosterlee [10] and was shown to have the exponential convergence rate and the linear computational complexity. Recently, this COS method was used to price discrete early-exercise options [11], and was extended to be based on Fourier series expansions [9]. Pricing American options is much harder than pricing European options. Because the essential difficulty lies in the problem that they are allowed to exercise at any time before the expiration date. And such an early exercise right has changed the problem into a free boundary problem mathematically, since the optimal exercise price prior to the expiration of the option is prompt time dependent and is part of the solution. Usually, the stock price is not continuously monitored, but just is monitored and then the option is exercised at some special time points. In this case, the discretely-monitored American option is also called the Bermudan option. If a barrier is set to such a Bermudan option, it becomes the Bermudan barrier option [13]. In this paper, the COS method is applied to price the Bermudan barrier option in which the pre-specified monitored dates are many times more than the pre-specified exercise dates. The corresponding numerical algorithm is presented for practical option pricing. This algorithm works very well and efficiently for different exponential Lévy asset models. Numerical experiments on this algorithm are also given to support the efficiency of this algorithm. This paper is structured as follows. After this introduction, the COS method is applied to derive the approximate pricing formula for Bermudan barrier options under the exponential Lévy asset models in Section 2. Then, the corresponding numerical algorithm is presented for practical Bermuda barrier option pricing in Section 3, as well as the error analysis of this algorithm is considered. Finally, via numerical experiments we compute the practical Bermudan barrier option prices under the Black-Scholes (BS) model and the CGMY model, respectively, in Section 4. The numerical results show that the presented algorithm is fast and efficient. 2 The COS Method Denote G(S) =(S K) + for call option and G(S) =(K S) + for put option, where K is the strike price, S is the spot price of the underlaying asset. Let T = t ml+l : m =0, 1,...,M 1, l=0, 1,...,L 1 }

3 DING Deng, et al. An efficient algorithm for Bermudan barrier option pricing 51 be the set of pre-specified monitored dates before the maturity T,andlet T e = t ml : m =1,...,M 1 } T be the set of pre-specified exercise dates, where 0 = t 0 <t 1 < <t ML = T with Δt = t k t k 1 = T/(ML) for any t k T. Consider a discretely monitored in T and discretely exercised in T e American barrier option, namely Bermudan barrier option, whose payoff is given by G(S tk )1 Stk <H} + R 0 1 Stk H}, t k T, where S t is the price process of the underlaying asset, which is given by the exponential Lévy models, H>Kis the constant barrier and R 0 is the contractual rebate. That is, this Bermudan barrier option is an up-and-out barrier option that cease to exist if the asset price S tk hits the barrier level H at one time t k T, and it can also be exercised at any time t k T e. Denote V (S, t k ), t k T e, the valuation process of this Bermudan barrier option, i.e., the value of this Bermudan barrier option at time t k and the spot price S tk = S. Withhelpofthe risk-neutral valuation formula, this price process can be computed recursively by the following backward induction: V (S, t ML ) = G(S)1 S<H} + R 0 1 S H}, E(S, t k ) = E [ e rδt V (S tk+1,t k+1 ) S tk = S ], t k T V (S, t k ) = E(S, t k )1 S<H} + e r(t tk) (1) R 0 1 S H}, t k T \ T e, V (S, t k ) = max G(S),E(S, t k ) } 1 S<H} + e r(t tk) R 0 1 S H}, t k T e, in specialty, the initial price is given V (S, t 0 )=E(S, t 0 )=E [ e rδt V (S, t 1 ) S t0 = S ]. (2) Here S is the spot price of underlaying asset, r>0 is the interest rate, and E[ ] is conditional expectation under the risk-neutral probability P. Let X t = log(s t /K) be the logarithm of the underlying asset price S t over the strike price K, and denote x =log(s/k) andh = log(h/k). Let f( x) be the condition density of X tk+1 given X tk = x for t k T. Set K(e x 1) +, for a call option, g(x) = K(1 e x ) + (3), for a put option. Then the backward induction (1) and the price formula (2) can be rewritten by v(x, t ML ) = g(x)1 x<h} + R 0 1 x h}, e(x, t k ) = e rδt v(y, t k+1)f(y x)dy, t k T, (4) v(x, t k ) = e(x, t k ))1 x<h} + R 0 1 x h}, t k T \ T e v(x, t k ) = max g(x),e(x, t k ) } 1 x<h} + e r(t tk) R 0 1 x h}, t k T e, and where v(x, t k )=V (Ke x,t k ) for any t k T. v(x, t 0 )=e(x, t 0 )=e rδt v(y, t 1 )f(y x)dy, (5) Since f(y x) decays to zero very quickly as y ± we may choose two bounds a and b such that R\[a,b] f(y x)dy < ε tol for some given tolerance ε tol without losing some significant

4 52 Appl. Math. J. Chinese Univ. Vol. 27, No. 1 accuracy. An useful formula is given by Fang and Oosterlee in [10,11]: [ [a, b] = (c 1 + x) δ c 2 + c 4, (c 1 + x 0 )+δ c 2 + ] c 4, (6) where c 1, c 2,andc 4 are the first, second, and fourth cumulants of the process X t, the constant δ depends on the tolerance level ε tol, and usually we choose δ = 10. Thus, we can use the approximation of the infinite integrals e(x, t k )in(4): b ē(x, t k )=e rδt v(y, t m+1 )f(y x)dy e(x, t k ), t k T. (7) a Note that the density f(y x) has the following Fourier-cosine expansion on [a, b]: f(y x) = 2 [ ( w j cos jπ y a ) b ( f(u x)cos jπ u a ) ] du, (8) j=0 a where w 0 = 1 2 and w j =1forallj =1, 2, 3,..., and denote V j (t k+1 )= 2 b ( v(y, t k+1 )cos jπ y a ) dy, j =0, 1, 2,... (9) a Then, substituting the expansion into (7) and using the characteristic function of f( x): φ(z; x) = f(u x)e izu du, z R, j=0 we can get the further approximations (e.g. one can refer to [10]): N 1 ( ẽ(x, t k )=e rδt w j Re exp i jaπ ) ( jπ )} φ ; x V j (t k+1 ) e(x, t k ), (10) j=0 where Re } denotes taking the real part of a complex number, and i = 1 is the imaginary unit. In special case, the approximation of initial price v(x, t 0 ) in (5) is given by N 1 ( ṽ(x, t 0 )=e rδt w j Re exp i jaπ ) ( jπ )} φ ; x V j (t 1 ). (11) j=0 On the other hand, from the theory of Lévy processes (e.g. [7,10]), the characteristic function φ(z; x) possesses the property: φ(z; x) =ϕ(z)e izx, z R, whereϕ(z) =φ(z; 0) is the characteristic function of the corresponding Lévy process. Hence, the approximations (10) and (11) can be simplified to N 1 ( ẽ(x, t k ) = e rδt w j Re exp ijπ x a ) ( jπ )} ϕ V j (t k+1 ), (12) N 1 ṽ(x, t 0 ) = e rδt j=0 ( w j Re exp ijπ x a Now, we summarize this approximation by the following algorithm: Algorithm 1. ) ( jπ )} ϕ V j (t 1 ). (13) 1) Compute the terminal value: ṽ(x, t ML )=g(x)1 x<h} + R 0 1 x h} ; 2) Compute the integral V j (t k ), t k T by V j (t k+1 )= 2 b ( ṽ(y, t k+1 )cos jπ y a ) dy, j =0, 1,...,N 1; (14) a

5 DING Deng, et al. An efficient algorithm for Bermudan barrier option pricing 53 3) Compute the series ẽ(x, t k ), t k T, by (12); 4) Calculate the values ṽ(x, t tk )by ṽ(x, t k ) = ẽ(x, t k )1 x<h} + e r(t tk) R 0 1 x h}, t k T \ T e, (15) ṽ(x, t k ) = max g(x), ẽ(x, t k ) } 1 x<h} + e r(t tk) R 0 1 x h}, t k T e. (16) 5) Compute the initial price ṽ(x, t 0 )thatisgivenby(13). 3 The Numerical Algorithm In order to use the approximate formulation to practically price the Bermudan barrier option, we still need to compute the integrals V j (t k ) in (14). For convenience we introduce the notions: for any a x 1 x 2 b and j =0, 1,...,N 1, 2 x2 C j (x 1,x 2 ; t k ) = D j (x 1,x 2 ) = 2R 0 x 1 x2 x 1 ( ẽ(x, t k )cos jπ x a ( cos jπ x a ) dx, (17) ) dx, (18) where ẽ(x, t ML )=g(x) is given in (3), and ẽ(x, t k ), t k T, are the series given in (12). We also denote x2 ( Φ j (x 1,x 2 )= e x cos jπ x a ) x2 ( dx and Ψ k (x 1,x 2 )= cos jπ x a ) dx, x 1 x 1 for any a x 1 x 2 b and j =0,...,N 1. By a simple integration, we have 1 [ ( Φ j (x 1,x 2 ) = 1+( jπ cos jπ x 2 a ) ( e x2 cos jπ x 1 a ) e x1 b a )2 + jπ ( sin jπ x 2 a ) e x2 jπ ( sin jπ x 1 a ] )e x1, (19) [ ( Ψ j (x 1,x 2 ) = sin jπ x 2 a ) ( sin jπ x 1 a )] jπ, (20) for j =0,...,N 1, with Ψ 0 (x 1,x 2 )=x 2 x 1.WefirstcalculateV j (t ML ), j =0, 1,...,N 1. We have V j (t ML )= C j (0,h; t ML )+D j (h, b), for a call option, C j (a, 0; t ML )+D j (h, b), for a put option, (21) for all j =0, 1,...,N 1. Moreover, by a simple calculation, j =0, 1,...,N 1, we have D j (x 1,x 2 ) = 2R 0 Ψ j(x 1,x 2 ), (22) C j (x 1,x 2 ; t ML ) = 2 αk( Φ j (x 1,x 2 ) Ψ j (x 1,x 2 ) ), (23) where α is a parameter such that α = 1 for a call option and α = 1 for a put option. Next, for each t k T \ T e,wehave V j (t k )=C j (a, h; t k )+e r(t tk 1) D j (h, b), j =0, 1,...,N 1. (24) Since the integral D j (x 1,x 2 ) has the analytic representation (22), we only need to calculate the integral C j (x 1,x 2 ; t k ). Fang and Oosterlee in [11] gave an efficient numerical algorithm which can approximate C j (x 1,x 2 ; t k ) by using FFT method with the operation cost O(N log 2 (N)).

6 54 Appl. Math. J. Chinese Univ. Vol. 27, No. 1 Finally, for each t k T e, we should find the value ṽ(x, t k ) in (16), or equivalently, to determine the early-exercise point x k at each time t k, which is the point where the continuation value is equal to the payoff, i.e., ẽ(x k,t k)=g(x k ). Let h k (y) =ẽ(y, t k ) g(y), t k T e. Then, the problem becomes to find the root x k of each equation h k(y) = 0. Note that the function ẽ(y, t k ), which is given in (12), is bounded and smooth, and the function g(y), which is defined in (3), is smooth except for y = 0 and bounded in [a, b]. We can use the Newton s method or the secant method to find the root x k.hereifx k is not in the interval [a, b], we set it in the nearest boundary point a or b. Once we find the early-exercise point x k, t k T e,we have two different cases for an up-and-out barrier option: Case 1: x k <h, which means the early-exercise point doesn t hit the up barrier. Thus, We have the authority to decide to execute the option now or reserve it to the next time point. So we can split the integral that defines V j (t k ) into three parts: [a, x k ], (x k,h)and[h, b]. We have C j (a, x k V j (t k )= ; t k)+c j (x k,h; t ML)+e r(t tk 1) D j (h, b), for a call option, C j (a, x k ; t ML)+C j (x k,h; t k)+e r(t tk 1) (25) D j (h, b), for a put option, for j =0, 1,...,N 1, where D j (h, b) is given in (22), and C j (a, x k ; t k)andc j (x k,h; t k)are approximated by C j (a, x k ; t k)and C j (x k,h; t k) calculated by the FFT algorithm. Case 2: x k h, which means the early-exercise point hits the up barrier. Thus, the option integral can be split into two parts: [a, h) and[h, b]: C j (a, h; t k )+e r(t tk 1) D j (h, b), for a call option, V j (t k )= C j (a, h; t ML )+e r(t tk 1) (26) D j (h, b), for a put option, for j = N 1,...,1, 0, where D j (h, b) andc j (a, h; t ML ) are given in (22) and (23), respectively, and C j (a, h; t k ) is approximated by C j (a, h; t k ) calculated by the FFT algorithm. Now, we can practically and numerically price an up-and-out Bermudan barrier option by the approximate formulation. We summarize this pricing process in the following algorithm: Algorithm 2: (To price an up-and-out Bermudan barrier option.) 1) Calculate V j (t ML ), j =0, 1,...,N 1 by (21), (22) and (23). 2) Take the following backward induction for k = ML 1,...,1: a) For each t k T \ T e, compute V j (t k ), j =0, 1,...,N 1, by the formula (24) until t k T e if k>l;gotostep3)untilk =1ifk<L. b) For each t k T e, find the root x k by the Newton method or the secant method; i) If x k <h,calculatev j(t k ), j =0, 1,...,N 1, by formula (25); ii) If x k h, calculatev j(t k ), j =0, 1,...,N 1, by formula (26). And then return to Step a). 3) Compute the option price: ṽ(x, t 0 ) by formula (13).

7 DING Deng, et al. An efficient algorithm for Bermudan barrier option pricing 55 4 Numerical Experiments In this section, we employ Algorithm 2 to do numerical tests for the following two underlying asset models: the BS model, in which the characteristic function ϕ(z) isgivenby ϕ(z) =exp irzδt 1 } 2 σ2 z 2 Δt, and the corresponding cumalants are given by c 1 =(r 1 2 σ2 )T, c 2 = σ 2 T,andc 4 =0;the CGMY model, in which the Characteristic function ϕ(z) isgivenby ϕ(z) =exp iμzδt 1 } 2 σ2 z 2 Δt φ CGMY (z,δt), where μ = r 1 2 σ2 + CTΓ( Y ) ( (M 1) Y M Y +(G +1) Y G Y ), φ CGMY (z,δt) =exp CΔtΓ( Y ) [ (M iz) Y M Y +(G +iz) Y G Y ]}, and cumalants c 1, c 2,andc 4 are given as follows c 1 = μt + CTΓ(1 Y )(M Y 1 G Y 1 ), c 2 = σ 2 T + CTΓ(2 Y )(M Y 2 + G Y 2 ), c 4 = CTΓ(4 Y )(M Y 4 + G Y 4 ). In these models, Δt = t k t k 1 = T/(ML), Γ( ) is the gamma function, and r, σ, C, G, M and Y are parameters, which are given in the following Table 1. The computer used for all numerical experiments has an Intel(R) Core(TM)2 Duo CPU 2.1GHz, and all codes in these numerical experiments are written in Matlab 7.5. Table 1: Parameters in BS model and CGMY model Models T r σ C G M Y BS CGMY First, we choose one method to find the early-exercise point x k. The Newton s method converges faster than the secant method theoretically (order 2 against 1.6). However, the Newton s method requires the evaluation of both h k (y) and its derivative h k (y) ateachstep, while the secant method only requires the evaluation of h k (y). In fact, we compare two methods via computing the prices of a Bermudan barrier put option with the parameters: S 0 = 100, K = 80, H = 120, R 0 =0,M = 10, and L = 1 under the CGMY model, we stop the iterations when h k (y n ) <ɛ 0 with ɛ 0 =10 8, 10 10, 10 12, respectively. Here we first set the initial points y 1 =0andy 2 = 0.1 (y 1 = 0 for Newton s method), and then set the initial points y 1 = x k, y 2 = x k 0.1 (y 1 = x k for Newton s method). Table 2 gives the CPU times under different N and different tolerance ɛ 0, which seems to support the claim that the secant method is faster in practice. Hence, we always use the secant method in our numerical experiments. Tables 3 and 4 give the errors of the Bermudan put option and the Bermudan barrier put option with different barrier levels, under the BS model and the CGMY model, respectively.

8 56 Appl. Math. J. Chinese Univ. Vol. 27, No. 1 Table 2: CPU time (s) to compute the option with Newton s and secant methods. Newton s method Secant method N e e e e e e e e e e e e e e e e e e 1 Here we consider the options with the parameters: S 0 = 100, R 0 =0,M =10andL =1, and set K = 110 for the BS model; K = 80 for the CGMY model. We use the option prices for N =2 12 as the corresponding exact prices. From these tables, we see that the results converge very fast. In fact, for both models, the values are quite accurate when N =2 8. The errors fluctuate randomly when N>2 8, since it reaches Matlab s highest accuracy. Hence, the algorithm is very efficient for the different models in financial markets. Table 3: Errors for the BS model against different N. Bermudan Bermudan barrier N H = 120 H = 130 H = 140 H = e e e e e e e e e e e e e e e e e e e e e e 15 Table 4: Errors for the CGMY model against different N. Bermudan Bermudan barrier N H = 120 H = 170 H = 220 H = e e e e e e e e e e e e e e e e e e e e e 14 Tables 5 and 6 give the prices of Bermudan put option and Bermudan barrier put option with different Ls under BS model and CGMY model, respectively. From these tables, we see that the prices of Bermudan and Bermudan barrier options are quite different, specially, with the increase of L. We also see that, as the barrier level H is increasing, the difference of two options is decreasing. Specially, the Bermudan barrier option price is tending to the Bermudan option price when L = 1. Here we take the higher barrier levels in CGMY model than ones in

9 DING Deng, et al. An efficient algorithm for Bermudan barrier option pricing 57 BS model because the volatility of the CGMY model is bigger than BS model. Table 5: Prices of two options when N =2 8 for BS model. Bermudan Bermudan barrier H = 120 H = 130 H = 140 H = L = L = L =5 Table 6: Prices of two options when N =2 8 for the CGMY model. Bermudan option Bermudan barrier option H = 120 H = 170 H = 220 H = L = L = L =5 From the tables above, we see that, no matter which model we choose, the algorithm converges very fast with the increasing of N. Hence, we conclude that the algorithm is a fast and efficient algorithm for the Bermudan barrier option pricing. Meanwhile, we must mention here, this algorithm also gives an approximation of the early-exercise prices: S t k = Ke x k, tk T e, which is very useful in the practical option pricing. For instance, Table 7 gives the such prices for the CGMY model under H = 220 with L =1,L =3,andL = 5, respectively. Table 7: Early-exercise prices for the CGMY model at H = 220 against different L. L =1 t 1 t 2 t 3 t 4 t 5 t 6 t 7 t 8 t 9 St k L =3 t 3 t 6 t 9 t 12 t 15 t 18 t 21 t 24 t 27 St k L =5 t 5 t 10 t 15 t 20 t 25 t 30 t 35 t 40 t 45 St k Acknowledgments. The authors are very grateful to the referee for his very valuable suggestions which helped in improving our paper. References [1] A D Andricopoulos, M Widdicks, P D Duck, P D Newton. Universal option valuation using quadrature methods, J Financ Econ, 2003, 67:

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