An efficient algorithm for Bermudan barrier option
pricing
DING Deng
1HUANG Ning-ying
2,∗ZHAO Jing-ya
1Abstract. 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 2009. 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 L´evy 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 math-ematics 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, Mer-ton’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: 2010-03-28.
MR Subject Classification: 42A10, 62P05, 65T40.
Keywords: American barrier option, Bermudan option, Fourier transform, Fourier-cosine expansion. Digital Object Identifier(DOI): 10.1007/s11766-012-2516-5.
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.
to partial integro-differential equations (PIDEs), Monte Carlo simulation techniques, and nu-merical 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 numer-ical integration methods are proposed. The QUAD method was introduced by Andricopouloset 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 algo-rithm works very well and efficiently for different exponential L´evy 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 L´evy 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
be the set of pre-specified monitored dates before the maturity T , and let Te=tmL: m = 1, . . . , M−1⊂ T
be the set of pre-specified exercise dates, where 0 = t0 < t1 < · · · < tML = T with Δt = tk − tk−1 = T /(M L) for any tk ∈ T. Consider a discretely monitored in T and discretely exercised in Te American barrier option, namely Bermudan barrier option, whose payoff is given by
G(Stk)1{Stk<H}+ R01{Stk≥H}, tk ∈ T,
where Stis the price process of the underlaying asset, which is given by the exponential L´evy models, H > K is the constant barrier and R0is the contractual rebate. That is, this Bermudan barrier option is an up-and-out barrier option that cease to exist if the asset price Stk hits the barrier level H at one time tk∈ T, and it can also be exercised at any time tk∈ Te.
Denote V (S, tk), tk ∈ Te, the valuation process of this Bermudan barrier option, i.e., the value of this Bermudan barrier option at time tk and the spot price Stk= S. With help of the risk-neutral valuation formula, this price process can be computed recursively by the following backward induction:⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ V (S, tML) = G(S)1{S<H}+ R01{S≥H}, E(S, tk) = Ee−rΔtV (Stk+1, tk+1)| Stk= S , tk∈ T V (S, tk) = E(S, tk)1{S<H}+ e−r(T −tk)R01 {S≥H}, tk∈ T \ Te, V (S, tk) = maxG(S), E(S, tk)1{S<H}+ e−r(T −tk)R01 {S≥H}, tk∈ Te, (1)
in specialty, the initial price is given
V (S, t0) = E(S, t0) =Ee−rΔtV (S, t1)| St0= S . (2) Here S is the spot price of underlaying asset, r > 0 is the interest rate, andE[· | ·] is conditional expectation under the risk-neutral probabilityP.
Let Xt= log(St/K) be the logarithm of the underlying asset price St over the strike price K, and denote x = log(S/K) and h = log(H/K). Let f (· | x) be the condition density of Xtk+1 given Xtk= x for tk∈ T. Set
g(x) =
K(ex− 1)+, for a call option,
K(1− ex)+, for a put option. (3)
Then the backward induction (1) and the price formula (2) can be rewritten by ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ v(x, tML) = g(x)1{x<h}+ R01{x≥h},
e(x, tk) = e−rΔt −∞∞ v(y, tk+1)f (y| x)dy, tk ∈ T, v(x, tk) = e(x, tk))1{x<h}+ R01{x≥h}, tk∈ T \ Te v(x, tk) = maxg(x), e(x, tk)1{x<h}+ e−r(T −tk)R01 {x≥h}, tk ∈ Te, (4) and v(x, t0) = e(x, t0) = e−rΔt ∞ −∞ v(y, t1)f (y| x)dy, (5) where v(x, tk) = V (Kex, tk) for any tk ∈ T.
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 < εtol for some given tolerance εtol without losing some significant
accuracy. An useful formula is given by Fang and Oosterlee in [10,11]: [a, b] = (c1+ x)− δ c2+√c4, (c1+ x0) + δ c2+√c4 , (6)
where c1, c2, and c4are the first, second, and fourth cumulants of the process Xt, the constant δ depends on the tolerance level εtol, and usually we choose δ = 10. Thus, we can use the approximation of the infinite integrals e(x, tk) in (4):
¯
e(x, tk) = e−rΔt b
a
v(y, tm+1)f (y| x)dy ≈ e(x, tk), tk ∈ T. (7) Note that the density f (y| x) has the following Fourier-cosine expansion on [a, b]:
f (y| x) = 2 b− a ∞ j=0 wjcos jπy− a b− a b a f (u| x) cos jπu− a b− a du , (8)
where w0=12 and wj= 1 for all j = 1, 2, 3, . . ., and denote Vj(tk+1) = 2 b− a b a v(y, tk+1) cos jπy− a b− a dy, j = 0, 1, 2, . . . . (9) Then, substituting the expansion into (7) and using the characteristic function of f (· | x):
φ(z; x) = ∞
−∞
f (u| x)eizudu, z∈ R, we can get the further approximations (e.g. one can refer to [10]):
˜ e(x, tk) = e−rΔt N−1 j=0 wjRe exp − i jaπ b− a · φ jπ b− a; x Vj(tk+1)≈ e(x, tk), (10) 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, t0) in (5) is given by
˜ v(x, t0) = e−rΔt N−1 j=0 wjRe exp − i jaπ b− a · φ jπ b− a; x Vj(t1). (11) On the other hand, from the theory of L´evy processes (e.g. [7,10]), the characteristic function φ(z; x) possesses the property: φ(z; x) = ϕ(z)eizx, z ∈ R, where ϕ(z) = φ(z; 0) is the charac-teristic function of the corresponding L´evy process. Hence, the approximations (10) and (11) can be simplified to ˜ e(x, tk) = e−rΔt N−1 j=0 wjRe exp ijπx− a b− a · ϕ jπ b− a Vj(tk+1), (12) ˜ v(x, t0) = e−rΔt N−1 j=0 wjRe exp ijπx− a b− a · ϕ jπ b− a Vj(t1). (13) Now, we summarize this approximation by the following algorithm:
Algorithm 1.
1) Compute the terminal value: ˜v(x, tML) = g(x)1{x<h}+ R01{x≥h}; 2) Compute the integral Vj(tk), tk∈ T by
Vj(tk+1) = 2 b− a b a ˜ v(y, tk+1) cos jπy− a b− a dy, j = 0, 1, . . . , N−1; (14)
3) Compute the series ˜e(x, tk), tk∈ T, by (12); 4) Calculate the values ˜v(x, ttk) by
˜ v(x, tk) = e(x, t˜ k)1{x<h}+ e−r(T −tk)R 01{x≥h}, tk ∈ T \ Te, (15) ˜ v(x, tk) = maxg(x), ˜e(x, tk)1{x<h}+ e−r(T −tk)R 01{x≥h}, tk ∈ Te. (16) 5) Compute the initial price ˜v(x, t0) that is given by (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 Vj(tk) in (14). For convenience we introduce the notions: for any a≤ x1≤ x2≤ b and j = 0, 1, . . . , N−1,
Cj(x1, x2; tk) = 2 b− a x2 x1 ˜ e(x, tk) cos jπx− a b− a dx, (17) Dj(x1, x2) = 2R0 b− a x2 x1 cos jπx− a b− a dx, (18)
where ˜e(x, tML) = g(x) is given in (3), and ˜e(x, tk), tk ∈ T, are the series given in (12). We also denote Φj(x1, x2) = x2 x1 excos jπx− a b− a dx and Ψk(x1, x2) = x2 x1 cos jπx− a b− a dx, for any a≤ x1≤ x2≤ b and j = 0, . . . , N−1. By a simple integration, we have
Φj(x1, x2) = 1 1 + (b−ajπ )2 cos jπx2− a b− a ex2− cosjπx1− a b− a ex1 + jπ b− asin jπx2− a b− a ex2− jπ b− asin jπx1− a b− a ex1, (19) Ψj(x1, x2) = sin jπx2− a b− a − sinjπx1− a b− a b− a jπ , (20)
for j = 0, . . . , N−1, with Ψ0(x1, x2) = x2− x1. We first calculate Vj(tML), j = 0, 1, . . . , N−1. We have
Vj(tML) =
Cj(0, h; tML) + Dj(h, b), for a call option,
Cj(a, 0; tML) + Dj(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
Dj(x1, x2) = 2R0 b− aΨj(x1, x2), (22) Cj(x1, x2; tML) = 2 b− aαK Φj(x1, x2)− Ψj(x1, x2), (23) where α is a parameter such that α = 1 for a call option and α =−1 for a put option.
Next, for each tk∈ T \ Te, we have
Vj(tk) = Cj(a, h; tk) + e−r(T −tk−1)Dj(h, b), j = 0, 1, . . . , N−1. (24)
Since the integral Dj(x1, x2) has the analytic representation (22), we only need to calculate the integral Cj(x1, x2; tk). Fang and Oosterlee in [11] gave an efficient numerical algorithm which can approximate Cj(x1, x2; tk) by using FFT method with the operation cost O(N log2(N )).
Finally, for each tk ∈ Te, we should find the value ˜v(x, tk) in (16), or equivalently, to determine the early-exercise point x∗k at each time tk, which is the point where the continuation value is equal to the payoff, i.e., ˜e(x∗k, tk) = g(x∗k). Let
hk(y) = ˜e(y, tk)− g(y), tk∈ Te.
Then, the problem becomes to find the root x∗k of each equation hk(y) = 0. Note that the function ˜e(y, tk), 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. Here if x∗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, tk ∈ Te, 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 Vj(tk) into three parts: [a, x∗k], (x∗k, h) and [h, b]. We have
Vj(tk) =
Cj(a, x∗k; tk) + Cj(xk∗, h; tML) + e−r(T −tk−1)Dj(h, b), for a call option,
Cj(a, x∗k; tML) + Cj(xk∗, h; tk) + e−r(T −tk−1)Dj(h, b), for a put option, (25)
for j = 0, 1, . . . , N−1, where Dj(h, b) is given in (22), and Cj(a, x∗k; tk) and Cj(x∗k, h; tk) are approximated by ˜Cj(a, x∗k; tk) and ˜Cj(x∗k, h; tk) 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]:
Vj(tk) =
Cj(a, h; tk) + e−r(T −tk−1)Dj(h, b), for a call option,
Cj(a, h; tML) + e−r(T −tk−1)Dj(h, b), for a put option, (26)
for j = N−1, . . . , 1, 0, where Dj(h, b) and Cj(a, h; tML) are given in (22) and (23), respectively, and Cj(a, h; tk) is approximated by ˜Cj(a, h; tk) 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 Vj(tML), j = 0, 1, . . . , N−1 by (21), (22) and (23). 2) Take the following backward induction for k = M L−1, . . . , 1:
a) For each tk ∈ T \ Te, compute Vj(tk), j = 0, 1, . . . , N−1, by the formula (24) until tk ∈ Teif k > L; go to Step 3) until k = 1 if k < L.
b) For each tk∈ Te, find the root x∗k by the Newton method or the secant method; i) If x∗k < h, calculate Vj(tk), j = 0, 1, . . . , N− 1, by formula (25);
ii) If x∗k ≥ h, calculate Vj(tk), j = 0, 1, . . . , N−1, by formula (26). And then return to Step a).
§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) is given by
ϕ(z) = exp
irzΔt−1 2σ
2z2Δt,
and the corresponding cumalants are given by c1 = (r− 12σ2)T , c2 = σ2T , and c4 = 0; the CGMY model, in which the Characteristic function ϕ(z) is given by
ϕ(z) = exp iμzΔt−1 2σ 2z2Δt· φ CGMY(z, Δt), where μ = r−1 2σ 2+ CT Γ(−Y )(M− 1)Y − MY + (G + 1)Y − GY,
φCGMY(z, Δt) = expCΔtΓ(−Y )(M− iz)Y − MY + (G + iz)Y − GY , and cumalants c1, c2, and c4 are given as follows
⎧ ⎪ ⎨ ⎪ ⎩ c1= μT + CT Γ(1− Y )(MY −1− GY −1), c2= σ2T + CT Γ(2− Y )(MY −2+ GY −2), c4= CT Γ(4− Y )(MY −4+ GY −4).
In these models, Δt = tk− tk−1 = T /(M L), 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 T6500 @2.1GHz 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 1 0.1 0.2 – – – –
CGMY 1 0.1 0 1 5 5 1.5
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 hk(y) and its derivative hk(y) at each step, while the secant method only requires the evaluation of hk(y). In fact, we compare two methods via computing the prices of a Bermudan barrier put option with the parameters: S0 = 100, K = 80, H = 120, R0= 0, M = 10, and L = 1 under the CGMY model, we stop the iterations when hk(yn) < 0with 0= 10−8, 10−10, 10−12, respectively. Here we first set the initial points y1 = 0 and y2 =−0.1 (y1= 0 for Newton’s method), and then set the initial points y1 = x∗k, y2= x∗k− 0.1 (y1= x∗k for Newton’s method). Table 2 gives the CPU times under different N and different tolerance 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.
Table 2: CPU time (s) to compute the option with Newton’s and secant methods.
Newton’s method Secant method
N 10−8 10−10 10−12 10−8 10−10 10−12
28 9.9542e − 2 1.0531e − 1 1.0835e − 1 9.1114e − 2 9.2299e − 2 9.5263e − 2 29 1.9313e − 1 2.0468e − 1 2.0638e − 1 1.7338e − 1 1.7755e − 1 1.8303e − 1 210 4.0092e − 1 4.1982e − 1 4.2310e − 1 3.6372e − 1 3.6964e − 1 3.7918e − 1
Here we consider the options with the parameters: S0 = 100, R0 = 0, M = 10 and L = 1, and set K = 110 for the BS model; K = 80 for the CGMY model. We use the option prices for N = 212 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 = 28. The errors fluctuate randomly when N > 28, 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 = 150
26 5.8214e− 05 6.1801e − 05 1.8165e − 05 9.1130e − 05 4.4563e − 05 27 1.4495e− 12 1.5660e − 11 7.6330e − 12 4.3752e − 12 1.7870e − 12 28 7.1054e− 15 0 7.1054e− 15 7.1054e − 15 2.1316e − 14 29 1.4211e− 14 7.1054e − 15 7.1054e − 15 0 7.1054e− 15 210 1.4211e− 14 7.1054e − 15 0 7.1054e− 15 7.1054e − 15
Table 4: Errors for the CGMY model against different N .
Bermudan Bermudan barrier
N H = 120 H = 170 H = 220 H = 270
26 1.3461e− 05 9.3043e − 03 5.3004e − 03 2.1493e − 03 2.9605e − 03 27 5.6179e− 10 7.3241e − 08 4.1628e − 08 3.9348e − 08 2.7914e − 08 28 2.1316e− 14 7.1054e − 15 0 7.1054e− 15 2.8422e − 14 29 4.2633e− 14 2.1316e − 14 1.4211e − 14 0 0 210 0 3.1974e− 14 3.5527e − 14 7.1054e − 15 2.1316e − 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
BS model because the volatility of the CGMY model is bigger than BS model.
Table 5: Prices of two options when N = 28for BS model.
Bermudan Bermudan barrier
H = 120 H = 130 H = 140 H = 150
10.47952012320 10.24498509994 10.44989087879 10.47695316176 10.47935297138 L = 1 −− 10.14611451110 10.42398742716 10.46532168741 10.46936227859 L = 3 −− 10.11165088729 10.41594867201 10.46306575972 10.46782445496 L = 5
Table 6: Prices of two options when N = 28for the CGMY model.
Bermudan option Bermudan barrier option
H = 120 H = 170 H = 220 H = 270
28.82978198901 19.62954741745 25.59734095601 27.59331042434 28.30521336917 L = 1 −− 19.42204685863 24.15394584645 24.62405691569 24.68015212608 L = 3 −− 19.93476082685 23.42152751020 23.55515289278 23.56245265804 L = 5
From the tables above, we see that, no matter which model we choose, the algorithm con-verges 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: St∗
k = Kex ∗
k, tk ∈ Te, 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, and L = 5, respectively.
Table 7: Early-exercise prices for the CGMY model at H = 220 against different L.
L = 1 t1 t2 t3 t4 t5 t6 t7 t8 t9 S∗ tk 19.364 20.198 21.200 22.422 23.951 25.925 28.606 32.565 39.534 L = 3 t3 t6 t9 t12 t15 t18 t21 t24 t27 S∗ tk 14.769 15.618 16.784 18.411 20.716 24.025 28.844 36.027 47.245 L = 5 t5 t10 t15 t20 t25 t30 t35 t40 t45 S∗ tk 13.526 14.495 15.890 17.877 20.685 24.648 30.270 38.354 50.299
Acknowledgments. The authors are very grateful to the referee for his very valuable sug-gestions 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: 447-471.
[2] O E Barndorff-Nielsen. Normal inverse Gaussian distributions and stochastic volatility modelling, Scand J Statist, 1997, 24 (1): 1-13.
[3] D S Bates. Jumps and stochastic volatility: exchange rate processes implicit in Deutsch mark options, Rev Financ Stud, 1996, 9: 69-107.
[4] F Black, M Scholes. The pricing of options and corporate liabilities, J Polit Econ, 1973, 81 (3): 637-654.
[5] P Carr, H Geman, D B Madan, M Yor. The fine structure of asset returns: An empirical investigation, J Bus, 2002, 75 (2): 305-332.
[6] P Carr, D Madan. Option valuation using the fast Fourier transform, J Comput Financ, 1999, 2: 61-73.
[7] R Cont, P Tankov. Financial Modelling with Jump Processes, Chapman & Hall/ CRC Press, 2003.
[8] C Dean, J F Lawless, G E Willmot. A mixed Poisson-Inverse-Gaussian regression model, Canad J Statist, 1989, 17 (2): 171-181.
[9] D Ding, S C U. Efficient option pricing methods based on Fourier series expansions, J Math Res Exposition, 2010, 31: 12-22.
[10] F Fang, C W Oosterlee. A novel pricing method for European option based on Fourier-cosine series expansions, SIAM J Sci Comput, 2008, 31: 826-848.
[11] F Fang, C W Oosterlee. Pricing early-exercise and discrete barrier options by Fourier-cosine series expansions, Numer Math, 2009, 114: 27-62.
[12] L Feng, V Linetsky. Pricing discretely monitored barrier options and defaultable bonds in L´evy process models: a fast Hilbert transform approach, Math Financ, 2008, 18: 337-384.
[13] B Gao, J Huang, M Subrahmanyam. The valuation of American barrier options using the de-composition technique, J Econ Dyn Control, 2000, 24: 1783-1827.
[14] S L Heston. A closed-form solution for options with stochastic volatility with applications to bond and currency options, Rev Financ Stud, 1993, 6: 327-343.
[15] S G Kou. A jump-diffusion model for option pricing, Manage Sci, 2002, 48: 1086-1101.
[16] R Lord, F Fang, F Bervoets, C W Oosterlee. A fast and accurate FFT-based method for pricing early-exercise options under L´evy processes, SIAM J Sci Comput, 2008, 30: 1678-1705.
[17] D B Madan, P Carr, E Chang. The variance gamma process and option pricing, Eur Financ Rev, 1998, 2: 79-105.
[18] C R Merton. Option pricing when underlying stock returns are discontinuous. J Financ Econ, 1976, 3: 125-144.
1 Department of Mathematics, University of Macau, Macao, China.
Email: [email protected], [email protected]
2 Statistics Department, China Banking Regulatory Commission, Beijing 100140, China.