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A NEW SPECTRAL ELEMENT METHOD FOR PRICING EUROPEAN OPTIONS UNDER THE BLACK-SCHOLES AND MERTON JUMP DIFFUSION MODELS FENG CHEN†, JIE SHEN‡ AND HAIJUN YU§

A BSTRACT. We present a new spectral element method for solving partial integro–differential equations for pricing European options under the Black-Scholes and Merton jump diffusion models. Our main contributions are: (i) using an optimal set of orthogonal polynomial bases to yield banded linear systems and to achieve spectral accuracy; (ii) using Laguerre functions for the approximations on the semi-infinite domain, to avoid the domain truncation; and (iii) deriving a rigorous proof of stability for the time discretizations of European put options under both the Black-Scholes model and the Merton jump diffusion model. The new method is flexible for handling different boundary conditions and non-smooth initial conditions for various contingent claims. Numerical examples are presented to demonstrate the efficiency and accuracy of the new method.

1. I NTRODUCTION AND M OTIVATION In the classical Black–Scholes model [4], the stock price is a standard Wiener process which is continuous in time. In order to describe phenomena such as unexpected economic turmoils, the jump diffusion model is introduced in [25]. In the jump diffusion model, the dynamics of the underline asset is a Lévy process, which, on the one hand, preserves many desirable properties of the Wiener process, such as independent and stationary increments, on the other hand, can capture the feature of volatility smile that is absent in the Black–Scholes model. Therefore the jump diffusion model remains one of the most active areas of financial mathematics (c.f., [10, 7, 3, 36, 35, 27, 40]). Option pricing is an important problem in financial mathematics. The pricing for the European style contingent claims has been extensively studied. Though analytic solutions are most preferred by practitioners (as indicated in [6]), they are not available for many existing financial derivatives on the market. In general, one can write the option price as the conditional expectation of the payoff function and compute it with Monte–Carlo simulations, which is widely used in financial industry. Another popular approach for pricing options, the transform method, was introduced in [8] and unified in [21]. It yields an elegant and fast solution to a wide variety of options. Its another advantage is the immediate extension to the multi-asset cases (c.f., [22, 18]). However, it requires analytic representations of characteristic functions of the stock price, which are not always available. Key words and phrases. spectral element, spectral–Galerkin, unbounded domain, Laguerre functions, option pricing, Black–Scholes, Merton jump diffusion. This work is partially supported by NSF DMS-0915066 and AFOSR FA9550-08-1-0416. †Department of Mathematics, Purdue University, West Lafayette, IN 47907-1957 ([email protected]). ‡ (Corresponding author) Department of Mathematics, Purdue University, West Lafayette, IN 47907-1957 ([email protected]). §LSEC, Institute of Computational Mathematics, Academy of Mathematics and System Sciences, Chinese Academy of Sciences, Beijing 100190, Peoples Republic of China ([email protected]). 1

2

FENG CHEN, JIE SHEN & HAIJUN YU

In this paper we focus on the PDE approach for the option pricing, and attempt to address three of the main difficulties in this approach. First, the infinite generator of the Lévy process is a nonlocal operator with an integral term. Thus a partial integro–differential equation (PIDE) for the option price is derived within the Black–Scholes framework. Second, many payoff functions possess slope discontinuities which might cause unwanted numerical oscillations in numerical solutions of option prices. Third, the PIDE is defined on an unbounded domain, which is often truncated in practice with potential large truncation errors. Therefore an accurate right boundary condition becomes crucial for the numerical computation. This is not a serious issue for the Black–Scholes model, in which the solution decays very fast and the truncated region does not need to be large. However, in some cases such as the Merton jump diffusion model we consider here, the decaying rate is slow, so a large domain with a large number of grid points have to be used with the domain truncation. Another way to get a correct boundary condition with the localization is to use the transparent boundary condition (cf. [1]), which is popular in computational electromagnetics. However, this requires a considerable amount of calculations which affects its flexibility. For the spatial discretization, the finite difference method and finite element method (c.f., [9, 14, 35, 2, 37]) have been used extensively in the community of computational finance and has achieved successes in many cases for its effectiveness and flexibility. However, as a low order method, a considerable number of grid points is needed to resolve the solution. Due to the recent thriving of algorithmic trading, the financial market becomes more sensitive to small bid–offer spread. It has at least two significant implications. First, the pricing process should be fast enough to allow a large amount of trading operations within a short time of period (i.e., high frequency trading). Secondly, it is beneficial to have a high order accuracy of the numerical solution. Therefore we consider in this paper a new spectral element method for the option pricing. It possesses both the spectral accuracy of the spectral method and the domain flexibility of the finite element method. Although the spectral element method is widely used in computational sciences and engineerings (cf. [19, 13]), its application to the computational finance is still limited. Its main advantage is that, for a given accuracy, much less grid points, and in many case much less computing times, are needed comparing to low order methods. In fact, to achieve a fixed accuracy, the numerical results presented in Section 5 show that our new spectral-element method is more efficient than a usual spectral-element method in [40] and can be orders of magnitude faster than a low-order finite difference method. It is well-known that option prices and their derivatives usually change dramatically near slope discontinuities of the payoff functions. By splitting the domain at the slope discontinuities, the solution can be effectively resolved by the spectral element method (c.f., [40, 38, 39]). Thus there is no need to artificially smooth the initial data as in [26]. However, previous spectral element implementations use the nodal Lagrange polynomial basis functions which result in linear system with dense block diagonals. In this paper, we use modal bases functions (c.f., [28, 30]) which lead to linear systems with banded block diagonals so it can be efficiently solved. Unlike in previous approach, we treat the unbounded domain directly by using an infinite element, avoiding the domain truncation error. We refer to [30] for the recent survey of spectral methods on unbounded domains, [23] for pricing game options by using Hermite polynomials, and [12] for evaluating bonds by using Laguerre polynomials. Another advantage of having a spectral representation is that the hedging strategies ∆ and Γ, which are first and second derivatives of the option price [20, 34], can be computed directly and accurately from the spectral representation. This is much more efficient and accurate than using a lower order method. Another difficulty in solving the PIDE is how to treat the integral term. Successful methods for implicit treatments include the fixed point iteration [14], the extrapolation [17], solving its associated heat

A NEW SPECTRAL ELEMENT METHOD FOR OPTION PRICING

3

equations [24], etc. We shall treat the integral term in the PIDE explicitly with a Crank-NicholsonLeap-Frog (CNLF) scheme which we shall prove is stable. Numerical experiments show this approach is very efficient. The rest of the paper is organized as follows. In Section 2 we briefly describe the Lévy driven market model and introduce the PIDE for the European option pricing under the Merton jump diffusion model. In Section 3 we describe our time advancing schemes and prove their stabilities. We then present in Section 4 our spectral element method together with the treatment of the integral. Numerical results of the European put option in the Black–Scholes model and the Merton jump diffusion model are given in Section 5. We end the paper with a few concluding remarks. 2. M ARKETS D RIVEN BY L E´ VY P ROCESSES √ Let Xt be a Lévy process defined on a probability space (Ω, F, P). Denote ι = −1. The Lévy– Khintchine representation theorem states that the characteristic function of Xt can be written as EP [e−ιzXt ] = e−tψ(z)

(2.1) with the Lévy–Khintchine exponent (2.2)

σ2z2 + ιγz + ψ(z) = − − 2

Z

∞

−∞

(eιzx − 1 − ιzxχ|x|61 )ν(dx) ,

where χΩ stands for the characteristic function for the set Ω. The triplet (σ, γ, ν) is known as the generating Lévy triplet with σ > 0, γ ∈ R, and ν being a positive measure satisfying Z 1 Z 2 (2.3) x ν(dx) < ∞, ν(dx) < ∞. −1

|x|>1

In the classic Black–Scholes model, Xt is a standard Brownian motion, so ν = 0. In the Merton jump diffusion model, Xt is the sum of a Brownian motion and a compound Poisson process: Xt = σWt +

Nt X

Ui , i=1 R∞ −∞ ν(dx)

(2.4)

< ∞ and Ui ’s are independent and where Nt is a Poisson process with intensity λ = identically distributed (i.i.d.) random variables with density ν(dx)/λ. In this case, (2.2) is reduced to Z ∞ σ2z2 + ιγ0 z + (eιzx − 1)ν(dx) . (2.5) ψ(z) = − − 2 −∞ Suppose the dynamics of the stock price is given by St = S0 ert+Xt ,

(2.6)

where r is the constant interest rate. The infinitesimal generator of the Markovian process St is an integro–differential operator defined as: E[f (x + St )] − f (x) t→0 t Z 2 2 σ x = fxx − rxfx − rf − (f (xey ) − f (x) − x(ey − 1)fx )ν(dy). 2 R

LSt f (x) = lim (2.7)

4


There exists an equivalent risk–neutral measure, Q, under which e−rt St becomes a martingale as long as the triplet (σ, γ, ν) satisfies the following conditions: Z ν(dy)ey < ∞, |y|>1 (2.8) Z σ2 γ = − − (e−1 − yχ|y|>1 )ν(dy). 2 Consider a contingent claim with a deterministic maturity T and payoff VT . According to the noarbitrage principle, its price V (S, t) at a given time t and St = S is V (S, t) = EQ [e−r(T −t) VT |St = S].

(2.9)

By applying Itô’s formula to V (St , t), we obtain the partial integro–differential equations (PIDE) on an unbounded domain: (2.10)

Vt (S, t) + LSt V (S, t) = 0,

S ∈ (0, ∞),

t>0

with the terminal condition V (S, T ) = VT .

(2.11)

Here we restrict the volatility σ(S, T ) to be non-vanishing. Viscosity solutions of the PIDE are considered in [11] for the pure jump model, i.e., σ(S, T ) ≡ 0. In the case of the European option under the Merton jump diffusion, Ui ’s in (2.4) are identically, independently and normally distributed. With the substitution v(S, τ ) = V (S, T − τ ), we obtain the following Cauchy problem: Z ∞ σ2S 2 (2.12) vτ = vSS + (r − λκ)SvS − (r + λ)v + λ v(ηS, τ )G(η)dη 2 0 with the integral kernel G(η) = √

(2.13)

(ln η−m)2 1 e− 2δ2 2πδη

and the initial condition for a put option v(S, 0) = max{K − S, 0},

(2.14)

where K is the strike price, m = EUi and κ = E(Ui − 1). The payoff function of (2.14) has a slope discontinuity at S = K. When λ = 0, the equation (2.12) reduces to the celebrated Black–Scholes equation: σ2S 2 vSS + rSvS − rv, 2 The hedging strategies are given by (2.15)

(2.16)

vτ =

∆ = vS ,

S ∈ (0, +∞),

τ > 0.

Γ = vSS .

While it is well-known that the solution of (2.10) goes to zero at the right infinity (cf, [11]), the rate of decay can be very slow in the Merton jump diffusion model. Consequently, when S is large, domain truncation may introduce significant errors. In the example (5.1), a truncation at S = 20K with a homogeneous Dirichlet boundary condition will introduce a truncation error in the order of 10−3 (see Figure 1). This is the main reason that we deal with the semi-infinite domain directly (see Section 4.1). Other kinds of options with different payoff functions and other markets driven by different


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x 10

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1

0.8

0.6

0.4

0.2 1000

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1700

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1900

2000

F IGURE 1. the European put option under the Merton jump diffusion model with parameters in (5.1); The x-axis represents the stock price, and y-axis the option price. Lévy processes can also be described by (2.10) and (2.11) with a decaying condition as S → +∞. Thus our method maybe used for pricing contingent claims with fixed maturities. 3. T IME D ISCRETIZATIONS We first rewrite (2.12) in a form which more convenient for numerical approximations: Z ∞ σ2 2 2 (3.1) vτ = (S vS )S + (r − λκ − σ )SvS − (r + λ)v + λ v(ηS, τ )G(η)dη. 2 0 We define differential operators σ2 2 (S vS )S + (r − σ 2 )SvS − rv, 2 (3.2) σ2 2 LM v , (S vS )S + (r − λκ − σ 2 )SvS − (r + λ)v, 2 and the integral operator Z ∞ (3.3) Rv(S) , v(ηS)G(η)dη. LBS v ,

0

With a substitution of ηS = Z in (3.3), we have Z ∞ ¯ (3.4) Rv(S) = v(Z)G(Z, S)dZ, 0

where Z

(3.5)

2

(ln −m) S 1 Z 1 ¯ G(Z, S) , G( ) = √ e− 2δ2 . S S 2πδZ

6


With the above notations, the Black-Scholes equation, i.e., (3.1) with λ = 0, becomes (3.6) and the PIDE (3.1) becomes

vt = LBS v, vt = LM v + Rv.

(3.7)

Consider the second order Crank–Nicholson scheme for (3.6): v k+1 − v k LBS v k + LBS v k+1 = . δt 2 At each time step, one needs to solve the following model problem, a linear elliptic equation with variable coefficients: ( − c1 (x2 v 0 (x))0 + c2 xv 0 (x) + c3 v(x) = f (x), x ∈ Ω = (0, +∞), (3.9) lim v(x) = 0,

(3.8)

x→∞

where c1 , c2 , and c3 are constants depending on σ, r, and δt. Note that the system (3.9) does not need any left boundary condition to guarantee its well-posedness due to the degenerate nature of x2 in the leading term. Theorem 3.1. The scheme (3.8) is unconditionally stable. More precisely, we have for all kδt 6 T , ( k+1 kv k 6 kv k k, if α 6 0, (3.10) k 2 2αT 0 2 kv k 6 Ce kv k , if α > 0, where α ,

(r−σ 2 )2 −2rσ 2 . 4σ 2

Proof. Taking the inner product of (3.8) with v k+1 + v k , we derive kv k+1 k2 − kv k k2 σ 2 kS(vSk+1 + vSk )k2 =− δt 2 2 k+1 k k+1 + vk ) (S(vS + vS ), v kv k+1 + v k k2 (3.11) + (r − σ 2 ) −r . 2 2 A simple manipulation leads to (3.12)

kv k+1 k2 − kv k k2 σ2 = − P k + αkv k+1 + v k k2 , δt 4

where r − σ 2 k+1 (v + v k )k2 . σ2 Therefore, we have kv k+1 k 6 kv k k, if α 6 0. When α > 0, we apply the triangle inequality on kv k+1 + v k k2 in (3.12) to get

(3.13)

P k = kS(vSk+1 + vSk ) −

kv k+1 k2 − kv k k2 6 2α(kv k+1 k2 + kv k k2 ), δt

(3.14) which implies (3.15)

kv k+1 k2 6

1 + 2αδt k 2 1 + 2αδt k+1 0 2 kv k 6 ( ) kv k 6 Ce2αT kv 0 k2 . 1 − 2αδt 1 − 2αδt


7

Next we consider the Crank-Nicholson Leap-Frog scheme for (3.7): (3.16)

v k+1 − v k−1 LM v k+1 + LM v k−1 = + λRv k , 2δt 2

∀k ≥ 1,

with v1 − v0 = LM v 1 + λRv 0 . δt Note that the form of (3.4) is suitable for the numerical implementation since the integral term is treated explicitly in the time discretization (see Section 4.2). In order to prove the stability for the scheme (3.16), we need the following lemma (c.f., [16]). (3.17)

Lemma 3.2 (heat kernel). Consider the one dimensional heat equation:  ut = uxx , (x, t) ∈ R × (0, T ],   u(x, 0) = g(x), (3.18)   lim u(x) = 0. x→±∞

Assume that g is bounded, continuous, nonnegative, and non-vanishing, then Z +∞ (x − y)2 1 exp[− √ ]g(y)dy. (3.19) u(x, t) = √ √ 2π 2t −∞ 2( 2t)2 Without loss of generality, we set m in (2.13) to be zero in the following analysis. When m 6= 0, one can shift the coordinate accordingly to get similar results. Next lemma provides the exact solution to a one dimensional advection-diffusion problem and establishes its connection with the integral kernel of our problem. 2

Lemma 3.3 (advection–diffusion). Assume m = 0, then Rv(S) = w(S, δ2 ), where w(S, t) is the solution to the following one dimensional advection-diffusion equation:   w = S 2 wSS + SwS , (S, t) ∈ R+ × (0, T ],   t w(S, 0) = v(S), (3.20)    lim w(S, t) = 0. S→∞

Proof. Consider the coordinate transformation x = ln S,

(3.21)

S = ex .

Define u(x) , w(S(x)) = w(ex ), then (3.22)

ut = wt ,

ux = SwS ,

uxx = SwS + S 2 wSS .

So the system (3.20) is transformed into  ut = uxx , (x, t) ∈ R × (0, T ],   u(x, 0) = v(ex ), (3.23)   lim u(x) = 0. x→±∞

According to Lemma 3.2, we have (3.24)

1 u(x, t) = √ √ 2π 2t

Z

+∞

exp[− −∞

(x − y)2 √ ]v(ey )dy. 2( 2t)2

8


Next we transform the coordinate back: S = ex ,

(3.25)

Z = ey .

The solution in (3.24) becomes (3.26)

1 w(S, t) = √ √ 2π 2t

Therefore Rv(S) = w(S,

δ2 2

Z

+∞

0

(ln S )2 v(Z) exp[− √Z ] dZ. 2( 2t)2 Z

) by comparing (3.26) with (3.4).

δ2

Lemma 3.4 (L2 –bound). kRv(S)k2 6 C(δ)kv(S)k2 , where C(δ) = e 2 . 2

Proof. According to Lemma 3.3, it is sufficient to estimate w(S, δ2 ). Taking the inner product of (3.20) with w, we have (3.27)

1d kwk2 = −kSwS k2 − (SwS , w). 2 dt

In addition, (3.28)

1 1 1 − kwk2 ) = − kwk2 . (SwS , w) = (S, (w2 )S ) = (Sw2 |S=+∞ S=0 2 2 2

Combining (3.27) and (3.28), we get (3.29)

d kwk2 6 kwk2 . dt 2

By using the Gronwall’s inequality on [0, δ2 ], we derive (3.30)

kRv(S)k2 = kw(S,

δ2 δ2 2 )k 6 e 2 kw(S, 0)k. 2

Theorem 3.5 (L2 –stability). There exists a constant β depending on α, λ and δ such that for all δt < β1 , we have (3.31)

kv m k2 6 Ckv 0 k2 ,

∀2 ≤ m ≤

T − 1, δt

where C is a generic constant depending on α, λ, δ and T . Proof. Taking inner product of (3.16) with v k+1 + v k−1 , we have (3.32)

kv k+1 k2 − kv k−1 k2 σ2 = − P k + αkv k+1 + v k−1 k2 + λ(Rv k , v k+1 + v k−1 ), 2δt 4

where (3.33)

P k = kS(vSk+1 + vSk−1 ) −

r − σ 2 k+1 (v + v k−1 )k2 . σ2


9

Applying the triangle inequality and Lemma 3.4, we derive

(3.34)

kv k+1 k2 − kv k−1 k2 6 2α(kv k+1 k2 + kv k−1 k2 ) 2δt λ + (C(δ)kv k k2 + 2kv k+1 k2 + 2kv k−1 k2 ) 2 1 6 (β1 kv k+1 k2 + β0 kv k k2 + β−1 kv k−1 k2 ) 2 β 6 (kv k+1 k2 + kv k k2 + kv k−1 k2 ), 6

where β = max{β1 , β0 , β−1 }. 3 Without loss of generality, we assume m to be an even number. Summing up (3.34) for odd k between 1 and m − 1, we derive

(3.35)

β1 = β−1 = 4α + 2λ,

β0 = C(δ)λ,

m−2 m−1 m β X k 2 X k 2 X k 2 kv m k2 − kv 0 k2 kv k ) kv k + kv k + 6 ( δt 3

(3.36)

6

k=2 k odd m X

k=1 k odd m−1 X

k=0 k odd m−2 X

k=2

k=1

k=0

β ( 3

6β

m X k=0

kv k k2 +

kv k k2 +

kv k k2 )

kv k k2 .

Rearranging terms in (3.36), we get (3.37)

m 2

kv k 6 δtβ

m X k=0

kv k k2 + kv 0 k2 .

We can then conclude (3.31) by applying the discrete Gronwall’s inequality (see, for instance, Lemma B.10 in [32]) to the above. 4. S PATIAL D ISCRETIZATIONS We describe in this section our spectral element method for (3.9) with special treatments for the integral term. 4.1. The Spectral Element Method. Consider the following function space: (4.1)

1 HB = {u ∈ H 1 (0, ∞) : lim u(x) = 0}. x→∞

1 such that The week formulation of (3.9) is: Find u ∈ HB

(4.2)

c1 (xu0 , xv 0 ) + c2 (xu0 , v) + c3 (u, v) = (f, v),

1 ∀v ∈ HB .

Next we split the domain Ω = (0, ∞) into M non-overlapping elements: (4.3)

Ω1 = (x0 , x1 ), · · · , Ωi = (xi−1 , xi ), · · · , ΩM = (xM −1 , xM ),

10


where 0 = x0 < x1 < · · · < xM = ∞. We define the approximation space (4.4)

ˆ n }, XN = {u ∈ C(Ω); u|Ωi ∈ Pni , 1 6 i 6 M − 1; u|ΩM ∈ P M

ˆ k = e−x/2 Pk . where N stands for the dimension of XN , Pk is the k-th degree polynomial space, and P Since the initial condition has a slope discontinuity at S = K, it is important that K is one of the −1 nodes {xi }M i=1 so that the solution can be well approximated in the approximation space XN . The spectral element method for (3.9) is: Find uN such that (4.5)

0 < AuN , vN >:= (xu0N , xvN ) + (xu0N , vN ) + (uN , vN ) = (IN f, vN ),

∀vN ∈ XN .

where IN : C(Ω) → XN is an interpolation operator based on Legendre-Gauss-Lobatto points at all subintervals Ωi . We now construct a set of basis functions for XN . We start with the interior modal basis functions −1 {φij (x)}16i6M 06j6ni −2 for Ωi , 1 ≤ i ≤ M − 1. ( Lij (x) − Lij+2 (x), x ∈ Ωi , i (4.6) φj (x) = 0, else, where Lij ’s are Legendre polynomials defined on the element Ωi , and ni ’s are numbers of Gaussian points assigned on Ωi . For the unbounded domain ΩM , the interior modal basis functions are φk = Lbk − Lbk+1 ,

(4.7)

where {Lbk }06k6nM −1 are Laguerre functions scaled on ΩM . The nodal basis functions at {xi }1≤i≤M −2 are  xi+1 − x  , x ∈ Ωi ,    2d i+1  (4.8) I i = x − xi−1 , x ∈ Ωi−1 ,   2di    0, otherwise, with di = (4.9)

xi −xi−1 . 2

Finally, the nodal basis functions at x0 and xM −1 are given by   x1 − x , x ∈ Ω1 , 0 2d1 I =  0, otherwise,

and

(4.10)

I M −1

 x−xM −1  e − 2 , x ∈ ΩM ,    = x − xM −1 , x ∈ ΩM −1 ,  2dM −1    0, otherwise.

It is easy to verify that the above basis functions form a basis for XN with the number of degrees of P freedom N = M i=1 (ni − 1) + (M − 1).

A NEW SPECTRAL ELEMENT METHOD FOR OPTION PRICING I0

I1

b

Ii−1

b

b

x1

x0

Ii

IK−1

b b

xK−1

xi

xi−1

11

xK = ∞

F IGURE 2. nodal functions defined on an unbounded domain. Plugging in uN =

(4.11)

M nX i −2 X

u1ki φij +

i=1 j=0

M −1 X

u2i I i

i=1

in (4.5) and running vN through all basis functions of XN , the equation (4.5) reduces to the following linear system A B1 u ¯1 f¯1 (4.12) = ¯ , B2 C u ¯2 f2 where with u ¯1 (resp. u ¯2 ) is the vector consisting the coefficients of {u1ki } (resp. {u2i }) in (4.11), f¯1 (resp. f¯2 ) is the vector consisting of < f, φ1j > (resp. < f, I i >), and A = diag(A1 , A2 , · · · , AM )

(4.13) with

Aikj =< φij , φik >,

Ckj =< Ij , Ik >,

B1 and B2 are rectangular matrices with entries < Aφij , Ik > and < AIj , φik >, respectively. Thanks to the properties of Legendre polynomial and Laguerre functions, the matrices {Ai }, C, B1 and B2 are sparsely banded (cf. Fig. 3). The linear system (4.12) can be efficiently solved by using a block stiffness 0

advection 0

mass 0

5

5

5

10

10

10

15

15

15

20

20

20

25

25 0

10 20 nz = 131

25 0

10 20 nz = 118

0

10 nz = 81

20

F IGURE 3. with 3 elements and 8 Gaussian points in each element Gaussian elimination as follows: I A−1 B1 A−1 f¯1 u ¯1 (4.14) = . u ¯2 0 C − B2T A−1 B1 f2 − B2T A−1 f¯1

12


Therefore, the computational complexity of our method is dominated by the transforms between P 2 physical values and spectral representation with O( M k=1 nk ) flops. This cost can be reduced to PM O( k=1 nk log nk ) (with a much large constant) if we use a Chebyshev-Legendre approach on each subdomain (cf. [15, 29, 31]). 4.2. Discretization of the Integral Term. Next we deal with the integral term in (2.12). Consider −1 first a direct approximation of Rv(S) by using Legendre-Gauss-Lobatto quadratures on {Ωi }M i=1 and Laguerre-Gauss-Radau quadrature on ΩM : (4.15)

M Z X

Rv(S) =

Ωi

i=1

¯ v(Z)G(Z, S)dZ ≈

M X i=1

Ji

ni X

i i ¯ ωji G(Z(ξ j ), S)v(Z(ξj )).

j=0

where Ji is the Jacobian of the affine mapping which maps Ωi to (−1, 1) or (0, ∞), wji are the quadrature weights, and Z(ξji ) are the mapped quadrature points on Ωi . 5

3

x 10

4

S=1e−5 3

x 10

S=1e−4

S=1e−3 3000

2

2

2000

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1

1000

0

0

1

2

3

0

0

2

4

−5

0

x 10 S=1e−2

S=1e−1

S=1e0

200

20

2

100

10

1

0.02

0.03

0

0

S=1e1

0.2

0.4

0

0.3

0

1 −3

S=1e2

0.4

3 −3

3

0.01

2 x 10

30

0

1

x 10

300

0

0

−4

0.03

3

0.02

2

0.01

1

x 10

2

3

2000

3000

S=1e3

0.2 0.1 0

0

10

20

30

0

0

100

200

300

0

0

1000

¯ F IGURE 4. G(Z, S) as a function of Z for different S. ¯ Note that G(Z, S) as a function of Z (see Figure 4) has a peak of magnitude O(S −1 ) in the interval (0, S]. In other words, there is a large gradient drop when S is small. Hence, a large number of quadrature points, much larger than what is needed to accurately represent the approximation solution, is needed in Ω1 to achieve the required accuracy. A simple fix for this difficulty is to use the overintegration technique proposed in [40]. Namely, we use n1 points to represent uN in Ω1 but n∗1 >> n1 points to over-integrate the integral term in Ω1 .


13

5. N UMERICAL R ESULTS We present in this section some numerical results for European put options under both the Black– Scholes model and the Merton jump diffusion model. The corresponding call option price can be calculated by using the put–call parity. We set the following benchmark parameters throughout all the implementations: T = 0.25, K = 100, r = 0.05, σ = 0.15, (5.1) λ = 0.1, δ = 0.45, m = −0.9. We recall that an analytic solution to the PIDE (2.12) is: ∞ −λ0 T X e (λ0 T )n (5.2) v= fn , n! n=0

m+ 21 δ 2

λ0

− 1, γ = ln(1 + κ), and fn is the Black–Scholes option price with where = λ(1 + κ), κ = e instantaneous variance σ 2 + nδ 2 /T and risk–free rate r − λκ + nγ/T . We will use (5.2) to validate our numerical methods. 5.1. Black-Scholes model. We use the scheme (3.8) with three elements for the spatial discretization: Ω1 = (0, K),

(5.3)

Ω2 = (K, 2K),

Ω3 = (2K, ∞).

Figure 5 shows the agreement between the numerical solution and the theoretical one with δt = 10−3 and the following number of quadrature points in each element: n1 = 20,

(5.4)

n2 = 20,

n3 = 6.

As is well-known (cf. [5]), when using a spectral method for unbounded domains, a proper scaling parameter can often significantly increase the efficiency. In the case of Laguerre functions for Ω3 , we can choose the scaling parameter as follows. Assuming that the solution is essentially zero for ˜ Then, if 2K + ξ is the largest Laguerre-Gauss-Radau point, we set the scaling parameter S > 2K + ξ. ˜ In this case, the corresponding scaling parameters for Ω3 are: to be α = ξ/ξ. (5.5) ξ˜ = K = 100, ξ ≈ 17.65, α ≈ 0.18. Next we examine the temporal and spatial accuracy. We first fix ni in (5.4) and change the time step delta

put

gamma 0.06

100

0

80

−0.2

60

−0.4

40

−0.6

0.04

0.02

−0.8

20

0

−1

0 0

100

200

300

0

100

200

300

0

100

200

300

F IGURE 5. Black-Scholes model. Line: exact; Dotted: numerical. δt gradually. We plot in Figure 6 the error calculated at the strike which is the point of most financial interest. The dotted lines in Figure 6 have the slope of 2 which indicates a second order convergence

14


in time. Next, we fix δt = 10−3 and plot in Fig. 7 the errors with different numbers of degrees of put

delta

0

gamma

0

log10(error)

0 −2

−2

−4

−4

−6

−6

−8

−8

−8

−10

−10

−10

−4

−3 log (dt)

−2 −4 −6

−2

−4

−3 log (dt)

10

−2

−4

−3 log (dt)

10

−2

10

F IGURE 6. Black–Scholes model: errors at the strike, with different time step sizes and fixed {ni } in (5.4). freedom (DoF). We observe an exponential convergence in space.

log10(error)

put

delta

gamma

−2

−2

−2

−4

−4

−4

−6

−6

−6

20

40

60 DoF

80

20

40

60 DoF

80

20

40

60 DoF

80

F IGURE 7. Black–Scholes model: L∞ –errors, with different {ni } and a fixed δt of 10−3 . 5.2. Merton jump diffusion model. We now consider the scheme of (3.16) and (3.17) for the Merton jump diffusion model. For the spatial discretization, we use the following four elements: (5.6)

Ω1 = (0, K),

Ω2 = (K, K),

Ω3 = (K, 4K),

Ω4 = (4K, ∞).

In Figure 8, we plot the numerical and exact solutions with the following computational parameters: (5.7)

δt = 10−3 ,

n1 = 4,

n2 = n3 = 36,

n4 = 6,

= 0.1.

The corresponding scaling parameters for Ω4 are α ≈ 0.09,

ξ ≈ 53.53,

ξ˜ = 10K.

In Figure 9, we plot the errors with fixed {ni } in (5.7) and different δt, while in Figure 10, we plot the errors with fixed δt = 10−3 . Again a second order convergence in time and exponential convergence in space are observed. Note that the saturation of the error when δt is very small in Figure 9 is due to the spatial discretization as more spatial points are needed to achieve better accuracy for computing ∆ and Γ, while the saturation of error with larger DoF in Figure 10 is due to the time discretization error for fixed δt.

A NEW SPECTRAL ELEMENT METHOD FOR OPTION PRICING put

delta

15

gamma 0.06

100

0

80

−0.2

60

−0.4

40

−0.6

0.04 0.02

−0.8

20

0

−1

0 0

500

1000

0

500

1000

0

500

1000

F IGURE 8. Merton jump diffusion model. Line: exact; Dotted: numerical. In Table 1, we list the computational time for the Merton jump diffusion model on a 2.5Ghz QuadCore (but only one core is used) AMD Opteron(tm) Processor. Parameters are fixed to be those in (5.1). To accurately compute the integral term, we further divide the first subdomain [0, K] into two subdomains [0, 0.1K] and [0.1K, K]. We fix the number of spatial points in [0, 0.1K] to be 4, and use 60 points to perform over-integration in [0, 0.1K]. In terms of the accuracy, our spectral-element method appears to be more efficient than a usual spectral-element method in [40] when comparing Table 1 in this paper with Table 3 in [40], and can be orders of magnitude faster than a low-order finite difference method according to Tables 1 and 2 in [40]. Note that the result in Table 3 of [40] is obtained by using the exact Dirichlet boundary condition at the truncated boundary. However, for most real applications, exact boundary condition is not available and is usually replaced by zero. This could introduce significant irreducible errors. Hence, our method should be more accurate when applied to real problems. computing time (ms.) L2w –error No. of time steps No. of spatial points 0.6 2.40e-3 40 35 = 16 × 2 + 3 1.7 7.67e-4 80 45 = 20 × 2 + 5 4.3 4.50e-5 160 58 = 26 × 2 + 6 11.7 4.25e-6 320 72 = 34 × 2 + 6 TABLE 1. computing times for the Merton jump diffusion model.

5.3. Local Volatility Model. Our scheme can be easily applied to jump diffusion models with local volatility. So as a final example, we compute the price of European put options with a space–time dependent local volatility function under the Merton jump diffusion model. Namely, instead of (3.1), we solve the following PIDE, Z ∞ 1 2 2 2 (5.8) vτ = (σ S vS )S + [rS − λκS − (σ S)S ]vS − (r + λ)v + λ v(ηS, τ )G(η)dη, 2 0 where the local volatility function (cf. [35]) is chosen to be (5.9)

σ = σ(S, τ ) = 0.15 + 0.15(0.5 + 2τ )

(S/100 − 1/2)2 . (S/100)2 + 1.44

16

FENG CHEN, JIE SHEN & HAIJUN YU put

delta

gamma

log10(error)

0 −2

−2

−4

−4

−2 −4

−6

−6

−6

−8 −8

−8 −4

−3 log (dt)

−2

−4

−3 log (dt)

10

−2

−4

−3 log (dt)

10

−2

10

F IGURE 9. Merton jump diffusion model: errors at the strike, with different time step sizes and the fixed {ni } in (5.7). put

delta

0

gamma

0

−1

log10(error)

−1 −2

−2

−2 −3 −3

−4

−4

−4 −6 40

60

80 DoF

100

120

−5 40

60

80 DoF

100

120

−5 40

60

80 DoF

100

120

F IGURE 10. Merton jump diffusion model: L∞ –errors, with different {ni } and a fixed δt of 10−3 .

Since no exact solution is available, we plot in Figure 11 the local volatility function at different time steps and the corresponding European put prices. Parameters are chosen as in (5.1) except for σ, and δt = 10−3 .

6. C ONCLUSIONS We developed an efficient and accurate spectral element scheme for the European option pricing with the PIDE approach. Our method enjoys the following advantages: (i) The unbounded domain is treated directly by using Laguerre functions to avoid the domain truncation error; (ii) the spectral element approach with interior modal basis functions is very efficient and flexible for payoff functions with slope discontinuities and various boundary conditions; (iii) the semi-implicit Crank-NicholsonLeap-Frog scheme with explicit treatment of integral terms is stable. The numerical results show that our scheme is of second-order in time and exponentially convergent in space, despite the slope discontinuity. The numerical results also indicate that our method would be more efficient and accurate for real applications where the solution at the right truncated boundary is unknown. While we have only considered the option pricing with one asset case here, we plan to use some of the techniques developed in this paper, together with the fast spectral sparse grid methods developed recently [33]) to deal with price contingent claims with two or more assets.


0.22

17

100

0.21 80 0.2 60

0.19 0.18

40

0.17 20 0.16 0.15

0 0

50

100

150

200

0

50

100

150

200

F IGURE 11. Left: The local volatility function at {τi } = {0, T /5, 2T /5, · · · , T } (from bottom to top). Different lines corresponding to different σ(S, τ ) as a function of S at different τi ’s. The horizontal axis is for the variable S. The vertical axis is for the value of σ(S, τ ). Right: option prices at {τi } (from bottom to top). The horizontal axis is the stock price S. The vertical axis is the put option price. R EFERENCES [1] Yves Achdou and Olivier Pironneau. Computational Methods for Option Pricing (Frontiers in Applied Mathematics) (Frontiers in Applied Mathematics 30). Society for Industrial and Applied Mathematics, Philadelphia, PA, USA, 2005. [2] Ariel Almendral and Cornelis W. Oosterlee. Numerical valuation of options with jumps in the underlying. Appl. Numer. Math., 53(1):1–18, 2005. [3] Erhan Bayraktar and Hao Xing. Pricing Asian options for jump diffusion. Mathematical Finance, 21(1):117–143, January 2011. [4] Fischer Black and Myron Scholes. The pricing of options and corporate liabilities. The Journal of Political Economy, c 1973 The 81(3):637–654, 1973. ArticleType: research-article / Full publication date: May - Jun., 1973 / Copyright University of Chicago Press. [5] John P. Boyd. Chebyshev and Fourier spectral methods. Dover Publications Inc., Mineola, NY, second edition, 2001. [6] René Carmona and Valdo Durrleman. Pricing and hedging spread options. SIAM Review, 45(4):627, 2003. [7] Peter Carr and Laurent Cousot. A PDE - approach to jump-diffusions. Quantitative Finance, 11(1):33, 2011. [8] Peter Carr, Dilip B. Madan, and Robert H Smith. Option valuation using the fast Fourier transform. Journal of Computational Finance, 2:61–73, 1999. [9] Huajie Chen, Fang Liu, Nils Reich, Christoph Winter, and Aihui Zhou. Two-Scale finite element discretizaitons for integrodifferential equations. J. Integral Equations Appl., 2011. [10] Rama Cont. Frontiers in quantitative finance: volatility and credit risk modeling. John Wiley and Sons, October 2008. [11] Rama Cont and Ekaterina Voltchkova. A finite difference scheme for option pricing in jump diffusion and exponential Lévy models. SIAM J. Numer. Anal., 43(4):1596–1626, 2005. [12] Javier de Frutos. A spectral method for bonds. Computers and Operations Research, 35:64–75, January 2008. ACM ID: 1268230. [13] M. O. Deville, P. F. Fischer, and E. H. Mund. High Order Methods for Incompressible Fluid Flow. Cambridge University Press, 1 edition, August 2002. [14] Y. d’Halluin, P. A. Forsyth, and K. R. Vetzal. Robust numerical methods for contingent claims under jump diffusion processes. IMA J Numer Anal, 25(1):87–112, January 2005. [15] W.S. Don and D. Gottlieb. The Chebyshev-Legendre method: implementing Legendre methods on Chebyshev points. SIAM J. Numer. Anal., 31:1519–1534, 1994.

18


[16] Lawrence C. Evans. Partial Differential Equations: Second Edition. American Mathematical Society, 2 edition, March 2010. [17] Liming Feng and Vadim Linetsky. Pricing options in jump-diffusion models: An extrapolation approach. OPERATIONS RESEARCH, 56(2):304–325, March 2008. [18] T. R. Hurd and Zhuowei Zhou. A Fourier transform method for spread option pricing. SIAM Journal on Financial Mathematics, 1:142–157, 2010. [19] George Em Karniadakis. Spectral/hp Element Methods for Computational Fluid Dynamics. Oxford University Press, 2005. [20] Damien Lamberton and Bernard Lapeyre. Introduction to Stochastic Calculus Applied to Finance, Second Edition. Chapman and Hall/CRC, 2 edition, November 2007. [21] Roger W. Lee. Option pricing by transform methods: extensions, unification and error control. The Journal of Computational Finance, 7(3), 2004. [22] C. C. W. Leentvaar and C. W. Oosterlee. Multi-asset option pricing using a parallel Fourier-based technique. J. Comput. Finance, 12(1):1–26, 2008. [23] Jin Ma, Jie Shen, and Yanhong Zhao. On numerical approximations of forward-backward stochastic differential equations. SIAM Journal on Numerical Analysis, 46:2636–2661, July 2008. ACM ID: 1405082. [24] A Mayo. Methods for the rapid solution of the pricing PIDEs in exponential and Merton models. Journal of Computational and Applied Mathematics, 222(1):128–143, 2008. [25] Robert C. Merton. Option pricing when underlying stock returns are discontinuous. Journal of Financial Economics, 3:125–144, 1976. [26] D. M. Pooley, K. R. Vetzal, and P. A. Forsyth. Convergence remedies for non-smooth payoffs in option pricing. J. Comput. Finance, 6:25–40, 2003. [27] Santtu Salmi and Jari Toivanen. An iterative method for pricing American options under jump-diffusion models. Applied Numerical Mathematics, 2011. [28] Jie Shen. Efficient spectral-Galerkin method I: direct solvers of second-and fourth-order equations using Legendre polynomials. SIAM J. Sci. Comput., 15(6):1489–1505, 1994. [29] Jie Shen. Efficient Chebyshev-Legendre Galerkin methods for elliptic problems. In A. V. Ilin and R. Scott, editors, Proceedings of ICOSAHOM’95, pages 233–240. Houston J. Math., 1996. [30] Jie Shen and Li lian Wang. Some recent advances on spectral methods for unbounded domains. Communicatons in Computational Physics, 5(2):195–241, 2009. [31] Jie Shen and Tao Tang. Spectral and High-Order Methods with Applications, volume 3 of Mathematics Monograph Series. Science Press, Beijing, 2006. [32] Jie Shen, Tao Tang, and Li-Lian Wang. Spectral Methods: Algorithms, Analysis and Applications, volume 41 of Springer Series in Computational Mathematics. Springer, 2011. [33] Jie Shen and Haijun Yu. Efficient spectral sparse grid methods and applications to high-dimensional elliptic problems. SIAM Journal on Scientific Computing, 32(6):3228, 2010. [34] Peter Tankov and Rama Cont. Financial Modelling with Jump Processes. Chapman and Hall/CRC, 1 edition, December 2003. [35] Jari Toivanen. Numerical valuation of European and American options under Kou’s jump-diffusion model. SIAM J. Sci. Comput., 30(4):1949–1970, 2008. [36] Jari Toivanen. A high-order front-tracking finite difference method for pricing American options under jump-diffusion models. The Journal of Computational Finance, 13(3), 2010. [37] Juergen Topper. Finite element methods in bond and option pricing. Society for Computational Economics, 131, 1999. [38] Wuming Zhu and David Kopriva. A spectral element approximation to price European options. II. the Black-Scholes model with two underlying assets. Journal of Scientific Computing, 39(3):323–339, June 2009. [39] Wuming Zhu and David Kopriva. A spectral element approximation to price European options with one asset and stochastic volatility. Journal of Scientific Computing, 42(3):426–446, March 2010. [40] Wuming Zhu and David A. Kopriva. A spectral element method to price European options. I. single asset with and without jump diffusion. J. Sci. Comput., 39(2):222–243, 2009.