A Numerical Method for Pricing Discrete Double Barrier Option by Legendre Multiwavelet
aa r X i v : . [ q -f i n . C P ] M a r A Numerical Method for Pricing Discrete DoubleBarrier Option by Legendre Multiwavelet
Amirhossein Sobhani a, ∗ , Mariyan Milev b a School of Mathematics, Iran University of Science and Technology, 16844 Tehran,Iran. b UFT-PLOVDIV, Department of Mathematics and Physics
Abstract
In this Article, a fast numerical numerical algorithm for pricing discrete doublebarrier option is presented. According to Black-Scholes model, the price of op-tion in each monitoring date can be evaluated by a recursive formula upon theheat equation solution. These recursive solutions are approximated by usingLegendre multiwavelets as orthonormal basis functions and expressed in opera-tional matrix form. The most important feature of this method is that its CPUtime is nearly invariant when monitoring dates increase. Besides, the rate ofconvergence of presented algorithm was obtained. The numerical results verifythe validity and efficiency of the numerical method.
Keywords:
Double and single barrier options, BlackScholes model, Optionpricing, Legendre Multiwavelete
1. Introduction
Barrier options play a key role in the price risk management of financialmarkets. There are two types of barrier options: single and double. In singlecase we have one barrier but in double case there are two barriers. A barrieroption is called knock-out (knock-in) if it is deactivated (activated) when the stock price touches one of the barriers. If the hitting of barriers by the stockprice is checked in fixed dates, for example weakly or monthly, the barrier optionis called discrete.Option pricing as one of the most interesting topics in the mathematicalfinance has been investigated vastly in the literature. Kamrad and Ritchken [1], Boyle and Lau [2], Kwok [3], Heyen and Kat [4], Tian [5] and Dai and Lyuu [6]used standard lattice techniques, the binomial and trinomial trees, for pricingbarrier options. Ahn et al. [7] introduce the adaptive mesh model (AMM) ∗ Corresponding author
Email addresses: [email protected], [email protected] (AmirhosseinSobhani), [email protected] (Mariyan Milev)
Preprint submitted to Elsevier March 29, 2017 hat increases the efficiency of trinomial lattices. The Monte Carlo simulationmethods were implemented in [8, 9, 10, 11, 12]. In [13, 14], numerical algorithms based on quadrature methods were proposed.Actually a great variety of semi-analytical methods to price barrier optionshave been recently developed which are based on integral transforms [15, 16, 17],or on the transition probability density function of the process used to describethe underlying asset price [13, 14, 18, 19, 20, 21, 22, 23]. These techniques are very high performing for pricing discretely monitored one and double barrieroptions and our computational results are in very good agreement with them.We would like to make the following essential remarks. An analytical solution forsingle barrier option is driven by Fusai et. al. in [15] where the problem of onebarrier is reduced to a Wiener-Hopf integral equation and a given z-transform solution of it. To derive a formula for continuous double barrier knock-out andknock-in options Pelsser inverts analytically the Laplace transform by a contourintegration [24]. Broadie et. al. have found an explicit correction formula fordiscretely monitored option with one barrier [19]. However, these three well-known methods [15, 19, 24] have not been still applied in the presence of two barriers, i.e. a discrete double barrier option. Farnoosh et al. [25, 26] presentednumerical algorithms for pricing discrete single and double barrier options withtime-dependent parameters. Also, in my last work [27] a numerical method forpricing discrete single and double barrier options by projection methods havebeen presented. This article is organized as follows. In Section 2, the process of finding priceof discrete double barrier option under the Black-Scholes model by a recursiveformula has bean explained. Definition and some features of Legendre multi-wavelets are given in section 3. In section 4, Legendre multi-wavelet expansionis implemented for pricing of discrete double barrier option. Finally, numerical results are given in section 5 to confirm efficiency of proposed method.
2. The Pricing Model
We assume that the stock price process follows geometric Brownian motion: dS t = ˆ rS t dt + σS t dB t where S , ˆ r and σ are initial stock price, risk-free rate and volatility respectively.We consider the problem of pricing knock-out discrete double barrier call option, i.e. a call option that becomes worthless if the stock price touches either loweror upper barrier at the predetermined monitoring dates:0 = t < t < · · · < t M = T. If the barriers are not touched in monitoring dates, the pay off at maturity timeis max ( S T − E, E is exercise price. The price of option is defineddiscounted expectation of pay off at the maturity time. P ( S, t, m −
1) as a func-tion of stock price at time t ∈ ( t m − , t m ), satisfies in the following partial dif-ferential equations[28] − ∂ P ∂t + ˆ rS ∂ P ∂S + 12 σ S ∂ P ∂S − ˆ r P = 0 , (1)subject to the initial conditions: P ( S, t ,
0) = ( S − E ) (max( E,L ) ≤ S ≤ U ) P ( S, t m ,
0) = P ( S, t m , m − ( L ≤ S ≤ U ) ; m = 1 , , ..., M − , where P ( S, t m , m −
1) := lim t → t m P ( S, t, m − z = ln (cid:0) SL (cid:1) the partial differential equation 1 and its initial condition is reduced as follows: − C t + µC z + σ C zz = ˆ rC (2) C ( z, t ,
0) = L (cid:16) e z − e E ∗ (cid:17) ( δ ≤ z ≤ θ ) C ( z, t m , m ) = C ( z, t m , m − (0 ≤ z ≤ θ ) ; m = 1 , , ..., M − C ( z, t, m ) := P ( S, t, m ); E ∗ = ln (cid:0) EL (cid:1) ; µ = ˆ r − σ ; θ = ln (cid:0) UL (cid:1) and δ = max { E ∗ , } . Next, by considering C ( z, t m , m ) = e αz + βt h ( z, t, m ) where: α = − µσ ; c = − σ β = αµ + α σ − ˆ r. the equation 2 is reduced to the well known heat equation: − h t + c h zz = 0 h ( z, t ,
0) = Le − αz (cid:16) e z − e E ∗ (cid:17) ( δ ≤ z ≤ θ ) ; m = 0 h ( z, t m , m ) = h ( z, t m , m − (0 ≤ θ ≤ z ) ; m = 1 , ..., M − h ( z, t, m ) = ( L R θδ k ( z − ξ, t ) e − αξ (cid:0) e ξ − e E ∗ (cid:1) dξ ; m = 0 R θ k ( z − ξ, t − t m ) h ( ξ, t m , m − dξ ; m = 1 , , ..., M − k ( z, t ) = 1 √ πc t e − z c t . (3)By assuming that monitoring dates are equally spaced, i.e; t m = mτ where τ = TM , h ( z, t m , m −
1) is a function of two variables z , m . Therefore, bydefining f m ( z ) := h ( z, t m , m − f ( z ) = Z θ k ( z − ξ, τ ) f ( ξ ) dξ (4)3 f m ( z ) = Z θ k ( z − ξ, τ ) f m − ( ξ ) dξ ; m = 2 , , ..., M (5)where f ( z ) = Le − αz (cid:16) e z − e E ∗ (cid:17) ( δ ≤ z ≤ θ ) . (6)by defining f m ( z ) := f m ( θz ) and k ( z, τ ) := θk ( θz, τ ) = 1 √ πc t e − ( θz )24 c t (7)we reach the following relations from 4,5 and 6: f ( z ) = Z k ( z − ξ, τ ) f ( ξ ) dξ (8) f m ( z ) = Z k ( z − ξ, τ ) f m − ( ξ ) dξ ; m = 2 , , ..., M (9)where f ( z ) = Le − αθz (cid:16) e θz − e E ∗ (cid:17) ( δθ ≤ z ≤ ) . (10)which helps us to use Legendre multiwavelete on interval [0 ,
3. Legendre Multiwavelet
Let L ([0 , ,
1] with the inner product < f, g > := Z f ( x ) g ( x ) dx and the norm k f k = √ < f, f > . An orthonormal multi resolution analysis (MRA) with multiplicity r of L ([0 , Definition 1.
A chain of closed functional subspaces V j , j ≥ of L ([0 , iscalled orthonormal multi resolution analysis of multiplicity r if: (i) V j ⊂ V j +1 , j ≥ . (ii) S j ≥ V j is dense in L ([0 , ,i.e. S j ≥ V j = L ([0 , . (iii) There exists a vector of orthonormal functions
Φ = [ φ , ..., φ r − ] T in L ([0 , , that is called multiscale vector, such that { φ lj,k := 2 j/ φ l (2 j x − k ); 0 ≤ l ≤ r − , ≤ k ≤ j − } form an orthonormal basis for V j . W j be subspace of V j +1 such that V j +1 = V j ⊕ W j and V j ⊥ W j , i.e. the orthogonal complement of V j in V j +1 , so we have V j = V ⊕ W ⊕ W ⊕ ...W j − (11) L ([0 , V ⊕ ∞ M j =0 W j . (12)The propertyiii of MRA shows that dim ( V j ) = dim ( W j ) = r j . Let the functionvector Ψ = [ ψ , ..., ψ r − ] be vector of orthonormal basis of W , that is calledmultiwavelet vector, then the structure of MRA implies that W j = span { ψ lj,k ; 0 ≤ l ≤ r − , ≤ k ≤ j − } , (13)where ψ lj,k := 2 j/ ψ l (2 j x − k ). According to iii and 11 for any V j we have two orthonormal basis set as follow:Φ j ( x ) = [ φ j, ( x ) , ..., φ r − j, ( x ) , ..., φ j, j − ( x ) , ..., φ r − j, j − ( x )] (14)Ψ j ( x ) = [ φ , ( x ) , ..., φ r − , ( x ) , ψ , ( x ) , ..., ψ r − , ( x ) , ...,ψ j − , ( x ) , ..., ψ r − j − , ( x ) , ..., ψ j − , j − − ( x ) , ..., ψ r − j − , j − − ( x )] (15)From relation 12 for any f ∈ L ([0 , f ( x ) = r − X l =0 c l φ l ( x ) + ∞ X j =0 2 j − X k =0 r − X l =0 c j,k ψ lj,k ( x ) (16)where c l = R f ( x ) φ l ( x ) dx and c j,k = R f ( x ) ψ lj,k ( x ) dx .Now we define orthonormal projection operator P J : L ([0 , → V J as follows: P J ( f ) := r − X l =0 c l φ l ( x ) + J − X j =0 2 j − X k =0 r − X l =0 c j,k ψ lj,k ( x ) (17)or equivalently P J ( f ) := J X k =0 r − X l =0 d J,k φ lJ,k ( x ) (18)where d j,k = R f ( x ) φ lj,k ( x ) dx . In order to simplify notation, we denote the i-thelement of Ψ j ( x ) by ψ i ( x ), so:Ψ j ( x ) = [ ψ ( x ) , ψ ( x ) , ..., ψ j ( x )] (19)5nd then we can rewrite 17: P J ( f ) := J X i =0 a i ψ i ( x ) = Ψ j ( x ) ′ F (20)where a i = R f ( x ) ψ i ( x ) dx and F = [ a , ..., a j ]. From relation 16 P J is conver-gence pointwise to identity operator I , i.e. ∀ f ∈ L [0 , θ ] lim J →∞ k P J ( f ) − f k = 0 . (21)We use Legendre polynomial to construct Legendre Multiwavelet that hasintroduced by Alpert in [30]. Legendre polynomial, p i ( x ), is defined as follows p ( x ) = 1 , p ( x ) = x with the following recurrence formula: p i ( x ) = xp i − ( x ) + (cid:18) ii + 1 (cid:19) ( xp i − ( x ) − p i − ( x ))The { p i ( x ) } ∞ i =0 is an orthogonal basis for L ([ − , V j as follows V j := { f | f be a polynomial of degree ≤ r on I i , ≤ i ≤ j } (22)where I i := [2 − j ( i − , − j i ). It is obvious that V j ⊂ V j +1 and S j ≥ V j = L ([0 , φ l be a Legendre multiscaling function, that is defined as φ l := (cid:26) √ l + 1 p l (2 x − x ∈ [0 , , , o.w, (23)and Φ := [ φ , ..., φ r − ] T be the multiscale vector. It is easy to verify that { φ lj,k := 2 j/ φ l (2 j x − k ); 0 ≤ l ≤ r − , ≤ k ≤ j − } , (24)forms an orthonormal basis for V j . Now let Ψ = [ ψ , ..., ψ r − ] be the Legendremultiwavelet vector. Because of W ⊂ V each ψ l could be expanded as follows: ψ l = r − X k =0 g l,k φ k (2 x ) + r − X k =0 g l,k φ k (2 x − , ≤ l ≤ r − W ⊥ V and 1 , x, .., x r − ∈ V , so the first r moment of { ψ l } r − l =0 vanish: Z ψ l ( x ) x i dx = 0 0 ≤ l, i ≤ r − Z ψ i ( x ) ψ j ( x ) dx = 0 0 ≤ i, j ≤ r − r unknown coefficients g i,j in 25, it is enough to solve 2 r equations 26 and 27. If f ∈ L ([0 , k times differentiable, the following theorem about bound of error is obtained [30]: Theorem 1.
Suppose that the real function f ∈ C r ([0 , . Then P J ( f ) approx-imates f with the following error bound: k P J ( f ) − f k ≤ ( − Jr +1) r r ! sup x ∈ [0 , | f r ( x ) | . (28)Legendre multiscaling and multiwavelet functions are presented for r = 4 asfollows [31]: φ ( x ) = 1 0 ≤ x < φ ( x ) = √ x −
1) 0 ≤ x < φ ( x ) = √ (cid:0) x − x + 1 (cid:1) ≤ x < φ ( x ) = √ (cid:0) x − x + 12 x − (cid:1) ≤ x < ψ ( x ) = − q (cid:0) x − x + 56 x − (cid:1) ≤ x < / q (cid:0) x − x + 296 x − (cid:1) / ≤ x < / ψ ( x ) = − q (cid:0) x − x + 270 x − (cid:1) ≤ x ≤ / q (cid:0) x − x + 2670 x − (cid:1) / ≤ x < / ψ ( x ) = − q (cid:0) x − x + 30 x − (cid:1) ≤ x < / q (cid:0) x − x + 450 x − (cid:1) / ≤ x < / ψ ( x ) = q (cid:0) x − x + 36 x − (cid:1) ≤ x < / q (cid:0) x − x + 804 x − (cid:1) / ≤ x < /
4. Pricing by Legendre Multiwavelet
Let operator K : L ([0 , → L ([0 , K ( f ) ( z ) := Z κ ( z − ξ, τ ) f ( ξ ) dξ. (31)where κ is defined in 7. Because κ is a continuous function, K is a boundedlinear compact operator on L ([0 , operator K , equations 8 and 9 can be rewritten as below: f = K f (32) f m = K f m − m = 2 , , ..., M (33)7e denote ˜ f ,J = P J K ( f ) (34)˜ f m,J = P J K (cid:16) ˜ f m − ,J (cid:17) = ( P J K ) m ( f ) , m ≥ . (35)where P J K is as follows: ( P J K )( f ) = P J ( K ( f )) . Since the continuous projection operators P J converge pointwise to identityoperator I , then operator P J K is also a compact operator andlim n →∞ k P J K − Kk = 0 (36)(see [34]). With attention to the following inequality k ( P J K ) m − K m k ≤ k ( P J K ) kk ( P J K ) m − − K m − k − k P J K − KkkKk m − (37)and relation 36 by induction we getlim n →∞ k ( P J K ) m − K m k = 0 . (38)Therefore, the following convergence result is concluded: (cid:13)(cid:13)(cid:13) ˜ f m,J − f m (cid:13)(cid:13)(cid:13) = k ( P J K ) m ( f ) − K m ( f ) k ≤ k ( P J K ) m − K m k k f k → as J → ∞ . (39)From 37 and 39, we infer that the rate of convergence ˜ f m,J to f m and P J K to K are the same. Using the relation 28 and properties of integral operator K , itis easy to confirm that k P J K − Kk ≤ ( − Jr +1) r r ! sup z,ξ ∈ [0 , | ∂κ ( z − ξ, τ ) ∂z r | . (40)Since, ˜ f m,J ∈ V J for m ≥
1, we can write˜ f m,J = r J X i =0 a mi ψ i ( z ) = Ψ ′ J ( x ) F m , where F m = [ a m , a m , · · · , a m j ] ′ . From equation 35 we obtain ˜ f m,J = ( P J K ) m − (cid:16) ˜ f ,J (cid:17) . (41)Since V J is a finite dimensional linear space, thus the linear operator P J K on V J could be considered as a r J × r J matrix K . Consequently equation 41 canbe written as following matrix operator form˜ f m,J = Ψ ′ J K m − F . (42)8or evaluation of the option price by 42, it is enough to calculate the matrixoperator K and the vector F . It is easy to check (see [27]) that: F = [ a , a , · · · , a r J ] ′ K = ( k ij ) r J × r J where a i = Z Z δ/θ ψ i ( η ) κ ( η − ξ, τ ) f ( ξ ) dξdη , ≤ i ≤ r J .k ij = Z Z ψ i ( η ) ψ j ( ξ ) κ ( η − ξ, τ ) dξdη . Therefore,the price of the knock-out discrete double barrier option can beestimated as follows: P ( S , t M , M − ≃ e αz + βt ˜ f M,J ( z /θ ) (43)where z = log (cid:0) S L (cid:1) and ˜ f M,n from 42. The matrix form of relation 42 im-plies that the computational time of presented algorithm be nearly fixed whenmonitoring dates increase. Actually, if we set N = r J the complexity of ouralgorithm is O ( N ) that dose not depend on number of monitoring dates.
5. Numerical Result
In the current section, the presented method in previous section for pric-ing knock-out call discrete double barrier option is compared with some othermethods. The numerical results are obtained from the relation 43 with r J basisfunctions. In the following we denote (cid:13)(cid:13)(cid:13) ˜ f m,J − f m (cid:13)(cid:13)(cid:13) by e ( J ) and L − error ( J ).As we discussed in the previous section, the rate of convergence ˜ f m,J to f m and P J K to K are the same. Therefore, e ( J − /e ( J ) must be about 2 r from 40.In addition, relation 40 implies that the slope of log ( L − error ( J )) be about α = − rlog (2). Source code has been written in Matlab
Example 1.
In the first example, the pricing of knock-out call discrete double barrier option is considered with the following parameters: r = 0 . , σ = 0 . , T = 0 . , S = 100 , E = 100 , U = 120 and L = 80 , , , , . . Intable 1, numerical results of presented method with Milev numerical algorithm[14], Crank-Nicholson [35], trinomial, adaptive mesh model (AMM) and quadra-ture method QUAD-K200 as benchmark [36] are compared for various number of monitoring dates. In addition, it can be seen that CPU time of presentedmethod is fixed against increases of monitoring dates. The L − error ( J ) aredemonstrated for L = 90 and M = 250 in Table 2 which results verify theconvergence rate of our algorithm. Fig.1 shows the plot of log ( L − error ( J )) for r = 3 , and it can be seen that the slope of log ( L − error ( J )) is near to α = − rlog (2) . l og ( L - e rr o r( J )) -18-16-14-12-10-8 Line with slope α = -3 log 2M=250M=125M=25M=5 (a) r = 3 J l og ( L - e rr o r( J )) -26-24-22-20-18-16-14-12-10 Line with slope α = -4 log 2M=250M=125M=25M=5 (b) r = 4Figure 1: log ( L − error ( J )) for example 1 with L=95 M L PresentedMethod(r=4 , J=5) Milev(200) Milev(400) Trinomial AMM-8 Benchmark80 2.4499 - - 2.4439 2.4499 2.449990 2.2028 - - 2.2717 2.2027 2.20285 95 1.6831 1.6831 1.6831 1.6926 1.6830 1.683199 1.0811 1.0811 1.0811 0.3153 1.0811 1.081199.9 0.9432 0.9432 0.9432 - 0.9433 0.9432CPU 0.25 s 1 s 5 s80 1.9420 - - 1.9490 1.9419 1.942090 1.5354 - - 1.5630 1.5353 1.535425 95 0.8668 0.8668 0.8668 0.8823 0.8668 0.866899 0.2931 0.2931 0.2931 0.3153 0.2932 0.293199.9 0.2023 0.2023 0.2023 - 0.2024 0.2023CPU 0.25 s 8 s 30 s80 1.6808 - - 1.7477 1.6807 1.680890 1.2029 - - 1.2370 1.2028 1.2029125 95 0.5532 0.5528 0.5531 0.5699 0.5531 0.553299 0.1042 0.1042 0.1042 0.1201 0.1043 0.104299.9 0.0513 0.0513 0.0513 - 0.0513 0.0513CPU 0.25 s 35 s 150 s80 1.6165 - - 1.8631 1.6163 1.616590 1.1237 - - 1.2334 1.1236 1.1237250 95 0.4867 - - 0.5148 0.4867 0.486799 0.0758 - - 0.0772 0.0759 0.075899.9 0.0311 - - - 0.0311 0.0311CPU 0.25 s
Table 1: Double barrier option pricing of Example 1: T = 0 . r = 0 . σ = 0 . S = 100, E = 100. e ( J ) e ( J − /e ( J ) e ( J ) e ( J − /e ( J )4 1.00241 e-4 - 7.45781 e -6 -5 1.22740 e-5 8.16 4.65569 e-7 16.016 1.50805 e-6 8.14 3.31567 e-8 14.047 1.90330 e-7 7.92 2.18567 e-9 15.168 2.29513 e-8 8.29 1.40662 e-10 15.53 Table 2: L − error of example 1 for L = 90 and M = 250. Example 2.
In this example, the parameters of knock-out call discrete doublebarrier option is considered as r = 0 . , σ = 0 . , T = 0 . , E = 100 , U = 110 and L = 95 . In table 3 the option price for different spot prices are evaluatedand compared with Milev numerical algorithm [14], Crank-Nicholson [35] and the Monte Carlo (MC) method with paths [37]. S Presented Method(r=4 , J=5) Crank-Nicolson(1000) Milev(1000) Milev(400) MC (st.error)with 10 paths95 0.174498 0.1656 0.174503 0.174498 -95.0001 0.174499 ≃ ≃ Table 3: Double barrier option pricing of Example 2: T = 0 . M = 5, r = 0 . σ = 0 . E = 100, U = 110 and L = 95. Example 3.
Due to the fact that the probability of crossing upper barrier duringoption’s life when U ≥ E is too small, the price of discrete single down-and-out call option can be estimated by double ones by setting upper barrier greaterthan E (for more details see[14]). Now, we consider a discrete single down- and-out call option with the following parameters: r = 0 . , σ = 0 . , T = 0 . , S = 100 , E = 100 and L = 95 , . , . . The price is estimated by doubleones with U = 2 . E . The numerical results are shown in table 4 and comparedwith Fusai ' s analytical formula [15], the Markov chain method (MCh)[38] andthe Monte Carlo method (MC) with paths [11] that shows the validity of presented method in this case. Fig.2 shows the plot of log ( L − error ( J )) for r = 3 , and it can be seen that the slope of log ( L − error ( J )) is near to α = − rlog (2) . l og ( L - e rr o r( J )) -14-12-10-8-6-4-20 M=5M=25M=125Line with slope α =-3 log 2 (a) r = 3 J l og ( L - e rr o r( J )) -18-16-14-12-10-8-6-4-2 M=5M=25M=125Line with slope α = - 4 log 2 (b) r = 4Figure 2: log ( L − error ( J )) for example 3 with L=95 L M PresentedMethod(r=4 ,J=6) PresentedMethod(r=4 ,J=7) Fusai AnalyticalMethod (IR17) MCh MC (st.error)with 10 paths95 25 6.63155 6.63156 6.63156 6.6307 6.63204 (0.0009)99.5 25 3.35559 3.35558 3.35558 3.3552 3.35584 (0.00068)99.9 25 3.00887 3.00887 3.00887 3.0095 3.00918 (0.00064)95 125 6.16864 6.16864 6.16864 6.1678 6.16879 (0.00088)99.5 125 1.96132 1.96130 1.96130 1.9617 1.96142 (0.00053)99.9 125 1.51019 1.51021 1.51068 1.5138 1.5105 (0.00046)CPU 0.48 s 0.83 s Table 4: Single barrier option pricing of Example 3: T = 0 . r = 0 . σ = 0 . S = 100, E = 100, U = 250.
6. Conclusion and remarks
In this article, we used the Legendre multiwavelet for pricing discrete sin- gle and double barrier options. In section 4 we obtained a matrix relation 42for solving this problem. Numerical results confirm that growth of computa-tional time is negligible when the number of monitoring dates increase. On theother hand, the rate of convergence of presented algorithm has been obtainedtheoretically and verified numerically .
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