A Second Order
L
0
Stable
Algorithm for Evaluating
European Options
Ruppa K. Thulasiram*
Department of Computer Science, University of Manitoba, Canada E-mail: [email protected] *Corresponding authorChen Zhen
Department of Mathematics,
University of Western Ontario, Canada E-mail: [email protected]
Amit Chhabra
Department of Computer Science, University of Manitoba, Canada E-mail: [email protected]
Parimala Thulasiraman
Department of Computer Science, University of Manitoba, Canada E-mail: [email protected]Abba B. Gumel
Department of Mathematics, University of Manitoba, Canada E-mail: [email protected]
Abstract: In this paper, we studythe option pricing problem,one of the prominent and challenging problems in computational finance. Using Pade approximation,we have developed a second orderL0 stable discrete parallel algorithm for experimentation on
advanced architectures. This algorithm is suitable for more complicated option pricing problems. For simulation purposes, we have implemented thesequential version of this algorithm and evaluated the European Options. Numerical results are compared with those obtained using othercommonly used numerical methods and shown that the new algorithm is robust and efficient than the traditional schemes.
Using explicit Forward Time Centered Spaace (FTCS) on the reduced Black-Scholes partial differential equation, we report pricing of European options. We have done our experiments on a shared memory multiprocessor machine using OpenMP and report a maximum speedup of 3.43 with 16 threads.
Keywords: European options; Pade approximation; finite-differencing; parallel computing.
Reference to this paper should be made as follows: Thulasiram, R.K., Zhen C., Chhabra, A., Thulasiraman, P. and Gumel, A.B. (2006) ”A Second OrderL0 Stable
Algorithm for Evaluating European Options”, Int. J. High Performance Computing and Networking, Vol. X, Nos. 1/2/3, pp.XX-XX.
Biographical notes:Thulasiram Ruppa K. is an Associate Professor and a research affiliate with the Institute of Industrial and Mathematical Sciences of the University of Manitoba. His research interests are in Computational Finance, Mobile Commerce, High Performance and Scientific Computing, Numerical Algorithms.
Chen Zhen is doing his Ph.D. in Mathematics and his general research interests are in partial differential equations.
Amit Chhabra is a Ph.D. student in Computer Science and his research interests are in derivative pricing and distributed and grid computing.
Parimala Thulasiram is an associate professor and her interests are in design of Parallel, Distributed and Multithreaded algorithms for various irregular applications.
Abba Gumel is a professor in Mathematics and research interests are in Mathematical
1 Introduction
Financial models used for evaluation and forecasting pur-poses typically lead to large nonlinear dynamical systems that have to be solved for a certain time span. Many of the problems in finance demand efficient algorithms and high performance computing capabilities??. In part, this is because, in most financial applications, there is a high premium on rapid solutions. Any delay in information processing can be translated into potential financial losses. From an academic point of view, the state of the art in financial modeling and analysis is evolving toward ever more mathematically complex models, presenting ever more formidable computational challenges both in terms of algorithm design and implementation. The emergence of complex financial instruments and the availability of powerful computers make the numerical solution of the mathematical model of the derivatives increasingly appealing to pricing?.
There are a multitude of financial instruments traded on the markets. The class of instruments known as European options are claims to payoffs at a time when the security expires. The solution for this must typically be performed numerically, and is usually a computationally intensive problem. To price an European option we may employ pricing models such as binomial tree approach ? for the asset price and then roll backward the tree. Discretization techniques such as finite differences, finite elements have been adopted to a visible extent in the study of financial instruments. These approaches are highly suitable for par-allel computing. In parpar-allel computing there are two types of embedded latencies: communication latency, due to re-mote accesses, and synchronization latency, due to data dependencies. Conventional message passing MPPs do not yield high performance if such latencies are frequent. Superscalar, superpipelined, VLIW, and prefetching tech-niques have all been used to hide or tolerate both commu-nication and synchronization latencies. However, the most general technique is multithreading. Multithreading tries to overlap computation with communication by means of
threads, thereby reducing latencies, where a thread is a set of instructions executed sequentially.
In this paper, we propose an efficient algorithms for a reduced Black-Scholes model for pricing European options using Pade approximation. Then we price a standard op-tion on a single underlying asset using explicit forward time centered space (FTCS) method??. We apply FTCS method to solve the Black-Scholes equation in its reduced form for financial derivatives.
The paper is organized as follows. We first introduce some basic definitions in the next subsection. Some liter-ature relating to the development of Black-Scholes model and related issues in solving the resulting partial differen-tial equations are reported in subsection 1.2. In section 1.2, the Pade Approximation of exponential functions is de-scribed briefly. A method on the (2,0) Pade approximation for option pricing problems is designed in section 1.2. In
section 1.2 this method is implemented to solve the math-ematical model of the European options. Parallel algo-rithms are proposed in section??. Numerical experiments and simulations are presented in section??. The final sec-tion??concludes our study.
1.1
Definitions
We start with the definitions of put and call options, as well as European style and American style options for the sake of completion and self sufficiency.
1. Call Option: A call option is a contract that gives the right to its holder (i.e. buyer) without creating an obligation, to buy a prespecified underlying asset at a predetermined price. Usually this right is created for a specific time period, e.g. six months, twelve months, or more. If the option can be exercised only at its ex-piration (i.e. the underlying asset can be purchased only at the end of the life of the option) the option is referred to as anEuropean style Call Option(or Euro-pean Call). If it can be exercised on any date before its maturity, then the option is referred to as an Amer-ican style Call Option (or American Call). Options can be written on numerous underlying assets, such as equity, precious metals, agricultural commodities, etc. The call option creates an obligation for its writer to fulfill the contract with the buyer. That is, when the option is exercised, the issuer of the option has to sell to the buyer (holder) of the option the asset at the contract price.
2. Put Option: A put option is a contract that gives to its holder the right without creating the obligation, to sell a prespecified underlying asset at a predetermined price. If the option can be exercised only at its expira-tion date (i.e. the underlying asset can be sold only at the end of the life of the option) the option is referred to as anEuropean style Put Option(orEuropean Put). If it can be exercised on any date before its maturity, then the option is referred to as anAmerican style Put Option(or American Put).
1.2
Black-Scholes Model
Option pricing is considered to be among the most mathe-matically complex problems in the area of finance because they need intensive application of stochastic calculus. In 1877, Charles Castelli introduced to the academic circles hedging and speculation aspects of options in his book: ”The theory of options in stocks and shares”. In 1900 Louis Bachelier proposed an analytical option valuation technique in his mathematics dissertation in Sorbonne, France. However, his method for generation of share prices was faulty in that it allowed negative prices of securities and prices of derivatives in excess of prices of the under-lying assets. A breakthrough occurred in 1962 when A. James Boness developed an option-pricing model in his work: ”A theory and measurement of stock option value”
that set the scene for further analysis. In 1973 Fischer Black and Myron Scholes expanded the model created by Boness by proving that the risk-free rate is the correct dis-count factor and that the assumptions regarding investor’s risk preferences are not needed.
Option pricing has received increasing attention from fi-nancial institutions ever since the pioneering work of Black and Scholes?. The Black-Scholes (B-S) model deals with option valuation where equilibrium option price can be found provided thatno arbitrage profits? argument holds. The B-S model yields deterministic formulas where all the inputs (current market values of stock prices, T-bill rates, the maturity of the instrument, exercise price, volatility etc.) needed for analysis are assumed to be known. How-ever, in practice precise quantification of this formula is rarely available. (Note: The readers are referred to? for a glossary of financial terms that are used but described in this paper).
Option pricing is a very challenging problem. Except for the simplest cases, the prices of the options can not be cal-culated with an analytical model, and therefore, numerical methods are sought to obtain a solution. Three popular numerical techniques employed in option pricing are finite difference techniques, lattice approaches, and Monte Carlo simulation. The first two methods usually need more com-putations for complicated models ? and a drawback of Monte-Carlo simulation is that it can not be used to price options which are continuously early-exercisable. Though lattice methods are perhaps the most widely used numer-ical method in finance, they are in fact simple explicit fi-nite difference schemes. It has been demonstrated that the standard lattice methods can not be applied directly to many more complex option pricing problem. An advan-tage of numerical PDE approaches (for example, finite dif-ference method) over lattice methods is in their generality. Numerical PDE approach is becoming a main stream re-search problem among the finance academician and practi-tioners alike to price a wide variety of exotic options?????. Various explicit and implicit schemes were used. Explicit methods usually need less computation because it does not entail solving a set of linear equations at each time step. Implicit methods, on the other hand, have better stability and convergence properties but are computationally more demanding. Many of the schemes used in finance are first order in time. Courtedon? developed a second order accu-rate finite difference method for valuing options. For sim-plest problems, one can generally transform the B-S model to the simple heat equation for which well developed solu-tion schemes are available. Clark? used Crank-Nicholson finite difference method for the Cray T3D supercomputer. However, the performance was considerably slower on a number of processing elements, due to the use of sequen-tial tridiagonal solver?. Mayo? evaluated the American options using fourth order implicit finite difference method. The algorithm gave fourth order accuracy in the log of the asset price and second order accuracy in time.
Mayo? developed a fourth order method in the log of asset prices to evaluate American options.
In general, it is often difficult to reduce the mathemat-ical severity of the finance model to a simpler form. In addition we need more accurate and effective schemes for option pricing problems so that the pricing could be ac-curately predicted and computation cost could be reduced at the same time. The focus of this paper is, therefore, to develop and implement an accurate and efficient finite difference method for option pricing. Parallel algorithms suitable for implementation on a state-of-the-art multipro-cessor architecture to solve the resulting linear algebraic system at every time step using complex arithmetic has been proposed and designed in this paper, as well.
2 Pade Approximation
The Method of Lines(MOL) semi-discretization ap-proach is often employed to solve second-order parabolic partial differential equations. By this method, the Ini-tial/Boundary Value Problem (IBVP) is transformed into a system of first-order ordinary differential equations which can be written in semi-discrete matrix-vector form as
dU(t)
dt =AU(t) +v(t), (1)
where A is a square matrix; v(t) results from the use of non-homogeneous boundary conditions andU(t) is the so-lution vector at timet. It is easy to show that the solution U(t) of eq.(1) satisfies the recurrence relation
U(t+k) =ekAU(t) +
Z t+k
t
e(t+k−s)Av(s)ds,
t= 0, k,2k, . . . , (2) wherek= ∆t. Whenv= 0 the recurrence relation eq.(2) takes the form U(t+k) = exp(kA)U(t). The develop-ment of numerical methods is then based on approximat-ing the exponential matrix function and integral term in these recurrence relations.
Several existing algorithms for the numerical solution are generated through rational approximations to the matrix exponential function. The most well known is arguably the approximation introduced by H. Pade in 1892. This approximation technique has been widely used in solving parabolic PDEs (see for example???) in science and engi-neering. We apply this approximation technique to study financial derivatives.
Because the Pade’s have complex poles, their use in ap-proximating matrix exponentials enable the solution of the finance models to be computed in parallel (using complex arithmetic) via a partial-fraction splitting technique?. Fur-thermore, the direct association of these matrix approxi-mations with the classical rational approximation of an-alytic functions gives a powerful tool for the analysis of stability of these approximations. More detailed discus-sion can be found in?.
A Pade approximationRm,k(z) to an analytic function
z = 0 given by Rm,k(z) = (Pk(z))/(Qm(z)) where Pk(z) and Qm(z) are polynomials in z of degrees k and m re-spectively having leading coefficients unity, and k, m are non-negative integers. For each pair of m and k, Pk(Z) andQm(Z) are those polynomials for which the Taylor se-ries expansions ofRm,k(z) about the origin agrees with the Taylor series off(z) for as many terms as possible?.
An important property of an (m,k) Pade where m > k
is that it leads toL0stability - thus, all frequency compo-nents and instabilities are not propagated from one step to the next.
3 Pade Approximation for Black-Scholes Model
Consider the Black-Scholes model,
∂u ∂t + σ2S2 2 ∂2u ∂S2 +rS ∂u ∂S −ru= 0 (3)
where u is the option price, t is time, σ is volatility, S
is the asset price, and r is the interest rate. The partial differential equation will be discretized as follows:
A finite number of equally spaced time steps between the current time (t = 0) and the maturity of the option, (t =T) are chosen. That is, ∆t= T /N and the total of
N+ 1 times are considered. Similarly, a finite number of equally spaced stock prices are also chosen. We suppose that Smax is a stock price which is sufficiently high that, when it is reached, the put option has virtually no value. We define ∆S =Smax/M and consider a total of M + 1 stock prices. One of these is assumed to be the current stock price. Then, by the above discretization, a grid con-sisting of a total of (M+ 1)(N+ 1) points is constructed. The (i, j) points on the grid is the point that corresponds to timei∆tand stock pricej∆S. We will use the variable,
uij, to denote the value of the option at the (i, j) point. Using central difference approximation for ∂S∂u and ∂S∂2u2, the Black-Scholes equation is written in discretized form as dU dt =−rj∆S ui+1,j+1−ui+1,j−1 2∆S −σ 2j2∆S2 2 ui+1,j+1+ui+1,j−1−2ui+1,j ∆S2 +rui+1,j (4)
Simplifying this, we can get ddtU =ajui+1,j−1+bjui+1,j+
cjui+1,j+1 whereaj = rj2 −σ 2j2 2 , bj =σ 2j2+r, andc j = −rj 2 − σ2j2
2 ; whereUdenotes the vector form ofu, i.e.,U= {u.,1, u.,2, . . . , u.,M}. With the corresponding boundary condition, we can write the equation into the matrix form
dU
dt =AU+C(t) whereC(t) arises from the use of a non-homogeneous boundary condition. Letk denote ∆tandh
denote ∆S. Using eq.( 2), the solution U(t) satisfies the
recurrence relation U(t+k) =ekAU(t) + Z t+k t e(t+k−s)AC(s)ds, t= 0, k,2k, . . . (5) The partial differential equation for the option pricing model is always associated with the terminal condition (condition at the maturity(t = T)) instead of the initial condition (t = 0). Therefore, in solving the difference equation, we should do a backward iteration instead of a forward iteration that results in standard initial/boundary value problems. Thus, equation (5) is rewritten as
U(t−k) =e−kAU(t) +
Z t−k
t
e(t−k−s)AC(s)ds, t= 0, k,2k, . . . (6) whereAis the tridiagonal matrix. If the boundary condi-tion is homogeneous, thenC(t) = 0. Various finite differ-ence methods results by choosing different Pade approxi-mation to e−kA. For example, a (0,1) Pade approxima-tion toe−kA leads to an explicit finite difference scheme, and (1,0) Pade approximation toe−kAleads to an implicit scheme. Generally, we can obtain higher order accurate difference schemes in t as we stated in section 1.2, if we use higher order Pade approximation to matrix e−kA. In other words, the approximation to matrixe−kA plays an important role in achieving higher order accuracy.
Note that we can apply the method to the general parabolic differential equations with suitable modification to the boundary conditions for the given options pricing model. In essence, we can say that a higher-order finite difference method can be derived for more general option pricing problems using appropriate rational approximation toe−kA.
4 Pade Algorithms and Implementation
A method based on the use of the (2,0) Pade approxima-tion is constructed and used to solve the vanilla European option in this section. The option price should satisfy the B-S equation. In addition to the variables introduced in section(1.2) we denote E as the strike price. Suppose the maturity of the option is att =T. Then the partial dif-ferential equation and the boundary conditions for a Put option are given by
∂u ∂t + σ2S2 2 ∂2u ∂S2 +rS ∂u ∂S −ru= 0 (7) u(S, T) =max(E−S,0) u(0, t) =Ee−r(T−t) lim S→∞u(S, t) = 0
4.1
A
(2
,
0)
Pade approximation Without
Transformation
With the discretization introduced in section 1.2, we can write the B-S model as
U(t−k) =e−kAU(t) +
Z t−k
t
e(t−k−s)AC(s)ds, t= 0, k,2k, . . . , (8) with the boundary and initial conditions given in (7), we have A= b1 c1 0 · · · 0 a2 b2 c2 ... . .. . .. . .. 0 .. . aM−1 bM−1 cM−1 0 · · · 0 bM cM , (9)
and C(t) = [a1u(0, t),0,· · · ,0]T. A (2,0) Pade approxi-mation toe−kAis
e−kA≈(I+kA+1 2k
2A2)−1+O(k3). (10)
The integral term can be approximated to second order, using the trapezoidal rule up to the second order. Thus, the whole approximation is second-order,
Z t−k t e(t−k−s)AC(s)ds≈ −1 2[C(t−k) +e −lAC(t)], t= 0, k,2k, . . . (11) Denoting matrixB= (I+kA+12k2A2) and substituting equations (10) and (11) into eq.(8) leads to
BU(t−k) =U(t)− l
2[BC(t−k) +C(t)],
t= 0, k,2k, . . . (12) Thus the B-S partial differential equation has been trans-formed into a discrete form which has the accuracy of
O(k2+h2); The aforementioned standard explicit or
im-plicit finite difference method are onlyO(k+h2) accurate. It is worth noting that a quin-diagonal solver is needed to implement eq.(12) because the matrixB is quin-diagonal. We denote eq.(12) to be a General Pade Discretization Al-gorithm 1 orGPDA1.
4.2
A
(2
,
0)
Pade approximation With
Transformation
Let S = Eex, t = T − τ σ2 2 , and u(S, t) = g(x, τ)v(x, τ), where g(x, τ) = Ee−12(k1−1)x− 1 4(k1+1) 2τ , in which, k1 = 2r σ2 ?.With this transformation, the B-S model can be reduced to
∂v ∂τ =
∂2v
∂x2 (13)
subject to the initial and boundary conditions (for the Eu-ropean Put option)
v(x,0) =max(e12(k1−1)x−e12(k1+1)x,0) (14) lim x→−∞v(x, τ) =e 1 2(k1−1)x+14(k1−1)2τ lim x→∞v(x, τ) = 0
It is, of course, easier to solve this diffusion equation (13) and (14) than the original B-S model. In other words, it is easier to find numerical solutions for the diffusion equa-tion and then, by a change of variables, convert the so-lutions into those of the B-S model. However, we should note that for complicated finance models, particularly the multi-factor models, it is not always feasible to reduce the problem to a constant coefficient diffusion equation like the one presented above. In this case, there is little choice but to use finite differences on the generalized (full) B-S model. This is the main reason for our elaboration on the more general non-transformed case in section(4.1).
A popular finite-difference method for solving PDEs is the Crank-Nicholson method which is based on (1,1) Pade approximation. Higher order Pade approximation give higher accuracy in time. However, this could lead to higher computational complexity. Therefore, a balance of accu-racy and available resources dictated our selection of (2,0) Pade to illustrate the higher order finite difference method in solving the option pricing problems. To our knowledge, this is a new contribution to the field of computational fi-nance. Moreover, the (2,0) method isL0-stable, and hence it has better stability than the Crank-Nicholson scheme which is only A0-stable. Moreover, with the L0-stability, the error remains bounded.
As we discussed in section(4.1), the Method of Lines reduces the partial differential equation (13) with the non-homogeneous boundary conditions (14) into a system of the form ddτV =A1V+C1(τ) where
A1= −2ρ ρ 0 · · · 0 ρ −2ρ ρ ... . .. . .. . .. 0 .. . ρ −2ρ ρ 0 · · · 0 ρ −2ρ , (15) in which, ρ = 1/h2, and C 1(τ) = [ρv(0, τ),0,· · ·,0]T
and V is the vector form of the finite difference form of the equation (13) if we use the same discretization as in section(1.2) and is given by V = {v.,1, v.,2, . . . , v.,M} A grid consisting of a total of (M + 1)(N+ 1) points is con-structed. If we denote k, h as time stepτ and space step in x respectively, where τ is from 0 to T and x is from
xmin toxmax, thenM×k=T andN×h=xmax−xmin. Note now the terminal conditiont =T is transformed to the initial conditionτ = 0.
rela-tion V(τ+k) =ekA1V(τ) + Z τ+k τ e(τ+k−m)A1C 1(m)dm (16)
Further, approximating the integral in eq.(??) by the trapezoidal rule gives, V(τ+k) = ekA1V(τ) +
l
2[C1(τ+k) +e
kA1C
1(τ)]. Using (2,0) Pade to
approx-imate the ekA1, ekA1 ≈ (I −kA1 + 1
2k 2A2 1)−1 Denote B1=I−kA1+12k2A2 1. Thus B1V(τ+k) =V(τ) + l 2[B1C1(τ+k) +C1(τ)] (17) We call the eq.(??) a General Pade Discretization Algo-rithm 2 or GPDA2. Similar to the GPDA1, a quin-diagonal solver is needed for implementation at every time step.
4.3
Pade Parallel Algorithm
The difficulties in implementing the high-order implicit method is the need to solve a set of linear equations at each time step. As mentioned above, we need to use a quin-diagonal solver if we use (2,0) Pade finite difference method. The situation becomes worse if we use higher order approximation. A more efficient approach is to em-ploy parallel computing. With the rapid development of computer technology, parallel computing provides a very promising solution to compute intensive problems in the emerging field of computational finance. Some of the op-tion pricing problems are well suited for multithreaded ar-chitectures ?, a state-of-the-art computing technology as mentioned before.
We discuss a parallel solution scheme in this section for the GPDA2 developed in section(4.2). Similar parallel algorithm can be constructed forGPDA1.
TheGPDA2is given by V(τ+k) = (I−kA1+1 2k 2A2 1)− 1V(τ) + (18) k 2[C(τ+k) + (I−kA1+ 1 2k 2A2 1)− 1C(τ)]
Using partial-fraction splitting,
V(τ+k) = [k1(I−r1kA1)−1+ k2(I−r2kA1)−1]V(τ) +k 2[k1(I−r1kA1) −1 +k2(I−r2kA1)−1]C(τ) +k 2C(τ+k) (19) wherer1, r2, k1, k2 are complex constants. ThenV(τ+k)
may now be obtained concurrently in a parallel architec-ture. Following is an example implementation methodol-ogy with two processors at the top level. The sub-tasks
within each processor could be assigned to multiple pro-cessors. On two processors:
processor1 : (I−r1kA1)w1=k1V(τ) (20) (I−r2kA1)w2=k2V(τ) processor2 : (I−r1kA1)z1=k1C(τ); (21) (I−r2kA1)z2=k2C(τ) V(τ+k) =w1+w2+ l 2[z1+z2+C(τ+k)] (22) Sincer1andr2 are conjugated, it follows that
processor1 : (I−r1kA1)w1=k1V(τ); q1= 2Re(w1) (23) processor2 : (I−r1kA1)z1=k1C(τ); q2= 2Re(w2) (24) V(τ+k) =q1+ l 2[q2+C(τ+k)] (25)
5 FTCS Algorithm and Implementation
The B-S equation is discretized as follows: A finite num-ber of equally spaced time steps between the current date (t = 0) and the maturity date of the option, (t = T) are chosen. That is, ∆t = T /N and the total of N + 1 times are considered. Similarly, a finite number of equally spaced asset prices (Nj) are also chosen. We suppose that
Smax is the maximum price an asset can reach. We de-fine ∆S=Smax/2Nj and consider a total of 2Nj+ 1 asset prices including the current asset price. By the above dis-cretization, a grid consisting of a total of (N+ 1)(2Nj+ 1) points is constructed as shown in figure??. The grid point (i, j) corresponds to time i∆t and price j∆S. We use the variable, ui,j to denote the value of the option at the grid point (i, j). The accuracy of the original system will be dictated by the convective term (∂u∂S) rather than the diffusion term (∂∂S2u2). Therefore, central-differencing for the convection term and forward or backward-differencing for the diffusion term would be sufficient. However, we applied central differencing to the diffusion term in the transformed Black-Scholes equation. The additional cost incurred for the diffusion term could be recovered with the multithreaded implementation of the scheme.
The solution scheme is marched in the time direction until it reaches a steady state as shown in the figure ??. With the help of this we calculate the relative errorerrin the value of the option. The valueuc
i,j denotes the option value at (i, j) grid point in cth computational time layer. To calculate the option values at thecth layer we use the values from the (c−1)th layer, which can be expressed as
Figure 1: A two dimensional grid for computing the Option Value
Hereρ=l/h2, wherel denotes time stepτ andhdenotes
space stepx. Ifτis from 0 toTandxis fromxmintoxmax, thenN×l=T and 2Nj×h=xmin−xmax. Therefore the terminal conditiont =T is reformed to the initial condi-tionτ= 0.T herelativeerriscalculatedaserr = uc
i,j−u c−1
i,j We stop our computation once the err falls below a cer-tain preset threshold value. Since our implementation
Figure 2: A three-dimensional mesh for computing option value with finite differencing method
uses an explicit scheme, by partitioning a given computa-tional layer among many processors, we achieve parallelism in computation and the computation of the option value in a particular layer is dependent only on the values in the previous layer (time step). We have implemented our algo-rithm in parallel on an symmetric multiprocessor machine with eight processors using OpenMP. All the processors are Intel Pentium III with 700MHz CPU and cache size of 1024KB. The total physical memory of the machine is over 7GB. Since there is no data dependency amongst the com-putational layers, we have used fine-grained parallelism to compute the values. That is, every thread computes one option value in the grid before it starts computing the next one. AssumingNc computational time steps for reaching steady state, then the order of our computation grid
be-Table 1: Explicit finite difference method for European Option without transformation
val err nt nS 6.472115 0.744271 10 20 Divergence 10 40 7.113129 0.103257 20 20 Divergence 20 40 7.083097 0.133289 40 20 Divergence 40 40 Divergence 40 80
comesNc∗N∗Nj. At any instance of time we just need two computational layers (the current layer and the one preceding it) for calculating theerr. That is, we save only two computational layers in memory, thereby, utilizing the available memory efficiently.
6 Experimental Results
We report the numerical experiments we performed and compare the (2,0) algorithm with two other finite differ-ence methods in this section, .
Specifically, we show the results of using several methods to value the European Put option with expirationT = 1 year, interest rate r = 0.1, volatility σ = 0.3 and strike priceE = 100.
The true value of this Put at the stock price S = 100 is determined using the binomial tree method with up to 2000 steps, and is given as 7.216386.
In Tables 1-3, we present the results of three different finite difference methods (explicit, implicit, and (2,0) Pade approximation) without transforming the B-S equation to heat equation. A uniform grid in space and times was used. In these tables nt is the number of time steps, andnS is the number of intervals in the S direction. Let Smin = 0,Smax= 200, and the error is the absolute error.
In Table 1, we present the results of using explicit finite difference method, we get rather good results, but we can also see from the table that the method is not robust enough. It may lead to divergence in certain cases.
In Tables 2 and 3, the results using implicit and (2,0) sequential finite difference methods are listed respectively. These numbers show that the new method GPDA1 is much better than the traditional implicit method. And to the extent of convergence, it is also better than explicit method.
In Tables 4,5 and 6, we present the results using explicit, implicit and (2,0) algorithm respectively employed on the transformed B-S model. The coordinates t, S are trans-formed to τ, xrespectively. Letxmin =−9.5, xmax = 0.5
Table 2: Implicit finite difference method for European Option without transformation
val err nt nS 6.917384 0.299002 10 20 7.047645 0.168741 10 40 6.985141 0.231245 20 20 7.112318 0.104068 20 40 7.019024 0.197362 40 20 7.144777 0.071609 40 40 7.175542 0.040844 40 80 7.183195 0.033191 40 160
Table 3: (2,0) Algorithm for European Option without transformation val err nt nS 7.046295 0.170091 10 20 7.170949 0.045437 10 40 7.051226 0.16516 20 20 7.175597 0.040789 20 40 7.052572 0.163814 40 20 7.176864 0.039522 40 40 7.207325 0.009061 40 80 7.214904 0.001482 40 160
corresponding to Smin ≈ 0, Smax = 160. τ is from 0 to 0.045 with σ= 0.3 andtvarying from 1 to 0.
The transformation leads to solving simpler heat equa-tion which eases the soluequa-tion processes as we can see from the above tables that the implementation is with-out much problem in terms of convergence. However, we need more steps (in both spatial and temporal directions) to get the same precision as the one obtained from non-transformed B-S equation employing same set of schemes. As we know, it is difficult to transform the generalized Black-Scholes equation to a simpler form in certain cases, we have to resort to numerical methods directly. In both cases, we can see from the tables that the new methods (GPDA1,GPDA2) perform better than the traditional explicit and implicit method. We can expect to get better results with higher order sub-diagonal Pade approxima-tions.
Now we present our experimental results obtained by varying the problem size and the number of threads on the eight processor shared memory machine.
We show the results for European Call Option with
ex-Table 4: Explicit finite difference method for European Option with transformation
val err nτ nS 6.328616 0.88777 10 40 8.664708 failed 10 80 7.050204 0.166182 20 80 Divergence 20 160 6.204828 1.011558 40 40 7.019965 0.196421 40 80 7.188730 0.027656 40 160 7.208365 0.008021 40 200
Table 5: Implicit finite difference method for European Option with transformation
val err nτ nS 6.007630 1.208756 10 40 6.866610 0.349776 10 80 6.928363 0.288023 20 80 7.103772 0.112614 20 160 6.124482 1.091904 40 40 6.959040 0.257346 40 80 7.132133 0.084253 40 160 7.152209 0.064177 40 200
Table 6: (2,0) Pade finite difference method for European Option with transformation
val err nτ nx 6.157258 1.059128 10 40 6.983973 0.232413 10 80 6.988059 0.228327 20 80 7.159135 0.057251 20 160 6.163922 1.052464 40 40 6.989179 0.227207 40 80 7.160113 0.056273 40 160 7.179975 0.036411 40 200
OV Threads 2 4 8 16 7.21031 N = 300 1.52 2.15 2.53 2.60 7.20845 N = 600 1.61 2.51 2.94 3.22 7.20784 N = 900 1.58 2.60 3.01 3.27 7.20752 N = 1200 1.52 2.62 3.03 3.28 7.20734 N = 1500 1.55 2.63 3.04 3.33 7.20721 N = 1800 1.58 2.70 3.09 3.43
Table 7: Option Value (OV) and Speedup achieved using 2-,4-,8-,16-Threads respectively at various
grid sizes withNj= 200.
piration time T = 1, interest rate r = 10% , volatility
σ= 30% and the strike priceE= 100.
Table ?? shows option value (OV) and relative speedup
Figure 3: Option Value
Figure 4: Execution Time
using 2, 4, 8 and 16 threads respectively withNj = 200. Speedup is the ratio of execution time of a single-thread program to that of multi-threaded parallel program. We achieved a speedup ranging between 1.52 to 3.43, max-imum being 3.43 using 16-threads with 1800 physical timesteps (N). Figure?? shows that the OV stabilizes as we increase theN. Figure??depicts the variance of the ex-ecution time with different number of threads. Threads in
all cases are evenly distributed among processors. With 16 threads (2 threads per processor) we get the most speedup. Our experiments show that, on further increasing the num-ber of threads, we don’t get any more speedup as the con-text switching and synchronization delays increase due to the fine-grained nature of the algorithm.
Figure 5: Execution Time
Figure 6: Speedup vs. Number of Threads
Doubling the number of asset values (Nj) for the calcula-tion of OV gave the maximum speedup of 3.40 atN= 1800 using 16 threads (refer to figure ??). We also notice the speedup increases with both the increase of threads and the problem size as shown in figure??. We assume that there are approximately 225 working days in a year with 8 working hours every day. Hence we takeN to the max-imum of 1800(225∗8) which gives the hourly precision of the option value.
7 Conclusions
A second order finite difference scheme for European op-tion pricing problem was constructed and implemented in this paper employing Pade approximations. An efficient parallel algorithm is also designed. Numerical verifica-tions demonstrated that the new method is more robust and efficient than the traditional explicit and implicit fi-nite difference method for option pricing problem. In ad-dition this algorithm lends itself for parallel implementa-tion which could reduce the overall computaimplementa-tional time
for a larger problem. Moreover, the (2,0) Pade algorithm can be extended to solve more complicated option pricing problems.
We can conclude that FTCS method is easy to imple-ment in a parallel environimple-ment though it does not seem to be efficient computationally due to slow convergence on the option values obtained using Pade approximation. To increase the accuracy of the computed option values we plan on studying the problem with implicit scheme.
Acknowledgment
The first author acknowledges the partial financial sup-port from University Research Grants Program (URGP) of The University of Manitoba and Natural Sciences and Engineering Research Council (NSERC) of Canada.
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