MATHEMATICS IN ENGINEERING, SCIENCE AND AEROSPACE MESA - www.journalmesa.com Vol. 1, No. 1, pp. 91-103, 2010 c CSP - Cambridge, UK; I&S - Florida, USA, 2010 °
Modeling the stochastic dynamics of Zakaat Fethi Bin Muhammad Belgacem1,? 1
Faculty of Information Technology, Arab Open University, P. O. Box 830, Al-Ardhia 92400, Kuwait.
?
Corresponding Author. E-mail:
[email protected]
Dedicated to: HRH Prince Talaal Bin Abd Al-Azeez, Founder of the Arab Open University. Abstract. In the Islamic faith, Zakaat is annually ordained on individuals whose fortunes exceed a given threshold called Nisaab, Z0 , to be distributed over the less fortunate (with fortunes, y(t) < Z0 ). For t in the time interval, I = [0, T ], with T being typically one lunar year, the Zakaat value taken from an individual with a fortune x(t), (x(t) ≥ Z0 , for all t in I), is proportional to the original fortune and typically prescribed at no less than, λx(T ) = x(T )/40. Usually the collection and redistribution of Zakaat monies from the Nisaab exeeding individuals, and redistribution over the less fortunate population sector is handled by a special ministry in the goverment or sanctioned agencies. To describe the dynamics of distribution of wealth among individuals of a population subject to the Zakaat system is by no means an easy task. Nevertheless, several types of models ranging from the discrete to the continuous, with theories emanating from multiparticle random walks, to systems of stochastic and periodic parabolic differential equations, can certainly be produced. In particular, one such model can be derived to yield a Black- Scholes mixed fortune-variables equation. In this work, using tools from Stochastic Calculus and Financial Mathematics concepts, we present a derivation and analysis of the mixed dual fortune-variables stochastic model for the Zakaat problem starting with the Stochastic System, dx (t) = a(x,t)dt + (1 − λ)σ(x,t)x(t)dw1 (t) ,
x(t) ≥ Z0 , 0 ≤ t ≤ T,
dy (t) = b(y,t)dt + λσ(x,t)x(t)dw2 (t) ,
y(t) < Z0 , 0 ≤ t ≤ T.
(0.1)
Here,for i = 1, 2, wi (t) is the standard Brownian Motion (see Belgacem [1-3]), satisfying the L2 relations, and for ρ, δ > 0, dw1 dw2 = ρdt, (dwi (t))2 = dt, 0
dtdwi (t) = dwi (t) dt = 0,
2008 Mathematics Subject Classification: 60H15; 58B99. Keywords: Stocastic Equations; Black-Scholes Equation; Zakaat Problem.
(dwi (t))2+δ = 0.
(0.2)
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In this paper, which in view of the newness of the problem is meant to be of expository nature we study the effect of the value λ, taken initially to be 1/40 = 0.025, on the dynamics of system (0.1) and the resulting analysis on the mixed fortune-variables (Black-Scholes Equation like) objective function, F(x, y,t) = F(x, y) = x(t) − y(t).
(0.3)
1 Introduction & Background In Islamic Shariaa (law), the third pillar requires that an individual, whose monetary fortune, x(t), reaches or exceeds a set amount or threshold, Z0 , labeled Nisaab, and stays above it for the period of one whole lunar year (T ), pay a known proportion of of his fortune, set at x(T )/40. This amount is called Zakaat and is expected in part to be shared with the less fortunate in the society, in particular those whose fortunes are way below the Nisaab (y(t) < Z0 ). In Islam practicing societies, Zakat is not a governmental tax, but is considered just as mandatory. Historical data shows that over centuries societies that firmly upheld the zakaat pillar, soon dispensed with any other form of usual taxes, governmental or otherwise. Nowadays this set up may seem to be far fetched, non-feasible, and idealistic at best. Our quest is to find supportive evidence for the contrary. Furthermore, under the empirical evidences yet seemingly repetitive pattern of eventual crumbling and recessions of interest based economies, studying alternative supportive structures to economies becomes, if not highly recommended, a must. A diligent search of the mathematical published literature does not seem to yield any serious attempt at modeling the dynamics of the Zakaat problem mathematically. Some mileage nevertheless can be perhaps obtained from the Jargon and the equations highly organized methodology summaries in Berck & Syndsaeter [7]. As far as we know, the approach presented in this work, may not only be new, but unique in its kind so far. Perhaps, the scarcity of financial mathematics literature about Zakaat comes from the difficulty of the mathematics needed to describe the financial dynamics of fortunes in a society subject to Zakaat practice. Indeed our motivation in this work, stems from a genuine attempt to lend mathematically quantitative and qualitative corroborative evidence in support of the claimed asymptotic idealism. An ultimate result in this regard, which in our opinion may be really hard to achieve, would be to show that the religiously prescribed Zakaat proportion λ = 1/40, is critical (ideal) in some realistic mathematical model of the Zakaat problem. Perhaps, trying to unravel the secret of Zakaat practicing societies success in achieving a high measure of equitability, is mathematically more accessible by direct means such as simulations, and dynamic programming. Game theory may also be another avenue under the light of which this problem may be considered. One can think of many directions in modeling this problem, the most natural of which is a multiparticle problem (see Gillespie [10] and references for ideas in this direction) each tagged with a fortune value randomly assigned at the start, stratified by an intermediate value (Nisaab), and observing the dynamics as the richer particles (xi ) after time increments, randomly yield to the less fortunate ones (yi ), a set proportion of their wealth so much so, that some of the less fortunate particles change sides (that is to say that after receiving Zakaat from several sources, a poor individual can actually become a Zakaat paying member by crossing up the threshold). The expectation is that, over time, the number of individuals in the society under the fixed threshold or Nisaab is highly minimized in comparison with those above.
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Using the essence of these ideas in stochastic models, and connecting with corresponding Fokker-Plank Equations is a favored approach in the Synergistics literature (see for instance Gardiner [9], and Risken [13]). This modeling technique was applied to great extent in Belgacem et al. [5-6]. In Financial Mathematics, however, deriving the related Black-Sholes equation is the way to go, a la Merton [12] for instance. Hybrids of these methods, will be considered in the application sections below, after introducing all necessary stochastic concepts for the buildup and analysis pertaining to our model.
2 Standard Wiener Processes An increasing number of articles in Financial Mathematics and Economics use continuous time mathematics to value financial claims. The sections below present a treatment of this subject and provides several examples of its applications in finance. The primary purpose is to present the rules of stochastic calculus so reader of is able to follow the applications to follow (without being intimidated by the mathematics). A more rigorous measure theoretic approach (a la Billingsley [8]), can also be set up but would render some of the mathematics rather cumbersome. To understand the economic implications of continuous trading, it is necessary to specify the properties of the time series of price changes in the societal environment. In the first section we briefly restate properties of Wiener and Ito processes and then turn to stochastic calculus. Ito’s Theory (see Gardiner [9], Mckean [11] and Risken [13], for more comprehensive treatments) is presented together with rules of differentiation. These rules collapse to simple rules of differentiation when uncertainty is removed (see Belgacem [1-3], and Gardiner [9], for various consequences and comparisons). When no explicit solutions can be obtained, numerical procedures approximate solutions. A stochastic process {w(t),t ≥ 0} is a Standard Wiener process if, (1) w(0) = 0. (2) the process {w(t),t ≥ 0} has stationary and independent increments.
(2.1)
(3) for every t > 0, w(t) is normal distributed with mean zero and variance t. Let ∆w be the change in the value of w(t) over a period of length ∆t. Then the expected change in value, E(∆w), is zero, while the variance, Var(∆w) is equal to the time increment, ∆t. Since the variance is given by, Var(∆w) = E[(∆w)2 ] − [E(∆w)]2 . (2.2) it follows that the second moment, E[(∆w)2 ], is given by, E[(∆w)2 ] = ∆t.
(2.3)
The standard Wiener process has the property that all higher moments are of a magnitude smaller then ∆t. That is, with O[(∆w)2 ] being of order (∆t)2 or smaller, E[(∆w)n ] = O[(∆w)2 ],
n > 2.
(2.4)
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The Wiener process has the property that, no matter how small ∆t is, the properties of the process are still maintained. Furthermore, it can be shown that in the limit, as ∆t tends to dt and ∆w becomes dw, the second and higher moments of the change, dw, can be viewed as deterministic in the sense that the probabilities of deviations from their means are negligible compared to their means. Indeed, for all practical purposes, only the first two moments play a meaningful role in characterizing the statistical evolution of the process and for all practical purposes (in the L2 sense) we can write, E(dw) = 0,
(dw)2 = dt + O[(dt)2 ],
dwdt = O[(dt)n ],
n > 2.
(2.5)
3 Ito Processes A stochastic process {dx(t),t > 0} is an Ito process if the random variable dx can be represented as, dx = µ(x,t)dt + σ(x,t)dw,
(3.1)
where µ(x,t) is the expected change in, x, at time, t, and, σ(x,t)dw, represents the uncertainty component in x. As an example, x could represent the stock price or some individual’s fortune at some point in time. The geometric Wiener process would be a special case with, µ(x,t) = µx,
σ(x,t) = σx.
(3.2)
We are interested in functions related to this type of process. For example, if x represents the stock price, we may be interested in valuing a contingent claim whose value depends on the stock price, x, and time,t, (or on the time to expiration, T −t). Let F(x,t) be continuously differentiable function of t,and twice continuously differentiable function of x, and denote the value of the claim. When x(t) is a deterministic function, then the differential dF is the limit of ∆F, where ∆F = F(x + ∆x,t + ∆t) − F(x,t),
(3.3)
and since by the Taylor’s Series Expansion Theorem we have, ∆F =
∂F ∂F ∆x + ∆t + O[(∆t)2 ]. ∂x ∂t
In the limit we obtain,
(3.4)
∂F ∂F dx + dt. (3.5) ∂x ∂t When x(t) is not certain but instead follows an Ito process, the above differential rule cannot be applied. Ito’s lemma is just the stochastic calculus equivalent of this differential rule. dF =
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4 The Ito Theory In this section we shall state the theory, provide a brief sketch of the proof, and then develop some rules of stochastic calculus (See Gardiner [9] and Mckean [11]). Theorem 1. Let x(t) be an Ito Process as defined by equation (3.1), and let F(x,t) be continuously differentiable function of t, and twice continuously differentiable function of x, then the total differential of Fis given by, σ2 (x,t) ∂F ∂F 1 ∂2 F dF(t) = dx + dt + σ2 (x,t) 2 dt = Fx dx + [Ft + Fxx ]dt. (4.1) ∂x ∂t 2 ∂x 2 Proof. From the properties of the Wiener process we recall that, (∆x)2 = [µ(x,t)∆t + σ(x,t)∆w]2 = µ2 (x,t)(∆t)2 + σ2 (x,t)(w)2 + 2µ(x,t)σ(x,t)∆w∆t = σ2 (x,t)∆t + O[(∆t)2 ], and, ∆x∆t = [µ(x,t)∆t + σ(x,t)∆w]∆t = O[(∆t)2 ]. Furthermore, using the Taylor’s series expansion, we have, ∆F =
∂F 1 ∂2 F 1 ∂2 F ∂F ∂2 F 2 ∆x + ∆t + ∆x∆t + (∆x) + (∆t)2 + O[(∆t)2 ]. ∂x ∂t 2 ∂x2 ∂x∂t 2 ∂t 2
Taking the limit after substituting the first two equations in the previous one, and rearranging terms, we obtain, ∂F ∂F 1 ∂2 F dx + dt + σ2 2 dt. dF = ∂x ∂t 2 ∂x u t We note that compared to equation (3.5), equation (4.1) contains an additional term which arises because not all second order effects in the Taylor expansion can be ignored, with regard to the Ito process. For instance, if dx = µdt + σdw, and F(x,t) = ln(x), then by Ito’s Theorem, dF = Fx dx + [Ft +
1 σ2 dx σ2 σ2 Fxx ]dt = dx + (0 − 2 )dt = − 2 dt, 2 x 2x x 2x
(4.2)
while if F(x,t) = ex , then dF = ex (µdt + σdw) +
σ2 ex σ2 dt = ex [(µ + )dt + σdw], 2 2
(4.3)
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5 Stochastic Differentiation Rules In this section, we establish the differentiation rules in stochastic calculus, [13]. Theorem 2. Let x(t) be an Ito Process as defined by equation (3.1), and let F(x,t) and G(x,t) are continuously differentiable functions of time t, and twice continuously differentiable functions for x, and x(t) is an Ito process, then we have the following rules: Addition Rule : d(F + G) = dF + dG, ∂F ∂G Multiplication Rule : d(FG) = FdG + GdF + σ2 dt , ∂x ∂x µ ¶ µ ¶ F (GdF − FdG) σ2 ∂G ∂F ∂G = − G − F dt. Division Rule : d G G2 G3 ∂x ∂x ∂x
(5.1) (5.2) (5.3)
∂F ∂G and ∂H ∂x = ∂x + ∂x , then applying the Ito Theorem, we have, µ ¶ µ ¶ ∂2 H ∂ ∂F ∂G ∂2 F ∂2 G ∂ ∂H = + = 2 + 2, = 2 ∂x ∂x ∂x ∂x ∂x ∂x ∂x ∂x
Proof. Let H = F + G. Since,
∂H ∂F ∂G ∂t = ∂t + ∂t ,
and,
µ ¶ ∂H σ2 ∂2 H ∂H dx + + dt. ∂x ∂t 2 ∂x2 The equation for dH can then be rewritten as, ¶ · µ µ ¶¸ ∂F ∂G σ2 ∂2 F ∂2 G ∂F ∂G + dx + + + dH = + 2 dt = dF + dG. ∂x ∂x ∂t ∂t 2 ∂x2 ∂x dH =
The multiplication and division rules can be proved by similar methods. Clearly, for σ = 0, the stochastic differentiation rules simplify to the known deterministic ones. u t In particular, letting F(x,t) = x, G(x,t) = ex , then, dG =
∂G ∂G 1 ∂2 G 1 dx + dt + σ2 2 dt = ex dx + σ2 ex dt ∂x ∂t 2 ∂x 2
(5.4)
and,
∂F 1 ∂2 F ∂F dx + dt + σ2 2 dt = dx. ∂x ∂t 2 ∂x x Setting, H(x,t) = y = xe = F(x,t)G(x,t), we get, dF =
dH = dF + dG + σ2 dt Letting, a = µx + µ +
∂G ∂F 1 = xex (dx + σ2 dt) + ex dx + ex σ2 dt. ∂x ∂x 2
σ2 x + σ2 = µ(1 + x) + σ2 (1 + x/2), & b = σx + σ = σ(1 + x), 2
(5.5)
(5.6)
(5.7)
then we can write, dH = ex (adt + bdw).
(5.8)
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6 Generalized Ito Calculus Theorem The Ito stochastic rules can be generalized to take into account valuation of claims on correlated Ito processes. Provided below are results regarding two correlated processes. Theorem 3. Let ρ be the instantaneous correlation between the processes, x1 (t), and x2 (t), and let F(x1 , x2 ,t) be continuously differentiable function of t, and twice continuously differentiable function of x1 and x2 , then the total differential of F is given by, · ¸ 2 ∂F ∂F ∂2 F ∂F 1 2 ∂2 F 2∂ F dF = dx1 + dx2 + dt + σ1 2 + σ2 2 + 2ρσ1 σ2 dt, (6.1) ∂x1 ∂x2 ∂t 2 ∂x1 ∂x2 ∂x1 ∂x2 where, for i = 1, 2, dxi = µi dt + σi dwi , (dwi )2 = dt, & dw1 dw2 = ρdt.
(6.2)
For instance, if for i = 1, 2, dxi = µi dt + σi dwi , and F(x1 , x2 ,t) = x1 x2t, then, · ¸ ∂F ∂F 1 ∂2 F 2∂2 F ∂2 F ∂F 2 2 dx1 + dx2 + dt + (dx ) + dx dx + (dx ) dF = 1 1 2 2 ∂x1 ∂x2 ∂t 2 ∂x12 ∂x1 ∂x2 ∂x22 = tx2 dx1 + tx1 dx2 + x1 x2 dt + tdx1 dx2
(6.3)
= (tµ1 x2 + x1 x2 + ρσ1 σ2 + tx1 µ2 )dt + tσ1 x2 dw1 + tσ2 x1 dw2 . If F(x1 ,t) and G(x2 ,t) are two twice continuously differentiable functions of x1 and x2 respectively and continuously differentiable functions of t, with x1 (t) and x2 (t) being Ito processes defined for i = 1, 2,by equation (6.2), then we have, Addition Rule : d(F + G) = dF + dG, Multiplication Rule : d(FG) = FdG + GdF + ρσ1 σ2
(6.4) ∂F ∂G , ∂x1 ∂x2
µ ¶ F (GdF − FdG) ∂G ∂F ∂G dt Division Rule : d = − σ22 F − 2 ρσ1 σ2 G . 2 G G ∂x ∂x1 ∂x2 G3
(6.5) (6.6)
Note that when x1 = x2 , these rules reduce to the rules for one variable (5.1-3).
7 The Black - Scholes Equation Ito’s Theory can be applied to a variety of problems in Financial Mathematics. Black and Scholes were first in applying Stochastic Calculus to the problem of valuing stock options. In this section, the reader is treated to the BS equation derivation (see Merton [12] for instance). Assume that the stock price can be represented by a geometric Wiener process, dS = µSdt + σSdw,
(7.1)
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Let the call price, C, be a twice continuously differentiable function of the stock price, S, and continuously differentiable function of time, t. Then using Ito’s theorem we have, 1 dC = Cs dS +Ct dt + σ2 S2Css dt 2
(7.2)
where,
∂C ∂C ∂2C , Ct = , and Css = 2 . ∂S ∂t ∂S Substituting (7.1) into (7.2), we obtain, · ¸ σ2 2 dC = Cs µS +Ct + S Css dt +Cs σSdw, 2 Cs =
and hence, setting µc =
Cs µS +Ct + σ2 S2Css /2 , C
and σc =
Cs σS , C
(7.3)
(7.4)
(7.5)
we get,
dC = µc dt + σc dw. (7.6) C We consider a portfolio, P, where λ is the fraction of wealth allocated to the option, and (1 − λ), is the fraction allocated to the stock. The instantaneous change in the portfolio, dP/P, can be expressed as, µ ¶ µ ¶ dC dS dp =λ + (1 − λ) . (7.7) P C S
Now substituting (7.1) and (7.3) into (7.7) and rearranging, we obtain, dP = [λµc + (1 − λ)µ] dt + [λσc + (1 − λ)σ] dw. P
(7.8)
The instantaneous return on this portfolio consists of a deterministic term in conjunction with a stochastic component. If we select a value for λ such that, λσc + (1 − λ)σ = 0, or alternatively, λ=
σ , (σ − σc )
(7.9)
(7.10)
then the portfolio becomes risk free. This being the case, the deterministic component must equate to the riskless rate of return, r, that is, λµc + (1 − λ)µ = r. (7.11) From equation (7.11) we have, (µ − r) (µc − r) = . σ σc
(7.12)
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Equation (7.12) says that the net rate of return per unit of risk must be the same for the two assets. Now substituting for µc and σc into (7.12), we obtain, (µ − r) σ2 Cs σS = [(Cs µS +Ct + S2Css )C − r]/ , σ 2 C
(7.13)
which simplifies to,
σ2 2 S Css + rSCs +Ct − rC = 0. (7.14) 2 This non-stochastic equation, together with the following call price boundary conditions, fully characterizes the call price, C(ST , T ) = Max(0, ST − X), St ≥ 0, 0 ≤ t ≤ T. (7.15) The solution of this differential equation is the Black–Scholes (BS) formula.
8 Stochastic Modeling of the Zakaat Problem Zakat is annually ordained on individuals whose fortunes x(t) exceed a given threshold called Nisaab, Z0 , for t in the time interval, I = [0, T ], with T being typically one lunar year. Taking a(x,t), σ1 (x,t), and λ to be the expected value of change, the relative standard deviation, and the portion of Zakaat payed respectively, we can then write the change in x(t) as the stochastic differential equation, dx (t) = a(x,t)dt + (1 − λ)σ1 (x,t)x(t)dw1 (t) ,
x(t) ≥ Z0 , 0 ≤ t ≤ T.
(8.1)
Similarly, since the Zakaat amount, λσ1 (x,t)x(t) is to be given to the individual with fortune y(t) < Z0 , having mean value change b(x,t) and standard deviation σ2 (x,t)y(t),the change in fortune of the individual under the threshold Z0 , can be described by the stochastic equation, dy (t) = b(x,t)dt + σ2 (x,t)y(t)dw2 (t) + λσ1 (x,t)x(t)dw2 (t) ,
y(t) < Z0 .
(8.2)
Together equations (8.1) and (8.2) represent a fairly general model for the Zakaat whose mathematical analysis in the first stage may be quite intricate. To render the model more tractable at this point, we make the following assumptions and natural simplifications. If we assume that σ2 = 0, σ1 = σ, a(x,t) = ax, and b(x,t) = by, where σ, a, and, b, are constants, then we obtain the following stochastic system, dx (t) = ax(t)dt + (1 − λ)σx(t)dw1 (t) ,
x(t) ≥ Z0 , 0 ≤ t ≤ T,
dy (t) = by(t)dt + λσx(t)dw2 (t) ,
y(t) < Z0 , 0 ≤ t ≤ T.
(8.3)
In fact there no loss of generality in taking, x(0) = X0 > Z0 > Y0 = y(0) > 0, and for 0 ≤ t ≤ T , x(t), y(t) > 0. In this case, system (8.3) may be rewritten as, dx (t) /x(t) = adt + (1 − λ)σdw1 (t) , dy (t) /y(t) = bdt + λσ(x(t)/y(t))dw2 (t) ,
(8.4)
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This means that on the average, (see for instance Belgacem-Karaballi [4]), x(t) = eat , y(t) = Y0 ebt+λσ
Rt
0 (x(s)/y(s))dw2 (s)
(8.5)
Now clearly since we expect, a > b, the values of y(t) depend directly on the constants λ, σ and the size of the ratio x/y in time. For instance without any Zakaat help, (λ = 0, (or if σ = 0)), then y(t) = Y0 ebt < Z0 , and in particular, b
= 0) seem to help not effect negatively the richer if they payed Zakaat and to greatly help the poorer individuals move towards becoming paying members as they approach the threshold Nisaab. On the other hand, it seems that Zakaat becomes more important the higher the value of x(t)/y(t). Indeed if the typical Zakaat is x(T )/40 ≥ Z0 /40, then this amount seems more sizable and more important to a individual with Y (T ), such that X(T )/Y (T ) = 100, than say to an individual with X(T )/Y (T ) = 2, or even 10, albeit it is helpful to all. In fact, it depends on how much larger the fortunes of the richer than the Nisaab are (X(T ) >> Z0 , or X(T ) ≥ Z0 ), and how much smaller than the Nisaab the fortunes of the poorer are (Y (T ) > y(T ), which means that we can assume (F(T ) ' x(T )). Case 1: y(T ) / x(T ), which means that we can write, [dF/dt]t=T ' [(1 + a) − (b + 2λ)]x(T ).
(9.11)
Hence, achieving equation (9.10) means that, λ≥
1+a−b . 2
(9.12)
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We see from equation (9.12) that λ ≥ 1/2, unless b ≥ a. On the other hand, assuming that λ = 1/40, and maintain the sign of equation (9.10), we must have, b ≥ a+
38 19 = a+ . 40 20
(9.13)
Case 2: y(T )
