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IOSR Journal of Mathematics (IOSR-JM) e-ISSN: 2278-5728,p-ISSN: 2319-765X, 6, Issue 6 (May. - Jun. 2013), PP 35-41 www.iosrjournals.org Cash Flow Valuation Mode Lin Discrete Time Olayiwola. M. A. and Oni, N. O. Department of Mathematics and Statistics, The Polytechnic, Ibadan, Saki Campus. Abstract: This research consider the modelling of each cash flow valuation in discrete time. It is shown that the value of cash flow can be modeled in three equivalent ways under same general assumptions. Also, consideration is given to value process at a stopping time and/ or the cash flow process stopped at some stopping times. I. Introduction It is observed that money has a time value that is to value an amount of money we get at some future date we should discount the amount from the future date back to today. In finance and economics, discounted cash flow analysis is a method of valuing a project, company, or asset using the concept of the time value of money. All future cash flows are estimated and discounted to give their present values. The sum of all future cash flows, both incoming and outgoing, is the net present value, which is taken as the value or price of the cash flows in question. Most future cash flows models in finance and economics are assumed to be stochastic. Thus, to value these stochastic cash flow we need to take expectations. 1.1 Probability Space Probability theory is concerned with the study of experiments whose outcomes are random; that is the outcomes cannot be predicted with certainty. The collection β¦ of all possible outcomes of a random experiment is called a sample space. Therefore, an element of π of β¦ is called a sample point. Definition 1: A collection ΰΈΏofβ¦is called a π-algebra if (i) β¦πΰΈΏ (ii) β¦ π ΰΈΏ, thenπ΄πΆ π ΰΈΏ (iii) For any countable collection π΄π , π β₯ 1 ο ΰΈΏ we have ππ β₯ 1, π΄π π ΰΈΏ Definition 2: Let β¦ be a sample space and ΰΈΏ is a π βalgebra of subsets of β¦. A function π β defined on ΰΈΏ and taking values in the unit interval 0,1 is called a probability measure if (i) π β¦ =1 (ii) π π΄ β₯0 βπ΄πΰΈΏ (iii) πππ at most countable family π΄π , π β₯ 1, of mutually disjoint events we have π ππ β₯1 , π΄π = π π΄π π β₯1 The triple (β¦,ΰΈΏ, β) is called probability space. The pair (β¦,ΰΈΏ)is called the probabilizable space. Definition 3: A probability space (β¦,ΰΈΏ, β)is called a complete probability space if every subset of a β- null event is also an event. 1.1.1 Filtered Probability Space Let (β¦,ΰΈΏ, β)be a probability space and F(ΰΈΏ) = ΰΈΏπ‘ = π‘π π, β , be a family of sub π βalgebraof ΰΈΏsuch that (i) (ii) (iii) ΰΈΏπ‘ : π‘π ΰΈΏcontained all the ββ null member of ΰΈΏ ΰΈΏπ ο ΰΈΏπ‘ whenever π‘ > π β₯ 0 where ΰΈΏπ is sub π βalgebra of ΰΈΏπ‘ F (ΰΈΏ) is right continuous in the sense that ΰΈΏπ‘+ = ΰΈΏπ‘ , π‘ β₯ 0 where ΰΈΏπ >π‘ =β© ΰΈΏπ >0 then F (ΰΈΏ) = π, β is called filtration of ΰΈΏ. The quadruplet(β¦,ΰΈΏ, β,F (ΰΈΏ)) is called a filter probability space. If F(ΰΈΏ) = ΰΈΏπ‘ : π‘π π, β is a filtration of ΰΈΏ, then ΰΈΏπ‘ is the information available about ΰΈΏ at time t while filtration F (ΰΈΏ) describes the flow of information with time. www.iosrjournals.org 35 | Page Cash Flow Valuation Mode Lin Discrete Time Definition 4: Let (β¦,ΰΈΏ, β,F(ΰΈΏ))be a filtered probability space with the filtration express as F (ΰΈΏ) = ΰΈΏπ‘ : π‘π π, β and let π π = π π‘ , π‘ π π be a stochastic process with values in R, then X(t) is called an adapted to the filtration F(ΰΈΏ)if X(t) is ΰΈΏπ‘ measurable for π‘ π π. Where X(t) is a random variable. 1.2 Mathematical Expectation Let π = π 0 (β¦ , π π ), then the mean or mathematical expectation of X is defined by πΈ π = π π π ππ ππβ¦ β¦ If X has probability density function ππ₯ which is absolutely continuous with respect to lebesgue measure then πΈ π = π π π₯ππ₯ π₯ ππ₯ For a random variable π = (π1 , π2 , . . . ππ ) πΈ π = πΈ(π1 , π2 , . . . ππ ) = πΈ π1 β πΈ π2 β β β πΈ(ππ ) 1.2.1 Conditional Expectation Let X1 and X2 be two random variables, then the conditional expectation of X1 given that X2 = x2 is defined as πΈ(π1 |π2 = π₯2 ) = πΈ(π1 |π₯2 ) = π₯1 π(π₯1 |π₯2 ) ππ₯1 π(π₯1 , π₯2 ) ππ₯1 π(π₯2 ) > 0 π(π₯2 ) Properties of Conditional Expectation (i) Let π , π π π , π, π πππ πΌπ β π£πππ’ππ ππππππ π£ππππππππ πππ F is a sub π βalgebra of ΰΈΏ then πΈ[(ππ + ππ)|F ] = πΈ(ππ|F) + E( bY|F) = π πΈ(π|F) + π πΈ( π|F) (ii) If π β₯ 0 π. π , π‘πππ πΈ(π|F )β₯ 0 π. π πΈ(β |F ) is positively preserving. = π₯1 (iii) E( Cβ£F ) = C where C is a constant value. (iv) If X is F - measurable, then E( Xβ£F ) = X (v) If X is F - measurable, then E(Xβ£F ) = E(X) (vi) If F1 and F2 are both π βalgebra ofΰΈΏ such that F1 ο F2 then πΈ(πΈ(π|F2β£F1) = E(Xβ£F1 ) a.s (vii) If π π β πΌ (β¦,ΰΈΏ, β) and ππ β π1 thenπΈ(ππ | F )β πΈ(π| F) in β πΌ (β¦,ΰΈΏ, β) (viii) If Ο: π β π is convex and Ο(x)π β πΌ (β¦,ΰΈΏ, β)such that E β£Ο(x)β£< β then E (Ο(x)β£F)β₯ ΟE( Xβ£F ) a.s (xi) If X is independent of F1then E(Xβ£F ) = E(X) 1.3 Stochastic Processes Definition 5: Let ππ‘ π , π π β¦ , π‘π π, β be a family of Rd β valued random variable define on the probability space (β¦,F, β) then the family π π‘, π π‘π π, β of X is called a stochastic process π π β¦ A stochastic process ππ‘ π‘, π , π π β¦ , π‘ π π is thus a function of two variables X: T X β¦ β π π , ππ‘ πππ π‘ β₯ 0 is said to be Ft β adapted if for every π‘ β₯ 0, Xt is Ft measurable. 1.4 Martingale Process Definition 6: A stochastic process ππ πβ₯1 is said to be a Martingale process if (i) πΈ[π β£ ππ ] < β (ii) πΈ[ππ+1 β£ π1 , π2 , . .. ππ ] = ππ Definition 7: Let π π‘. π = π π‘ , π‘ π π be an adapted R β valued stochastic process on a filtered probability space (β¦,F, (Ft)t ππ ,P) then X is called martingale if for each π‘ π π ο R E[X(t)β£F] = X(s) a.s s < t. II. Model Formulation The discounted cash flow formular is derived from the future value formular for computing the time value of money and compounding returns. πΆ = π(1 + π)π www.iosrjournals.org 36 | Page Cash Flow Valuation Mode Lin Discrete Time Discounted present value (DPV) for one cash flow (FV) in one future period is expressed as πΆ π= = πΉπ(1 β π)π (1 + π)π Where, ο V is the discounted present value; ο C is the norminal value of a cash flow amount in a future periods; ο r is the interest rate, which reflects the cost of tying up capital and may also allow for the risk that the payment may not be received in full; π ο d is the discount rate, which is (1+π) ο n is the time in years before the future cash flow occurs. For multiple cash flows in multiple time periods to be discounted it is necessary to sum them and take the expectation. β πΆ π0 = πΈ . . . . . . . . (1) (1 + π)π π=1 We take the expectation because we assume that most cash flows models are stochastic. Introduction of the value at time π‘ β₯ 0 we have a dynamic model β πΆπ ππ‘ = πΈπ‘ (1 + π)πβπ‘ π=π‘+1 Here πΈπ‘ (β) is the expectation given information up to and including time t. 1 πππ‘π‘πππ ππ = 1+π π β π0 = πΈ πΆπ ππ π=1 2.1 General definitions Definition 2.1 : A cash flow (πΆπ‘ ) π‘ π π is a process adapted to the filtration Ft and that for each π‘ π π, πΆπ‘ < β π. π A cash flow process that is non-negative a.s will be referred to as a dividend process. Definition 2.2 A discount process is a process π: π × π × β¦ β π satisfying (i) 0 < π π , π‘ < β π. π β π , π‘ π π (ii) π π , π‘, π ππ Fmax(s,t) β measurable β π , π‘ π π πππ (iii) π π , π‘ = π π , π’ π π’, π‘ π. π β π , π’, π‘ π π A discount process fulfilling ππΌ 0 < π π , π‘ < 1 π. π β π , π‘ π π π€ππ‘π π β€ π‘will be referred to as normal discount process π π , π‘ is interpreted as the stochastic value at time s of getting one unit of currency at time t, while π π‘, π will be referred to as the growth of one unit of currency invested at time t, at time s. We write ππ‘ = π 0, π‘ , π‘ β π as a short-hand notation. Definition 2.3 The discount rate or instantaneous rate at time t implied by discount process denoted by r t for t = 1, 2, . . . is defined as 1 π¬ π‘β1 ππ‘ = β1= β1 π(π‘ β 1, π‘) π¬ π‘ Where β§ is the deflator associated with m. The relationship between the discount process and rate process is deducted from the following Lemma. Lemma 1: Let m be a discount process and let r be the discount rate implied by m. Then the following holds: (i) β1 < ππ‘ < β, π‘πβ (ii) π β₯ 0 πππ π ππ ππππππ πππ πππ’ππ‘ ππππππ π (iii) For any given π β₯ π we have for π‘ π β 0 < π΄ < ππ πππ 0 < ππ‘ β€ π βπ‘πΌπ 1+π (iv) The instantaneous rate process and discount process uniquely determine each other. III. Valuation of Cash Flow Definition 3: Given a cash flow process Ct for π‘ π β and a discount process π π , π‘ for π , π‘ π π we define for π‘ π β the value process as ππ‘ = πΈ[ βπ=π‘+1 πΆπ π(π‘, π)|Ft ] www.iosrjournals.org 37 | Page Cash Flow Valuation Mode Lin Discrete Time The value process is defined as ex β dividend if cash flows from time t+1 and onwards is included in the value at time t. It is otherwise considered as cum β divided if the cash flow is included at time t. The following lemma gives a sufficient condition as to when is the value process will be finite a.s Lemma 2: If C is a cash flow process and πΈ | βπ=1 πΆπ ππ | < β π. π Then |Vt| Λ β a.s for all π‘ π β Proof: Since πΈ | βπ=1 πΆπ ππ | < β for π‘ π β |Vt| = πΈ[| βπ=π‘+1 πΆπ π(π‘, π)|Ft ] = πΈ[| βπ=π‘+1 πΆπ π(π‘, 0)π(0, π)|Ft ] β€ π(π‘, 0)πΈ[| βπ=π‘+1 πΆπ π(0, π)|Ft ] 1 β€ π (0,π‘) πΈ[| βπ=π‘+1 πΆπ ππ |Ft ] 1 πΈ[| βπ=1 πΆπ ππ + βπ=π‘+1 πΆπ ππ ππ‘ 1 πΈ[| βπ=1 πΆπ ππ |Ft + | π‘π=1 πΆπ ππ |]< β ππ‘ β€ β π‘ π=1 πΆπ ππ |Ft ] β€ π. π Thus, if C is a discount process such that V0 Λ β , then Vt Λ β a.s for every π‘ π β Note that if X is a random variable with E|X| Λ β then E[X|Ft] for t = 1, 2, . . . , is a uniformly integrable (U.I) martingale. Hence, if [| βπ =1 πΆ π ππ |] Λ β a.s then πΈ [ βπ =0 πΆπ ππ |Ft ] is a U.I martingale. Below proposition gives the characteristics of value process Proposition 2: Let C and m be a cash flow and discount process respectively. If [| βπ =1 πΆ π ππ |] Λ β then the discounted value process (π π‘ ππ‘ ) can be written as π π‘ ππ‘ = ππ‘ β π΄ π‘ πππ π‘ π β Where M is a U.I martingale and A an adopted process. Furthermore, lim π π‘ ππ‘ = 0 π .π π‘ ββ Proof: Note that [| πΏππ‘ π =0 πΆ π ππ |] Λ β π . π π πππππΈ ππ‘ = πΈ [ π =0 πΆ π ππ |Ft] for π‘ π β β β [| π =0 πΆ π ππ |] β Λβ π‘ π΄π‘ = πΆ π ππ , π‘ π β π =0 It is then immediate that π π‘ ππ‘ = ππ‘ β π΄ π‘ . Since [| βπ =1 πΆ π ππ |] Λ β, M is a U.I martingale and A ia adapted. As π‘ β β, the UI martingale ππ‘ = πΈ [ βπ =0 πΆ π ππ |Ft] = βπ =0 πΆ π ππ = π΄ β π . π This implies that as π‘ β β lim π π‘ ππ‘ = lim ππ‘ = lim π΄ π‘ = 0 π‘ ββ π‘ ββ π‘ ββ Since πβ = π΄ β = βπ =0 πΆ π ππ is finite a.s . Now let C be dividend process. Then A is an increasing process and πΈ π΄ β = πΈ [ βπ =0 πΆ π ππ ] < β by assumption. Thus: π π‘ ππ‘ = πΈ [ βπ =π‘ +1 πΆ π ππ |Ft] = πΈ π΄ β Ft ] - At Characteristics relation between C (cash flow process), m(discount process) and V(value process) is given in the following theorem. Theorem 3: Let C be a cash flow process and m a discount process such that E[|C k|mk] <β. Then the following three statements are equivalent. (i) For every π‘ π β π π‘ = πΈ [ βπ =π‘ +1 πΆ π π(π‘ , π )|Ft] (ii) For every π‘ π β π‘ ππ‘ = π π‘ ππ‘ + πΆ π ππ π =1 is a UI martingale (b) π π‘ ππ‘ β 0 π . π π€ πππ π‘ β 0 (iii) For every π‘ π β π π‘ = πΈ [π(π‘ , π‘ + 1)(πΆ π‘ +1 + π π‘ +1 )|Ft] limπ ββ πΈ [π π‘ , π π π |Ft] = 0 (a) (b) Proof: Note that πΈ | show that π =1 πΆ π ππ | β β€ πΈ[ π =1 |πΆ π |ππ ] β < β which implies that | www.iosrjournals.org π =1 πΆ π ππ | β < β π . π Now to 38 | Page Cash Flow Valuation Mode Lin Discrete Time (i) ο (ii) and (i) ο (iii) (i) ο (ii) The if part follows from the proposition. For the only if part the expression in (ii) a is written as βππ +1 πΆ π +1 = ππ +1 π π +1 β ππ π π β ππ +1 ππ And take the sum from t to T β 1 π ππ πΆ π = ππ π π β ππ‘ π π‘ β ππ + ππ‘ β π =π‘ +1 If we let π β β the term ππ π π β 0 π . π by the assumption and ππ β πβ a.s from the convergence result of UI martingales. Thus, we have β π π‘ ππ‘ = πΆ π ππ β πβ + ππ‘ . .. .. . . . . (1) π =π‘ +1 It is observed that πΈ πβ Ft ] = Mt from the convergence result of UI martingale. Taking conditional expectation with respect to Ftof (1) πΈ π π‘ ππ‘ Ft ] = πΈ [ βπ =π‘ +1 πΆ π ππ |Ft] - πΈ πβ Ft ] + πΈ ππ‘ Ft ] π π‘ ππ‘ = πΈ [ βπ =π‘ +1 πΆ π ππ |Ft] -ππ‘ + ππ‘ π π‘ ππ‘ = πΈ [ βπ =π‘ +1 πΆ π ππ |Ft] 1 π ππ‘ = πΈ [ βπ =π‘ +1 πΆ π ππ |Ft] = πΈ [ βπ =π‘ +1 πΆ π π |Ft] ππ‘ ππ‘ = πΈ [ βπ =π‘ +1 πΆ π π(π‘ , π )|Ft] Note: (i) πβ = π΄ β = βπ =1 πΆ π ππ πΈ πβ Ft ] = πΈ [ βπ =1 πΆ π ππ |Ft] = ππ‘ (ii) πΈ ππ‘ Ft ] = πΈ [ βπ =1 πΆ π ππ |Ft | Ft] = πΈ [ βπ =1 πΆ π ππ |Ft] Regarding (i) ο (iii) We begin with the if part. Fix π‘ π β π π‘ = πΈ [π(π‘ , π‘ + 1)πΆ π‘ +1 + βπ =π‘ +2 πΆ π π(π‘ , π ) |Ft] π π‘ = πΈ [π(π‘ , π‘ + 1)πΆ π‘ +1 + π(π‘ , π‘ + 1) βπ =π‘ +2 πΆ π π(π‘ + 1, π ) |Ft] π π‘ = πΈ [π(π‘ , π‘ + 1)πΆ π‘ +1 + βπ =π‘ +2 πΆ π π(π‘ + 1, π ) |Ft] π π‘ = πΈ [π(π‘ , π‘ + 1)πΆ π‘ +1 + π π‘ +1 |Ft] πππ€ πππ‘ π β₯ π‘ . πΉπππ π π = βπ =π +1 πΆ π π(π , πΎ ) |Ft We have π π = πΈ [ βπ =π +1 πΆ π π(π , π )π(π‘ , π ) |FT] π(π‘ ,π ) = πΈ [ βπ =π +1 πΆ π π(π‘ ,π ) |FT] π π π(π‘ , π ) = πΈ [ β π =π +1 πΆ π π(π‘ , π ) |FT] . . . . . . . . . . . . . (2) Taking conditional expectation with respect to FT of (2) πΈ [π π π(π‘ , π )/πΉ π ] = πΈ [ βπ =π +1 πΆ π π(π‘ , π ) |FT|FT] = πΈ [ βπ =π +1 πΆ π π(π‘ , π ) |FT] 1 = πΈ [ βπ =π‘ +1 πΆ π ππ |Ft] ππ‘ Since βπ =1 πΆ π ππ β€ βπ =1 |πΆ π |ππ And the last random variable is integrable by assumption we get for every π‘ π β and A π Ft lim πΈ π π‘ , π π π 1π΄ = lim π π‘ , π π π 1π΄ = 0 π ββ π ββ To prove the only if part we iterate (iii) a to get π π‘ = πΈ [ ππ =π +1 πΆ π π π‘ , π + π(π‘ , π )π π |Ft] 1 ππ‘ = πΈ [ βπ =π‘ +1 πΆ π ππ |Ft] + πΈ [π(π‘ , π )π π |Ft] ππ‘ π΄π π β β πππ πΈ [π(π‘ , π )π π |Ft] β 0 a.s from (iii)b since π β πΆ π ππ β€ π =π‘ +1 |πΆ π |ππ π =1 And the last random variable is integrable by assumption. Thus, we get π π = πΈ [ βπ =π‘ +1 πΆ π π(π‘ , π )|Ft] using the theorem of dominated convergence. IV. Stopping the cash flow and value process Theorem3: Show that the three equivalent representations of the value process at deterministic times. The theorem also assumes that cash flow stream is defined for all π‘ β₯ 0. In the stream, we shall consider the value process stopping time and / or the cash flow process stopped at some stopping time. Using the fact that the martingale www.iosrjournals.org 39 | Page Cash Flow Valuation Mode Lin Discrete Time ππ‘ = π π‘ ππ‘ + π‘ π =1 πΆ π ππ from theorem 3, is uniformly integrable we can get the following results. Proposition 4: Let C and m be cash flow process and discount process respectively such that πΈ [ βπ =1 |πΆ π |π(π‘ , π ) < β for every π‘ π β . Further let π πππ π be (Ft ) stopping times such that π β€ π π . π then the following two statements are equivalent. (i) We have π π = πΈ [ ππ =π +1 πΆ π π π , π + π π π(π , π )1π <β |FΟ] {Ο Λ β} (ii) (a) for every π‘ π β π‘ ππ‘ = π π‘ ππ‘ + πΆ π ππ π =1 is a UI martingale, and (b) π π‘ ππ‘ β 0 π . π ππ ππ π‘ β β Proof: We start with (ii) ο (i) considering the stopping times π β§ π Where π π β because the stopping time π may be unbounded we get π β§π ππ β§π = π π β§π ππ β§π + πΆ π ππ . . . . . . . . . . . (3) π =1 ππ π β β, π π β§π ππ β§π β π π ππ 1π <β π . π Since π π ππ 1π <β β 0 π . π assuming we let π β β Thus, we have from (3) π π π ππ 1π <β + πΆ π π =1 π π Since M is uniformly integrable we take conditional expectation of ππ with respect to the π βalgebraFΟ to get on {Ο Λ β} πΈ ππ FΟ ] = πΈ [ ππ =1 πΆ π ππ + π π ππ 1π <β |FΟ] Note that πΈ ππ FΟ ] = ππ Thus, ππ = πΈ [ ππ =1 πΆ π ππ + ππ =π +1 πΆ π ππ + π π ππ 1π <β |FΟ] ππ = ππ =1 πΆ π ππ + πΈ [ ππ =π +1 πΆ π ππ + π π ππ 1π <β |FΟ] ππ = π ππ = πΈ [ ππ = π π =1 πΆ π ππ + πΈ [ π π =π +1 πΆ π πΆ π ππ + π π ππ 1π <β π =1 ππ + π π ππ 1π <β |FΟ] It implies π π π π =1 πΆ π ππ + π π ππ 1π <β = π =1 πΆ π ππ + πΈ π =π +1 πΆ π ππ + π π ππ 1π <β FΟ] Hence, π π ππ 1π <β = πΈ [ ππ =π +1 πΆ π ππ + π π ππ 1π <β |FΟ] π π π π = πΈ [ ππ =π +1 πΆ π ππ + π π ππ 1π <β |FΟ] ππ = πΈ[ π π =π +1 πΆ π π π π(π , π ) + π π π(π , π )1π <β |FΟ] This is our desired result. To prove (i) ο (ii) Let π = β πππ π = π‘ πππ π‘ π β ππ π π . π π π = πΈ [ ππ =π +1 πΆ π π(π , π ) + π π π(π , π )1π <β |FΟ] π π‘ = πΈ [ βπ =π‘ +1 πΆ π π(π‘ , π ) + π β π(π‘ , β)|Ft] π π π π‘ = πΈ [ βπ =π‘ +1 πΆ π ππ + π β πβ |Ft] π π‘ ππ‘ = πΈ [ π‘ π =π‘ +1 πΆ π π‘ ππ + π βπβ|Ft] πππ‘π π‘ π ππ‘ π π‘ ππ‘ β 0 π . π π‘ β β π . π π π‘ ππ‘ = πΈ [ βπ =π‘ +1 πΆ π ππ |Ft] π π‘ ππ‘ = πΈ [ π‘π =1 πΆ π ππ + βπ =π‘ +1 πΆ π ππ β π‘π =1 πΆ π ππ |Ft] π π‘ ππ‘ = πΈ [ π‘π =1 πΆ π ππ β π‘π =1 πΆ π ππ |Ft] π π‘ ππ‘ = πΈ [ βπ =1 πΆ π ππ |Ft] β π‘π =1 πΆ π ππ Since πΈ [ βπ =1 πΆ π ππ |Ft] is a UI martingale i.e ππ‘ Thus, we have β π‘ π π‘ ππ‘ = ππ‘ β Hence, ππ‘ = π π‘ ππ‘ + π‘ π =1 πΆ π ππ as desired. πΆ π ππ π =1 www.iosrjournals.org 40 | Page Cash Flow Valuation Mode Lin Discrete Time V. Conclusion We have been able to show that cash flow models can be written in equivalent forms using a suitable discount factor. We proceed to consider the value process stopping at stopping time and / or cash flow process stopped at some stopping times with the use of a proposition. References: [1]. Akepe A. O (2008): Option Pricing on Multiple Assets M.Sc. Project University of Ibadan [2]. Armerin F. (2002): Valuation of Cash flow in Discrete Time working paper Ayoola E. O: Lecture note on Advance Analysis Lebesgue Measure University of Ibadan, Unpublished [3]. Ugbebor O. O (1991): Probability Distribution and Elementary Limit Theorems. [4]. University of Ibadan External Studies Programme University Of Ibadan [5]. Elliott R. J (1982): Stochastic Calculus and Applications, Springerverlag [6]. Pratt, S. et al (2000): Valuing a Business. McGraw-Hill Professional. McGraw Hill [7]. Williams, D. (1999): Probability with Martingales, Cambridge University Press Ξ¦Ksendal, B. (1998): Stochastic differential Equation, Fifth Edition, Springer - verlag www.iosrjournals.org 41 | Page