Int. J. Anal. Appl. (2022), 20:54 Direct Solution of Black-Scholes-Merton European Put Option Model on Dividend Yield With Modified-Log Payoff Function S.E. Fadugba1,∗, A.A. Adeniji2, M.C. Kekana2, J.T. Okunlola3, O. Faweya4 1Department of Mathematics, Ekiti State University, Ado Ekiti, 360001, Nigeria 2Department of Mathematics, Tshwane University of Technology, Pretoria, South Africa 3Department of Mathematical and Physical Sciences, Afe Babalola University, Ado Ekiti, Nigeria 4Department of Statistics, Ekiti State University, Ado Ekiti, 360001, Nigeria ∗Corresponding author: sunday.fadugba@eksu.edu.ng Abstract. This paper proposes a framework based on the celebrated transform of Mellin type (MT) for the direct solution of the Black-Scholes-Merton European Put Option Model (BSMEPOM) on Divi- dend Yield (DY) with Modified-Log Payoff Function (MLPF) under the geometric Brownian motion. The focal goal of this paper is to use MT to obtain a valuation formula for the European Put Option (EPO) which pays a DY with MLPF. By means of the MT and its inversion formula, the price of EPO on DY was expressed in terms of integral equation. The valuation formula of EPO was obtained with the help of the convolution property of MT and final time condition. MT was tested on an illustrative example in order to measure its performance, effectiveness and suitability. The MLPF was compared with other existing payoff functions. Hence, the effect of DY on the pricing of EPO with MLPF was also investigated. 1. Introduction Option valuation has become extremely popular in computational finance. This popularity has been displayed as one of the key major areas in derivative security. In other words, option valuation has contributed greatly to the financial markets. There is a massive growth in trading activities on derivatives globally from the inception of the Black-Scholes pricing formula [1,2]. It is noteworthy to say that the Black-Scholes models for linear payoff function has been used by many researchers and as well as become one of the utmost areas in financial markets over the last few decades. Immediately Received: Aug. 17, 2022. 2010 Mathematics Subject Classification. 91B24, 91B28, 91G20. Key words and phrases. convolution property; dividend yield; final time condition; Mellin transform. https://doi.org/10.28924/2291-8639-20-2022-54 ISSN: 2291-8639 © 2022 the author(s). https://doi.org/10.28924/2291-8639-20-2022-54 2 Int. J. Anal. Appl. (2022), 20:54 after the huge success recorded by the Black-Scholes model for vanilla option flavours, several other valuation formula were developed for options pricing with different payoff functions such as Mellin transform, binomial model, finite difference method, Monte Carlo method, e.t.c; see [3] – [6]. For mathematical framework, some implementations of transform methods of different types in financial markets; see [7]– [15]. Ghevariya [16] solved the classical Black-Scholes European put option model for Modified-Log payoff function with the help of the MT. In this paper, a direct solution of BSMEPOM via the celebrated transform of Mellin type is proposed in the sense of DY and MLPF. The remaining part of the paper is listed as follows, Section 2 captures the brief concepts of MT. A new result that captures the governing model for EPO on DY with MLPF and the solution of BSMEPOM on DY with MLPF is stated and proved in Section 3. An illustrative example on the application of MT to EPO is captured by Section 4. Section 5 is the concluding part of the paper. 2. Mellin transform This section captures some definitions of terms based on the framework of the Mellin transform. 2.1. Definitions of Terms. Definition 2.1. Let f (x) be a locally Lebesgue integrable function. The Mellin transform of f (x) is defined as M[f (x),ω] := f̃ (ω)= ∫ ∞ 0 f (x)xω−1dx (2.1) The Mellin transform variable ω is a complex number, ω = <(.)+ i=(.), where <(.) is the real part, i is the imaginary unit and =(.) is the imaginary part. Definition 2.2. If f (x) is an integrable function with fundamental strips (a,b), then if c is such that a < c < b and {f̃ (ω) : ω = c + it} is integrable, the inverse Mellin transform is defined as M−1[f̃ (ω)]= f (x)= 1 2πi ∫ c+i∞ c−i∞ f̃ (ω)x−ωdω (2.2) Remark 2.1. For more details on the condition that ensures the existence of MT; see [17]. Remark 2.2. The fundamental operational properties of the Mellin transforms such as scaling, shifting, derivatives, integrals, convolution, multiplicative convolution and Parseval’s formula are well detailed in [9,12,17,18]. 3. The Solution of BSMEPOM on DY with MLPF Ghevariya derived BSM formula on non-dividend yield for ML-payoff function [16]. In this section, Black-Scholes-Merton formula on dividend yield with MLPF is derived via the MT in the following result. Int. J. Anal. Appl. (2022), 20:54 3 Theorem 3.1. Consider the BSMEPOM on DY with MLPF of the form ∂PE(St,t) ∂t +(r −q)St ∂PE(St,t) ∂St + (σSt) 2 2 ∂2PE(St,t) ∂(St) 2 − rPE(St,t)=0 (3.1) subject to the boundary conditions lim St→∞ PE(St,t)=0 on [0,T) (3.2) lim St→0 PE(St,t)= K er(T−t) on [0,T) (3.3) and MLPF PE(ST ,T)= [ ST ln ( K ST )]+ on [0,∞) (3.4) where PE(St,t), t, T, St, K, σ, r and q are the price of EPO, current time, time to expiry, underlying asset price, strike price, volatility, risk-free interest rate and DY, respectively, then the valuation formula for BSMEPOM on DY with MLPF is given by PE(St,t)= Ste −qτ [σ√τη(d)− (D1 +D2τ)N(−d)] (3.5) with D1 = ln ( St K ) , (3.6) D2 = ( r −q + σ2 2 ) , (3.7) d = D1 +D2τ σ √ τ , (3.8) τ = T − t,η(κ)= 1 √ 2π e− κ2 2 ,N(κ)= ∫ κ −∞ η(κ)dκ. (3.9) Proof. Taking the MT of (3.1) and using its linearity, independence of time derivatives and shifting properties and rearranging terms, one obtains ∂P̃E(ω,t) ∂t =− σ2 2 (ω2 +ω(1−α1)−α2)P̃E(ω,t) (3.10) where α1 = 2(r −q) σ2 ,α2 = 2r σ2 Solving (3.1), yields P̃E(ω,t)= m(ω)e −1 2 σ2(ω2+ω(1−α1)−α2)t (3.11) But m(ω)=M(PE(ST ,T),ω)e 1 2 σ2(ω2+(1−α1)ω−α2)T (3.12) which is equivalent to m(ω)= g̃(ω)e 1 2 σ2(ω2+(1−α1)ω−α2)T (3.13) 4 Int. J. Anal. Appl. (2022), 20:54 Substituting (3.13) into (3.11), yields P̃E(ω,t)= g̃(ω)e 1 2 σ2(ω2+(1−α1)ω−α2)τ (3.14) with τ = T − t. By means of (2.2), (3.14) yields PE(St,t)= 1 2πi ∫ c+i∞ c−i∞ g̃(ω)e 1 2 σ2(ω2+(1−α1)ω−α2)τS−ωt dω (3.15) which is the integral equation for governing equation (3.1). Let ξ(St)= 1 2πi ∫ c+i∞ c−i∞ e σ2 2 (ω2+ω(1−α1)−α2)S−ωt dω (3.16) Using the fact that e 1 2 σ2(ω2+(1−α1)ω−α2)τ = e−β1(β 2 1+α2)+β1(ω+β2) 2 (3.17) where β1 = σ2τ 2 ,β2 = 1−α1 2 (3.18) Thus ξ(St)= e−β1(β 2 1+α2) 2πi ∫ c+i∞ c−i∞ eβ1(ω+β2) 2 S−ωt dω (3.19) Using the transformation given by [19]. eφω 2 = 1 2 √ π ∫ ∞ 0 1 √ φ exp ( −(lnSt)2 4φ ) (St) ω−1dSt, <(φ)≥ 0 (3.20) yields ξ(St)= e −β1(β21+α2) Sβ2 σ √ 2πτ e −1 2 ( ln(S) σ √ τ )2 (3.21) Similarly, ξ ( St v ) = e−β1(β 2 1+α2) (St v )β2 σ √ 2πτ e −1 2   ln ( St v ) σ √ τ  2 (3.22) Using the terminal condition (3.4), then g(St)=M−1(g̃(ω))= [ ST ln ( K ST )]+ (3.23) Also g(v)= [ v ln ( K v )]+ (3.24) With the help of the convolution property of MT, (3.15) becomes PE(St,t)= ∫ ∞ 0 g(v)ξ ( St v ) 1 v dv (3.25) Substituting (3.22) and (3.24) into (3.25), one gets PE(St,t)= ∫ ∞ 0 [ v ln ( K v )]+ e−β1(β 2 1+α2) (St v )β2 σ √ 2πτ e −1 2   ln ( St v ) σ √ τ  2 1 v dv (3.26) Int. J. Anal. Appl. (2022), 20:54 5 PE(St,t)= e −β1(β21+α2) S β2 t σ √ 2πτ ∫ K 0 [ v ln ( K v )] e −1 2   ln ( St v ) σ √ τ  2 1 v dv (3.27) Simplifying further, yields PE(St,t)= e −β1(β21+α2) S β2 t σ √ 2πτ ∫ K 0 ln(K) 1 vβ2 e −1 2   ln ( St v ) σ √ τ  2 dv −e−β1(β 2 1+α2) S β2 t σ √ 2πτ ∫ K 0 ln(v) 1 vβ2 e −1 2   ln ( St v ) σ √ τ  2 dv (3.28) PE(St,t)= e −β1(β21+α2) S β2 t σ √ τ [ln(K)G1 −G2] (3.29) where G1 = 1 √ 2π ∫ K 0 1 vβ2 e −1 2   ln ( St v ) σ √ τ  2 dv (3.30) G2 = 1 √ 2π ∫ K 0 ln(v) vβ2 e −1 2   ln ( St v ) σ √ τ  2 dv (3.31) Let y = ln ( S v ) σ √ τ (3.32) Thus G2 = σ √ τS −β2+1 t e β1(β2−1)2[σ √ τJ1 − ln(St)J2] (3.33) where J1 = 1 √ 2π ∫ ln( Stv ) σ √ τ ∞ ye− 1 2 (y−σ √ τ(β2−1))2dy (3.34) J2 = 1 √ 2π ∫ ln( Stv ) σ √ τ ∞ e− 1 2 (y−σ √ τ(β2−1))2dy (3.35) Taking t = y −σ √ τ(β2 −1),d = ln ( St K ) −σ2τ(β2 −1) σ √ τ = ln ( St K ) + ( r −q + σ 2 2 ) τ σ √ τ (3.36) Equations (3.34) and (3.35) become J1 =−[η(d)+σ √ τ(β2 −1)N(−d)] (3.37) and J2 =−N(−d) (3.38) 6 Int. J. Anal. Appl. (2022), 20:54 respectively. Substituting (3.37) and (3.38) into (3.33), yields G2 =−σ √ τS −β2+1 t e β1(β2−1)2[σ √ τη(d)+(σ2τ(β2 −1)− ln(St))N(−d)] (3.39) Similarly, G1 = σ √ τS −β2+1 t e β1(β2−1)2[N(−d)] (3.40) Using (3.39), (3.40), the values of α2, β1, β2 and (3.29), the result follows. Hence, this completes the proof � 4. Numerical Example Consider the valuation of the EPO on a DY with MLPF via the MT using the following parameters S,K,r,σ,q,T in Table 1. Table 1. The parameters Parameters Values S in dollars 100 K in dollars 100, 110, 120, 130, 140, 150 r 8% σ 0.5 q 0,5%,20%,60%,100% T in months 6 The results obtained are displayed in Tables 2 and 3. Table 2. The comparative study of MLPF, Log Payoff [20] and Linear Payoff [1] with q =0 K MLPF Log Payoff [20] Linear Payoff [1] 100 0 0 0 110 9.5310 0.0953 10 120 18.2322 0.1823 20 130 26.2364 0.2624 30 140 33.6472 0.3365 40 150 40.5465 0.4055 50 Int. J. Anal. Appl. (2022), 20:54 7 Figure 1. The Plots of Table 2. Table 3. The Effect of DY on the price of EPO with MLPF K/q 0 0.05 0.2 0.6 1 100 9.5684 10.3063 12.6497 19.3538 25.5118 110 13.7482 14.6425 17.3986 24.6869 30.6874 120 18.4538 19.4684 22.5140 30.0284 35.6212 130 23.5158 24.6103 27.8197 35.2583 40.2814 140 28.7839 29.9193 33.1776 40.3074 44.6658 150 34.1365 35.2780 38.4899 45.1414 48.7877 Figure 2. Physical Interpretation of the Effect of Dividend Yield on the Price of EPO using Table 3. 8 Int. J. Anal. Appl. (2022), 20:54 5. Conclusion A direct solution of BSMEPOM via the celebrated transform of Mellin type in the sense of DY and MLPF has been proposed in this paper. The MT has the ability of handling complex functions by means of its fundamental properties and it is closely related to other well-known transforms such as Laplace and Fourier types. The integral equation for the representation of the price of EPO with DY was obtained. The closed form approximation formula for EPO was also obtained via MT with the help of its convolution property and final time condition. Moreover, the MT was tested on some parameters to show its performance, effectiveness, and suitability. From Table 2, it is clearly seen that the MLPF used in this present paper performed better than the log payoff function used in [20] and also was found to be very close to the linear payoff function of plain vanilla [1]. It is observed from Table 3, that the holder is more beneficial to enter into a European put option. In other words, however, the benefits of these cash flows are given to the holder of a put option. Table 3 shows that increase in DY leads to increase in the prices of the EPO with MLPF. The effect of DY is captured in Figure 2. 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Mellin transform 2.1. Definitions of Terms 3. The Solution of BSMEPOM on DY with MLPF 4. Numerical Example 5. Conclusion References