Int. J. Anal. Appl. (2023), 21:29 Analytic Solution of Black-Scholes-Merton European Power Put Option Model on Dividend Yield with Modified-Log-Power Payoff Function S.E. Fadugba1,2,3, A.A. Adeniji4, M.C. Kekana4, J.T. Okunlola5, O. Faweya6 1Department of Mathematics, Ekiti State University, Ado Ekiti, 360001, Nigeria 2Department of Physical Sciences, Mathematics Programme, Landmark University, Omu-Aran, Nigeria 3Landmark University SDG 4 (Quality Education Research Group), Omu-Aran, Nigeria 4Department of Mathematics, Tshwane University of Technology, Pretoria, South Africa 5Department of Mathematical and Physical Sciences, Afe Babalola University, Ado Ekiti, Nigeria 6Department of Statistics, Ekiti State University, Ado Ekiti, 360001, Nigeria ∗Corresponding author: sunday.fadugba@eksu.edu.ng, fadugba.sunday@lmu.edu.ng Abstract. This paper proposes a framework based on the celebrated transform of Mellin type (MT) for the analytic solution of the Black-Scholes-Merton European Power Put Option Model (BSMEPPOM) on Dividend Yield (DY) with Modified-Log-Power Payoff Function (MLPPF) under the geometric Brownian motion. The MT has the capability of tackling complex functions by means of its funda- mental properties and it is closely related to other well-known transforms such as Laplace and Fourier types. The main goal of this paper is to use MT to obtain a valuation formula for the European Power Put Option (EPPO) which pays a DY with MLPPF. By means of MT and its inversion formula, the price of EPPO on DY was expressed in terms of integral equation. Moreover, the valuation formula of EPPO was obtained with the help of the convolution property of MT and final time condition. The MT was tested on an illustrative example in order to measure its performance, effectiveness and suitability. The MLPPF was compared with other existing payoff functions. Hence, the effect of DY on the pricing of EPPO with MLPPF was also investigated. 1. Introduction The popularity of option pricing in financial mathematics has been displayed as one of the key major areas in derivative security. In other words, option valuation has contributed greatly to the financial Received: Jan. 9, 2023. 2010 Mathematics Subject Classification. 91B24, 91G20. Key words and phrases. convolution property; final time condition; integral representation; Mellin transform. https://doi.org/10.28924/2291-8639-21-2023-29 ISSN: 2291-8639 © 2023 the author(s). https://doi.org/10.28924/2291-8639-21-2023-29 2 Int. J. Anal. Appl. (2023), 21:29 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 model for linear payoff 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 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. Fadugba et al. [17] obtained a direct solution of the Black-Scholes-Merton European put option model on dividend yield with modified-log payoff function via a framework based on MT. In this paper, an analytic solution of BSMEPPOM via the celebrated transform of Mellin type is proposed in the sense of DY and MLPPF. The remaining part of the paper is listed as follows; Section 2 captures the brief concepts of MT. The governing model for EPPO on a DY with MLPPF is presented in Section 3. Section 4 captures the solution of BSMEPPOM with DY and MLPPF. An illustrative example on the application of MT to EPPO is captured in Section 5. Section 6 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 and its inversion formula [18]. 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. It is clearly seen that the Mellin transform M[f (x),ω] and the inverse Mellin transform M−1[f̃ (ω)] are linear integral operators. Remark 2.2. For more details on the condition that ensures the existence of MT; see [18,19]. Int. J. Anal. Appl. (2023), 21:29 3 Remark 2.3. 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 [9,12,18,20]. 3. The BSMEPPOM on a DY with MLPPF The BSMEPPOM on a DY with MLPPF is given by ∂Pρ(S ρ t ,t) ∂t +ρ ( r −q + (ρ−1)σ2 2 ) S ρ t ∂Pρ(S ρ t ,t) ∂S ρ t + (σS ρ t) 2 2 ∂2Pρ(S ρ t ,t) ∂(S ρ t) 2 = rPρ(S ρ t ,t) (3.1) subject to the boundary conditions lim S ρ t→∞ Pρ(S ρ t ,t)=0 on [0,T) (3.2) lim S ρ t→0 Pρ(S ρ t ,t)= K er(T−t) on [0,T) (3.3) and MLPPF Pρ(S ρ T ,T)= [ S ρ T ln ( K S ρ T )]+ on [0,∞) (3.4) where Pρ(S ρ t ,t), ρ, t, T, S ρ t , K, σ, r and q are the price of EPPO, power of the option, current time, time to expiry, underlying asset price, strike price, volatility, risk-free interest rate and DY, respectively. 4. Solution of the Black-Scholes-Merton European Put Option Model with MLPPF Ghevariya derived BSM formula on non-dividend yield for ML-payoff function [16]. In this section, analytic solution of BSMEPPOM with dividend yield for ML-power payoff function is obtained via the MT as follows. Taking the MT of (3.1) and using its linearity, independence of time derivatives and shifting properties and rearranging terms, one obtains ∂P̃ρ(ω,t) ∂t =− σ2ρ2 2 (ω2 +ω(1−b1)−b2)P̃ρ(ω,t) (4.1) where b1 = ρ−1 ρ + 2(r −q) ρσ2 , b2 = 2r ρ2σ2 (4.2) Solving (3.1), yields P̃ρ(ω,t)= l(ω)e −1 2 ρ2σ2(ω2+ω(1−b1)−b2)t (4.3) But l(ω)=M(Pρ(ST ,T),ω)e 1 2 ρ2σ2(ω2+(1−b1)ω−b2)T (4.4) 4 Int. J. Anal. Appl. (2023), 21:29 which is equivalent to l(ω)= f̃ (ω)e 1 2 ρ2σ2(ω2+(1−b1)ω−b2)T (4.5) Substituting (4.5) into (4.3), yields P̃ρ(ω,t)= f̃ (ω)e 1 2 ρ2σ2(ω2+(1−b1)ω−b2)τ (4.6) where τ = T − t. By means of (2.2), (4.6) yields Pρ(S ρ t ,t)= 1 2πi ∫ c+i∞ c−i∞ f̃ (ω)e 1 2 ρ2σ2(ω2+(1−b1)ω−b2)τ(S ρ t) −ωdω (4.7) which is the integral equation for governing equation (3.1). Let ξ(S ρ t)= 1 2πi ∫ c+i∞ c−i∞ e ρ2σ2 2 (ω2+ω(1−b1)−b2)(S ρ t) −ωdω (4.8) Using the fact that e 1 2 ρ2σ2(ω2+(1−b1)ω−b2)τ = e−α1(α 2 2+b2)+α1(ω+α2) 2 (4.9) where α1 = ρ2σ2τ 2 ,α2 = 1−b1 2 (4.10) Thus ξ(S ρ t)= e−α1(α 2 2+b2) 2πi ∫ c+i∞ c−i∞ eα1(ω+α2) 2 (S ρ t) −ωdω (4.11) By means of the transformation given by [21], one obtains eφω 2 = 1 2 √ π ∫ ∞ 0 1 √ φ exp ( −(lnSρt) 2 4φ ) (S ρ t) ω−1dS ρ t , <(φ)≥ 0 (4.12) Therefore, ξ(S ρ t)= e −α1(α22+b2) (S ρ t) α2 ρσ √ 2πτ e −1 2 ( ln(S ρ t ) ρσ √ τ )2 (4.13) Similarly, ξ ( S ρ t v ) = e−α1(α 2 2+b2) ( S ρ t v )α2 ρσ √ 2πτ e −1 2   ln ( S ρ t v ) ρσ √ τ   2 (4.14) Using (3.4), then h(S ρ t)=M −1(f̃ (ω))= [ S ρ T ln ( K S ρ T )]+ (4.15) Thus, h(v)= [ v ln ( K v )]+ (4.16) With the help of the convolution property of MT, (4.7) becomes Pρ(S ρ t ,t)= ∫ ∞ 0 h(v)ξ ( S ρ t v ) 1 v dv (4.17) Int. J. Anal. Appl. (2023), 21:29 5 Using (4.14) and (4.16), (4.17) becomes Pρ(S ρ t ,t)= ∫ ∞ 0 [ v ln ( K v )]+ e−α1(α 2 2+b2) ( S ρ t v )α2 ρσ √ 2πτ e −1 2   ln ( S ρ t v ) ρσ √ τ   2 1 v dv (4.18) Pρ(S ρ t ,t)= e −α1(α22+b2) (S ρ t) α2 ρσ √ 2πτ ∫ K 0 [ v1−α2 ln ( K v )] e −1 2   ln ( S ρ t v ) ρσ √ τ   2 1 v dv (4.19) Simplifying further, yields Pρ(S ρ t ,t)= e −α1(α22+b2) (S ρ t) α2 ρσ √ 2πτ ∫ K 0 ln(K) 1 vα2 e −1 2   ln ( S ρ t v ) ρσ √ τ   2 dv −e−α1(α 2 2+b2) (S ρ t) α2 ρσ √ 2πτ ∫ K 0 ln(v) 1 vα2 e −1 2   ln ( S ρ t v ) ρσ √ τ   2 dv (4.20) Pρ(S ρ t ,t)= e −α1(α22+b2) (S ρ t) α2 ρσ √ τ [ln(K)H1 −H2] (4.21) where H1 = 1 √ 2π ∫ K 0 1 vα2 e −1 2   ln ( S ρ t v ) ρσ √ τ   2 dv (4.22) H2 = 1 √ 2π ∫ K 0 ln(v) vα2 e −1 2   ln ( S ρ t v ) ρσ √ τ   2 dv (4.23) Let x = ln ( Sρ v ) ρσ √ τ (4.24) Thus H2 = ρσ √ τ(S ρ t) −α2+1eα1(α2−1) 2 [ρσ √ τG1 − ln(S ρ t)G2] (4.25) where G1 = 1 √ 2π ∫ ln(Sρtv ) ρσ √ τ ∞ xe− 1 2 (x−ρσ √ τ(α2−1))2dx (4.26) G2 = 1 √ 2π ∫ ln(Sρtv ) ρσ √ τ ∞ e− 1 2 (x−ρσ √ τ(α2−1))2dx (4.27) 6 Int. J. Anal. Appl. (2023), 21:29 Once again, by variable transformation y = x −ρσ √ τ(α2 −1), dρ = ln ( S ρ t K ) −ρ2σ2τ(α2 −1) ρσ √ τ = ln ( S ρ t K ) + ( r −q + ( ρ− 1 2 ) σ2 ) τ ρσ √ τ (4.28) Equations (4.26) and (4.27) become G1 =−[η(dρ)+ρσ √ τ(α2 −1)N(−dρ)] (4.29) and G2 =−N(−dρ) (4.30) respectively, with η(κ)= 1 √ 2π e− κ2 2 ,N(κ)= ∫ κ −∞ η(κ)dκ (4.31) Substituting (4.29) and (4.30) into (4.25), yields H2 =−ρσ √ τ(S ρ t) −α2+1eα1(α2−1) 2 [ρσ √ τη(dρ) +(ρ2σ2τ(α2 −1)− ln(S ρ t))N(−dρ)] (4.32) Similarly, H1 = ρσ √ τ(S ρ t) −α2+1eα1(α2−1) 2 [N(−dρ)] (4.33) Using (4.32) and (4.33) and the values of b2, α1 and α2, (4.21) becomes Pρ(S ρ t ,t)= S ρ te [r(ρ−1)−ρq+12ρ(ρ−1)σ 2]τ [ ρσ √ τη(dρ)− (Q1 +Q2τ)N(−dρ) ] (4.34) with Q1 = ln ( S ρ t K ) , (4.35) Q2 = ρ ( r −q + ( ρ− 1 2 ) σ2 ) , (4.36) dρ = Q1 +Q2τ ρσ √ τ , (4.37) τ = T − t. (4.38) Remark 4.1. Setting q =0 in (4.34), yields the fundamental valuation formula for BSMEPPOM on non DY with MLPPF Pρ(S ρ t ,t)= S ρ te [r(ρ−1)+12ρ(ρ−1)σ 2]τ [ ρσ √ τη(dρ)− (R1 +R2τ)N(−dρ) ] (4.39) with R1 = ln ( S ρ t K ) , (4.40) Int. J. Anal. Appl. (2023), 21:29 7 R2 = ρ ( r + ( ρ− 1 2 ) σ2 ) , (4.41) dρ = R1 +R2τ ρσ √ τ , (4.42) τ = T − t. (4.43) Remark 4.2. Setting ρ =1 in (4.34), yields the fundamental valuation formula for plain EPO on DY P1(St,t)= Ste −qτ [σ√τη(d1)− (D1 +D2τ)N(−d1)] (4.44) with D1 = ln ( St K ) , (4.45) D2 = ( r −q + σ2 2 ) , (4.46) d1 = D1 +D2τ σ √ τ , (4.47) τ = T − t. (4.48) Remark 4.3. Setting ρ = 1 and q = 0 in (4.34), yields the fundamental valuation formula for plain EPO on non DY P1(St,t)= St [ σ √ τη(d1)− (B1 +B2τ)N(−d1) ] (4.49) with B1 = ln ( St K ) , (4.50) B2 = ( r + σ2 2 ) , (4.51) d1 = B1 +B2τ σ √ τ , (4.52) τ = T − t. (4.53) 8 Int. J. Anal. Appl. (2023), 21:29 5. Numerical Example Consider the valuation of the EPPO on a DY with MLPPF via the MT using the following parameters in Table 1. The results obtained are displayed in Figures 1 - 10. Table 1. The parameters Parameters Values S in dollars 100 K in dollars 100, 110, 120, 130, 140, 150 r 8% σ 50% q 0,5%,20%,60%,100% T in years 1 2 Figure 1. The Effect of DY on the price of EPPO with ρ =1, q =0 Int. J. Anal. Appl. (2023), 21:29 9 Figure 2. The Effect of DY on the price of EPPO with ρ =1, q =0.05 Figure 3. The Effect of DY on the price of EPPO with ρ =1, q =0.2 10 Int. J. Anal. Appl. (2023), 21:29 Figure 4. The Effect of DY on the price of EPPO with ρ =1, q =0.6 Figure 5. The Effect of DY on the price of EPPO with ρ =1, q =1.0 Int. J. Anal. Appl. (2023), 21:29 11 Figure 6. The Comparative Study of EPPO Price with different DY. Figure 7. The Plots of Linear Payoff. 12 Int. J. Anal. Appl. (2023), 21:29 Figure 8. The Plots of MLPPF for ρ =1. Figure 9. The Plots of Log Payoff. Int. J. Anal. Appl. (2023), 21:29 13 Figure 10. The comparative study of Linear Payoff, MLPPF and Log Payoff. 6. Conclusion An analytic solution of BSMEPPOM via the celebrated transform of Mellin type in the sense of DY and MLPPF has been proposed in this paper. The integral equation for the representation of the price of EPPO with DY was obtained. The closed form approximation formula for EPPO 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 and suitability. The effect of DY is captured in Figures 1 -5. From Figure 6, it is observed that increase in DY leads to increase in the prices of the EPPO with MLPPF. It also is observed from Figures 6, that the holder is more beneficial to enter into a European power put option. In other words, however, the benefits of these cash flows are given to the holder of a put option. 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