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The Black-Scholes equation provides a model for certain types of transac- tions in financial markets. Mathematically, it corresponds to a linear reaction- advection-diffusion evolution partial differential equation, which can be writ- ten in the form ди .2 + Rx Ru. (8.67) %3D In its simplest formulation, D and R are non-negative constants. In its time-independent form, equation (8.67) becomes + Rr dy dx dy Ry = 0, (8.68) dx2 where y replaces u and (R-5 (8.69) D This is a second-order Cauchy-Euler differential equation. A corresponding discrete model is provided by the difference equation (k(k + 1)A?yk + (Řk)Ayk – Ryk = 0. (8.70) Comparison with equation (8.56) shows that a = R, b= -R. (8.71) Substitution of these values into equation (8.64) gives the following roots to the characteristic equation ri = 1, r2 = -R, (8.72) which produces the general solution (Yk = A(k + r1– 1)"1 + B(k + r2 – 1)2, (8.73) where A and B are arbitrary constants. From equation (8.58d), we obtain (k +r1 – 1)"1 = k² = k (8.74)

The Black-Scholes equation provides a model for certain types of transac- tions in financial markets. Mathematically, it corresponds to a linear reaction- advection-diffusion evolution partial differential equation, which can be writ- ten in the form ди .2 + Rx Ru. (8.67) %3D In its simplest formulation, D and R are non-negative constants. In its time-independent form, equation (8.67) becomes + Rr dy dx dy Ry = 0, (8.68) dx2 where y replaces u and (R-5 (8.69) D This is a second-order Cauchy-Euler differential equation. A corresponding discrete model is provided by the difference equation (k(k + 1)A?yk + (Řk)Ayk – Ryk = 0. (8.70) Comparison with equation (8.56) shows that a = R, b= -R. (8.71) Substitution of these values into equation (8.64) gives the following roots to the characteristic equation ri = 1, r2 = -R, (8.72) which produces the general solution (Yk = A(k + r1– 1)"1 + B(k + r2 – 1)2, (8.73) where A and B are arbitrary constants. From equation (8.58d), we obtain (k +r1 – 1)"1 = k² = k (8.74)

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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8.3.1 Example A The Black-Scholes equation provides a model for certain types of transac- tions in financial markets. Mathematically, it corresponds to a linear reaction- advection-diffusion evolution partial differential equation, which can be writ- ten in the form du + Rx Ru. (8.67) Ət In its simplest formulation, D and R are non-negative constants. In its time-independent form, equation (8.67) becomes dy + Rr dy Ry = 0, (8.68) dx? dx where y replaces u and R (R= (8.69) D' This is a second-order Cauchy-Euler differential equation. A corresponding discrete model is provided by the difference equation (k(k + 1)A?yk + (Rk)Ayk – Ryk = 0. (8.70) Comparison with equation (8.56) shows that a = R, b= -R. (8.71) Substitution of these values into equation (8.64) gives the following roots to the characteristic equation r1 1, 2 = -R, (8.72) which produces the general solution (yk A(k + r1 1)"1 + B(k + r2 – 1)"2, (8.73) where A and B are arbitrary constants. From equation (8.58d), we obtain (k + r1 – 1)"1 = k² = k (8.74) and R/D R (k + r2 – 1)"2 = (k + D I (k + 5 – 1) I (k + 5 – 1- 5 +1) I (k+ 5) T(k) R (8.75) Therefore, the general solution to this form of the discrete (time-independent) Black-Scholes equation is ВГ (к + 3) (k – 1)! Yk = Ak + (8.76)
Transcribed Image Text:8.3.1 Example A The Black-Scholes equation provides a model for certain types of transac- tions in financial markets. Mathematically, it corresponds to a linear reaction- advection-diffusion evolution partial differential equation, which can be writ- ten in the form du + Rx Ru. (8.67) Ət In its simplest formulation, D and R are non-negative constants. In its time-independent form, equation (8.67) becomes dy + Rr dy Ry = 0, (8.68) dx? dx where y replaces u and R (R= (8.69) D' This is a second-order Cauchy-Euler differential equation. A corresponding discrete model is provided by the difference equation (k(k + 1)A?yk + (Rk)Ayk – Ryk = 0. (8.70) Comparison with equation (8.56) shows that a = R, b= -R. (8.71) Substitution of these values into equation (8.64) gives the following roots to the characteristic equation r1 1, 2 = -R, (8.72) which produces the general solution (yk A(k + r1 1)"1 + B(k + r2 – 1)"2, (8.73) where A and B are arbitrary constants. From equation (8.58d), we obtain (k + r1 – 1)"1 = k² = k (8.74) and R/D R (k + r2 – 1)"2 = (k + D I (k + 5 – 1) I (k + 5 – 1- 5 +1) I (k+ 5) T(k) R (8.75) Therefore, the general solution to this form of the discrete (time-independent) Black-Scholes equation is ВГ (к + 3) (k – 1)! Yk = Ak + (8.76)
with the dropping of the common factor (k +r – 1)". Equation (8.63) is the characteristic equation for equation (8.56), and its two solutions are r1,2 = |(1 – a) + V(1- a)2 – 4b (8.64) 11 A discrete, finite difference model of the Cauchy–Euler equation is the following k(k +1)A?yk + akAyk + byk = 0, (8.56) | where the following continuous-to-discrete correspondences have been made 12
Transcribed Image Text:with the dropping of the common factor (k +r – 1)". Equation (8.63) is the characteristic equation for equation (8.56), and its two solutions are r1,2 = |(1 – a) + V(1- a)2 – 4b (8.64) 11 A discrete, finite difference model of the Cauchy–Euler equation is the following k(k +1)A?yk + akAyk + byk = 0, (8.56) | where the following continuous-to-discrete correspondences have been made 12
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