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)

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