(iv) If r is not an integer, then I(k +1) I(k – r + 1)' (8.58d)

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Extracted text: (iv) If r is not an integer, then I(k +1) I(k – r + 1)' (8.58d)


Extracted 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 ди du + Rx Ru. (8.67) In its simplest formulation, D and R are non-negative constants. In its time-independent form, equation (8.67) becomes dy x2. dx2 + Ræ - Ry = 0, dx (8.68) where y replaces u and R R = D' (8.69) 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)Ayr – 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) and R/D R 1 (k + r2 – 1)"2 = ( k + D I (k + – 1) I (k + 5 – 1- 5 +1) T (k + #) I(k) R D (8.75) Therefore, the general solution to this form of the discrete (time-independent) Black-Scholes equation is BT (k + 5, Ak + R Yk (8.76) (k – 1)!