( (k + r1 – 1)"1 = k² = k R/D (*+*-1) I (k + – 1) (k+ 용-1-B+ 1) r (k + #). R (k + r2 – 1)"2 = (k + D T(k) general solution to this form of the discrete (time-indep equation is ВГ (k + 3) (k – 1)! Yk = Ak +

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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Show me the steps of determine blue and theequationis there

(iv) If r is not an
integer,
then
I(k +1)
I(k – r + 1)'
(8.58d)
Transcribed Image Text:(iv) If r is not an integer, then I(k +1) I(k – r + 1)' (8.58d)
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)!
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 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)!
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