Page 326 - Numerical Methods for Chemical Engineering

Problems 315

as wit inr z y
cnent A at
artia ressre p A iid
inet
L

θ

g as at interace
iid c = H A p A
A
b z
v z (y)
iid y
tet
Figure 6.32 Reaction-enhanced diffusion into a falling film.

The current value (fair price) V of the option should depend upon the current asset
price S and the time remaining to expiry, T − t. The Black-Scholes equation (derived in
Chapter 7) states that V (S, t) satisfies
2
∂V 1 2 2 ∂ V ∂V
+ σ S + rS − rV = 0 (6.244)
∂t 2 ∂S 2 ∂S
with the final and boundary conditions
final condition V (S, T ) = max{S − E, 0}
2 2 (6.245)
∂ V ∂ V
spatial boundary conditions 0 = =
∂S 2 ∂S 2
0 S max max(S(t),E)
r is the (risk-free) interest rate used to define the time value of money. σ is the volatility of
the underlying asset, and measures how vigorously the price varies with time,
N s N s
2
2 k=1 (R k − R ) S(t k + t) − S(t k ) k=1 R k
σ = R k = R = (6.246)
(N s − 1)( t) S(t k ) N s
This model is based upon a treatment of the asset price as a random walk, the mathematics
of which will be discussed in Chapter 7. While an analytical solution exists for this “plain
vanilla option,” more realistic cases generally require numerical solution. Write a program
that computes V (S, t) for a European call option. For more on this subject, consult Wilmott
(2000).

6.C.1. Consider a system with a reversible reaction A + B ⇔ C + D, with kinetics r R =
k(c A c B − K −1 c C c D ) occurring inside a catalyst pellet that is the shape of a long, narrow
eq
cylinder of radius R.Given R, k, K eq , the effective binary diffusivities D j within the catalyst
pellet, and the surface concentrations c jS of each species j = A, B, C, D, write a program
that computes the net reaction rate per unit volume of catalyst. Report your results for the
following parameter values (make sure to use proper concentration units that agree with the
other terms in your equations),
l 3
−7
−3
2
R = 0.1cm D j = 10 cm /s k = 10 K eq = 10
mol s (6.247)
c AS = 1M c BS = 1M c CS = c DS = 0