Mathematical Models in Finance and Industry
Resit Project 14/15
Submission deadline: Thu 20 August 2015 16:00
1. Consider the situation of a concentration u of a toxic pollutant in a (one dimensional) river.
The pollutant moves in water under the action of advection and diffusion. Additionally,
the pollutant degrades to other, non-toxic substances over time at a degradation rate c > 0
that is proportional to the concentration u. Let k denote the positive diffusion coefficient
and v the velocity of the water. Derive the following equation for u:
ut − kuxx + (vu)x + cu = 0,
x ∈ R, 0 < t < T.
(1)
2. Suppose that the effects of diffusion and degradation are neglectably small and the water
flow is constant with velocity a < 0. Then we can approximate the problem (1) by studying
the following equation,
x ∈ R, 0 < t < T.
ut + aux = 0,
The initial condition shall be given by u(x, 0) = u0 (x), x ∈ R, where u0 is twice differentiable with bounded derivatives. This equation can be approximated by the following
numerical scheme:
un+1
− unj
unj+1 − unj
T
j
+a
= 0, j ∈ Z, 0 ≤ n ≤
,
∆t
h
∆t
u0j = u0 (xj ),
j ∈ Z.
Prove that there exists a constant C such that
sup |u(xj , tn ) −
j∈Z
unj |
≤ C(∆t + h)
T
for all 0 ≤ n ≤
,
∆t
(2)
provided that |a|∆t/h ≤ 1. You can proceed as follows:
(a) Let
u(xj+1 , tn ) − u(xj , tn )
u(xj , tn+1 ) − u(xj , tn )
+a
∆t
h
n
and use Taylor expansion to show that |Lj | ≤ CL (∆t + h), where CL may depend on
a and sup |uxx |, sup |uxt |, sup |utt | in R × [0, T ].
Lnj :=
1
(b) Show that the error enj = u(xj , tn ) − unj satisfies
en+1
− enj
j
∆t
+a
enj+1 − enj
= Lnj
h
and use induction on n to prove
sup |enj |
j∈Z
≤ n∆tCL (∆t + h)
T
for all 0 ≤ n ≤
∆t
and then conclude (2) holds.
3. Solve the following problem
(
ut + ((x − 1)u)x + cu = 0, x ∈ [0, 1], 0 < t < T,
u(x, 0) = 1,
x ∈ [0, 1].
(Hint: First find a transformation of the form ũ = f (t)u with a suitable function f (t) to
reduce the problem to a pure advection equation.)
4. Derive the Black-Scholes Formula for a European put option without making use of the
put-call-parity. Proceed along the lines of the corresponding calculations for the European
call option.
5. Write a Matlab program (also submit the Matlab code) to evaluate the Black-Scholes
formula for a European Put option using appropriate special functions of Matlab.
(a) Graph the option price for E = 10, σ = 0.2, r = 5% (per annum), and five months
to maturity.
(b) Repeat the calculations for the parameter choices
• E = 8 and E = 12, keeping the others fixed,
• σ = 0.1 and σ = 0.4, keeping the others fixed,
• r = 2% and r = 8%, keeping the others fixed.
Describe how the option price changes in each of these cases. Try to give a financial
and a mathematical explanation for the changes.
6. Write a Matlab program (also submit the Matlab code) to implement the numerical
method presented in the lecture notes for valuing an American Put option.
(a) Graph the option price for E = 10, σ = 0.2, r = 5% (per annum), and four months
to maturity.
(b) Graph the free boundary t 7→ S ∗ (t) for T − 3 months < t < T using the above
parameters.
2