Applied Mathematical Finance and its Object Oriented Implementation.
Prof. Dr. Christian Fries, Dr. Alessandro Gnoatto
Exercise Handout 4
Wintersemester 2014/2015
Exercise 1
Let X = σW denote a scaled Brownian motion.
(a) Check numerically the that Y = eX where Y (t) = Y (0) +
Rt
0
dY with
1 2
σ Y dt + σY dW .
2
dY =
(b) P
Use the class BrownianMotion for a Monte-Carlo sampling of ∆W and approximate
∆Y with
1
∆Y = σ 2 Y ∆t + σY ∆W .
2
Print the results for selected paths and analyze them in a spreadsheet.
Rt
0
dY by
Exercise 2
(a) Prove (analytically) that
RT
0
σ(t)dWt is normally distributed with mean 0 and variance
RT
0
σ 2 (t)dt.
(b) Perform the following implementation:
(i) Create an interface which represents σ(t) in an abstract way, e.g.
VolatilityModelInterface {
public double getValue(double time);
/*
* other member functions
*/ }
(ii) Create classes (at least two) implementing the volatility model interface (say σ(t) = at or
σ(t) = exp(at) etc.)
(iii) Check proposition (a) for your classes.
Exercise 3
Let dXt = µ(t, ω)dt + dWt be a 1-dimensional Ito process and assume for simplicity that µ is bounded.
Then from Lemma 53 (see section 2.8 “Martingales” of the lecture scripts) we know that unless µ = 0 the
process Xt is not an Ft -martingale. However, we can construct an Ft -martingale from Xt by multiplying
by a suitable exponential martingale. Define Yt = Xt Mt where
Z t
Z
1 t 2
Mt = exp −
µ(s, ω)dWs −
µ (s, ω)ds .
2 0
0
Use Ito’s formula to prove that Yt is an Ft -martingale.