MSF 526: Final Exam
Prof. Matthew Dixon
Dec 7th, 2020
This exam consists of four questions. You are required to answer three
questions in two hours. Each question carries equal marks. If you answer
all four questions, your best three answers shall be used. Write your name
clearly on each sheet of paper that you upload to Blackboard and show all
intermediate calculations. This is an open book exam. Use of laptops and
calculators is permitted. Communication via instant messaging, email or
any medium is strictly forbidden.
1 Question 1
The following Heston two-factor model prescribes evolution equations for
the underlying S(t) and the variance V (t) of the form
dSt
St
= µdt+
√
VtdW
1
t , (1)
dVt = κ(θ − Vt)dt+ σdW 2t . (2)
and the two Wiener processes are correlated with correlation ρ1,2.
• Write the above system of equations in the canonical form, a : R2 ×
R+ → R2 and b : R2 ×R+ → R2×2 by giving the form of a and b in
[5]
dXt = a(Xt, t)dt+ b(Xt, t)dWt, Xt := (St,Vt)
• Show that the Euler scheme can be written as
Xt−1 −Xt = a(Xt, t)∆t+ b̃(Xt, t)�t, �t ∼ N(0, I2×2)
and give the form of the matrix b̃. [5]
1
• Using a two-step Euler approximation, simulate five paths of the un-
derlying and variance and estimate the mean, standard deviation and
standard error of the underlying at time t2 = 1.0. Use the sequence
of pairs of iid standard Gaussian random numbers in the table below
and assume that ρ1,2 = −0.9, µ = 0.1, κ = 1.8, θ = 0.25, ∆t = 0.5,
and σ = 0.12. The initial underlying and variance at time t0 = 0 is
S0 = 1.0 and V0 = 0.01. Show all intermediate calculations. [10]
Sample (�1,�2)
1 (-0.86361717, -1.32958849)
2 (0.2072987, 0.40024067)
3 (-1.82467762,1.04039457)
4 (0.17777817, -0.30351275)
5 (-4.50286929, 1.24437217)
Table 1: Uncorrelated standard Gaussian random numbers for use at the
first step.
Sample (�1,�2)
1 (0.1212719, 0.2952849)
2 (0.1472987, 0.40124017)
3 (-1.82467762,0.24039157)
4 (2.17774812, -0.07341225)
5 (0.20236921, -0.2434211)
Table 2: Uncorrelated standard Gaussian random numbers for use at the
second step.
2 Question 2
A portfolio consists of a stock and an at-the-money call option struck
at K = 100 on the same stock and expiring in two days. Assume that
the weight in the stock is w1 = 1/3 and the weight in the option is
w2 = 2/3.
Using Monte-Carlo simulation, evaluate the 90% quantile of the two-
day portfolio loss distribution for the portfolio. The daily return of
the stock is normal with mean µ1 = 0.1, variance σ
2
1 = 0.01 and the
stock price is currently 100. You should use the following ten pairs
2
of standard normal random numbers for your simulation. You may
assume that the returns are simple, i.e. St+1 = St(1 + rt). [17] Is the
portfolio loss distribution Gaussian? Briefly explain your answer. [3]
t1 t2
0.01 -2.0
1.2 0.8
-0.23 2.54
0.45 0.17
0.22 -0.14
0.81 0.02
0.21 0.49
-3.42 0.57
-0.71 0.11
0.03 -1.67
Table 3: uncorrelated std. Gaussian samples
3 Question 3
The table below shows the simulated values of the underlying at times
t = 3m, t = 6m and t = 9m. Use the Least-Squares Monte-Carlo
time Path 1 Path 2 Path 3 Path 4 Path 5
t = 3m 19.2533 34.7788 35.6238 39.5507 36.1251
t = 6m 18.1637 25.4520 24.2960 34.0307 29.6701
t = 9m 28.1132 31.2341 23.1940 38.1234 31.211
method to calculate the price of an up-and-out barrier call option
with an american exercise feature. The option expires in 9m, is struck
at $25 , has au upper barrier level of 35 and pays zero dividend. The
spot price of the underlying stock is $25 and the risk-free rate is 1%.
Tabulate or show all intermediate calculations. The fitted regression
coefficients for a model with Laguerre basis functions at time t = 6m
is L0(x) = 1, L1(x) = 1 − x, L2(x) = 12(x
2 − 4x + 2) are α̂0 = 0.01,
α̂1 = 0.25 and α̂2 = 0.05. The fitted regression coefficients at time
t = 3m are α̂0 = 0.02, α̂1 = 0.35 and α̂2 = −0.05. [20]
3
4 Question 4
Using Monte-Carlo simulation, estimate the expected loss of an equally
weighted portfolio (i.e. weights are 0.5) on two zero coupon corporate
bonds, each with principals of $1. The first bond is issued by IBM
and expires in 5 years and the second bond is issued by Zynga and
expires in 10 years. The risk free rate is 1%. The hazard rate for the
IBM bond is 0.1 and the recovery rate is 0.45. The hazard rate for
the Zynga bond is 0.8 and the recovery rate is 0.3. The correlation of
default is 0.6.
Use a Gaussian Copula model with an exponential model for the de-
fault times τ = F−1τ (ΦZ(z)) where Z ∼ N(0,Σ) is zero-mean bivari-
ate std. Gaussian – the diagonals of Σ are 1 (i.e. std.devs are 1).
ΦZ(z) =
1√
2π
exp(−z2/2) denotes the marginal CDF of a bivariate
std. Gaussian distribution.
You are given ten realizations u ∈ U([0, 1]) as
u = (0.46659097, 0.34000663, 0.2746701, 0.39228212, 0.92687015,
0.01410045, 0.09450884, 0.48553626, 0.07740258, 0.47256735)
and you should take the first 5 realizations for modeling the default
times of the IBM bond and the latter five for the Zynga bond.
5 Cheat sheet
Euler Integration: Given the following general stochastic differen-
tial equation
dX = a(t,X)dt+ b(t,X)dW.
The Euler scheme is given by
Y (ti+1)− Y (ti) = a(ti, Yi)∆t+ b(ti, Yi)∆W
Cholesky: The Cholesky lower factor matrix for a covariance matrix
4
Σ =
[
σ21 ρ1,2σ1σ2
ρ1,2σ1σ2 σ
2
2
]
takes the form
L =
[
σ1 0
ρ1,2σ2
√
1− ρ21,2σ2
]
.
Correlated random number generation: If Z is an n-vector of
univariate standard Gaussian random variables, then
Y = LZ
is an n-vector of multivariate Gaussian random variables with mean 0
and n× n covariance matrix Σ.
Credit modelling The inverse of an exponential distribution with
constant parameter λ is
τ = −
1
λ
ln(1− u)
The expected loss of a credit risky portfolio with N instruments and
principal Vi is approximated with Monte Carlo over M paths as
EQ[L] ≈
1
M
M∑
j=1
N∑
i=1
wiVi(1−Ri)× exp(−r × τ
(j)
i )
5