Description
Problem 1. Comparing different Monte Carlo schemes. Consider the
Black-Scholes setup (geometric Brownian motion) with r = 6%, δ = 0.03, σ =
20%, S0 = 100, and assume we want to price an European option with strike
K = 100 and maturity T = 1.
(a) Implement a simple Monte Carlo scheme using m simulation trials for European Call and Put options. This should be a function of n (number of
time steps) and m. In all practical applications you should use at least 300
time steps and at least 1 million simulated paths. Furthermore, implement
a calculation of the standard error of the estimate of the option price and
a way to time the simulation routine. PLease output the results obtained
when varying n (300 -700) and m (1-5 million). Pick just a few values in
each range.
(b) Implement a Monte Carlo scheme for European call and put options using the antithetic variates method (see section 4.3 of the textbook), the
delta–based control variate (section 4.5 of the textbook) with β1 = −1, and
the combined antithetic variates with delta-based control variate method.
Report the values obtained in four columns: Monte Carlo (MC), MC with
Antithetic Variates, MC with Delta-based Control Variate, and MC with
both Antithetic Variates and Delta-based Control Variate. Report the estimated option values, the corresponding standard deviations, as well as the
time it takes to obtain each result. Write a paragraph comparing the results
you obtained. Discuss the methods implemented.
Problem 2. Multiple Monte Carlo Processes
Assume a portfolio of 10 million dollars is invested as follows:
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• 40% in IBM stock
• 30% in a 10 year Treasury bill
• 30% in Chinese Yuan (which depends on the Yuan Dollar Exchange Rate)
We assume that the processes X, Y, Z decribed next are modeling the evolution
of the IBM share price, unit of Treasury Bill and number of Yuan obtained for
$1 respectively
dXt = 0.01Xtdt + 0.3XtdW1
t
, X0 = 80
dYt = 100(90000 + 1000t − Yt)dt +
p
YtdW2
t
, Y0 = 90000
dZt = 5(6 − Zt)dt + 0.01p
ZtdW3
t
, Z0 = 6.1
1. Calculate the number of shares and the amount in the Yuan that the
portfolio contains when it is started.
2. Assume the Brownian motions are independent and perform Monte Carlo
simulations for al assets for 10 days (t = 10/252). Use 3 million simulations
and use ∆t = 0.001. Calculate VAR for the portfolio (α = 0.01, N = 10
days)
3. Calculate the CVAR (conditional value at risk)
Problem 3. Generating correlated BM and pricing basket options.
(a) Given a correlation matrix A,
A =
1.0 0.5 0.2
0.5 1.0 −0.4
0.2 −0.4 1.0
,
perform a Cholesky decomposition of the matrix A, which is a decomposition
of the form A = LLT
, when A is symmetric and positive definite matrix.
The function you construct should return the lower triangular matrix, L.
Consider three assets, starting with S(0) = [100, 101, 98]. The assets are
assumed to follow a standard geometric Brownian motion of the form
dSi(t) = µiSi(t)dt + σiSi(t)dWi(t).
We assume µ = [0.03, 0.06, 0.02], the volatility σ = [0.05, 0.2, 0.15], and the
BM’s have the correlation matrix A (e.g., d < W1, W2 >= a12dt = 0.5dt).
(b) We take maturity T = 100 days and the number of simulated paths is
m = 1000, Consider one day sampling frequency, i.e. ∆t = 1/365. Generate
a 3−dimensional matrix where each row represents a time step, each column
represent a separate simulation for a specific asset and the 3-rd dimension
represents different assets in the basket. Plot one realization (sample path)
for this 3−dimensional process.
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(c) Basket options are options on a basket of assets. A commonly traded basket
option is a vanilla call/put option on a linear combination of assets. To
clarify, suppose Si(t), i = 1, . . . , N are the prices of N stocks at time t and
let ai
, i = 1, . . . , N are real constants. Set
U(t) = X
N
i=1
aiSi(t)
A vanilla basket option is simply a vanilla option on U(T). Specifically, on
the exercise date T, the payoff of the option is max{α(U(T)−K), 0}, where
K is the exercise price and α = 1 for a call and α = −1 for a put. Price an
European call option and an European put option with K = 100 on the 3
asset basket given in part (b), using a Monte Carlo simulation. Consider a
simple average basket a1 = a2 = a3 = 1/3.
(d) Please price an exotic option on the basket in part (b), described using the
following conditions where we use B = 104, and K = 100:
(i) If the asset 2 (S2) hits the barrier B < S2(t) for some t then the payoff
of the option is equal to an European Call option written on the asset
2;
(ii) If maxt∈[0,T] S2(t) > maxt∈[0,T] S3(t), then the payoff of the option is
(S
2
2
(T) − K)+;
(iii) Take Ai(0, T) := PT
t=1 Si(t), the average of the daily values for stock
i. If A2(0, T) > A3(0, T), then the payoff is (A2(0, T) − K)+;
(iv) otherwise, the option is a vanilla call option on the basket, similar to
part (c) of this problem.
Problem 4. (BONUS) Simulating the Heston model. Consider
the Heston stochastic volatility model with parameters: S0 = 100, V0 =
0.010201, κ = 6.21, θ = 0.019, σ = 0.61, ρ = −0.7, r = 3.19%.
(a) Apply the Euler discretization schemes as presented in Table 1 of the
paper [1]. Please implement all the five schemes listed there and use
Monte Carlo simulations to price a call option with strike K = 100 and
maturity T = 1. The exact call option price (benchmark) is in this case
C0 = 6.8061. Provide a table listing the estimated call option price, the
bias, the root mean square error (RMSE), and the computation time in
seconds. Report the results for each of the five schemes in one table.
(b) Quadrature methods can be used to price options under the Heston
Model. The paper [2] provides a good implementation of the Heston
Model and the corresponding pricing problem. Compute numerically
the price of the option from part (a) using the formula (1.4) in [2], with
the parameters provided in the same paper in Section 6. Please note
that in order to calculate the option price C(S0, K, V0, t, T) one needs
to evaluate the integral in equation (1.5) in [2]. Use Simpsons Rule for
this purpose, with tolerance ε = 10−4
.
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(c) Compare and contrast the two approaches.
References
[1] Lord, Roger, Remmert Koekkoek, and Dick Van Dijk. A comparison of biased simulation schemes for stochastic volatility models. Quantitative Finance 10.2 (2010): 177-194.
[2] Mikhailov, Sergei and Nogel, Ulrich. Hestons stochastic volatility model:
Implementation, calibration and some extensionsWilmott Journal, 2004.
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