IEOR 4706 : Final (April 10, 12 pm – April 11, 12 pm) Name : SID : 1 Exercice 1: Super–replication under interest rate and volatility uncertainty We consider a one–period binomial, that is to say we...

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Answer To: IEOR 4706 : Final (April 10, 12 pm – April 11, 12 pm) Name : SID : 1 Exercice 1: Super–replication...

Himanshu answered on Dec 23 2021
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Exercise 1
1.
The covariance matrix of a multivariate randomized parameter is occasionally
unknown in statistical and must be approximated. The subject of how to estimate the
real covariance matrix using a sample from a multivariate distribution is addressed in
the section on approximation of covariance matrices. The sample covariance matri
x
can be used to deal with simple scenarios when all observations are complete.
Returns
BAC 2.2%
MNRA 4%
AAPL 5.2%
2.
Minimum variance 08% with 0.1% return
3.
Minimum variance 14%
4.
Risk parity portfolio
Risk parity aims to provide equity-like returns for low-risk portfolios. This portfolio,
which is entirely made up of stocks, carries a 15% risk. The predicted return is the
same as the unleveraged portfolio, but the annualised risk is just 12.7 percent. This
equates to a 15 percent reduction in risk.
Exercise 2
1.
a.
A ordinary differential equation (ODE)
dx(t)
dt
= f(t, x) , dx(t) = f(t, x)dt , (1)
with initial conditions x(0) = x0 can be written in integral form
x(t) = x0 +
∫ t
0
f(s, x(s))ds , (2)
where x(t) = x(t, x0, t0) is the solution with initial conditions x(t0) = x0. An
example is given as
dx(t)
dt
= a(t)x(t) , x(0) = x0
b.
When we take the ODE (3) and assume that a(t) is not a deterministic parameter
but rather a stochastic parameter, we get a stochastic differential equation (SDE).
The
stochastic parameter a(t) is given as
a(t) = f(t) + h(t)ξ(t) , (4)
where ξ(t) denotes a white noise process.
Thus, we obtain
dX(t)
dt
= f(t)X(t) + h(t)X(t)ξ(t) . (5)
When we write (5) in the differential form and use dW(t) = ξ(t)dt, where dW(t)
denotes differential form of the Brownian motion,we obtain:
dX(t) = f(t)X(t)dt + h(t)X(t)dW(t)
c.
In general an SDE is given as
dX(t, ω) = f(t, X(t, ω))dt + g(t, X(t, ω))dW(t, ω) , (7)
where ω denotes that X = X(t, ω) is a random variable and possesses the initial
condition X(0, ω) = X0 with probability one. As an example we have already
encountered
dY (t, ω) = µ(t)dt + σ(t)dW(t, ω) .
Furthermore, f(t, X(t, ω)) ∈ R, g(t, X(t, ω)) ∈ R, and W(t, ω) ∈ R. Similar as
in (2) we may write (7) as integral equation
X(t, ω) = X0 +
∫ t
0
f(s, X(s, ω))ds +
∫ t
0
g(s, X(s, ω))dW(s, ω) .
2.
a. For the calculation of the stochastic integral ∫ T
0
g(t, ω)dW(t, ω), we assume that
g(t, ω) is only changed at discrete time points ti (i = 1, 2, 3, ..., N − 1), where
0 = t0 < t1 < t2 < . . . < tN−1 < tN < T . We define the integral
S =
∫ T
0
g(t, ω)dW(t, ω) , (9)
as the Riemannßum
SN(ω) = ∑
N
i=1
g(ti−1, ω)
(
W(ti, ω) − W(ti−1, ω)
)
. (10)
with N → ∞ .
b.
A random variable S is called the Itˆo integral of a stochastic process g(t, ω) with
respect to the Brownian motion W(t, ω) on the interval [0, T] if
lim
N→∞
E
[(S −

N
i=1
g(ti−1, ω)
(
W(ti, ω) − (W(ti−1, ω)
)] = 0 , (11)
for each sequence of partitions (t0, t1, . . . , tN) of the interval [0, T] such that
maxi(ti − ti−1) → 0. The limit in the above definition converges to the stochastic
integral in the mean-square sense. Thus, the stochastic integral is a random
variable,
the samples of which depend on the individual realizations of the paths W(., ω).
c.
The simplest possible example is g(t) = c for all t. This is still a stochastic
process, but a simple one. Taking the definition, we actually get
∫ T
0
c dW(t, ω) = c lim
N→∞

N...
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