1. (20 points) Throughout the course we repeatedly used the property that the stochastic integral

Z t

0

φ(s) · dW(s) (W is a Brownian motion)

is a martingale under suitable conditions. In this problem we study an example where the stochastic

integral is not a martingale.

Consider a 3-dimensional Brownian motion

B(t) = (B1(t), B2(t), B3(t)).

That is, the components of B are independent Brownian motions. We assume that B1(0) = B2(0) =

B3(0) = 1.

(a) (5 points) Consider the function

u(x) = 1

p

x

2

1 + x

2

2 + x

2

3

.

Show that u is a harmonic function on R

3 \ {0}. That is,

∂

2u

∂x2

1

(x) + ∂

2u

∂x2

2

(x) + ∂

2u

∂x2

3

(x) = 0

for x 6= 0.

(b) (5 points) Write |B(t)|

−1 = u(B(t)) as a stochastic integral with respect to the Brownian motion.

(c) (5 points) Find the mean and standard deviation of B2

1

(t) + B2

2

(t) + B2

3

(t).

(d) (5 points) Using the results of (a)-(c), argue heuristically that u(B(t)) → 0 as t → ∞, and explain

why this suggests that u(B(t)) is not a martingale. (To get some intuition, you may simulate

paths of the process and see what happens.)

2. (30 points) Consider the Vasicek model where the short rate process satisfies the dynamics

dr = (b − ar)dt + σdW(t)

under the risk neutral probability measure Q. The unit of time is year.

(a) (10 points) Suppose at time t = 0 we observe that the short rate is r(0) = 0.03, and we also

observe the following term structure:

p(0, 0) = 1, p(0, 0.1) = 0.997, p(0, 0.5) = 0.982, p(0, 1) = 0.965,

p(0, 2) = 0.931, p(0, 5) = 0.8845, p(0, 10) = 0.87.

Here p(t, T) is the price of the T-bond at time t. Calibrate a Vasicek model based on the data,

and explain carefully what you do. Comment on the fit of the model.

(b) (10 points) Price, using Monte-Carlo for example, a European call option on a 7-bond with

maturity date at time 5, with strike price 0.9. (The underlying bond matures in 7 years, but the

option expires in 5 years.)

(c) (10 points) Discuss the sensitivity of the call price with respect to the accuracy of the parameters.

This is an open-ended question and requires programming.

3. (30 points) Consider two stocks whose dynamics are given by

dS1(t) = S1(t)(µ1dt + σ1dW1(t)),

dS2(t) = S2(t)(µ2dt + σ2dW2(t)),

where µ1, µ2, σ1, σ2 are constants, and W1 and W2 are independent standard Brownian motions.

Assume S1(0) = S2(0) = 1.

(a) (5 points) Consider a constant-weighted portfolio (w, 1 − w) of the two stocks, where w ∈ R is

the weight of stock 1. Derive an explicit formula for the value Vw(t) of the constant-weighted

portfolio depending on the weight w. We assume that Vw(0) = 1.

(b) (5 points) Fix t. If the paths of the Brownian motions are fixed, what is the value of w which

maximizes Vw(t)?

(c) (20 points) We have a collection of strategies indexed by the weight w ∈ R. Suppose at time 0

you have $1. Suppose at time 0 we distribute the money among all these portfolios according to

the standard normal distribution: That is,

– We invest $ √

1

2π

e

− w2

2 dw in the portfolio with weight w.

So we we are holding a portfolio of portfolios (think of each strategy as a stock). At time t, the

amount of money with weight w becomes

$

1

√

2π

e

− w2

2 ×

Vw(t)

Vw(0) × dw,

and our entire portfolio has value

V¯ (t) = Z ∞

−∞

1

√

2π

e

− w2

2

Vw(t)

Vw(0)dw.

If you consider the distribution of your capital, you have a probability distribution. Show that

√

1

2π

e

− w2

2 ×

Vw(t)

Vw(0)

V¯ (t)

is the density of a normal distribution, and find the corresponding mean and variance in terms

of the Brownian motions. (This is similar to computing the posterior distribution in Bayesian

statistics.)

If you need hints, look up the paper Asymptotically Optimal Portfolios by Jamshidian (1992).

4. (40 points) Consider a stochastic optimal control problem. The state process is one-dimensional and

is given by

dXt = µ(t, Xu

t

, ut)dt + σ(t, Xu

t

, ut)dWt,

where Wt is a one-dimensional Brownian motion.

Consider the gain functional given by

Hu

(t, x) = Et,x ”

exp Z T

t

F(s, Xu

s

, us)dt + G(Xu

T

)

!# .

The value function is given by

H(t, x) = sup

u∈At,T

Hu

(t, x).

Note that this is different from the usual stochastic control problem we considered as there is an

exponential inside the expectation.

(a) (10 points) Formulate a dynamic programming principle (DPP) along the lines of Section 20 of

the notes.

(b) (10 points) Using the argument in Section 21 of the notes, derive the HJB equation for this

problem. Show that it is given by

∂tH(t, x) + sup

u

(H(t, x)F(t, x, u) + L

uH(t, x)) = 0, H(T, x) = e

G(x)

.

(c) (20 points) Using the DPP and HJB derived above, solve the problem

sup

u

E

h

e

R T

0

u

2

t dt+X2

T

i

,

where the state process evolves according to the SDE

dXu

t = (x + u)dt + dWt.

When solving the resulting HJB, use the ansatz

H(t, x) = e

A(t)x

2+B(t)

where A and B are deterministic functions of time. By solving, I mean finding the value function

for all t, x and finding the optimal control.

What is opportunity? Is it total right? In the event that we make this inquiry to individuals around us, we will get around a hundred unique answers, in light of the fact that everybody will comprehend this wonder in his or her own specific manner. Flexibility can be characterized beginning from various perspectives, as indicated by various societies, from various perspectives and even from various convictions. Everybody was conceived absolutely free and every one of them must have the privilege to live openly. In view of a few articles, we can see that out there individuals need to be free and autonomous from each other, however is this idea conceivable and satisfactory by the general public? Flexibility is the privilege to do what we need, live where we need, eat what we need, realize what we need to realize, for instance we pick what religion we need to accept, without thinking about others judgments! Truly, I don't know whether this in of right truly exists since we were educated in a way where we are permitted to pick, however toward the end somebody will carry out that activity (picking) for us. So would we say we are extremely free?! It is safe to say that we are following the meaning of "Opportunity"? As I would see it, flexibility is a nonattendance of weight and requirement from other individuals, where individuals is allowed to do what laws and social traditions don't disallow. We can barely say that all social orders are free similarly. By and by, I trust that we, Human creatures are God animals. God is the person who made us and gave us the privilege to life. As indicated by the Bible, God began to make the sky and the earth, at that point the morning and the night where He permitted the presence of life, beginning from flying creatures to every one of the creatures and till the end the production of the Man, called Adam. Since Adam was so alone, God chose to give him an organization, an accomplice. In light of this, God made a general public for Humans and with it, we were educated to obey and take after principles. Would we be able to endeavor to envision what the world may transform into if each one of us could do whatever we wantedâ€¦ without any restrictions and regard towards the guidelines and others? All things considered, how might we live unreservedly? From my perspective, we can live unreservedly by regarding others rights to live uninhibitedly as well, by contemplating other individuals rights. At the end of the day, the principle thought behind the word opportunity is to be deferential and capable to the general public. Opportunity is a critical thing to each person in light of the fact that as we had experience in the previous decades a considerable measure of verbal confrontations have begun debating about this point. For instance, I might want to discuss the right to speak freely. I consider flexibility discourse is the most imperative part when we endeavor to perceive what the significance behind opportunity is. As we probably am aware, out there is many individuals that are indicted for their inventive personality and even, some of them are covered in light of the fact that the general public can't acknowledge their method for communicating their sentiments. In my examination, somebody said that opportunity of articulation will never bring about antagonistic vibe among individuals on the off chance that they regard each other's thoughts. As I specified previously, opportunity isn't a flat out right, and there are excessively numerous impediment on it. As a matter of first importance, the general public right including national security which is vital, security is the most essential thing in flexibility. Our flexibilities ought to never cause to debilitating our national security. I have expressed a portion of my assessments about flexibility, now I might want to portray the word opportunity by a definition. As per Wikipedia, "opportunity is intrinsic to human and that he will battles as long as he lives to reestablish his inborn appropriate to be free man...", I like this definition particularly in light of the fact that I think it is valid, our flexibilities are settled by us, nobody can set it! Since God gave us flexibility for nothing, we should do our best to keep up and keep this right. I think the most mortifying approaches to rebuff a human, is to deny him or her from opportunity, for instance when somebody goes to imprison, that individual is deny from flexibility thus he is enduring. Flexibility isn't outright, it must be constrained. We can't simply do what we like and say "I am a liberated individual!". We can't slaughter, pirate sedates or disregard the laws and tenets for flexibility, they ought to be regarded and I absolutely concur with this sentence "the opportunity of anybody closes where the opportunity of another starts!". For me, each adolescent ought to be instructed to be free, planting in them the adoration for flexibility, yet they may be administered, it ought not be unbounded. Flexibility in teenager's state is essential for building a decent character and a steady identity. Youngsters need to stay nearby with companions, do whatever they like, or don't care for, they need to encounter everything. On the off chance that they are given opportunity without any breaking points, they may get lost, so guardians should disclose to them what is correct or wrong, so there is no restriction or confinement. That is the reason guardians' direction is extremely essential for our reality. Just flexibility with no direction will prompt an awful activities and cause obliteration in the general public. Flexibility is an express, a procedure of association of people and gatherings with the earth, when the general public is changing or a man rolls out a few improvements so that there seem more chances of decision and acknowledgment of imperative purposes and qualities. The most essential thing that we should think about is making the most of our opportunity without hurting our security. For conclusion, flexibility is an inborn right, something that people are conceived with. Nothing in this planet can undermine our opportunity, deny us from our rights. As say previously, our privilege of opportunity should never make hurt some other individual, straightforwardly or by implication. We should regard flexibility as everybody on the planet/society has it, in the event that you need others to regard your opportunity, right off the bat you should figure out how to regard them. Also, following this ways, we can appreciate and carry on with our flexibility through our life. Last however not the minimum, I need to cite a sentence that my educator in secondary school used to state "Quanto mais liberdade, mais responsabilidade.", it implies the more flexibility you have, the more duties you confront. All things considered, this sentence is extremely intelligent and I trust that that individuals out there have understand this little mystery!>

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