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IEOR 4706: Foundations of financial engineering.

创建日期：2024-05-18 17:07:35

所属分类：金融Finance

标签： IEOR 4706

Foundations of financial engineering.

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IEOR 4706: Foundations of financial engineering.

Instructions:
• The only document allowed is your two–sided cheat sheet (US paper format).
Pocket calculators are not allowed.
• Despite the fact that within an exercise questions may be linked to each other, ev-
ery question in the exam can be done by itself, provided you admit the necessary
results from earlier questions.
• Please SHOW YOUR WORK. This is the easiest way for us to award you
points.
• Questions marked by a (?) are either longer or more difficult than the other
ones. Notice also that exercises are given in increasing order of difficulty.
• Please be also aware that it is not necessary for you to do all the questions of
all the exercises to get 100 points. We do not expect you to finish absolutely ev-
erything, and there are more questions to give you more opportunities to answer
as many of them as possible.
• Please write down your name on your blue books IMMEDIATELY. Once the
end of the exam has been announced, you must put down your pens, stand up,
pass your answer sheets to the side for us to recover them, and wait until we
are done before exiting. Failure to comply with this rule will be sanctioned by 10
points subtracted from your grade.
• For the sake of fairness between the students, we will not answer questions
during the exam. If you think there is a mistake or a typo (there should not be,
as the whole team proof–checked everything, but this still could happen), signal
it on your answer sheet and proceed accordingly.
• If you see an exercise who resembles closely one in your practice midterms or
finals, be aware that it will be graded more strictly than other exercises you have
not encountered before.
• You CANNOT turn this page around until all the exams have been distributed
and we give you explicitly the authorisation to start. Failure to comply with this
rule will be sanctioned by 10 points subtracted from your grade.
1
Exercice 1: Super–replication under interest rate and volatility uncertainty
We consider a one–period binomial, that is to say we work on the space Ω := {ωu, ωd}, which we endow with the
σ−algebra F := {∅,Ω, {ωu}, {ωd}}. The filtration considered is F := (F0,F1), with F0 := {∅,Ω}, and F1 := F . The
market contains one risky asset whose price evolves as follows
S0
dS0
P[{ω d}] = 1− p
uS0
P[{ω
u}] = p
where p ∈ (0, 1) and 0 < d < u. There is also one non–risky asset, whose price at time t is given by S0t := (1 + r)t, t =
0, 1. However, unlike the model considered in the lectures, we assume that the true values of the parameters r, u and
d are not known perfectly, and that the only information that we have is that they lie in intervals with known bounds.
More precisely, we have
0 ≤ r ≤ r ≤ r, 0 < u ≤ u ≤ u, and 0 < d ≤ d ≤ d,
where the bounds are known explicitly. We moreover assume that
u > d.
To take this uncertainty into account, we consider a family of probability measures on (Ω,F), denoted by P, such
that
P := {P : ∃(r, d, u) ∈ [r, r]× [d, d]× [u, u], P[{S1 = uS0}] = p, P[{S1 = dS0}] = 1− p, P[{S01 = 1 + r}] = 1}.
Every P ∈ P thus represents one possible binomial model with parameters (r, d, u) ∈ [r, r]× [d, d]× [u, u].
1) Prove that if d < 1 + r, and 1 + r < u, then we have that for any P ∈ P, the usual Condition (NA) holds under the
measure P.1 We will assume that these inequalities hold throughout the rest of the exercise.
2) Let us consider an option with payoff h(S1). Our goal will be to compute its super–replication price, defined in this
context by
p
(
h(S1)
)
:= inf
{
x ∈ R : ∃∆ ∈ R,P[{Xx,∆1 ≥ h(S1)}] = 1, ∀P ∈ P}.
Show that if the self–financing portfolio Xx,∆ super–replicates the option, then we necessarily have that for any
(r, d, u) ∈ [r, r]× [d, d]× [u, u] {
∆uS0 + (1 + r)(x−∆S0) ≥ h(uS0),
∆dS0 + (1 + r)(x−∆S0) ≥ h(dS0).
3) Deduce that if the self–financing portfolio Xx,∆ super–replicates the option, then we necessarily have that for any
(r, d, u) ∈ [r, r]× [d, d]× [u, u]
h(uS0)− (1 + r)x
S0(u− 1− r) ≤ ∆ ≤
(1 + r)x− h(dS0)
S0(1 + r − d) ,
and then that x ≥ max
(r,d,u)∈[r,r]×[d,d]×[u,u]
f(r, d, u), where we defined
f(r, d, u) := 11 + r
(
1 + r − d
u− d h(uS0) +
u− 1− r
u− d h(dS0)
)
.
1We recall that Condition (NA) under P stipulates that for any ∆ ∈ R, P[X0,∆1 ≥ 0] = 1 =⇒ P[X0,∆1 = 0] = 1.
2
4) Prove that for any (d, u) ∈ [d, d]× [u, u]
max
r∈[r,r]
f(r, d, u) =
{
f(r, d, u), if dh(uS0) ≥ uh(dS0),
f(r, d, u), if dh(uS0) < uh(dS0).
We denote the corresponding point where the maximum is attained by r?(d, u), with
r?(d, u) :=
{
r, if dh(uS0) ≥ uh(dS0),
r, if dh(uS0) < uh(dS0).
5)(?) Assume now that there exists a pair (u?, d?) ∈ [u, u]× [d, d] such that
x = f
(
r?(d?, u?), d?, u?
)
.
Prove that the strategy with initial capital x and with a number of risky assets held at time 0 given by
∆ := h(u
?S0)− h(d?S0)
(u? − d?)S0 ,
super–replicates the option.
6) Conclude that p
(
h(S1)
)
= x.
7) For any probability measure Pr,d,u ∈ P associated to some given (r, d, u) ∈ [r, r]× [d, d]× [u, u], we can associate an
equivalent probability measure Qr,d,u such that
Qr,d,u[{ωu}] = 1−Qr,d,u[{ωd}] := 1 + r − d
u− d .
Check, and comment, that for any (r, d, u) ∈ [r, r] × [d, d] × [u, u], we have that (St/(1 + r))t=0,1 is an (F,Qr,d,u)–
martingale, and that the following super–hedging duality holds
p

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