IEOR 4706-英语代写
时间:2022-12-21
IEOR 4706: Foundations of financial engineering.
Fall 2019, Professor Possamaï.
Final: Wednesday, December 18, 2019.
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.
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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.
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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.
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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
(
h(S1)
)
= max
(r,d,u)∈[r,r]×[d,d]×[u,u]
EQr,d,u
[
h(S1)
1 + r
]
.
8)(?) Compute explicitly p
(
h(S1)
)
when h is the payoff of a Call option with maturity 1 and strike K > 0, that is to say
h(S1) = (S1 −K)+.
Exercise 2: Asian option in Black–Scholes model
We consider the one–dimensional Black–Scholes model seen in class, with time horizon T > 0, such that the unique
risky asset in the market has the following dynamics under the unique risk–neutral measure Q
St = S0 +
∫ t
0
rSsds+
∫ t
0
σSsdBQs , t ∈ [0, T ],
where r ≥ 0 is the interest–rate, σ > 0 the volatility of S, S0 > 0 its initial value, and BQ is a Q–Brownian motion.
The goal of the exercise is to evaluate the price of an Asian Call option on S, with maturity T and strike K > 0,
whose payoff at maturity is given by
ΦT :=
(
1
T
∫ T
0
Ssds−K
)+
.
We denote by ACt(T,K;S) the value at any time t ∈ [0, T ] of such an option, and as usual, by Ct(T,K;S) the value
at time t ∈ [0, T ] of the call option with strike K, maturity T and underlying S.
1) Consider another option with maturity T , whose payoff is defined by
ΨT :=
(
exp
(
1
T
∫ T
0
log(Ss)ds
)
−K
)+
The value at any time t ∈ [0, T ] of the option with payoff ΨT is denoted by GACt(T,K;S).
3
a) Prove that
ΦT =
(
1
T
∫ T
0
(
Ss −K
)
ds
)+
,
and then that
ΦT ≤ 1
T
∫ T
0
(
Ss −K
)+ds.
b) Prove that
exp
(
1
T
∫ T
0
log(Ss)ds
)
≤ 1
T
∫ T
0
Ssds,
and deduce that
ΨT ≤ ΦT .
Hint: It would be profitable to admit the so–called Jensen’s inequality, which states that for any convex function
f : R −→ R, and any integrable function g : [0, T ] −→ R, we have
f
(
1
T
∫ T
0
g(s)ds
)
≤ 1
T
∫ T
0
f
(
g(s)
)
ds.
c) Deduce then that
GAC0(T,K;S) ≤ AC0(T,K;S) ≤ 1
T
∫ T
0
e−r(T−s)C0(u,K;S)du.
2)a) Show that ∫ T
0
BQs ds =
∫ T
0
(T − s)dBQs .
2)b) Deduce that the random variable 1/T
∫ T
0 log(Ss)ds has a Gaussian distribution under Q, with mean m and variance
v2, where
m = log(S0) +
(
r − σ
2
2
)
T
2 , v
2 := σ
2T
3 .
2)c) Show then (this should remind you of Black–Scholes formula) that
GAC0(T,K;S) = S0e−
T
2 (r+
σ2
6 )N (d1)−Ke−rTN (d0),
where
d0 :=
1
σ

T/3
log
(
S0e
T
2 (r−σ
2
2 )
K
)
, d1 := d0 + v.
3) We now define a new process
Yt = Y0 +
∫ t
0
Sudu, t ∈ [0, T ],
where Y0 > 0 is a given constant.
a) Prove that the pair (S, Y ) is a Markovian diffusion, that is to say that you can find maps b : (0,+∞)2 −→ R2
and Σ : (0,+∞)2 −→ R2×2 (where R2×2 is the set of 2× 2 matrices) such that(
St
Yt
)
=
(
S0
Y0
)
+
∫ t
0
b(Ss, Ys)ds+
∫ t
0
Σ(Ss, Ys)
(
dBQs
dBQs
)
, t ∈ [0, T ].
b) We consider an option with maturity T and payoff (1/TYT −K)+. We are looking for a self–financing portfolio
Xx,∆ which replicates this option and takes a very specific form, namely
Xx,∆t = u(t, St, Yt), t ∈ [0, T ],
4
where the map u : [0, T ]×(0,+∞)2 −→ R is supposed to be as smooth as necessary. Using the two–dimensional
Ito¯’s formula (see Footnote 2 in the next exercise), show that necessarily, the map u must then satisfy the PDE
∂u
∂t
(t, x, y) + rx∂u
∂x
(t, x, y) + x∂u
∂y
(t, x, y) + σ
2
2 x
2 ∂
2u
∂x2
(t, x, y)− ru(t, x, y) = 0, (t, x, y) ∈ [0, T )× (0,+∞)2,
u(T, x, y) =
(
y
T
−K
)+
, (x, y) ∈ (0,+∞)2.
We will assume that this PDE has a unique non–negative solution with bounded derivatives with respect to x
and y.
c) Deduce a replicating strategy for the option with payoff ΦT , and prove as well that
AC0(T,K;S) = u(0, S0, 0).
4) The PDE derived in the previous question is not easy to deal with numerically, so we will try to reduce its dimension.
a)(?) Prove that an option with maturity T and payoff 1/T
∫ T
0 Ssds−K can be replicated by a self–financing portfolio
(x?,∆?), where
x? := S0
1− e−rT
rT
−Ke−rT , ∆?t :=
1− e−r(T−t)
rT
, t ∈ [0, T ].
b) We define Zt := Xx
?,∆?
t S
−1
t , t ∈ [0, T ]. Show that
dZt =
(
∆?t − Zt
)
σ
(
dBQt − σdt
)
.
c) Let us consider the probability measure QS whose density with respect to Q is given by
dQS
dQ := e
−rT ST
S0
.
Show that QS is well–defined, that the process BS , defined by BSt := BQt − σt, t ∈ [0, T ], is a QS–Brownian
motion, and deduce that Z is a QS–martingale.
d) Show that
AC0(T,K;S) = S0EQ
S
[Z+T ],
and deduce then that AC0(T,K;S) = S0V (0, Z0), where the function V is assumed to be smooth and solves
the PDE 
∂V
∂t
(t, z) + σ
2
2
(
∆?t − z
)2 ∂2V
∂z2
(t, z) = 0, (t, z) ∈ [0, T )× (0,+∞),
V (T, z) = z+, z ∈ (0,+∞).
Exercise 3: Variance swap hedging by using liquid Calls
Fix some horizon T > 0. We consider a filtered probability space (Ω,F ,F = (Ft)0≤t≤T ,Q), where Q is directly
assumed to be a risk–neutral measure, under which the dynamics of the unique risky asset S is given by
St = S0 +
∫ t
0
SsσsdBQs , t ∈ [0, T ],
where S0 > 0, where BQ is an (F,Q)–Brownian motion, and where the process (σt)t∈[0,T ] is an F–adapted process
satisfying σ ≤ σt ≤ σ, t ∈ [0, T ], for some constants 0 < σ ≤ σ < +∞. We assume that the interest rate is constant
and equal to 0 for simplicity. The first goal of this exercise is to price and hedge the random leg of a so–called variance
swap with maturity T , whose payoff is given by
JT :=
1
T
∫ T
0
σ2t dt.
5
1) By using Ito¯’s formula, show that
JT = − 2
T
log
(
ST
S0
)
+ 2
T
∫ T
0
σtdBQt ,
and verify that
2
T
∫ T
0
σtdBQt =
∫ T
0
φ(St)dSt,
for some function φ : R+ −→ R that you will explicitly determine.
2) Deduce from the previous question a replication strategy for an option with maturity T and with payoff 2T
∫ T
0 σtdB
Q
t .
3) We assume in this question that it is possible to sell in the market, at the price p, a European option with maturity
T and with payoff log(ST ) (this is called a log–contract). How can you then replicate and hedge JT ? Prove that the
price of this hedge is given by
2
T
(
log(S0)− p
)
.
4) We assume from now on that we can buy and sell dynamically a European call with maturity T ′ > T and strike S0.
Explain why its price Pt at any time t ≤ T is given by
Pt = EQ
[
(ST ′ − S0)+
∣∣Ft].
5)(?) We assume now that the dynamics of σ is of the form
σt = σ0 +
∫ t
0
ϕ(Ss, σs)
(
dBQs + ηdWQs
)
,
where σ0 > 0, η > 0, ϕ : R+×R+ −→ R+ is a continuous and bounded function, andWQ is another (F,Q)–Brownian
motion, independent of BQ. Explain why there exist functions p : [0, T ]× R+ × R+ and v : [0, T ]× R+ × R+ such
that
Pt = p(t, St, σt), EQ
[
log(ST )
∣∣Ft] = v(t, St, σt), t ∈ [0, T ].
6)(?) We assume that the functions p and v of the previous question are smooth. Verify first that if we define the following
two–dimensional vectors
Xt :=
(
St
σt
)
, and BQt :=
(
BQt
WQt
)
, t ∈ [0, T ],
we can then write
Xt =
(
S0
σ0
)
+
∫ t
0
(
X1sX
2
s 0
ϕ(X1s , X2s ) ηϕ(X1s , X2s )
)
dBQs , t ∈ [0, T ],
where for i = 1, 2, Xit denotes the ith coordinate of the vector Xt.
Then, using the multidimensional Ito¯’s formula2, prove that for any t ∈ [0, T ]
p(t, St, σt) = p(0, S0, σ0) +
∫ t
0
(
∂p
∂t
(s, Ss, σs) +
1
2S
2

2
s
∂2p
∂S2
(s, Ss, σs) +
1 + η2
2 ϕ
2(Ss, σs)
∂2p
∂σ2
(s, Ss, σs)
)
ds
+
∫ t
0
Ssσsϕ(Ss, σs)
∂2p
∂S∂σ
(s, Ss, σs)ds+
∫ t
0
(
Ssσs
∂p
∂S
(s, Ss, σs) + ϕ(Ss, σs)
∂p
∂σ
(s, Ss, σs)
)
dBQs
+
∫ t
0
ηϕ(Ss, σs)
∂p
∂σ
(s, Ss, σs)dWQs ,
2If X and Y are two Ito¯ processes of the form
dXt = bXt dt+ σXt dB
Q
t , dYt = b
Y
t dt+ σYt dB
Q
t + η
Y
t dW
Q
t ,
then for any smooth function v(t, x, y), we have
dv(t,Xt, Yt) =
∂v
∂t
(t,Xt, Yt)dt+
∂v
∂x
(t,Xt, Yt)dXt +
∂v
∂y
(t,Xt, Yt)dYt +
1
2
∂2v
∂x2
(t,Xt, Yt)(σXt )2dt+
1
2
∂2v
∂y2
(t,Xt, Yt)
(
(σYt )2 + η2t )dt
+ ∂
2v
∂x∂y
(t,Xt, Yt)σXt σYt dt
6
and
v(t, St, σt) = v(0, S0, σ0) +
∫ t
0
(
∂v
∂t
(s, Ss, σs) +
1
2S
2

2
s
∂2v
∂S2
(s, Ss, σs) +
1 + η2
2 ϕ
2(Ss, σs)
∂2v
∂σ2
(s, Ss, σs)
)
ds
+
∫ t
0
Ssσsϕ(Ss, σs)
∂2v
∂S∂σ
(s, Ss, σs)ds+
∫ t
0
(
Ssσs
∂v
∂S
(s, Ss, σs) + ϕ(Ss, σs)
∂v
∂σ
(s, Ss, σs)
)
dBQs
+
∫ t
0
ηϕ(Ss, σs)
∂v
∂σ
(s, Ss, σs)dWQs ,
7) Explain why (p(t, St, σt))t∈[0,T ] and (v(t, St, σt))t∈[0,T ] are (F,Q)–martingales, and deduce that
p(t, St, σt) = p(0, S0, σ0) +
∫ t
0
(
Ssσs
∂p
∂S
(s, Ss, σs) + ϕ(Ss, σs)
∂p
∂σ
(s, Ss, σs)
)
dBQs +
∫ t
0
ηϕ(Ss, σs)
∂p
∂σ
(s, Ss, σs)dWQs ,
v(t, St, σt) = v(0, S0, σ0) +
∫ t
0
(
Ssσs
∂v
∂S
(s, Ss, σs) + ϕ(Ss, σs)
∂v
∂σ
(s, Ss, σs)
)
dBQs +
∫ t
0
ηϕ(Ss, σs)
∂v
∂σ
(s, Ss, σs)dWQs ,
8)(?) Deduce that in order to replicate the payoff log(ST ), one should hold at each time t ∈ [0, T ] a quantity Ψt of options
with payoff (ST ′ − S0)+, where
Ψt :=
∂v
∂σ (t, St, σt)
∂p
∂σ (t, St, σt)
,
as well as ∆t risky assets S, where
∆t :=
∂v
∂S
(t, St, σt) +
ϕ(St, σt)
Stσt
∂v
∂σ
(t, St, σt)−Ψt
(
∂p
∂S
(t, St, σt) +
ϕ(St, σt)
Stσt
∂p
∂σ
(t, St, σt)
)
.
9) How can we hedge dynamically JT by using the risky asset and the call (and thus not the log–contract)? Prove that
the price of this hedge in terms of S0 and v(0, S0, σ0) is given by
2
T
(
log(S0)− v(0, S0, σ0)
)
.
What partial differential equation does the function v satisfy?
10) We consider now a so–called weighted variance swap with maturity T and continuous weight function w : R+ −→ R+,
for which the random leg’s payoff is given by
JwT :=
1
T
∫ T
0
σ2tw(St)dt.
Define for some fixed x0 > 0 the function F : R+ −→ R by F (x) :=
∫ x
x0
∫ y
x0
2w(z)
Tz2 dzdy. Prove that
JwT = F (ST )− F (S0)−
∫ T
0
F ′(St)dSt.
11) How can you hedge JwT , without knowing σ, by simply using cash, Call options, Put options and the risky asset S?
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