Chapter 3 4 5 are to be known for the exam. It starts at 4 pm NYC time and ends at 6:30. I will...

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Chapter 3 4 5 are to be known for the exam. It starts at 4 pm NYC time and ends at 6:30. I will therefore need someone to do the exam on the spot in two hours.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 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. 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  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− σ2 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 Itō’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 := B Q t − 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) =

Himanshu · Accepted Answer

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 matrix 
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.

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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