M.I.T.
Sloan School of Management
15.450-Fall 2010
Professor Leonid Kogan
Solutions for Practice Problems
1. Consider a 3-period model with t = 0, 1, 2, 3. There are a stock and a risk-free asset.
The initial stock price is $4 and the stock price doubles
Black-Scholes Formula
Brandon Lee
15.450 Recitation 2
Brandon Lee
Black-Scholes Formula
Expectation of a Lognormal Variable
Suppose X N , 2 . We want to know how to compute E e X .
This calculation is often needed (e.g., page 30 of Lecture Notes 1)
becaus
Nonlinear Least Squares
Applications to MIDAS and Probit Models
Brandon Lee
15.450 Recitation 11
Brandon Lee
Nonlinear Least Squares
Nonlinear Least Sqaures
Consider the model
yt = h (xt , ) + t
Here we assume that we know the functional form of h (xt , )
Stochastic Calculus
Brandon Lee
15.450 Recitation 3
Brandon Lee
Stochastic Calculus
Brownian Motion
Dening properties of Brownian motion Zt are
1
Z0 = 0.
2
It has continuous paths.
3
It has independent increments: if t1 < t2 s1 < s2 , then
Zt2 Zt1 and Zs2
Stochastic Calculus II
Brandon Lee
15.450 Recitation 4
Brandon Lee
Stochastic Calculus II
Kolmogorov Backward Equation
Consider a stochastic process Xt . Xt is a martingale if for s > t ,
Et [Xs ] = Xt
In other words, conditional expectation of future val
Introduction to Econometrics
Brandon Lee
15.450 Recitation 9
Brandon Lee
Introduction to Econometrics
Law of Large Numbers
Suppose xt are IID and E [xt ] = . Then the Law of Large
Numbers states that
plim
1
T
T
xt =
t =1
Intuitively, the LLN says that a
M.I.T.
Sloan School of Management
15.450-Fall 2010
Professor Leonid Kogan
Problem Set 6
1. (To be solved in a group) Suppose we observe returns of N trading strategies over the
same time period, xn , n = 1, ., N , t = 1, ., T . We want to develop a test o
Applications of Risk-neutral Pricing
Brandon Lee
15.450 Recitation 5
Brandon Lee
Applications of Risk-neutral Pricing
Outline
We are going to study two examples that illustrate the
concepts we have learned in the lectures.
The rst example will go over pri
M.I.T.
Sloan School of Management
15.450-Fall 2010
Professor Leonid Kogan
Practice Problems
1. Consider a 3-period model with t = 0, 1, 2, 3. There are a stock and a risk-free asset.
The initial stock price is $4 and the stock price doubles with probabili
Fundamental Theorem of Asset Pricing
Brandon Lee
15.450 Recitation 1
Brandon Lee
Fundamental Theorem of Asset Pricing
No Arbitrage
Roughly speaking, an arbitrage is a possibility of prot at zero
cost.
Often implicit is an assumption that such an arbitrage
M.I.T.
Sloan School of Management
15.450-Fall 2010
Professor Leonid Kogan
Problem Set 2
1. (To be solved in a group) This problem illustrates how to extend a static model like
CAPM to a dynamic setting, allowing us to price risky streams of cash ows.
Time
Last Name
First Name
M.I.T.
Sloan School of Management
ID Number
15.450-Spring 2010
Professor Leonid Kogan
Final Exam
Instructions: Carefully read these instructions! Failure to follow them may lead to deduc
tions from your grade.
1. Immediately write you
M.I.T.
Sloan School of Management
15.450-Spring 2010
Professor Leonid Kogan
Problem Set 5
1. (To be solved individually)
Consider a commodity with the price process Pt following
Pt P = (Pt1 P ) + t ,
t = 1, 2, ., T ,
| < 1
2
where t are IID N (0, ) shocks
M.I.T.
Sloan School of Management
15.450-Fall 2010
Professor Leonid Kogan
Problem Set 3
1. (To be solved in a group) Consider the Black-Scholes model for stock returns, with
constant interest rate r and constant drift and diusion of stock returns, and .
Y
M.I.T.
Sloan School of Management
15.450-Fall 2010
Professor Leonid Kogan
Problem Set 1
1. (To be solved individually) Consider a one-period model of the market. Assume there
are three possible states at time t = 1: 0, 1, and 2, all equally likely. There
M.I.T.
Sloan School of Management
15.450-Spring 2010
Professor Leonid Kogan
Problem Set 4
1. (To be solved in a group) Consider the Black-Scholes model with stock returns following
dSt
= dt + dZt
St
Assume that the interest rate is r and the stock pays no
M.I.T.
Sloan School of Management
15.450-Fall 2010
Professor Leonid Kogan
Examples of Dynamic Programming Problems
Problem 1 A given quantity X of a single resource is to be allocated optimally among N
production processes. Each process produces an output
M.I.T.
15.450-Fall 2010
Sloan School of Management
Professor Leonid Kogan
Dynamic Programming: Justication of the Principle of
Optimality
This handout adds more details to the lecture notes on the Principle of Optimality. As
in the notes, we are dealing w
M.I.T.
15.450-Spring 2010
Sloan School of Management
Professor Leonid Kogan
Handout: Crossing Probabilities of the Brownian Motion
Consider the process Xt , dXt = dZt . Assume that X0 = 0. We want to compute the
probability that Xt reaches a > 0 before re
Key Points: Derivatives
Leonid Kogan
MIT, Sloan
15.450, Fall 2010
c
Leonid Kogan ( MIT, Sloan )
Key Points: Derivatives
15.450, Fall 2010
1/1
Discrete Models
Denitions of SPD () and risk-neutral probability (Q).
Absence of arbitrage is equivalent to exis