Stochastic Methods of Asset Pricing
Lecture 22
Review of the Principles of Option Pricing
Andrew Lyasoff
November 8, 2016
Lecture 22
2016 by Andrew
Lyasoff
Foreword
The Role of ELMM
Arbitrage
The Role and the Meaning of the Equivalent Local Martingale Me
Notes on Stochastic Calculus
Jerome Detemple and Marcel Rindisbacher
June 2013
1
Part I: Introduction to Ito Calculus
1.1 Stochastic processes
1.1.1 Definitions
Probability space (, F, P )
set of states of nature
F sigma algebra of observable events
P
Mean-Variance Analysis and the CAPM
Marcel Rindisbacher
October 7, 2014
1
Mean-Variance Analysis without Risk Free Asset
If there is no riskfree asset the mean variance frontier the global risk minimal portfolio is not any
longer the portfolio where all t
Boston University School of Management
MF 730
Exercise V: Dynamic Portfolio Choice
Marcel Rindisbacher
October 20, 2014
(i) Consider a setting with one stock, constant market price of risk, constant volatility and a
Vasicek model for the short rate. What
Boston University School of Management
MF 730
Exercise III: Malliavin Calculus
Marcel Rindisbacher
October 20, 2014
(i) Clarke-Ocone formula underRa change
of measure: Consider the density process dQ|FT =
ZT dP|FT where ZT = E 0 s dWs T for some progress
Boston University School of Management
MF 730
Exercise II: No arbitrage pricing
Marcel Rindisbacher
September 29, 2014
(i) Girsanov theorem in Black-Scholes market: In a market setting with one risky and one risk
free asset, constant volatility, constant
Boston University School of Management
MF 730
Exercise IV: Mean-Variance Analysis
Marcel Rindisbacher
October 22, 2014
(i) (Minimum Variance Frontier, Tangency Portfolio and Mean-Variance Portfolio)
(a) Show that both the mean-variance and optimal portfol
Boston University School of Management
MF 730
Exercise I: Ito Calculus
Marcel Rindisbacher
September 22, 2014
(i) Demonstrate the following properties of the stochastic exponential:
1. Show that if M is of bounded total variation, dMt = mt dt, then E (M )
Proof of FFTAP I: Density process of local martingale
measure for arbitrary numeraires
Marcel Rindisbacher
1
FTAPI
Consider a continuous time financial market model with Brownian filtration one risk free asset
and n risky asset. Suppose the nth asset is t
Proof of FFTAP II
Marcel Rindisbacher
1
Ito process setting: FFTAPII
Consider a standard market model with Brownian filtration where is not necessarily square.
Risky assets have dynamics
dSit
= rt dt + it0 (t dt + dWt )
Sit
i = 1, . . . , n
where W is d-d
Lecture 0: History of Investments
Marcel Rindisbacher
September 2016
History of Investment: Institutional Perspective (until
1950)
1550 First joint-stock companies (East Indian Company)
1600 Development of public markets in the Netherlands
1759 First pens
Lecture 7: Dynamic Portfolio Choice: Optimal
Portfolios and Bonds
Marcel Rindisbacher
September 2013
Marcel Rindisbacher
Dynamic Portfolio Choice
1 / 64
Alternative decomposition of portfolio
Unobserved short rate: substitute information in term structure
Lecture 11: Active Portfolio Management
Marcel Rindisbacher
September 2013
Marcel Rindisbacher
Dynamic Portfolio Choice
1 / 51
Introduction
Is there value to active fund management?
Value created if non-redundant return
Ability to extract new information
Lecture 8: Portfolio Choice and Human Capital
Marcel Rindisbacher
September 2013
Marcel Rindisbacher
Mean-Variance Portfolio Choice
1 / 56
Introduction: Big issues
1
How should individuals save and invest?
During active cycle of life (working phase)
Durin
Lecture 4: Mean-Variance Portfolio Choice
Marcel Rindisbacher
September 2013
History and Introduction
Harry Markowitz 1952 article on mean-variance analysis marks
the beginning of modern finance.
A mathematical formulation (and solution) of the question o
Lecture 1: Stochastic Calculus Review
Marcel Rindisbacher
September 2013
Stochastic basis and stochastic processes
Probability space (, F , P)
set of states of nature
F sigma algebra of observable events
P probability measure on (, F )
Stochastic proce
Lecture 3: Introduction to Malliavin Calculus
Marcel Rindisbacher
September 2013
Stochastic calculus of variation
Malliavin calculus is a calculus of variations for stochastic processes
Applies to Brownian functionals: random variables and stochastic pro
Lecture 2: No Arbitrage Pricing
Marcel Rindisbacher
September 2013
Finanical market model
Asset structure:
Money market account:
dS0t = S0t rt dt; S00 = 1
m
R
R
t
S0t
= E 0 rs ds t = exp 0 rs ds
d risky assets:
dSit = Sit (it dt + it dWt ) ;
m
R
iv
Sit
Lecture 9: Dynamic Asset Liability Management
Marcel Rindisbacher
September 2013
Marcel Rindisbacher
Dynamic Portfolio Choice
1 / 56
Introduction and Motivation
Basic problem:
Aggregate pension valuation of companies in the S&P 500
index dropped from a $2
7-24-2005
Volumes of Revolution by Slicing
Start with an area a planar region which you can imagine as a piece of cardboard. The cardboard
is attached by one edge to a stick (the axis of revolution). As you spin the stick, the area revolves and
sweeps out
7-27-2005
Work
The work required to raise a weight of P pounds a distance of y feet is P y foot-pounds. (In m-k-s
units, one would say that a force of k newtons exerted over a distance of y feet does k y newton-meters, or
joules, of work.)
Example. If a 1
2-5-2008
Vector Spaces
Vector spaces and linear transformations are the primary objects of study in linear algebra. A
vector space (which Ill dene below) consists of two sets: A set of objects called vectors and a eld (the
scalars).
Denition. A vector spa
4-29-2013
Unitary Matrices and Hermitian Matrices
Recall that the conjugate of a complex number a + bi is a bi. The conjugate of a + bi is denoted
a + bi or (a + bi) .
In this section, Ill use ( ) for complex conjugation of numbers of matrices. I want to
10-31-2005
Substitution
You can use substitution to convert a complicated integral into a simpler one. In these problems, Ill
let u equal some convenient x-stu say u = f(x). To complete the substitution, I must also substitute
du
du
for dx. To do this, co
7-18-2005
Trig Substitution
Trig substitution reduces certain integrals to integrals of trig functions. The idea is to match the
given integral against one of the following trig identities:
1 (sin )2 = (cos )2
1 + (tan )2 = (sec )2
(sec )2 1 = (tan )2
If
1-23-2006
Trigonometric Integrals
For trig integrals involving powers of sines and cosines, there are two important cases:
1. The integral contains an odd power of sine or cosine.
2. The integral contains only even powers of sines and cosines.
I will look
4-13-2008
Coordinate Transformations
A coordinate transformation of the plane is a function f : R2 R2 . Ill usually assume that f has
continuous partial derivatives, and that f is essentially one-to-one in the region of interest. (A function is
one-to-one