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 Rol
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 st
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 minima
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 v
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
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
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)
(
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
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 f
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
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 Developme
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 n
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 in
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 ma
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 e
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 Br
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 ri
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).
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
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
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
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 substituti
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
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 cont
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-o