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