Math 622
Notes to Lecture 4, Part II
Spring 2013
An addendum to Part I of Lecture 4. We state without proof an extension
of Theorem 4 of Part I.
t
Theorem 1 Let X(t) = X(0) +
t
(s) ds +
0
(s) dW (s). Assume that b > X(0)
0
and let Tb be the rst time X hit

Math 622
Lecture Notes I
Spring 2013
Jump Process Models: Part I
I. Overview of price modeling in continuous time
Let cfw_F(t); t 0 be a ltration modeling the accumulation of market information
available to investors as time progresses. A simple paradigm

Mathematical Finance II
Spring 2013
Lecture Notes for Lectures 9 and 10
These notes review setting up of multi-asset models and the mathematical techniques for their analysis. As an example, a model of a market with a tradeable foreign
currency is given.

642:622; Notes for Lecture 7; Spring 2013
Asian Options
I. Asian Options: Denitions and Examples.
Let S(t) denote the price of an underlying asset. Its average price over the time
interval [0, T ] is
1 T
Save (T ) =
S(u) du.
T 0
Options whose payo depends

Math 622; Lecture Notes 7; American Options; Spring 2013
I. Denitions and Examples.
American options are options which the owner may exercise at any time between
date of purchase (t = 0) and expiry (t = T ).
Consider an underlying asset with price cfw_S(t

Math 622
Notes to Lecture 4, Part I
Spring 2013
I. Useful review and denitions for the theory of stopping times.
(a) -algebras and measurability.
A collection G, of subsets of a set , is a -algebra if it is closed under the
operations of complementation,

Math 622
Notes to Lecture 5, Part II
Spring 2013
4. Pricing a lookback option for the Black-Scholes price.
Lookback options are options whose payos depend on the maximum or minimum
of the underlying assets price over the lifetime of the option. For exampl

Math 622
Notes to Lecture 6
Spring 2013
1. An extension of Itos rule.
Let cfw_S(t) represent an asset price and let Y (t) := maxcfw_S(u); 0 u t denote
its running maximum. This notation will be used throughout the lecture.
When the process S is continuous

Mathematical Finance II
Spring 2013
Lecture Notes for Lecture 5
1
1. The reection principle for Brownian motion.
Let W be a Brownian motion with a ltration cfw_F(t); t 0, and let be a
stopping time. Dene
B (t) =
W (t),
if t ;
W ( ) [W (t) W ( )], if t >

59W; 1”thqu Lgdwe .
. Chane._££measure_ _ _ 7‘47 _
_ Let [R beﬂrobabi‘; _wleaswe on our-J (at
7 E71.” clexuzﬁa ex dnhvyyiiy mMLE-W_
WiW gagfandom vmiable. such that {P(Z7IOI=L
m_,_Epii]ji—}Aﬁnﬂ- a nugmmm QLLQ.33_93L_
' m ‘ Q (RX : = ﬁfe? f i