STAC70 Statistics &
Finance I
Lecture 22
1
Risk-Neutral Pricing of
Cashflow
Consider asset generating random cashflows
with a total of C(t) is paid from time 0 to t
Want to create portfolio X(t) s.t. its final
discounted value at T is equal to the total
T

STAC70 Statistics & Finance I
Assignment 5 Solutions
These problems cover sections 5.1 to 5.6 from the textbook.
1. (Exercise 4.18 from textbook: this is from chapter 4, but really relates to risk-neutral pricing
from chapter 5 ) Let a stock price be a ge

ACTB47 Introductory
Life Contingencies
Lecture 1
1
Administrivia
Instructor: Sotirios Damouras
Contact Info:
Pronounced Sho-tee-ree-os or Sam
email: sdamouras@utsc.utoronto.ca
Office hours: MO 1-3pm & FRI 11am-1pm
@ IC 344, or by appointment (email)
Cours

ACTB47 Introductory
Life Contingencies
Lecture 5
1
Options as Insurance
Suppose you own a house
and buy home insurance
pay premium to insurance
company
company will compensate you
in case of damage to house
Profit from insurance is
similar to long put!
Pr

STAC70 Statistics &
Finance I
Lecture 2
1
Binomial Option Pricing
Put-call parity gives relationship that prices
must satisfy, but it doesnt tell us how to find
option prices
In order to calculate option prices, we need to
model the behavior of the underl

1. Let W (t ) be a Brownian motion (BM) and X (t ) exp 5W (t ) 4t 3 . Find processes (t ) & (t ) such
that dX (t ) (t ) dW (t ) (t )dt . Write (t ) & (t ) in terms of W (t ) and t .
be two equivalent probability measures on (,
&
2. Let Q
d
( ) ( ) . Let

Math 5632 Notes
Black-Scholes Formula
Review of normal distribution N (, 2 ).
Black-Scholes Formula
Assumptions:
The return on the stock is normally distributes and independent overtime.
The volatility of the return is a constant .
The dividend yield

STAC70 Statistics & Finance I
Assignment 4 Solutions
These problems cover sections 4.4 to 4.6 and 5.1 to 5.3 from the textbook.
1. (Exercise 4.5 from textbook: Solving the generalized geometric Brownian motion equation) Let
S (t ) be a positive stochastic

ACTB47 Introductory
Life Contingencies
Lecture 4
1
Options
Non-binding agreement (right but not
obligation) to buy/sell an asset in the future
at a price set today
Option buyer (holder/long) holds right to buy/sell
asset if he chooses to do so
Option sell

STAC70 Statistics & Finance I
Assignment 3
Due Friday, Feb 28 (in class)
These problems cover sections 4.1 to 4.5 (inclusive) from the textbook.
1. Prove directly from the definition of It integrals that
t
t
0
0
udW (u) tW (t ) W (u)du .
(Hint: show it h

Spring, 2012
Brownian Motion and Stochastic Differential Equations
Math 425
1
Brownian Motion
Mathematically Brownian motion, Bt 0 t T , is a set of random variables, one for each value of the
real variable t in the interval [0, T ]. This collection has t

624 Chapter 20. Brownian Motion and Ito's Lemma 1"
Problems 625
The last term in equation (20.43) is the expected change in the option price conditional on 203 Use Itos L
the jump times the probability of the jump. emma t0 evaluate d S 1.
, . 20.4 U A
The

ACTB47 Introductory
Life Contingencies
Lecture 9
1
Forwards vs Futures
Forwards have no capital commitments today
E.g. Can enter into either 1 or 1000 contracts
In both cases, no party pays anything today
In the second case however, gain/loss is magnified

ACTB47 Introductory
Life Contingencies
Lecture 6
1
Put-Call Parity
Payoff of synthetic &
genuine forwards are
equal their value
today (t=0) must also
be equal
synthetic fwd
Cash Flows
Position
t=0
t=T
Long Call
C
+ (ST K)+
Short Put
+P
(K ST)+
PV(KF0,T)

2/1/2013
Why Do Firms Manage Risk?
Hedging can be optimal if extra $ in good times
is worth less than extra $ in bad times
Lecture 8
Profits is such cases are concave
hedging (i.e. reducing uncertainty)
can increase expected cash flow
Concave profits ca

STAC70 Statistics &
Finance I
Lecture 21
1
Forwards vs Futures
Forwards have no capital commitments
E.g. Can enter into either 1 or 1000 contracts:
Nothing paid today, but in 2nd case gain/loss is
magnified by 1000
Forwards have high credit risk
Futures

STAC70 Statistics &
Finance I
Lecture 21
1
Forwards vs Futures
Forwards have no capital commitments
E.g. Can enter into either 1 or 1000 contracts:
Nothing paid today, but in 2nd case gain/loss is
magnified by 1000
Forwards have high credit risk
Futures

ACTB47 Introductory
Life Contingencies
Lecture 3
1
Forward Contract
Example: Bank A has agreed to buy one
share of stock XYZ from bank B on Jan 11,
2014 at a price of $100 per share
Characteristics
Notation
Underlying asset: Stock XYZ
Expiration date: Jan

(20.8) dX (t ) dt dZ (t ), where Z (t ) is BM
(20.9) dX (t ) X (t ) dt dZ (t )
dX (t )
dt dZ (t )
X (t )
dS (t )
(20.32) If
(r )dt dZ (t ), then
S (t )
(20.11)
S (T ) a S (0) a e[ a ( r ) 2 a ( a 1)
1
where dZ S dZ Q dt
(20.38) Under the risk-neutral

ACTB47 Introductory
Life Contingencies
Lecture 11
1
Forwards vs Futures
Because of marking-to-market, forward and
futures have different cashflows
Forwards have single cashflow at expiration
Futures can have daily cashflows till expiration
Moreover, futur

ACTB47 Introductory
Life Contingencies
Lecture 2
1
Trading Derivatives
1.
Two main ways to trade in derivatives
Over-the-counter (OTC)
2.
Contracts are negotiated and traded directly
between two parties, without an intermediary
Through a derivatives excha