Arbitrage and the Creation of New Securities
As a simple thought experiment, suppose that the only
bonds now available have semiannual coupons and a
principal of $100. Their maturity dates are six months
apart, with the first maturity date being six month
90. Important Formulae
One cannot escape the feeling that these formulae have an independent existence and
an intelligence of their own, that they are wiser than we are, wiser even than their
discoverers, that we get more out of them than was originally p
77. Problem Set 9
1.
Consider a probability experiment consisting of picking a card at random from a
deck of 52 playing cards.
a.
b.
How many elements are there in the sample space?
What are the probabilities of the following events:
Let the following eve
79. Inference
God is subtle but not malicious. He never would have created a world with beings who
wanted to learn and things impossible to know.
Einstein
Consider a population of numbers:
The relative frequency of each distinct number in the population c
75. OPTION VALUATION
European Option Formulae
1.Non-dividend paying stock
C XN (d1 ) Ee r (T t ) N (d 2 ) , P Ee r (T t ) N ( d 2 ) XN ( d1 ),
C P X Ee r (T t ) ,
where
2
X
ln (r )(T t )
E
2
d1
d 2 d1 T t
Tt
N '( Z )
1 12 Z 2
e
2
C
N (d1 ); P ( E X )
80. Black-Scholes Differential Equation
Why does the value, C, of a call option satisfy the Black-Scholes Differential Equation
rC rX
C
C
1
X
X
t
2
2
2
2C
X 2
First it is useful to think of stock prices behaving as per random walk with a positive
drift.
Probability and Statistics
God created the integers; the rest is the work of man.
Kronecker
66. DESCRIPTIVE STATISTICS
The truth is by no means given to us. Given to us are the data of
our consciousness.
Einstein
Numbers, numbers everywhere! What do they
AnIntroductiontoBinomial
TreeModels
LiAn
HKUST,2016
Summary
Option pricing models
One-step binomial models for European
options
Risk-neutral valuation
Two-step binomial models
European vs. American options
Hedge ratio
Binomial trees in practices
2
81. Problem Set 10
Problem1:
Betty and Bob know that the P-E Multiples of a large collection of growth
stocks is mound shaped with a mean of 20 and standard deviation of 5.
Their portfolio consists of 25 randomly selected stocks from this
collection.
a.
W
78. Project 9
Problem 1:
Exponential Moving Averages smooth price data over time. Consider the
prices as per
P1
P2
P3
P4
P5
.
Pn
The definition is:
EMA1 =
EMAn =
=
P1
EMA n-1 + K (Pn EMAn-1)
KPn + (1-K) EMAn-1
K is the smoothing constant; 0 < K < 1.
Probl
Duration
Duration is a measure of the sensitivity of a bond to
changes in interest rates.
P = bond price
y = bond yield
For small changes in y,
dP
P y
dy
dP
P dy
y
P P
dP
=
dollar duration
dy
dP
dy
=
duration
P
For a given principal, duratio
Interest Rate Swaps
How could we create a bond that has a long maturity and
a small duration?
Solution: Let coupons change as interest rates change
Suppose we create a default-free semiannual floating
rate bond with principal of $1
ri
=
1
2
3
r1
2
r2
2
r3
Forward Rate Agreements
Business firms frequently want to lock in today a rate for
future short term borrowing or lending. Financial firms
meet this need with our first derivative, the forward rate
agreement (FRA).
An FRA contract specifies a notional pri
Commodity Forwards
Storable commodities are different from our other asset classes in fundamental
ways. Stocks pay dividends and domestic and foreign bonds pay interest to their
owners. All owners receive the same payment per share or per bond. The
owners
Futures Contracts
Futures contracts are similar to forwards and serve the
same purposes, but they arent identical.
They are similar in that with both
1) a long or short position can be taken with no initial
cost
2) the long side has no claim on income gen
Currency Swaps
A currency swap is very similar to an ordinary swap
except that the principals of the two sides are in different
currencies.
The amounts of the two principals are chosen to have
equal value on the initial date. Because of changes in the
exc
Forward Contracts
The buyer of a forward contract agrees to buy a
designated underlying asset for a specified price on a
specified future date. The seller agrees to sell on the
same terms.
The specified price is called the contract price.
The specified fu
Equity Swaps
Equity swaps are used to transform the returns from
floating rate investments into the returns from equity
investments and vice versa. Their potential benefits
include:
giving a quick and convenient way to increase or
decrease equity exposure
Introduction
Derivatives are securities whose future payoffs are tied by
contract to some underlying variable.
The underlying variable is usually the price of a traded
asset
equities
interest rates
currencies
commodities
but can also be a property of asse
Risk Neutral Pricing
Black-Scholes Formula
Lecture 19
Dr. Vasily Strela
(Morgan Stanley and MIT)
Risk Neutral Valuation: Two-Horse Race Example
One horse has 20% chance to win another has 80%
chance
$10000 is put on the first one and $50000 on the secon
84. Additional Problems
Problem 1:
a.
Betty has an account with an initial value of $100,000. All cash
balances earn 5% per annum compounded annually. She invests $
50,000 from the account in Miracle Stock Fund and sells it for $
75,000 after six months.