Derivatives
Lecture 1:
Introduction to Derivatives &
Related Mathematical Concepts
Otto Khatamov
2010/2011
Course Rules
You must read the module handbook that
contains important course information. It is
your responsibility to look at the information.
I will post by e-mail the lectures notes two
days before the actual day. You are expected
to read them.
I would like to encourage you to ask questions
during the course.
Course Rules
Your success in this course will depend on
you achieving the learning goals
(and it‟s hard work).
If you want to contact me, please send me an
e-mail to make an appointment.
You do not need to buy a lot of books, if you
want to buy a book get „Hull‟. You can consult
the other books at the library.
Today’s Learning Outcomes
1. What is this course about?
2. Basics of derivative markets:
a) Who are the actors?
b) What products are traded?
3. What mathematical tools are needed
to understand derivatives?
Agenda
What is a Derivative?
Who are the Actors?
What are the Market Structures?
Introduction to Mathematics of
Derivatives.
Reading for today’s lecture
REQUIRED READING:
Hull (2008), chapter 1
RECOMMENDED READING
Jarrow, R. and S. Turnbull (2000), chapter 1
Neftci, N., (2000), chapters 3 and 5
Watsham and Parramore (1997), chapters 3
and 4
McDonald, R. (2006), chapter 1 and appendix
B
What are Derivatives?
A derivative is an instrument whose value
depends on, or is derived from, the value of
another asset.
Literally a derivative instrument is a byproduct
of another (underlying) financial instrument.
Derivatives fulfill different needs for the
economy.
They are at the core of “financial engineering”.
Examples: futures, forwards, swaps, options,
exotics…
Why Using Derivatives?
To hedge risks.
To speculate by using leverage.
To lock in an arbitrage profit.
To change the nature of a liability.
To change the nature of an investment
without incurring the costs of selling one
portfolio and buying another.
Basic Types
Forward
For example interest rate, foreign exchange…
Future
For example oil, bonds…
Swap
For example interest rate, foreign exchange…
Option
For example stocks, bonds…
Securitization
For example mortgages, corporate debt…
Example
On May 24, 2010 the treasurer of a
corporation enters into a long forward
contract to buy £1 million in six months
at an exchange rate of 1.4422
This obligates the corporation to pay
$1,442,200 for £1 million on November
24, 2010
What are the possible outcomes?
Derivative Markets
Exchange-traded markets
Standardized contracts
Electronic trading or open outcry (a trading
floor)
Limited credit risks
Over-the-counter (OTC) markets
Network of dealers
Provide quotes for contracts tailored to specific
needs of clients
Non-standardized contracts
Trades subject to credit risks
Size of OTC & Exchange-Traded
Markets
Source: Bank for International Settlements. Chart shows total principal
amounts for OTC market and value of underlying assets for exchange market
Who Trades Derivatives
Companies Hedging
Banks Hedging/Speculation/Issue
Mutual Funds Speculation
Hedge Funds Arbitrage/Speculation
Are Derivatives Dangerous?
”In my view, derivatives are financial
weapons of mass destruction, carrying
dangers that, while now latent, are
potentially lethal.” Warren Buffett 2002
Leverage Effect
profit or losses can be very high.
Complex
Non-linear
Sometimes the structure is opaque
The pricing involves elaborate mathematics
Leverage
If you invest 100$ in a stock, the
maximum that you can loose on this
investment is 100$.
If you sell a 100$ future you can
loose much more than the initial
investment.
Lehman Brother’s
Lehman Brothers‟ leverage before
their bankruptcy was 25 times.
It means that the company could in
theory loose 25 times their capital.
You know the end of the story!
Complexity
Without entering into details an Asset
Backed Security (ABS) is a packaged
of debt assets.
Each asset element is complex to
evaluate on its own.
Some time one element of an ABS is
an other ABS.
Example: hedge risks
An airline company need to buy fuel
for their jets. The company knows
how much it needs for the next
year.
Buying forward contracts on fuel
can hedge them against the price of
oil.
Example:
Speculate by Using Leverage
An investor believed that the price of oil is
going to go up in a month time, but do not
want to buy a lot of oil today and keep it for a
month.
This investor can buy today a call option on oil
with maturity one month. The price of the
option is a small fraction of the underlying
price.
What are the possibilities?
Example:
change the nature of a liability
A company has a debt which is on
its balance sheet.
This company can package this
debt with an asset, let say a factory
and sell it to an investor.
Example:
Portfolio Management
A company has too much
exposure on the automobile
industry and wish to reduce it.
Selling the portfolio and buying a
new one would be extremely costly
(broker fees, bid/offer).
Selling an index on automobile
industry is a much cheaper option.
Mathematics
Complexity of cash flows requires
elaborate mathematics.
You cannot predict the future, you can
just calculate the probabilities of what
is likely to happen.
Mathematics are at the core of modern
finance, whether in academia or in the
financial industry.
Basics
I assume that you all know:
Sets and Numbers.
Functions theory
Exponential logarithm functions
Limit of functions
Basic probability
If you do not have those basics
you should either spend a lot of time catching up,
or you should not take this course!
Differentiation
Definition of a differentiation
It is useful to describe trend
Integration
Definition of integration
Basis of probability theory
Differential Equations
Finance is based on differential
equations of the form:
A solution is a function f that satisfies this
equation
Probability
Bayesian probability is a measure of the
likelihood of an event compared with all
possible states of the world.
For example the likelihood that a dice
rolls on the number 6 is:
Expectation
Expectation is a way of deciding what is the
probability of gain.
For example a coin toss were you win 1 if it is
head and -1 if it is tail is:
So if the coin is fair then the expectation of gain is
0.
If the coin is not fair and there is a probability of
head is 55% then the expectation of gain is 0,55
Normal Density Function
Fundamental tool for derivatives
The probability density function of a
continuous random variable X that is
normally distributed with mean μ and
variance σ2 is:
Markov Processes
The past “path” followed by the variable is
independent of the probability distribution of the
value in the future.
Only the current value of the variable is relevant
for predicting the future.
That is why non-Markov processes are known as
path dependent.
Modeling financial prices as Markov stochastic
processes implies that we cannot tell anything
about the asset's price tomorrow by using its
price history up to today.
Markov stochastic processes are thus consistent
with the weak form of the market efficiency
hypothesis.
Wiener Processes
A Wiener process (W) is a particular type of
Markov process that has the following
properties:
is almost surely continuous
has independent increments with
distribution
Stochastic Processes
A stochastic process, also called random
process, with state space X is a collection of
X-valued random variables indexed by a set T
(time).
Stochastic processes can also be categorized
as discrete-variable or continuous-variable:
a continuous-variable process: the variable can take
any value within a certain range
a discrete-variable stochastic process: only certain
values are possible
Martingale
A discrete-time martingale is a discrete-
time stochastic process (i.e., a sequence of
random variables) X1, X2, X3, ... that
satisfies for all n:
This can be interpreted as: The best
approximation of the future value of a
random variable is the current value.
Option Equation
We note the value at maturity of
the underling and K the strike
The cash flow at maturity for a call
option is:
Interest Calculation
You want to invest an amount A for T
years at an interest rate of r.
Simple Interest Calculation
Compound Interest Calculation
Modeling Asset Prices
No one knows the future, but it
is possible to calculate the
expected probability of gain.
Modeling Asset Prices
What is the difference between a
bookmaker and a derivatives
trader?