EC3070 FINANCIAL DERIVATIVES
SPECULATION
Speculation Speculation entails running the risk of a loss in the expectation of
a high reward. A pure investment is devoid of risk and, therefore, of any speculative element. Financial speculation involves the buy
EC3070 FINANCIAL DERIVATIVES
CONTINUOUS-TIME STOCHASTIC PROCESSES
Discrete-Time Random Walk The concept of a Wiener process is an extrapolation of that of a discrete-time random walk. A standardised random
walk is a process that is dened over the set of i
EC3070 FINANCIAL DERIVATIVES
PRESENT VALUES
The Initial Value of a Forward Contract. One of the parties to a forward
contract assumes a long position and agrees to buy the underlying asset at a
certain price on a certain specied future date denoted t = .
EC3070 FINANCIAL DERIVATIVES
OPTIONS
Options consist of rights to buy or rights to sell, which can be exercised or
foregone at the discretion of the holder. The right to sell is called a put option
and the right to buy is called a cal l option. In either
EC3070 FINANCIAL DERIVATIVES
OPTIMAL HEDGE RATIO
If an asset has been hedged, then the movements in its spot price and in the
accompanying short hedge should constitute compensating variations. The
optimum size of the hedge will be a function of the varia
EC3070 FINANCIAL DERIVATIVES
The Mean Value Theorem
Rolles Theorem. If f (x) is continuous in the closed interval [a, b] and
dierentiable in the open interval (a, b), and if f (a) = f (b) = 0, then there
exists a number c (a, b) such that f (c) = 0.
When
EC3070 FINANCIAL DERIVATIVES
MARKING TO MARKET
Imagine a futures contract that has been written at time t = 0, which obliges
a party to supply an asset or a commodity at time for a settlement price
of F |0 . This party, which takes the short position, mig
EC3070 FINANCIAL DERIVATIVES
SPECULATION AND CRISES
Speculation Speculation entails running the risk of a loss in the expectation of a high reward. Financial speculation involves the buying, holding,
selling, and short-selling of stocks, bonds, commoditie
EC3070 FINANCIAL DERIVATIVES
BINOMIAL OPTION PRICING MODEL
A One-Step Binomial Model Suppose that a portfolio consists of N
units of stock as assets, with a spot price of S0 per unit, together with a
liability of one call option. The initial value of the
EC3070 FINANCIAL DERIVATIVES
CONTINUOUS-TIME STOCHASTIC PROCESSES
Discrete-Time Random Walk . A standardised random walk dened
over the set of integers cfw_t = 0, 1, 2, . . ., is a sequence by w(t) =
cfw_wt ; t = 0, 1, 2, . . . in which
wt+1 = wt + t+1 ,
EC3070 FINANCIAL DERIVATIVES
FUTURES: MARKING TO MARKET
The holder of a futures contract will be required to deposit with the brokers
a sum of money described as the margin, which will be calculated at a
percentage of the current spot price S0 of the asse
EC3070 FINANCIAL DERIVATIVES
ITOS LEMMA
Preliminaries Itos lemma enables us to deduce the properties of a wide variety of continuous-time processes that are driven by a standard Wiener process
w(t).
We may begin an account of the lemma by summarising the
EC3070 FINANCIAL DERIVATIVES
COMPOUND INTEREST
Investments
Imagine that a sum of y0 = 100 is invested at an annual rate of interest
of r = 5% per annum. After one year has elapsed, the sum will have grown to
y0 (1 + r) = 105; and the opportunity will aris
EC3070 FINANCIAL DERIVATIVES
HEDGING VIA FORWARD CONTRACTS
Example 1. Hedging a Long Forward Contract. Acme Metals buys 10 Comex
gold contracts at 100 ounces each for June delivery at 11.a.m Monday at a
futures prices of F |t = $400 per ounce. They do thi
EC3070 FINANCIAL DERIVATIVES
GLOSSARY
Ask price The bid price.
Arbitrage An arbitrage is a nancial strategy yielding a riskless prot and
requiring no investment. It commonly amounts to the successive purchase and
sale, or vice versa, of an asset at dierin
EC3070 FINANCIAL DERIVATIVES
FORWARD CONTRACTS
In a forward contract, a party agrees to buy or sell an asset at a given price
at a future date . The party that agrees to buy the asset, is taking a long
position. The party that is selling is taking a short
EC3070FINANCIALDERIVATIVES
Exercise3TradingStrategies
Table1below(Hull:Table1.2,p.7)containsinformationonthe
pricesdifferentcallandputoptionsonanIntelstockwiththesame
expirationdate(October)hadonthe12thofSeptember2006:
Table1
Strike
Price
Oct
Call
Oct
Put
EC3070 FINANCIAL DERIVATIVES
Exercise 2
1. A stock price is currently S0 = 40. At the end of the month, it will be
u
d
either S1 = 42 or S1 = 38. The risk-free rate of continuously compounded
interest is 8% per annum. What is the value c1|0 of a one-month
EC3070 FINANCIAL DERIVATIVES
Exercise 1
1. A credit card company charges an annual interest rate of 15%, which is
eective only if the interest on the outstanding debts is paid in monthly
instalments. Otherwise, the interest charges are compounded with the
EC3070 FINANCIAL DERIVATIVES
PUTCALL PARITY
Upper Bounds Let p |0 be the current price at time t = 0 of a put option
eective at time , and let c |0 be the current price of a corresponding call
option. Also, let K |0 be the strike price of the options at t
EC3070 FINANCIAL DERIVATIVES
BLACKSCHOLES OPTION PRICING
The Dierential Equation The Black-Scholes model of option pricing assumes that the price St of the underlying asset has a geometric Brownian
motion, which is to say that
dS = S dt + S dw,
(1)
where
EC3070 FINANCIAL DERIVATIVES
THE BINOMIAL THEOREM
Pascals Triangle and the Binomial Expansion. To prove the binomial
theorem, it is necessary to invoke some fundamental principles of combinatorial calculus. We shall develop the necessary results after der
EC3070 FINANCIAL DERIVATIVES
BINOMIAL OPTION PRICING MODEL
A One-Step Binomial Model The Binomial Option Pricing Model is a simple device that is used for determining the price c |0 that should be attributed
initially to a call option that gives the right
EC3070 FINANCIAL DERIVATIVES
BIBLIOGRAPHY
Textbooks
Chance, D.M. and R. Brooks, (2008), An Introduction to Derivatives and Risk
Management: Seventh Edition, Thompson South-Western Publishers.
Hull, J., (2006), Options, Futures and Other Derivatives, Prent
EC3070 FINANCIAL DERIVATIVES
Yesterday, upon the stair,
I saw a man who wasnt there.
He wasnt there again today.
I wish, I wish, hed go away.
Antigonish (1899), by William Hughes Mearns (18751965)
They sought it with thimbles, they sought it with care;
Th
EC3070 FINANCIAL DERIVATIVES
Exercise 2
u
1. A stock price is currently S0 = 40. At the end of the month, it will be either S1 = 42
d
or S1 = 38. The risk-free rate of continuously compounded interest is 8% per annum.
What is the value c1|0 of a one-month
EC3070 FINANCIAL DERIVATIVES
Exercise 1
1. A credit card company charges an annual interest rate of 15%, which is eective only
if the interest on the outstanding debts is paid in monthly instalments. Otherwise, the
interest charges are compounded with the
Hull: Chapter 10
Options, Futures, and Other Derivatives, 7th
1
Take
a position in the option and the
underlying.
Take
a position in 2 or more options of
the same type (A spread) .
Combination: Take
a position in a
mixture of calls & puts (A combinatio
Introduction
Options, Futures, and Other Derivatives, 7th
Edition, Copyright John C. Hull 2008
1
Derivative: a
financial instrument whose
value depends (or derives from) the
values of other, more basic, underlying
values (Hull, p. 1).
2
To hedge risks.
T
EC3070 FINANCIAL DERIATIVES
TAYLORS THEOREM AND SERIES EXPANSIONS
Taylors Theorem. If f is a function continuous and n times dierentiable in
an interval [x, x + h], then there exists some point in this interval, denoted by
x + h for some [0, 1], such that