Discrete Financial Mathematics Wim Schoutens

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Discrete Financial Mathematics
Wim Schoutens
Leuven, 2004-2005
Lecture Notes to the Course
(G0Q20a) Discrete Financial Mathematics
Abstract
The aim of the course is to give a rigorous yet accessible introduction to the
modern theory of discrete financial mathematics. The student should already
be comfortable with calculus and probability theory. Prior knowledge of basic
notions of finance is useful.
We start with providing some background on the financial markets and the
instruments traded. We will look at different kinds of derivative securities, the
main group of underlying assets, the markets where derivative securities are
traded and the financial agents involved in these activities. The fundamen-
tal problem in the mathematics of financial derivatives is that of pricing and
hedging. The pricing is based on the no-arbitrage assumptions. We start by dis-
cussing option pricing in the simplest idealised case: the Single-Period Market.
Next, we turn to Binomial tree models. Under these models we price European
and American options and discuss pricing methods for the more involved exotic
options. Monte-Carlo issues come into play here.
Finally, we set up general discrete-time models and look in detail at the
mathematical counterpart of the economic principle of no-arbitrage: the exis-
tence of equivalent martingale measures. We look when the models are complete,
i.e. claims can be hedged perfectly. We discuss the Fundamental theorem of
asset pricing in a discrete setting.
To conclude the course, we make a bridge to continuous-time models. We
look at them as limiting cases of discrete models. The discrete models will guide
us in the analysis of continuous-time models in the Continuous Mathematical
Finance Course.
2
Contents
1 Derivative Background 1
1.1 Financial Markets and Instruments . . . . . . . . . . . . . . . . . 1
1.1.1 Basic Instruments . . . . . . . . . . . . . . . . . . . . . . 1
1.1.2 The Bank Account . . . . . . . . . . . . . . . . . . . . . . 6
1.1.3 Derivative Instruments . . . . . . . . . . . . . . . . . . . . 9
1.1.4 Markets . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.1.5 Contract Specifications . . . . . . . . . . . . . . . . . . . 14
1.1.6 Types of Traders . . . . . . . . . . . . . . . . . . . . . . . 15
1.1.7 Modelling Assumptions . . . . . . . . . . . . . . . . . . . 17
1.2 Arbitrage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.3 Arbitrage Relationships . . . . . . . . . . . . . . . . . . . . . . . 23
1.3.1 The Put-Call Parity . . . . . . . . . . . . . . . . . . . . . 23
1.3.2 The Forward Contract . . . . . . . . . . . . . . . . . . . . 26
1.3.3 Dividends . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
1.3.4 Currencies . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
1
1.3.5 Commodities . . . . . . . . . . . . . . . . . . . . . . . . . 32
1.3.6 The Cost of Carry . . . . . . . . . . . . . . . . . . . . . . 32
2 Binomial Trees 34
2.1 Single Period Market Models . . . . . . . . . . . . . . . . . . . . 34
2.2 Two-Step Binomial Trees . . . . . . . . . . . . . . . . . . . . . . 43
2.2.1 European Call . . . . . . . . . . . . . . . . . . . . . . . . 43
2.2.2 Matching Volatility with u and d . . . . . . . . . . . . . . 46
2.3 Binomial Trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
2.3.1 European Call and Put Options . . . . . . . . . . . . . . 49
2.3.2 American Options . . . . . . . . . . . . . . . . . . . . . . 53
2.4 Moving towards The Black-Scholes Model . . . . . . . . . . . . . 59
3 Mathematical Finance in Discrete Time 62
3.1 Information and Trading Strategies . . . . . . . . . . . . . . . . . 63
3.2 No-Arbitrage Condition . . . . . . . . . . . . . . . . . . . . . . . 67
3.3 Risk-Neutral Pricing . . . . . . . . . . . . . . . . . . . . . . . . . 70
3.4 Complete Markets . . . . . . . . . . . . . . . . . . . . . . . . . . 74
3.5 The Fundamental Theorem of Asset Pricing . . . . . . . . . . . . 75
3.5.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
4 Exotic Options 82
4.1 Monte Carlo Pricing . . . . . . . . . . . . . . . . . . . . . . . . . 83
4.2 Lookback Options . . . . . . . . . . . . . . . . . . . . . . . . . . 83
4.3 Barrier Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
2
4.4 Asian Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
5 The Black-Scholes Option Price Model 100
5.1 Continuous-Time Stochastic Processes . . . . . . . . . . . . . . . 101
5.1.1 Information and Filtration . . . . . . . . . . . . . . . . . . 101
5.1.2 Martingales . . . . . . . . . . . . . . . . . . . . . . . . . . 102
5.2 Brownian Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
5.3 Itoˆ’s Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
5.3.1 Stochastic Integrals . . . . . . . . . . . . . . . . . . . . . 107
5.3.2 Itoˆ’s Lemma . . . . . . . . . . . . . . . . . . . . . . . . . 109
5.4 Stochastic Differential Equations . . . . . . . . . . . . . . . . . . 110
5.5 Geometric Brownian Motion . . . . . . . . . . . . . . . . . . . . . 112
5.6 The Market Model . . . . . . . . . . . . . . . . . . . . . . . . . . 114
5.7 Equivalent Martingale Measures and Risk-Neutral Pricing . . . . 117
5.7.1 The Pricing of Options under the Black-Scholes Model . . 120
5.7.2 Hedging . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
5.8 The Greeks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
5.9 Drawbacks of the Black-Scholes Model . . . . . . . . . . . . . . . 129
6 Miscellaneous 132
6.1 Decomposing Options into Vanilla Position . . . . . . . . . . . . 132
6.2 Variance Swap . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
3
Chapter 1
Derivative Background
1.1 Financial Markets and Instruments
A market typically consist out of a riskfree Bank account and some other risky
assets. On these basic instruments other financial contracts are written; these
financial contracts are so-called derivative securities. This text is on the (risk-
neutral) pricing of derivative securities.
This section provides the institutional background on the main group of
underlying assets, the related derivative securities, the markets where derivatives
securities are traded and the financial agents involved in these activities.
1.1.1 Basic Instruments
Next, we highlight some of the most common underlying securities.
1
Stocks – Equity
The basis of modern economic life are companies owned by their shareholders;
the shares provide partial ownership of the company, pro rata with investment.
Shares have value, reflecting both the value of the company’s real assets and
the earning power of the company’s dividends. With publicly quoted companies,
shares are quoted and traded on the Stock Exchange. Stock is the generic term
for assets held in the form of shares. Stock markets date back to at least 1531,
when one was started in Antwerp, Belgium. Today there are over 150 stock
exchanges.
Interest Rates – Fixed-Income
The value of some financial assets depends solely on the level of interest rates,
e.g. Treasury notes, municipal and corporate bonds. These are fixed-income
securities by which national, state and local governments and large companies
partially finance their economic activity. Fixed-income securities require the
payment of interest in the form of a fixed amount of money at predetermined
points in time, as well as the repayment of the principal at maturity of the secu-
rity. Interest rates themselves are notional assets, which cannot be delivered. A
special fixed-income product is the bank account, which we typically assume to
be riskfree. We go into more detailon possible bank account models in Section
1.1.2.
2
Currencies – Foreign Exchange
A currency is the denomination of the national units of payment (money) and
as such is a financial asset. Companies may wish to hedge adverse movements
of foreign currencies and in doing so use derivative instruments.
A foreign currency has the property that the holder of the currency can earn
interest at the risk-free interest rate prevailing in the foreign country. We thus
have to kind of interest rates, the domestic and the foreign interest rate.
Commodities
Commodities are a kind of physical products like gold, oil, cattle, fruit juice.
Trade in these assets can be for different purposes: for using them in the pro-
duction process or for speculation. Derivative instruments on these asset can
be used for hedging and speculation. Special care has to be taken with com-
modities because of storage costs (see Section 1.3.5) In Figure 1.1, one sees the
some prices (of the market on the 16th of October 2004) of (futures on) energy,
metal, livestock and other commodities.
3
Figure 1.1: Commodities future prices on 16-10-2004
Miscellaneous
Indexes
An index tracks the value of a basket of stocks (FTSE100, S&P500, Dow
Jones Industrial, NASDAQ Composite, BEL20, EUROSTOXX50, ...), bonds,
and so on. Derivative instruments on indices may be used for hedging if no
4
derivative instruments on a particular asset in question are available and if the
correlation in movement between the index and the asset is significant. Further-
more, institutional funds (such as pension funds), which manage large diversi-
fied stock portfolios, try to mimic particular stock indices and use derivative on
stock indices as a portfolio management tool. On the other hand, a speculator
may wish to bet on a certain overall development in a market without exposing
him/herself to a particular asset.
In Figure 1.2, one sees the Belgian Bel-20 Index over a period of more than
4 years.
Figure 1.2: BEL-20
A new kind of index was generated with the Index of Catastrophe Losses
(CAT-index) by the Chicago Board of Trade (CBOT) lately. The growing num-
ber of huge natural disasters (such as hurricanes and earthquakes) has led the
insurance industry to try to find new ways of increasing its capacity to carry
risks. Currently investors are offered options on the CAT-index, thereby taking
5
in effect the position of traditional reinsurance.
Credit Risk Market
Credit risk captures the risk on a financial loss that an institution incurs
when it lends money to another institution or person. This financial loss real-
izes whenever the borrower does not meet all of its obligations under its borrow-
ing contract. Because credit risk is so important for financial institutions the
banking world has developed instruments that allows them to evacuate credit
risk rather easily. The most commonly known and used example is the credit
default swap. These instruments can best be considered as tradable insurance
contracts. This means that today I can buy protection on a bond and tomorrow
I can sell that same protection as easily as I bought it. Credit default swaps
work just as an insurance contract. The protection buyer (the insurance taker)
pays a fee and in exchange he gets reimbursed his losses if the company on which
he bought protection defaults.
In Figure 1.3, one sees credit default swap bid and offer rates for major
aerospace/transport and auto (parts) companies.
1.1.2 The Bank Account
In the next Chapter we will start building models for the stock or asset price
process. Here we focus on one instrument which we typically assume to be
available in all the later on encountered market models: the bank account. There
are two (quite related) regimes under which we will work: discrete or continuous
compounding. This has all to do with when and how frequently the interest
6
Figure 1.3: Credit Default Swap rates
gained on the invested money is paid out. Typically the discrete compounding
will be only used in discrete time models; the continuous compound can be used
in almost any situation.
Consider an amount A is invested for n years at an interest rate of R per
annum. If the rate is compounded once per annum, the terminal value of the
investment is
A(1 +R)n.
If it is compounded m times per annum, the terminal value of the investment is
A
(
1 +
R
m
)mn
.
We note that there is a difference. Indeed, take A = 100 euro and n = 1, when
7
the interest rate is 10 percent a year. The first regime leads to 110 euros after
one year. However, with quarterly payments (m = 4), i.e. with payments every
three month, we have after one year 100× (1.0025)4 = 110.38 euros. In Figure
1.4 one sees the effect of increasing the compounding frequency from a yearly
compounding to a daily compounding.
0 50 100 150 200 250 300 350
110
110.1
110.2
110.3
110.4
110.5
110.6
discrete compound interest rates (A=100, n=1, r=0.10)
compounding frequency (m)
en
d v
alu
e
Figure 1.4: Discrete compounding
The limit as m tends to infinity is known as continuous compounding. We
have
lim
m→∞
A
(
1 +
R
m
)mn
= A exp(nR).
With such a continuous compounding the invested amountA grows toA exp(nR)
after n years. Note that it is because of the above discussion, important to state
the units/frequency in which the interest rate is measured/compounded. For
example, an interest rate of 10 percent continuous compounding is the same as
8
(1− exp(0.10))/100 = 10.517 percent annual compounding.
Throughout the text we will make use of a bank account on which we can
put money and borrow money on a fixed continuously compounded interest rate
r. This means that 1 euro on the bank account at time 0 will give rise to ert
euro on time t > 0. Similarly, if we borrow 1 euro now, we have to pay back ert
euro at time t > 0. Or equivalently, if we borrow now e−rt euro we have to pay
back 1 euro at time t.
One euro on the bank account will grow over time; at some time t we denote
its value by B(t). Note thus that we set B(0) = 1. We call B = {B(t), t ≥ 0}
the bank account price process or bond price process.
Related to all this is the time value of money. An investor will prefer 100 euro
in his pocket today to 100 euro in his pocket one year from now. The interest
paid on the riskless bank account expresses this. Using continuous compounding
with a rate r = 0.10, 100 euro is equivalent with 110.517 euros in one year. If
we receive a cash-flow X at some future time T , the equivalent now is equal to
exp(−rT )X . 100 euro in one year is equivalent with 90.484 euros now. This
procedure is called discounting and exp(−rT ) is the discounting factor.
1.1.3 Derivative Instruments
In practitioner’s terms a ’derivative security’ is a security whose value depends
on the value of other more basic underlying securities. We adopt the more
precise definition:
A derivative security, or contingent claim, is a financial contract whose value
9
at expiration date T (more briefly, expiry) is determined exactly by the price
process of the underlying financial assets (or instruments) up to time T.
Derivative securities can be grouped under three general headings: Options,
Forwards and Futures, and Swaps. During this text we will mainly deal with
options although our pricing techniques may be readily applied to forwards,
futures and swaps as well.
Options
An option is a financial instrument giving one the right but not the oblig-
ation to make a specified transaction at (or by) a specified date at a specified
price.
A lot of different type of options exists. We give here the basic types. Call
options giveone the right to buy. Put options give one the right to sell. European
options give one the right to buy/sell on the specified date, the expiry date,
on when the option expires or matures. American options give one the right
to buy/sell at any time prior to or at expiry. Asian options depend on the
average price over a period. Lookback options depend on the maximum or
minimum price over a period and barrier options, depend on some price level
being attained or not.
The price at which the transaction to buy/sell the underlying assets (or
simply the underlying), on/by the expiry date (if exercised), is made is called
the exercise price or strike price. We usually use K for strike price, time t = 0,
for the initial time (when the contract between the buyer and the seller of the
10
option is struck), time t = T for the expiry or final time.
Consider, say, an European call option, with strike price K; write St for the
value (or price) of the underlying at time t. If St > K, the option is in the
money, if St = K, the option is said to be at the money and if St < K, the
option is out the money. This terminology is of course motivated by the payoff,
the value of the option at maturity, from the option which is
ST −K if ST > K and 0 otherwise
(more briefly written as (ST − K)+). This payoff function for K = 100 is
visualized in Figure 1.5
80 85 90 95 100 105 110 115 120
0
2
4
6
8
10
12
14
16
18
20
Payoff of European Call (K=100)
stock price at maturity
Pa
yo
ff
Figure 1.5: Payoff of Call Option (K=100)
There are two sides to every option contract. On one side there is the person
who has bought the option (the long position); on the other side you have the
11
person who has sold or written the option (the short poistion). The writer
receives cash up front but has potential liabilities later.
In Figure 1.6, one can see that by investing in an option one can make huge
gains, but also if markets goes the opposite direction as anticipate, it is possible
to loose all money one has invested.
Figure 1.6: Stock Prices and European Call Option at time t = 0 and t = T .
In 1973, the Chicago Board Options Exchange (CBOE) began trading in
options on some stocks. Since then, the growth of options has been explosive.
Risk Magazine (12/1997) estimated $35 trillion as the gross figure for worldwide
derivatives markets in 1996.
In Figure 1.7, one sees some of the prices of call options written on the SP500-
index. The main aim of this text is to give a basic introduction to models for
determining these kind of option prices.
12
Forwards, Futures
A forward contract is an agreement to buy or sell an asset at a certain future
date T for a certain price K. It is usually between two large and sophisticated
financial agents (banks, institutional investors, large corporations, and broker-
age firms) and not traded on an exchange. The agents who agrees to buy the
underlying asset is said to have a long position, the other agent assumes a short
position. The payoff from a long position in a forward contract on one unit of
an asset with price ST at the maturity time T of the contract is
ST −K.
Compared with a call option with the same maturity and strike price K we see
that the investor now faces a downside risk, too. He has the obligation to buy
the asset for price K.
A futures contract, like a forward contract, is an agreement to buy or sell an
asset at a certain future date for a certain price. The difference is that futures
are traded. As such, the default risk is removed from the parties to the contract
and borne by the clearing house.
Swaps
A swap is an agreement whereby two parties undertake to exchange, at known
dates in the future, various financial assets (or cash flows) according to a pre-
arranged formula that depends on the value of one or more underlying assets.
Examples are currency swaps (exchange currencies) and interest-rate swaps (ex-
13
change of fixed for floating set of interest payments) and the nowadays popular
credit default swaps as in Figure 1.3.
1.1.4 Markets
Financial derivatives are basically traded in two ways: on organized exchanges
and over-the-counter (OTC). Organized exchanges are subject to regulatory
rules, require a certain degree of standardization of the traded instruments
(strike price, maturity dates, size of contract, etc.). Examples are the Chicago
Board Options Exchange (CBOE), the London International Financial Futures
Exchange (LIFFE).
The exchange clearinghouse is an adjunct of the exchange and acts as an
intermediary in the transactions. It garantuess the performanjce of the parties
to each transaction. Its main task is to keep track of all the transactions that
take place during a day so it can calculate the net poistions of each of its
members.
OTC trading takes between various commercial and investments banks such
as Goldman Sachs, Citibank, Deutsche Bank.
1.1.5 Contract Specifications
It is very important that the financial contract specifies in detail the exact nature
of the agreement between the two parties. It must specify the contract size (how
much of the asset will be delivered under one contract), where delivery will be
made, when exactly the delivery is made, etc. When the contract is traded at
14
an exchange, it should be made clear how prices will be quoted, when trade is
allowed, etc.
Financial assets in derivatives are generally well defined and unambiguous,
e.g. it is clear what a Japanese Yen is. When the asset is a commodity, there
may be quite a variation in the quality and it is important that the exchange
stipulates the grade or grades of the commodity that are acceptable.
Most contracts are refered to by its delivery month and year. The contract
must specify in detail the period of that month when delivery can be made. For
some future the delivery period is the entire month. For other contract delivery
must be at a special day, hour, etc.
Some contracts are in terms of a so-called settlement price. For example
derivatives on indices (like the SP-500). The settlement price is calculated by
the exchange by a very detailed algorithm. It can be e.g. averages of the index
taken every five minutes during one hour, but also just the closing price of the
asset.
Other specification by the exchange deal with movement limits. Trade will be
halted if these limits are exceeded. The purpose of price limits is to prevent large
movements from occurring because of speculative excesses, extremal situation
(11th of September), ...
1.1.6 Types of Traders
We can classify the traders of derivatives securities in three different classes.
15
Hedgers
Successful companies concentrate on economic activities in which they to best.
They use the market to insure themselves against adverse movements of prices,
currencies, interest rates etc. Hedging is an attempt to reduce exposure to risk.
Hedgers prefer to forgo the chance to make exceptional profits when future un-
certainty works to their advantage by protecting themselves against exceptional
loss.
Speculators
Speculators want to take a position in the market – they take the opposite
position to hedgers. Indeed, speculation is needed to make hedging possible, in
that a hedger, wishing to lay off risk, cannot do so unless someone is willing to
take it on.
In speculation, available funds are invested opportunistically in the hope of
making a profit: the underlying itself is irrelevant to the speculator, who is only
interested in the potential for possible profit that trade involving it may present.
Arbitrageurs
Arbitrageurs try to lock in riskless profit by simultaneously entering into trans-
actions in two or moremarkets. An arbitrage opportunity exists, for example, if
a security can be bought in New York at one price and sold at a slightly higher
price in London. The underlying concept of the here presented theory is the
absence of arbitrage opportunities.
16
1.1.7 Modelling Assumptions
We will discuss contingent claim pricing in an idealized case. We will not allow
market frictions; there is no default risk, agents are rational and there is no
arbitrage. More concrete this means
• no transaction costs
• no bid/ask spread
• no taxes
• no margin requirements
• no restrictions on short sales
• no transaction delays
• same interest for borrowing and lending
• market participants act as price takers
• market participants prefer more to less
We develop the theory of an ideal – frictionless – market so as to focus
irreducible essentials of the theory and as a first-order approximation to real-
ity. Understanding frictionless markets is also a necessary step to understand
markets with frictions.
The risk of failure of a company – bankruptcy – is inescapably present in
its economic activity: death is part of life. Moreover those risks also appear
at the national level: quite apart from war, recent decades have seen default
17
of interest payments of international debt, or the threat of it (see for example
the 1998 Russian crisis). We ignore default risk for simplicity while developing
understanding of the principal aspects.
We assume financial agents to be price takers, not price makers. This implies
that even large amounts of trading in a security by one agent does not influence
the security’s price. Hence agents can buy or sell as much of any security as
they wish without changing the security’s price.
To assume that market participants prefer more to less is a very weak as-
sumption on the preferences of market participants. Apart from this we will
develop a preference-free theory.
The relaxation of all these assumptions is subject to ongoing research.
We want to mention the special character of the no-arbitrage assumption.
It is the basis for the arbitrage pricing technique that we shall develop, and we
discuss it in more detail below.
1.2 Arbitrage
The essence of arbitrage is that it should not be possible to guarantee a profit
without exposure to risk. Were it possible to do so, arbitrageurs would do so,
in unlimited quantity, using the market as a money-pump to extract arbitrarily
large quantities of riskless profit. This would, for instance, make it impossible
for the market to be in equilibrium. We shall see that arbitrage arguments
suffice to determine prices - the arbitrage pricing technique.
18
To explain the fundamental arguments of the arbitrage pricing technique we
use the following:
Example: Consider an investor who acts in a market in which only three
financial assets are traded: (riskless) bonds B (bank account), stocks S and
European Call options C with strike K = 100 on the stock S. The investor may
invest today, time t = 0, in all three assets, leave his investment until time
t = T and gets his returns back then. We assume the option C expires at time
t = T . We assume the current prices (in euro, say) of the financial assets are
given by
B(0) = 1, S(0) = 100, C(0) = 20
and that at t = T there can be only two states of the world: an up-state with
euro prices
B(T, u) = 1.25, S(T, u) = 175, and therefore C(T, u) = 75,
and a down-state with euro prices
B(T, d) = 1.25, S(T, d) = 75, and therefore C(T, d) = 0.
Now our investor has a starting capital of 25000 euro from which he buy the
following portfolio,
Portfolio I:
Asset Number Total amount in euro
Bond 10000 10000
Stock 100 10000
Call option 250 5000
19
Depending of the state of the world at time t = T the value of his portfolio
will differ: In the up state the total value of his portfolio is 48750 euro:
Asset Number × Price Total amount in euro
Bond 10000 × 1.25 12500
Stock 100 × 175 17500
Call option 250 × 75 18750
TOTAL 48750
whether in the down-state his portfolio has a value of 20000 euro:
Asset Number × Price Total amount in euro
Bond 10000 × 1.25 12500
Stock 100 × 75 7500
Call option 250 × 0 0
TOTAL 20000
Can the investor do better ? Let us consider the restructured portfolio with
initial investment of 24600 euro:
Portfolio II:
Asset Number Total amount in euro
Bond 11800 11800
Stock 70 7000
Call option 290 5800
We compute its return in the different possible states. In the up-state the total
20
value of his portfolio is again 48750 euro:
Asset Number × Price Total amount in euro
Bond 11800 × 1.25 14750
Stock 70 × 175 12250
Call option 290 × 75 21750
TOTAL 48750
and in the down-state his portfolio has again a value of 20000 euro:
Asset Number × Price Total amount in euro
Bond 11800 × 1.25 14750
Stock 70 × 75 5250
Call option 290 × 0 0
TOTAL 20000
We see that this portfolio generates the same time t = T return while costing
only 24600 euro now, a saving of 400 euro against the first portfolio. So the
investor should use the second portfolio and have a free lunch today!
In the above example the investor was able to restructure his portfolio, re-
ducing the current (t = 0) expenses without changing the return at the future
date t = T in both possible states of the world. So there is an arbitrage pos-
sibility in the above market situation, and the prices quoted are not arbitrage
prices. If we regard (as we shall do) the prices of the bond and the stock
(our underlying) as given, the option must be mispriced. Let us have a closer
look between the differences between Portfolio II, consisting of 11800 bonds, 70
stocks and 29 call options, in short (11800, 70, 290) and Portfolio I, of the form
21
(10000, 100, 250) The difference is the portfolio, Portfolio III say, of the form
(11800, 70, 290)− (10000, 100, 250) = (1800,−30, 40).
Asset Number Total amount in euro
Bond 1800 1800
Stock -30 -3000
Call option 40 800
So if you sell short 30 stocks, you will receive 3000 euro from which you
buy 40 options, put 1800 euro in your bank account and have a gastronomic
lunch of 400 euro. But what is the effect of doing that ? Let us consider the
consequences in the possible states of the world. We see in both cases that
the effects of the different positions of Portfolio III offset themselves: In the
up-state:
Asset Number × Price Total amount in euro
Bond 1800 × 1.25 2250
Stock -30 × 175 -5250
Call option 40 × 75 3000
TOTAL 0
In the down state:
Asset Number × Price Total amount in euro
Bond 1800 × 1.25 2250
Stock -30 × 75 -2250
Call option 40 × 75 0
TOTAL 0
22
But clearly the portfolio generates an income at t = 0 of which you had a free
lunch, and a good one. Therefore it is itself an arbitrage opportunity.
If we only look at the position in bonds and stocks, we can say that this
position covers us against possible price movements of the option, i.e. having
1800 euro in your bank account and being 30 stocks short has the opposite time
t = T value as owning 40 call options. We say that the bond/stock position is
a hedge against the position in options.
Let us emphasize that the above arguments were independent of the prefer-
ences and plans of the investor.
1.3 Arbitrage Relationships
We will in this section use arbitrage-based arguments to develop general bounds
on the value of options. In our analysis here we use non-dividend paying stocks
as the underlying, with price process S = {St, t ≥ 0}. We assume we have a
risk-free bank account available which uses continuously compounding with a
fixed interest rate r.
1.3.1 The Put-Call Parity
Next, we will deduce afundamental relation between put and call options, the
so-called put-call parity. Suppose there is a stock (with value St at time t), with
European call and put options on it, with value Ct and Pt respectively at time
t, with expiry time T and strike-price K. Consider a portfolio consisting of one
stock, one put and a short position in one call (the holder of the portfolio has
23
written the call); write Πt for the time t value of this portfolio. So
Πt = St + Pt − Ct.
Recall that the payoffs at expiry are
for the call : CT = max{ST −K, 0} = (ST −K)+,
for the put : PT = max{K − ST , 0} = (K − ST )+.
For the above portfolio we hence get at time T the payoff
if ST ≥ K : ΠT = ST + 0− (St −K) = K,
if ST ≤ K : ΠT = ST + (K − St)− 0 = K.
This portfolio thus guarantees a payoff K at time T . How much is it worth
at time t? The riskless way to guarantee a payoff K at time T is to deposit
Ke−r(T−t) in the bank at time t and do nothing (we assume continuously com-
pounded interest here). Under the assumption that the market is arbitrage-free
the value of the portfolio at time t must thus be Ke−r(T−t), for it acts as a
synthetic bank account and any other price will offer arbitrage opportunities.
Let us explore these arbitrage opportunities.
If the portfolio is offered for sale at time t too cheaply–at price Πt <
Ke−r(T−t) – we can buy it, borrow Ke−r(T−t) from the bank, and pocket a
positive profit Ke−r(T−t)−Πt > 0. At time T our portfolio yields K, while our
bank debt has grown to K. We clear our cash account – use the one to pay off
the other – thus locking in our earlier profit, which is riskless.
If on the other hand the portfolio is priced at time t at a too high price –
at price Πt > Ke
−r(T−t) – we can do the exact opposite. We sell the portfolio
24
short – that is,we buy its negative: buy one call, write one put, sell short one
stock, for Πt and invest Ke
−r(T−t) in the bank account, pocketing a positive
profit Πt −Ke−r(T−t) > 0. At time T , our bank deposit has grown to K, and
again we clear our cash account – use this to meet our obligation K on the
portfolio we sold short, again locking in our earlier riskless profit.
We illustrate the above with so-called arbitrage tables. In such a table we
simply enter the current value of a given portfolio and then compute its value
in all possible states of the world when the portfolio is cashed in. In the case
Πt < Ke
−r(T−t):
Transactions Current cash flow Value at expiry
ST < K ST ≥ K
buy 1 stock −St ST ST
buy 1 put −Pt K − ST 0
write 1 call Ct 0 −ST +K
borrow Ke−r(T−t) −K −K
TOTAL
Ke−r(T−t) − St
−Pt + Ct > 0
0 0
Thus the rational price for the portfolio at time t is exactly Ke−r(T−t). Any
other price presents arbitrageurs with an arbitrage opportunity (to make and
lock in a riskless profit) – which they will take ! Therefore
Proposition 1 We have the following put-call parity between the prices of the
underlying asset and its European call and put options with the same strike price
25
and maturity on stocks that pay no dividends:
St + Pt − Ct = Ke−r(T−t).
The value of the portfolio above is the discounted value of the riskless equiv-
alent. This is a first glimpse at the central principle, or insight, of the entire
subject of option pricing. Arbitrage arguments allow one to calculate precisely
the rational price – or arbitrage price – of a portfolio. The put-call parity
argument above is the simplest example of the arbitrage pricing technique.
1.3.2 The Forward Contract
Next, we will deduce a fair price (based on the no-arbitrage assumption) for the
following forward contract: The contract states that party A (the buyer) must
buy from party B (the seller) the (non-dividend paying) stock at time T at the
price K (the strike price).
We claim that F = S0 − exp(−rT )K is the correct initial price of this
derivative which party A will pay to party B at time t = 0. Indeed, suppose
you are party B, so you sold the forward contract and has received at time
t = 0, the amount F . At time zero, you do the following:
• buy 1 stock for the price S0;
• borrow exp(−rT )K.
To buy the stock you need S0. You already have from the forward F = S0 −
exp(−rT )K and receives from your loan exp(−rT )K. So you spent all the
available money and at time t = 0 you have the following portfolio:
26
• long 1 stock.
• short 1 forward;
• short exp(−rT )K bonds.
Look what happens at time T . You must deliver the stock to party A. You
give away your stock in your portfolio, for this you receive K. The forward
contract ends and you pay back your bank. You have to pay back the amount
exp(rT ) exp(−rT )K = K. This you can do exactly with the money you received
from party A. In the end everything is settled, you have no gain, no lost. Note
that your initial investment is also zero.
Note that any other price for the forward would have led to an arbitrage
situation. Indeed, suppose you received Fˆ > F . Then by following the above
strategy you pocket at time t = 0 the difference Fˆ −F > 0 which you can freely
spent. At time T you just close all the position as described above. Without
any initial investment and risk, you have then spent at time 0, Fˆ −F > 0. This
is clearly an arbitrage opportunity (for party B). In case Fˆ < F , party A con
set up a portfolio will always leads to an arbitrage opportunity (check this !).
Forward/future contracts in practice are almost always struck at the price
K, such that F = S0 − exp(−rT )K = 0, i.e.
K = exp(rT )S0.
By doing this entering (in both ways: long or short) a forward contract is at
zero cost. For this reason, exp(rT )S0 is called the time T forward price of the
stock.
27
Finally, note that the put-call parity can be simply rewritten in terms of
the call price, the put price and the forward contract price as: C − P = F (all
derivatives have the same strike and time to maturity). From this one can see
that at the forward price of the stock, i.e. in case K = exp(rT )S0 and hence
F = 0, call and puts have the same value: C = P .
1.3.3 Dividends
Up to now, we have assumed that the risky asset pays no dividends, however
in reality stocks can pay some dividend to the stock holders at some moments.
We assume that the amount and timing of the dividends during the live of an
option can be predicted with certainty. Moreover, we will assume that the stock
pays a continuous compound dividend yield at rate q per annum. Other ways
of dividend payments can be considered and techniques are described in the
literature to deal with this.
Continuous payment of a dividend yield at rate q means that our stock is
following a process of the form:
St = exp(−qt)S¯t,
where S¯ is describing the stock prices behavior not taking into account divi-
dends. A stock which pays continuously dividends and an identical stock that
not pays dividends should provide the same overall return, i.e. dividends plus
capital gains. The payment of dividends causes the growth of the stock price to
be less than it would otherwise be by an amount q. In other words, if, with a
continuous dividend yield of q, the stock price grows from S0 to ST at time T ,
28
then in absence of dividends it would grow from S0 to exp(qt)ST . Alternatively,
in absence of dividends it would grow from exp(−qt)S0 to ST . This argumen-
tation brings us to the fact that we get the same probability distribution for
the stock price at time T in the following cases: (1) The stock starts at S0
and pays a continuous dividend yield at rate q and (2) the stock starts at price
exp(−qt)S0 and pays no dividend yield.
The put-call parity for a stock with dividend yield q can be obtained from the
put-call parity for non-dividend-payingstocks. With no dividends we obtained
:
St + Pt − Ct = K exp(−r(T − t)).
If we now take into account dividends, the change comes down to replacing St
with St exp(−q(T − t)). We have:
exp(−q(T − t))St + Pt − Ct = K exp(−r(T − t)).
This relation can be proven by considering the portfolio consisting of exp(−q(T−
t)) number of stocks, one put option and minus one call option. We reinvest
the dividends on the shares instantaneously in additional shares, i.e. at some
future time point t ≤ s ≤ T , we have exp(−q(T − s)) number of stock; at the
expiry date of the option we own one stock, one put and minus one call. The
value of the portfolio at that time thus always equals K. By the no-arbitrage
argument the time T value of the portfolio must equal K exp(−r(T − t)), the
time t-value of a future payment (at time T ) of K.
If our asset is an index, the dividend yield is the (weighted) average of the
29
dividends yields on the stocks composing the index.In practice, the dividend
yield can be determined from the forward price of the asset. It is the agreement
to buy or sell an asset at a certain future time for a certain price, the delivery
price. At the time the contract is entered into, the delivery price is chosen so
that the value of the forward is zero. This means that it costs nothing to buy or
sell the contract. For an asset paying a continuous yield at rate q, the delivery
price of a forward contract expiring at time T , is given by (proof this yourself !)
F = S0 exp((r − q)T ). (1.1)
Assuming the short rate r and the delivery price of the forward as given, q can
easily be obtained.
1.3.4 Currencies
If the underlying is not a stock but a currency, we must take into account the
domestic as well as the foreign interest rate. Let us continuous compounding
and denote these interest rates by rd and rf , respectively.
We are in Europe so our domestic currency is the euro. Consider a forward
contract on the USD: you must buy N USD at some point in the future T for
the price of K EUR/USD. Assume the currence exchange rate is S0 EUR/USD.
What is the value of this future contract and for what value of K such that
the contract has a zero value (the forward price of the USD). It will turn out
now
K = exp((rd − rf )T )S0.
30
Indeed, suppose K > exp((rd − rf )T )S0. An investor can then do the following
(at time 0).
• Borrow N × S0 exp(−rfT ) EUR at rate rd.
• Use this cash to buy N × exp(−rfT ) USD and put this on an USD-
bankaccount at rate rf
• Short the forward contract.
Then the holding of the foreign currency grows to N because of the interest (rf )
earned. Under the terms of the contract this holding is exchanged for N ×K
at time T . An amount exp(rdT )N × S0 exp(−rfT ) is required to repay the
borrowing. Hence a net profit of N × (K −S0 exp((rd − rf )T )) > 0 is, therefor,
made at time T . In case K < exp((rd − rf )T )S0 you can the following
• Borrow N × exp(−rfT ) USD at rate rf .
• Use this cash to buy N × S0 exp(−rfT ) EURD and put this on an EUR-
bankaccount at rate rd
• Take a long position in the forward contract.
then the domestic currency grows to N × S0 exp((rd − rf )T )), you pay N ×K
to receive N USD and uses these dollars to pay the loan. In total you earned
N × (S0 exp((rd − rf )T ))−K) which in this case was assumed to be positive.
31
1.3.5 Commodities
We now consider the cas of commodities. Important here is the impact of storage
costs. If the storage costs incurred at any time are proportional to the price of
the commodity, they can be regarded as providing a negative dividend yield. In
this case from equation (1.1),
F = S0 exp((r + u)T ),
where u is the storage costs per annum as a proportion of the spot price.
1.3.6 The Cost of Carry
The relationship between all above future/forward prices and spot prices can be
summerized in terms of what is known as the cost of carry. This measures the
storage cost plus the interest that is paid to finance the asset less the income
earned on the asset. For a non-dividend paying stock, the cost of carry is r since
there are no storage costs and no income is earned; for a stock index, it is r− q
since income is earned at rate q on the asset; for a currency it is rd − rf ; for a
commodity with storage costs that are a proportion u of the price, it is r + u;
and so on.
Define the cost of carry as c. For an investment asset, the future price is
F = S0 exp(cT ).
32
Figure 1.7: Call options on S&P 500 Index
33
Chapter 2
Binomial Trees
2.1 Single Period Market Models
Our aim here is to show in the simplest possible non-trivial model how the
theory based on the principle of no-arbitrage works.
Example
Let our financial market consist of two financial assets, a riskless bank account
(or bond) B and a risky stock S, with today’s price S0 = 20 euro. We look at
a single-period model and assume that starting from today (t = 0) the world
can only be in one of two states at time t = T : the stock price will either be
ST = 22 euro or ST = 18 euro. We are interested in valuing a European call
option to buy the stock for 21 euro at time t = T . At time t = T , this option
can have only two possible values. It will have value 1 euro, if the stock price
34
Figure 2.1: One-period binomial tree example
is 22 euro; if the stock price turns out to be 18 euro at time t = T , the value of
the option will be zero. The situtation is illustrated in Figure 2.1.
It turns out that we can price the option by the assumption that no arbitrage
opportunities exist. We set up a portfolio of the stock and the option in such a
way that there is no uncertainty about the value of the portfolio at the time of
expiry, t = T . We then argue that, because the portfolio has no risk, the return
earned on it must equal the risk-free interest rate of the bank account. This
enables us to work out the cost of setting up the portfolio and, therefore, the
option’s price.
Consider a portfolio consisting of a long postion in ∆ shares of the stock and
a short position in one call option. We calculate the value of ∆ that makes the
portfolio riskless. If the stock price moves up from 20 to 22 euro, the value of
the shares is 22∆ and the value of the option is 1 euro, so that the total value
of the portfolio is 22∆ − 1 euro. If the stock price moves down from 20 to 18
euro, the value of the shares is 18∆ euro and the value of the option is zero, so
that the total value of the portfolio is 18∆ euro. The portfolio is riskless if the
35
value of ∆ is chosen so that the final value of the portfolio is the same for both
alternatives. This means
22∆− 1 = 18∆
or
∆ = 0.25
A riskless portfolio is, therefore,
• Long 0.25 shares.
• Short 1 option.
If the stock price moves up to 22 euro, the value of the portfolio is
22× 0.25− 1 = 4.5.
If the stock price moves down to 18 euro, the value of the portfolio is
18× 0.25 = 4.5.
Regardless of whether the stock price moves up or down, the value of the port-
folio is always 4.5 euro at the end of the life of the option.
Riskless portfolios must, in the absence of arbitrage opportunities, earn the
risk free rate of interest. Suppose that in this case the risk-free rate is 12 percent
per annum and that T = 0.5, i.e. six months. It follows that the value of the
portfolio today must be the present value of 4.5 euro, or
4.5e−0.12×0.5 = 4.238
36
The value of the stock today is known to be 20 euro. Suppose the option price
is denoted by f . The value of the portfolio today is
20× 0.25− f = 5− f
It follows that
5− f = 4.238
or
f = 0.762.
This shows that, in the absence of arbitrage opportunities, the current value of
the option must be 0.762. If the valueof the option were more than 0.762 euro,
the portfolio would cost less than 4.238 euro to set up and would earn more than
the risk-free rate. If the value of the option were less than 0.762 euro, shorting
the portfolio would provide a way of borrowing money at less than the risk-free
rate.
In other words, if the value of the option were more than 0.762 euro, for
example 1 euro, you can borrow for example 42380 euro and buy 10000 times
the above portfolio at a cost of
10000(0.25× 20− 1) = 40000euro.
You pocket 2380 euro and after 6 months, you sell 10000 portfolio and cashes in
45000, because the value of one portfolio is always 4.5 euro. With this money
you pay back the bank for the money you borrowed plus the interests on it, i.e.
you pay the bank an amount of 42380 × e0.12×0.5 = 45000 euro. At the end
37
Figure 2.2: General one-period binomial tree
of all this you earned 2380 euro without taking any risk and without an initial
capital. If the value of the option were less than 0.762 euro, you do the opposite.
Generalization
We can generalize the argument just presented by considering a stock whose
price is initially S0 and an option on the stock whose current price is f . We
suppose that the option lasts for time T and that during the life of the option
the stock can move up from S0 to a new level, S0u or down from S0 to a new
level, S0d (u > 1; 0 < d < 1). If the stock price moves up to S0u, we suppose
that the payoff from the option is fu; if the stock price moves down to S0d, we
suppose the payoff from the option is fd. The situation is illustrated in Figure
2.2.
As before, we imagine a portfolio constisting of a long position in ∆ shares
and a short position in one option. We calculate the value of ∆ that makes the
portfolio riskless. If there is an up movement in the stock price, the value of the
38
portfolio at the end of the life op the option is
S0u∆− fu.
If there is a down movement in the stock price, the value becomes
S0d∆− fd.
The two are equal when
S0u∆− fu = S0d∆− fd,
or
∆ =
fu − fd
S0u− S0d . (2.1)
In this case, the portfolio is riskless and must earn the riskless interest
rate. If we denote the risk-free interest rate by r, the present value of the
portfolio is
(S0u∆− fu)e−rT = (S0d∆− fd)e−rT .
The cost of setting up the portfolio is
S0∆− f.
It follows that
(S0u∆− fu)e−rT = S0∆− f,
or
f = S0∆− (S0u∆− fu)e−rT .
39
Substituting from equation (2.1) for ∆ and simplifying, this equation reduces
to
f = e−rT [pfu + (1− p)fd] (2.2)
where
p =
erT − d
u− d (2.3)
Remark 2 If we assume that u > erT , together with u > 1 and 0 < d < 1,
one can easily show that the value of p given in (2.3) satisfies 0 < p < 1. Note
that it is natural to assume that u > erT , because it means that after a time T ,
you can gain more (a factor u) by investing in the risky stocks, than you can
earn with a riskless investment in bond (a factor erT ). If this was not the case
no one would invest in stocks. Ofcourse, you can also lose money (d factior by
investing in stocks.
Remark 3 Equation (2.1) shows that ∆ is the ratio of the change in the option
price to the change in the stock price.
Remark 4 The option pricing formula in (2.2) does not involve the probabili-
ties of the stock moving up or down. This is suprising and seems counterintu-
itive. The key reason is that the probabilities of future up or down movements
are already incorporated into the price of the stock.
Risk-Neutral Valuation
Although we do not need to make any assumptions about the probabilities of
an up and down movement in order to derive Equation (2.2), it is natural to
40
interpret the variable p in Equation (2.2) as the probability of an up movement
in the stock price. The variable 1−p is then the probability of a down movement,
and the expression
pfu + (1− p)fd
is the expected payoff from the option. With this interpretation of p, Equation
(2.2) then states that the value of the option today is its expected future value
discounted at the risk-free rate.
We now investigate the expected return from the stock when the probability
of an up movement is assumed to be p. The expected stock price at time T ,
Ep[ST ], is given by
Ep[ST ] = pS0u+ (1− p)S0d
= pS0(u− d) + S0d.
Substituting from (2.3) for p, this reduces to
Ep[ST ] = S0e
rT (2.4)
showing that the stock price grows, on average, at the risk-free rate. Setting the
probability of an up movement equal to p is therefore, equivalent to assuming
that the return on the stock equals the rsik-free rate. In a risk-neutral world
the expected return on all securities is the risk-free interest rate. Equation (2.4)
shows that we are assuming a risk-neutral world when we set the proability of
an up movement to p. Equation (2.2) shows that the value of the option is its
expected payoff in a risk-neutral world discounted at the risk-free rate.
41
This result is an example of an important genereal principle in option pricing
known as risk-neutral valuation. The principle states that it is valid to assume
the world is risk neutral when pricing options. The resulting option prices
are correct not just in a risk-neutral world, but in the real world as
well.
The Single-Period Example Revisited
We now turn back to the numerical example in Figure 2.1 to illustrate that risk-
neutral valuation gives the same answers as no-arbitrage arguments. In Figure
2.1, the stock price is currently 20 euro and will move either up to 22 euro or
down to 18 euro at the end of six months. The option considered is a European
call option with strike price of 21 euro and an expiration date in six months.
The risk-free interest rate is 12 percent per annum.
We define p as the probability of an upward movement in the stock price in
a risk-neutral world. (We know from the analysis given earlier in this section
that p is given by Equation (2.3). However, for the purpose of this illustration
we suppose that we do not know this.) In a risk-neutral world the expected
return on the stock must be the risk-free rate of 12 percent. This means that p
must satisfy
22p+ 18(1− p) = 20e0.12×0.5
or
p =
20e0.12×0.5 − 18
4
= 0.8092
At the end of the six months, the call option has a 0.8092 probability of being
42
worth 1 euro and a 0.1908 probability of being worth zero. Its expected value
is, therefore,
0.8092× 1 + 0.1908× 0 = 0.8092
In a risk-neutral world, this should be discounted at the risk-free rate. The
value of the option today is, therefore,
0.8092e−0.12×0.5 = 0.7620
This is the same value as the value obtained earlier, illustrating that no-arbitrage
arguments and risk-neutral valuation give the same answer.
2.2 Two-Step Binomial Trees
We can extend the analysis to a two-step binomial tree. The objective of the
analysis is to calculate the option price at the initial node of the tree. This can
be done by repeatedly applying the principles established earlier in the chapter.
2.2.1 European Call
We can first apply the analysis to a two-step binomial tree. Here the stock price
starts at 20 euro and in each of the two time steps may go up by 10 percent or
down by 10 percent. We suppose that each time step is six months long and
the risk-free interest rate is 12 percent per annum.
We consider a European call option with a strike price of 21 euro. Figure
2.3 shows the tree with both the stock price and the option price at each node.
(The stock price is the upper number and the option price is the lower number.)
43
Figure 2.3: Two-period binomial tree example
The option prices at the final nodes of the tree are easily calculated. They are
the payoffs from the option. At node D, the stock price is 24.2euro and the
option price is 24.2− 21 = 3.2 euro; at nodes E and F, the option is out of the
money and its value is zero.
At node C, the option price is zero, because node C leads to either node E
or node F and at both nodes the option price is zero. Next, we calculate the
option price at node B.
Using the notation introduced earlier in the chapter, u = 1.1, d = 0.9,
r = 0.12, and T = 0.5 so that p = 0.8092. Equation (2.2) gives the value of the
option at node B as
e−0.12×0.5[0.8092× 3.2 + 0.1908× 0] = 2.4386
It remains for us to calculate the option at the initial node, A. We do so by
focusing on the first step of the tree. We know that the value of the option at
node B is 2.4386 and that at node C it is zero. Equation (2.2), therefore, gives
44
Figure 2.4: General two-period binomial tree
the value at node A as
e−0.12×0.5[0.8092× 2.4386 + 0.1908× 0] = 1.8583
The value of the option is 1.8583 euro.
We can generalize the case of two time steps by considering the situation in
Figure 2.4.
The stock price is initially S0. During each step, it either moves up to u
times its value or moves down to d times its value. The notation for the value
of the option is shown on the tree. For example, after two up movements, the
value of the option is fuu. We suppose that the risk-free interest rate is r and
the length of the time step is ∆t years.
Repeated application of Equation (2.2) gives
fu = e
−r∆t[pfuu + (1− p)fud] (2.5)
fd = e
−r∆t[pfud + (1− p)fdd] (2.6)
f = e−r∆t[pfu + (1− p)fd] (2.7)
45
Substituting the first two equations in the last one, we get
f = e−2r∆t[p2fuu + 2p(1− p)fud + (1− p)2fdd]. (2.8)
This is constistent with the principle of risk-neutral valuation mentioned earlier.
The variable p2, 2p(1 − p), and (1 − p)2 are the probabilities that the upper,
middle, and lower final nodes will be reached. The option price is equal to
its expected payoff in a risk-neutral world discounted at the risk-free
interest rate.
As we add more steps to a binomial tree, the risk-neutral valuation principle
continues to hold. The option price is always equal to the present value (dis-
counting at the risk-free interest rate) of its expected payoff in a risk-neutral
world.
2.2.2 Matching Volatility with u and d
In practice, when constructing a binomial tree to represent the movements in
a stock price, we choose the parameters u and d to match the volatility of the
stock price. To see how this is done, suppose that the expected return on a stock
in the real world is µ: The expected stock price at the end of the first time
step is S0(1+µ∆t). The volatility of a stock price, σ, is defined so that σ
2∆t is
the variance of the return in a short period of time of length ∆t. Suppose from
empirical data we estimated that the probability of an up movement in the real
world is equal to q. In order to match the expected return on the stock, we
46
must therefore, have
qS0u+ (1− q)S0d = S0(1 + µ∆t),
or
q =
(1 + µ∆t)− d
u− d (2.9)
The variance of the stock price return is
qu2 + (1− q)d2 − [qu+ (1− q)d]2.
In order to match the real world stock price volatility we must therefore have
qu2 + (1− q)d2 − [qu+ (1− q)d]2 = σ2∆t.
or equivalently
q(1− q)(u− d)2 = σ2∆t. (2.10)
Substituting from Equation (2.9)into Equation (2.10) we get
((1 + µ∆t)− d) (u− (1 + µ∆t)) = σ2∆t
When terms in (∆t)2 and higher powers of ∆t are ignored (remember ∆t is
supposed to be small), one solution to this equation is
u = (1 + σ
√
∆t) (2.11)
d = (1− σ
√
∆t) (2.12)
47
Indeed,
((1 + µ∆t)− d) (u− (1 + µ∆t))
= −(1 + µ∆t)2 + (1 + µ∆t)(u+ d)− ud
= −1− 2µ∆t− (µ∆t)2 + 2(1 + µ∆t)− (1 + σ
√
∆t)(1− σ
√
∆t)
= −(µ∆t)2 + σ2∆t
Another setting is
u = eσ
√
∆t (2.13)
d = e−σ
√
∆t, (2.14)
which is, because ∆t is supposed to be small, approximatelly the same as (2.11).
These are the values proposed by Cox, Ross and Rubinstein. Note that in both
cases the values of u and d are independent of µ, which implies that if we move
from the real world to the risk-neutral world the volatility on the stock remains
the same (at least in the limit as ∆t tends to zero). This is an illustration
of an important general result known as Girsanov’s theroem. When we move
from a world with one set of risk preferences to a world with another set of risk
preferences, the expected growth rates change, but their volatilities remain the
same. Moving from one set of risk preferences to another is sometimes referred
to as changing the measure.
48
2.3 Binomial Trees
The above one- and two-steps binomial trees are very imprecise models of reality
and are used only for illustrative purposes. Clearly an analyst can expect to
obtain only a very rough approximation to an option price by assuming that
the stock movements during the life of the option consist of one or two binomial
steps. When binomial trees are used in pratice, the life of the option is typically
divided into 30 or more time steps of length ∆t. In each time step there is
a binomial stock movement. With 30 time steps this means that 31 terminal
stock prices and 230 possible stock price paths are considered.
2.3.1 European Call and Put Options
Consider the evaluation of an option on a non-dividend-paying stock. We start
by dividing the life of the option into a large number of small intervals of length
∆t. We assume that in each time interval the stock price moves from its initial
value S to one of two new values Su and Sd. In general, u > 1 and 0 < d < 1.
The movement from S to Su is, therefore, an ”up” movement and the movement
from S to Sd is a ”down” movement. In the above sections we introduced what
is known as the risk-neutral valuation principle. This states that any security
which is dependent on a stock can be valued on the assumption that the world
is risk neutral. It means that for the purposes of valuing an option, we can
assume:
• The expected return from all traded securities is the risk-free interest rate.
49
• Future cash flows can be valued by discounting their expected values at
the risk-free interest rate.
We make use of this when using a binomial tree. The tree is designed to represent
the behavior of a stock price in a risk-neutral world. In this risk-neutral world
the probability of an up movement will be denoted by p. The probability of a
down movement is 1− p; as seen above in (2.3):
p =
er∆t − d
u− d .
As mentioned above, a popular way of chosing the parameters u and d is
u = eσ
√
∆t
d = e−σ
√
∆t
Figure 2.5 illustrates the tree of stock prices over 5 time periods that is
considered when the binomial model is used.
At time zero, the stock price S0 is known. At time ∆t there are two possible
stock prices, S0u and S0d; at time 2∆t, there are three possible stock prices,
S0u
2, S0ud, and S0d
2; and so on. In general, at time i∆t, i+ 1 stock prices are
considered. These are
S0u
jdi−j , j = 0, . . . , i.
European call and put options are evaluated by starting at the end of the tree
(time T ) and working backward. The value of the option is known at time T .
For example, a European put option is worth max{K − ST , 0} and a European
call option is worth max{ST −X, 0}, where ST is the stock price at time T and
50
Figure 2.5: General binomial tree for stock price
K is the strike price. Because a risk-neutral world is being assumed, the value
at each node at time T −∆t can be calculated as the expected value at time T
discounted at rate r for a time period ∆t. Similarly, the value at each node at
time T −2∆t can be calculated as the expected value at time T −∆t discounted
for a time period ∆t at rate r, and so on. Eventually, by working back through
all the nodes,the value of the option at time zero is obtained. This procedure
is illustrated in Figure 2.6.
Another way of calculating the option prices is by directly taking the dis-
counted value of the expected payoff of the option in the risk-neutral world. For
example the European put, with strike price K and maturity T has a value:
e−rT
N∑
j=0

 N
j

max{K − S0ujdN−j , 0}pj(1− p)N−j
For more complex options, but where the payoff only depends on the final stock
price, i.e. the payoff is a function of ST , g(ST ) say, a similar expression can be
51
Figure 2.6: General binomial tree for stock price
derived; the current value of the option is then given by:
e−rTEp[g(ST )] = e−rT
N∑
j=0

 N
j

 g(S0ujdN−j)pj(1− p)N−j ,
where Ep denotes the expectation in the risk-neutral world, i.e. with a probabil-
ity p given by (2.3) of an up-move of size u , and a probability of a down-move
of (1− p), or equivalently, with a probability
 N
j

 pj(1− p)N−j (2.15)
of ending with a time T stock price of S0u
jdN−j . The distribution (2.15) is
called the Binomial distribution.
52
2.3.2 American Options
If the option is American, the procedure only changes slightly. It is necessary
to check at each node to see whether early exercise is preferable to holding the
option for a further time period ∆t. Eventually, again by working back through
all the nodes the value of the option at time zero is obtained.
American put option
Consider a five-month American put option on a non-dividend-paying stock
when the current stock price is 50 euro, the strike price is also 50 euro, the risk-
free interest rate is 10 percent per annum, and the volatility is 40 percent per
annum. With our usual notation, this means that S0 = 50, K = 50, r = 0.10,
σ = 0.40, and T = 152/365 = 0.416. Suppose that we divide the life of the
option into five intervals of length one month (= 0.0833 year) for the purposes
of constructing a binomial tree. Then
∆t = 0.0833
u = eσ
√
∆t = 1.1224
d = e−σ
√
∆t = 0.8909
p = (er∆t − d)/(u− d) = 0.5073
Figure 2.7 shows the related binomial tree.
At each node there are two numbers. The top one shows the stock price
at the node; the lower one shows the value of the option at the node. The
53
Figure 2.7: Binomial tree for American put option
probability of an up movement is always 0.5073; the probability of a down
movement is always 0.4927. The stock price at the jth node (j = 0, 1, . . . , i) at
time i∆t (i = 0, 1, 2, 3, 4, 5) is calculated as S0u
jdi−j .
The option prices at the final nodes are calculated as max{K −ST , 0}. The
option prices at the penultimate nodes are calculated from the option prices at
the final nodes. First, we assume no exercise of the option at the nodes. This
means that the option price is calculated as the present value of the expected
option price one step later. For example at node C, the option price is calculated
as
(0.5073× 0 + 0.4927× 5.45)e−0.10×0.0833 = 2.66
54
whereas at node A it is calculated as
(0.5073× 5.45 + 0.4927× 14.64)e−0.10×0.0833 = 9.90
We then check to see if early exercise is preferable to waiting. At node C,
early exercise would give a value for the option of zero because both the stock
price and the strick price are 50 euro. Clearly it is best to wait. The correct
value for the option at node C is, therefore, 2.66 euro. At node A, it is a different
story. If the option is exercised, it is worth 50 − 39.69 = 10.31 euro. This is
more than 9.90. If node A is reached, the option should therefore, be exercised
and the correct value for the option at node A is 10.31 euro.
Option prices at earlier nodes are calculated in a similar way. Note that it
is not always best to exercise an option early when it is in the money. Consider
node B. If the option is exercised, it is worth 50− 39.69 = 10.31 euro. However,
if it is held, it is worth
(0.5073× 6.38 + 0.4927× 14.64)e−0.10×0.0833 = 10.36
The option should, therefore, not be exercised at this node, and the correct
option value at the node is 10.36 euro.
Working back through the tree, we find the value of the option at the initial
node to be 4.49 euro. This is our numerical estimate for the option’s current
value. In practice, a smaller value of ∆t, and many more nodes, would be used.
It can be shown that with 30, 50, and 100 time steps we get values for the option
of 4.263, 4.272, and 4.278.
In general suppose that the life of an American put option on a non-dividend-
55
paying stock is divided into N subintervals of length ∆t. We will refer to the
jth node at time i∆t as the (i, j) node. Define fi,j as the value of the option
at the (i, j) node. The stock price at the (i, j) node is S0u
jdi−j . Because the
value of an American put at its expiration date is max{K − ST , 0}, we know
that
fN,j = max{K − S0ujdN−j , 0}, j = 0, 1, . . . , N
There is a probability, p, of moving from the (i, j) node at time i∆t to the
(i+1, j+ 1) node at time (i+1)∆t, and a probability 1− p of moving from the
(i, j) node at time i∆t to the (i + 1, j) node at time (i + 1)∆t. Assuming no
early exercise, risk-neutral valuation gives
fi,j = e
−r∆t(pfi+1,j+1 + (1− p)fi+1,j)
for 0 ≤ i ≤ N −1 and 0 ≤ j ≤ i. When early exercise is taken into account, this
value for fi,j must be compared with the option’s intrinsic value, and we obtain
fi,j = max
{
K − S0ujdi−j , e−r∆t(pfi+1,j+1 + (1− p)fi+1,j)
}
Note that, because the calculations start at time T and work backward, the
value at time i∆t captures not only the effect of early exercise possibilities at
time i∆t, but also the effect of early exercise at subsequent times. In the limit as
∆t tends to zero, an exact value for the American put is obtained. In practice,
N = 30 usually gives reasonable results.
56
It is never optimal to exercise an American call option
We are now going to proof that for a non-dividend paying stock the price of a
European call and an American call are the same. This means that an early
exercise of an American call is never optimal. To prove this striking result we
first proof
Proposition 5 The current price C of a European (and American) call option,
with strike price K and time to expiry T , on a non-dividend paying stock with
current price S satisfies :
C ≥ max{S − e−rTK, 0}.
Proof: That C ≥ 0 is obvious, otherwise ’buying’ the call would give a riskless
profit now and no obligations later.
To prove the remaining lower bound, we setup an arbitrage table (Table 2.1)
to examine the cash flows of the following portfolio:
sell 1 stock short, buy 1 call, invest in bank account e−rTK.
Assuming the condition C ≥ S− e−rTK is violated, i.e. C < S− e−rTK we
get the arbitrage Table 2.1.
So in all possible states of the world at expiry we have a non-negative return
for a portfolio, which has a positive current cash flow. This is clearly an arbitrage
opportunity and hence our assumption was wrong. •
Suppose now that the American call is exercised at some time t strictly less
than expiry T , i.e. t < T . The financial agent thereby realises a cash-flow
57
Portfolio Current cash flow Value at expiry
ST ≤ K ST > K
Short 1 stock S −ST −ST
Buy 1 call −C 0 ST −K
Bank account −e−rTK K K
Balance S − C − e−rTK ≥ 0 K − ST ≥ 0 0
Table 2.1: Arbitrage table for bounds on calls
St−K. From the above proposition we know that the value of the call must be
greater or equal to St− e−r(T−t)K, which is greater than St−K. Hence selling
the call would have realised a higher cash-flow and the early exercise of the call
was suboptimal. In conclusion:
CA = CE
There are two reasons why an American call should not be exercised early.
• Insurance: An investorwhich holds the call option does not care if the
share price falls far below the strike price - he just discards the option -
but if he held the stock, he would. Thus the option insures the investor
against such a fall in stock price, and if he exercises early, he loses this
insurance.
• Interest on the strike price: When the holder exercises the option, he buys
the stock and pays the strike price, K. Early exercise at time t < T
58
deprives the holder of the interest on K between times t and T : the later
he pays out K, the better.
Notice how this changes when we consider American puts in place of calls:
The insurance aspect above still holds, but the interest aspect above is reversed
(the holder receives cash K at the exercise time, rather than paying it out).
2.4 Moving towards The Black-Scholes Model
By creating a tree with more and more time steps, that is by taking smaller and
smaller time-steps, we can get finer and finer graduations at the final stage and
thus hopefully a more accurate price. However, we have to be a little careful
about how we do this in order to get the prices to converge to a meaningful
value. Which limiting price we obtain will depend on how we make the tree
finer - this essentially comes down to assumptions we make about the random
process the asset follows.
Let us try to price an option with payoff function f(ST ) and we will refine
the Cox-Ross-Rubenstein model with choices
u = eσ
√
∆t (2.16)
d = e−σ
√
∆t. (2.17)
Taking N time steps we have that risk-neutral probaility of moving upwards
equals:
pN =
exp(rT )− exp(−σ√T/N)
exp(σ
√
T/N − exp(−σ√T/N))
59
Let us now investigate the risk-neutral limiting distribution of ST :
ST = S0
N∏
j=1
eZjσ
√
T/N = S0 exp

σ√∆t N∑
j=1
Zj

 ,
where Zj are independent random variables taking the values −1 and 1, with
probabilities pN and 1− pN respectively, for j = 1, . . . , N .
In other words:
logST = logS0 + σ
√
T/N
N∑
j=1
Zj .
Now we can apply the Central Limiting Theorem (CLT).
Theorem 6 (CLT) Assume X1, X2, . . . is a series of independent random varoables,
all with the same distribution as X of which the second moment is finite. Then
∑N
j=1−NE[X ]√
NVar[X ]
→D N ,
with N a standard Normal distributed variable (with mean zero and variance
equal to one).
We note that
E[Zj ] = 2pN − 1;
Var[Zj ] = 4pN(1− pN ).
Hence, a simple calculation, using
NE[Zj ]
√
T/Nσ → (r − (1/2)σ2)T ;
√
Var[Zj ]N
√
T/Nσ → σ
√
T .
60
leads to
logST →D logS0 + σ
√
TN +
(
r − 1
2
σ2
)
T
whenN → +∞. The distribution of the logarithm of the stock price thus follows
a Normal distribution with mean
(
r − 12σ2
)
T and variance σ2T ; the stock price
itself is thus lognormally distributed.
The price of the derivative in the limit will be given by
lim
N→∞
exp(−rT )EpN [g(ST )] = exp(−rT )E
[
g
(
S0 exp(σ
√
TN +
(
r − 1
2
σ2
)
T
)]
.
In case of the European call option with strike K and time to maturity T , one
can with a little effort show that its initial price is given by:
C(K,T ) = S0N(d1)−K exp(−rT )N(d2),
where
d1 =
ln(S0/K) + (r +
σ2
2 )T
σ
√
T
(2.18)
d2 =
ln(S0/K) + (r − σ22 )T
σ
√
T
= d1 − σ
√
T (2.19)
and N(x) is the cumulative probability distribution function for a variable that
is standard normal distributed. This is the famous Black-Scholes formula. This
lognormal model (the Black-Scholes model), will be studied in detail in the
course ”Continuous Financial Mathematics”.
61
Chapter 3
Mathematical Finance in
Discrete Time
Any variable whose value changes over time in an uncertain way is said to
follow a stochastic p.rocess. Stochastic processes can be classified as discrete-
time or continuous-time. A discrete-time stochastic process is one where the
value of the variable can change only at certain fixed points in time, whereas
a continuous-time stochastic process is one where changes can take place at
any time. Stochastic processes can also be classified as continuous-variables or
discrete-variables. In a continuous-variable process, the underlying variable can
take any value within a certain range, whereas in a discrete-variable process,
only certain discrete values are possible. Binomial tree models belong to the
discrete-time, discrete-variable stochastic processes.
62
In this chapter we study so-called finite markets, i.e. discrete-time models
of financial markets in which all relevant quantities take a finite number of val-
ues. We specify a time horizon T , which is the terminal date for all economic
activities considered. For a simple option pricing model the time horizon typi-
cally corresponds to the expiry date of the option. We thus work with a finite
probability space (Ω, P ), with a finite number |Ω| of possible outcomes ω, each
with a positive probability: P ({ω}) > 0.
3.1 Information and Trading Strategies
Access to full, accurate, up-to-date information is clearly essential to anyone
actively engaged in financial activity or trading. Indeed, information is arguably
the most important determinant of success in financial life. We shall confine
ourselves to the situation where agents take decisions on the basis of information
in the public domain, available to all. We shall further assume that information
once known remains known and can be accessed in real time.
Our financial market contains two financial assets. A risk-free asset (the
bond) with a deterministic price process Bi, and a risky assets with a stochastic
price process Si. We assume B0 = 1 (we reckon in units of the initial value of
the bond) and Bi > 0; we say it is a numeraire. 1/Bi is called the discounting
factor at time i.
As time passes, new information becomes available to all agents. There
exists a mathematical object to model this information flow, unfolding with
63
time: filtrations. The concept filtration is not that easy to understand. The full
theory will lead us too far. In order to clear this out a bit, we explain the idea
of filtration in a very idealized situation. We will consider a stochastic process
X which starts at some value, zero say. It will remain there until time t = 1,
at which it can jump with positive probability to the value a or to a different
value b. The process will stay at that value until time t = 2 at which it will
jump again with positive probability to two different values: c and d say if is
was at time t = 1 at a and f and g say if the process was at time t = 1 at state
b. From then on the process will stay in the same value. The universum of the
probability space consists of all possible paths the process can follow, i.e. all
possible outcomes of the experiment. We will denote the path 0 → a → c by
ω1, similarly the paths 0 → a → d, 0 → b → f and 0 → b → g are denoted by
ω2, ω3 and ω4 respectively. So we have Ω = {ω1, ω2, ω3, ω4}.
In this situation we will take the following flow of information, i.e. filtrations:
Ft = {∅,Ω} 0 ≤ t < 1;
Ft = {∅,Ω, {ω1, ω2}, {ω3, ω4}} 1 ≤ t < 2;
Ft = D(Ω) = F 2 ≤ t.
We set here F = D(Ω), the set of all subsets of Ω.
To each of the filtrations given above, we associate resp. the following par-
64
titions (i.e. the finest possible one) of Ω:
P0 = {Ω} 0 ≤ t < 1;
P1 = {{ω1, ω2}, {ω3, ω4}} 1 ≤ t < 2;
P2 = {{ω1}, {ω2}, {ω3}, {ω4}} 2 ≤ t.
At time t = 0 we only know that some event ω ∈ Ω will happen, at time
t = 2 we will know which event ω∗ ∈ Ω has happened. So at times 0 ≤ t < 1 we
only know that some event ω∗ ∈ Ω. At time point after t = 1 and strictly before
t = 2, i.e. 1 ≤ t < 2, we know to which state the process has jumped at time
t = 1: a or b. So at that time