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Phillip Monin
MATH 605
Introduction to
Financial
Mathematics
VERSION
1.0: 2020-02-01 19 : 4 5 : 3 1 Z
Copyright © 2020 Phillip Monin
Contents
1
Options
2
Modeling Asset Prices
3
Brownian Motion
4
Stochastic Integration
5
Ito’s Formula
6
Introduction to Black-Scholes-Merton
7
Introduction to Partial Differential Equations
8
The Black-Scholes-Merton Formula
9
Consequences of Black-Scholes
10
The Greeks
11
Special Topics
Index
7
127
17
25
35
47
97
107
87
59
83
69
Introduction
This book is a collection of lecture notes specifically made for the students in M605: Introduction to Financial Mathematics at Georgetown
University, which I have taught annually since 2018. It is very much a
work in progress and will continue to evolve.
The impetus for these notes was the unique nature the students
in the graduate mathematics and statistics program at Georgetown.
These students are extraordinary, and are phenomenally gifted in
mathematics. They benefit from a serious introduction to stochastic
processes and quantitative finance, but have little background in the
field of finance itself. I have yet to find a book that combines graduate mathematics with introductory finance in a way that accounts for
the exposure to probability and computational methods that the students taking M605 obtain in their prerequisites and is flexible enough
to allow the course to steer itself naturally; hence these notes.
These lecture notes are sketches and are not meant to be the sole
source for the topics they cover. The material is borrowed from a
variety of sources, including books by Wilmott et al. and Steele. I
have tried to cite the borrowed material properly along the way. I am
responsible for all errors. Again, these notes are specifically tailored
to M605, and they should not be distributed beyond the students in
that class.
I want to thank the students of M605 for inspiring me week in
and week out with their intellectual curiosity, energy, and diligence.
While they deserve far more, these notes are dedicated to them.
1
Options
This course will approach the topic of financial mathematics and
financial modeling by considering the Black-Scholes-Merton model,
one of the most elegant and widely known financial models. The
Black-Scholes-Merton model provides a theoretical framework for
pricing options. Options belong to a class of financial assets called
derivatives.
A derivative is a financial instrument that derives its value from
that of another financial instrument, called an underlying asset.
Derivatives can be used for a wide variety of purposes:
• hedging: reduce overall risk by taking offsetting positions
• investment: take calculated risks based on fundamentals or reasoned views
• speculation: take outsize risks akin to gambling
Derivatives are widespread and essential to modern finance. As
with many things in finance and in life, there are several benefits and
drawbacks of derivatives.
Benefits associated with derivatives:
Derivatives provide a means for risk sharing, that is, the slicing and
dicing of risks across events and time, so that (in theory) the market
participants most willing and best able to bear these risks actually
do so. In this way, derivatives help lead to an efficient use of capital,
which promotes economic growth.
Drawbacks associated with derivatives:
Derivatives can be complex, depend nonlinearly on the underlying,
and encourage reckless risk-taking, potentially leading to problems
or catastrophes. In 2002, Warren Buffett called derivatives “financial
weapons of mass destruction.”
8
1.1
math 605 introduction to financial mathematics
Options
An option gives its holder the right, but not the obligation, to buy
or sell an underlying financial asset at a pre-specified price over a
pre-specified period of time or at a pre-specified date. An option
is a derivative because its value derives from the value of another
financial asset, the underlying. There are several types of options. We
will focus on some of the more elementary types in this course.
Options terminology:
• The pre-specified price is called the strike price or an exercise
price
• The end date of the contract is called the expiry, the expiration
date, or the maturity
• The holder, or buyer, of the option is said to exercise the option
if and when he or she exercises his or her right to buy or sell the
underlying according to the terms of the options contract.
• The other party in the contract is called the writer of the option.
In contrast to the option holder, the option writer has a potential
obligation: he or she may be required to buy the asset from, or
sell the asset to, the option holder. This requires compensation;
the buyer of the option must pay the option writer. This is called
the option’s price or premium. We will consider the following
questions at length: What is the value of the option, i.e. the right
to buy or sell an asset in the future? How can the writer of the
option minimize his or her risk?
• An American option allows the holder to exercise the option at
any time before the expiration date. A European option allows
exercise only at the expiration date.
• Contracts that provide the option to buy the asset at a pre-specified
price are called call options (“call away the asset”) and contracts
that provide the option to sell the asset at a pre-specified price are
called put options (“put the asset to someone else”).
Payoff diagrams
Today is time t, and a stock has current price St . Consider a call
option with strike K and maturity T. The payoff is
(
CT =
ST − K,
0,
ST ≥ K
= (ST − K )+
ST < K
(1.1)
options
where
(
x+ ≡
x,
0,
x≥0
x<0
(1.2)
Figure 1.1: Payoff diagram for a long
call option.
Similarly, for a put option,
(
K − ST ,
PT =
0,
K ≥ ST
= (K − ST )+
K < ST
(1.3)
Link to option chain for Apple:
https://finance.yahoo.com/quote/AAPL/options/
1.1.1 Example: Obtaining Leveraged Exposure with a Call Option
First, a primer on financial leverage. Leverage refers to using borrowed money to buy something. For example, you want to buy a
$500,000 house. To avoid having to purchase mortgage insurance,
you want to put down 20%, or $100,000, and borrow the $400,000
balance in the form of a mortgage loan to buy the house. You have
used borrowed money to purchase an asset, and are said to be 5-to-1
leveraged or have a leverage ratio of 5-to-1.
One of the most important aspects about leverage is that it magnifies returns. Suppose the house appreciates 5% to $525,000. For simplicity, suppose the mortgage rate is 0%. What is your equity in the
9
10
math 605 introduction to financial mathematics
Figure 1.2: Payoff diagram for a long
put option.
house? What is the return on your initial capital of $100,000? If you
sell the house for $525,000 and pay off the $400,000 mortgage, then
you are left with $125,000 on net in equity, which is a 25% return on
your initial investment of $100,000. So the asset appreciates 5% but
your equity in it goes up 25% because you are 5-to-1 leveraged. However, it should be easy to see that this also works in reverse: if the
house instead depreciates 5% to $475,000, then your equity in it will
be $75,000, which is a -25% return on your capital.
Leverage can also refer to a situation where you may not explicitly
borrow money to buy an asset, but the nature of the asset, usually a
derivative, permits exposure “as if” you had borrowed money. That
is, one position in some asset is equivalent to some other position in
other assets wherein money was borrowed. This is called embedded
leverage, and will be a general theme of the analysis with options.
Today is January 10th. We’d like to value a certain call option. This
call option contract stipulates that on November 18th the holder has
the right, but not the obligation, to purchase one share of XYZ for
$2.00 per share. (Options are most often written in lots of 100 shares,
but to keep things simple we usually only consider an option for one
share.) What is the price of this option?
Suppose that on November 17th the share price of XYZ is $2.20.
The holder of the option can purchase XYZ for only $2.00, when in
the market it is selling for $2.20. What does the holder do?
options
She exercises the option, buying the share for $2.00 and immediately sells it in the market for $2.20, making a cool profit of $0.20 per
share.
On the other hand, if the share price of XYZ is only $1.80, then it
would not be sensible to exercise the option. Why?
Exercising the option means that she can buy it for 2.00, but it’s
cheaper in the market.
Suppose a simplified world where XYZ only takes the values $1.80
and $2.20 on November 18th, and does so with equal probability.
Then the expected profit for the option is
0.5 ∗ 0 + 0.5 ∗ $0.20 = $0.10
Ignoring interest, it seems reasonable that the value of the option
might be close to $0.10. Things aren’t this simple really, as this course
will show, but assume for the moment that the price was indeed
$0.10. Then the net profit if XYZ raises to $2.20 is $0.20 - $0.10 =
$0.10, a 100% increase. However, the net loss if XYZ declines to $1.80
is $0 - $0.10 = - $0.10, a 100% decrease.
A 100% increase or decrease is huge, especially when you consider
what would’ve happened if the holder just bought the stock. If he
or she bought the stock at $2.00 and sold it at $2.20, then a profit
of $0.20 on an initial investment of $2.00 yields a 10% gain; if XYZ
dropped to $1.80, then a loss of $0.20 yields a 10% loss.
What have we learned?
• Option prices can respond in exaggerated ways to price changes
in the underlying. This is variously called embedded leverage,
economic leverage, gearing, and represents a use for options as
a way to get higher amounts of exposure to the underlying. Buy
stock, get ±10%. Buy option get ±100%. 10x or 10-to-1 leverage.
• Dependence of the option’s price on today’s price of the underlying. The greater (lesser) the share price of XYZ on November
18th, the greater (lesser) the profit. While we do not know what
XYZ will be in the future, we do know it now, and it seems reasonable that the higher the share price is now, the higher it will be in
the future. Thus the price of the option today depends on today’s
share price.
• Dependence of the option’s price on the strike. The dependence
of the value of the option on the strike price is clear. Why? Same
asset but different strikes yield different payoffs.
• Dependence of the option’s price on the time to maturity. On
November 17th, one day before the option expires, there is little
11
12
math 605 introduction to financial mathematics
time left for the underlying’s price to change by much. However,
today, on January 10th, there is a lot of time for the underlying’s
price to change. Thus it seems that the option price should depend
on the time to maturity of the option.
• Dependence of the option’s price on the volatility. We will see later
that the option price depends on a property of the “randomness”
of the asset price called the volatility. The volatility is intuitively a
measure of how jagged the graph of the asset price is against time.
This clearly affects the distribution of the asset price at expiration,
and thus should affect the option price.
• Dependence of the option’s price on interest rates. Finally, the
option price should depend on interest rates. The option is paid
for up-front and any potential payoff doesn’t come until later. The
price should thus reflect income that would’ve otherwise been
made by investing the premium in the bank.
1.1.2 Example: Hedging Downside Risk with a Put Option
Today is January 10th. We’d like to understand how to value a certain put option. This put option contract stipulates that on November
18th the holder has the right, but not the obligation, to sell one share
of XYZ for $3.00 per share.
Suppose that on November 17th the share price of XYZ is $2.70.
The holder of the option can sell XYZ for $3.00, when in the market
the price is $2.70. What does the holder do?
She exercises the option, selling the share for $3.00 and immediately buys it in the market for $2.70, making a cool profit of $0.30 per
share.
On the other hand, if the share price of XYZ is higher at $3.30, then it
would not be sensible to exercise the option. Why?
Buy at 3.30 sell at 3, 0.30 loss.
Suppose a simplified world where XYZ only takes the values $2.70
and $3.30 on November 18th, and does so with equal probability.
Then the expected profit for the option is
0.5 ∗ 0 + 0.5 ∗ $0.30 = $0.15
Ignoring interest, it seems reasonable that the value of the option
might be close to $0.15. Again, things aren’t this simple in reality, but
assume for the moment that the price was indeed $0.15. Then the net
profit if XYZ drops to $2.70 is $0.30 - $0.15 = $0.15, a 100% increase.
However, the net loss if XYZ increases to $3.30 is $0 - $0.15 = - $0.15,
a 100% decrease. So, like the call example, a put option lets its holder
obtain leveraged exposure to the underlying.
options
13
In a departure from the last example, let’s use the parameters
of this example to consider a popular use of options: hedging. We
will consider the use of a put option to hedge downside risk. The
construction is called a protective put and involves the following
portfolio:
own a share (ie long) of XYZ plus long a put option.
Suppose I bought the share of XYZ at the same time as I entered the
put contract, when the price of XYZ was $3.00 per share. Now, if
the price of XYZ increases to $3.30 per share, then my total position
is worth (we ignore the option premium for now to illustrate the
hedge):
($3.30 − $3.00) + 0 = $0.30
This reflects the price appreciation in the value of the share of XYZ
and the fact that when the price goes up I will decline to exercise the
option and it will expire worthless.
Suppose instead that the price of XYZ falls to $2.70 per share.
Now I have lost $0.30 on my position in the stock, but because I hold
a put option with a strike at $3.00, I can exercise it and partially offset
my loss, for a combined position worth:
($2.70 − $3.00) + $0.30 = $0
Figure 1.3: Payoff diagram for a protective put.
14
math 605 introduction to financial mathematics
Algebraically, a protective put has payoff
(
K, K > S
+
S + (K − S) =
S, S > K
1.2
(1.4)
Exercises
Figure 1.4: Payoff diagram for a long
call option.
The payoff for a long call option, i.e. for an investor who buys a
call option, with strike K at expiration is given by CT = (ST − K )+
and that the corresponding payoff diagram is given by Figure 1.4.
It is useful to observe that the x-axis in the payoff diagram is the
value of the stock at expiration, ST , and the y-axis is the payoff of the
option, CT . Also note that, for simplicity, we have ignored the option
premium paid by the buyer of the call option at initiation. Please
ignore option premia paid throughout this exercise.
Similarly, the payoff for a long put option, i.e. for an investor who
buys a put option, with strike K at expiration is given by PT = (K −
ST )+ and the corresponding payoff diagram is given by Figure 1.5.
Exercise 1.1
Write a mathematical expression for the payoff of a short call option with strike K, i.e. for an investor who sells (writes) a call option,
at expiration. Draw the corresponding payoff diagram.
options
Figure 1.5: Payoff diagram for a long
put option.
Exercise 1.2
Write a mathematical expression for the payoff of a short put option with strike K, i.e. for an investor who sells (writes) a put option,
at expiration. Draw the corresponding payoff diagram.
Exercise 1.3
Suppose I buy a call option with strike K and sell a put option
with the same strike on the same underlying stock and with the
same expiration date. What is the payoff MT at expiration for this
portfolio? (Hint: It is a simple function of S and K.) Draw the corresponding payoff diagram.
Exercise 1.4
Use the definition of x + to simplify (ST − K )+ − (K − ST )+ . This
is the payoff of a position composed of a long call option and a short
put option. What do you notice?
Exercise 1.5
Suppose the share price of Amazon (AMZN) is $1200. Consider
two call options on AMZN with the same expiration, one with strike
$1250 and one with strike $1300. Which option is likely to be more
expensive, i.e. have a higher premium? Why?
15
16
math 605 introduction to financial mathematics
Exercise 1.6
Suppose you buy a call with strike price K1 and sell a call on the
same stock with the same expiration date but with a strike price of
K2 , where K2 > K1 . Write an expression for the payoff of this options
portfolio and draw the payoff diagram (ignoring the premiums on
the two options). Is the initial cost of this portfolio positive or negative?
2
Modeling Asset Prices
The Black-Scholes-Merton (BSM) model is our entree into the world
of financial modeling. The model produces the theoretical price of a
European option given certain dynamics for the option’s underlying,
i.e. the stock. Like all models, the BSM model makes certain assumptions, and one of these assumptions is that the dynamics of the stock
are taken as exogenous to the model and are given by a geometric Brownian motion (GBM). In this section we pursue a heuristic
derivation of a GBM. Along the way we get glimpses of Brownian
motion, stochastic processes, stochastic integrals, and Ito’s lemma,
all of which we will come back to with more rigor in a subsequent
section.
As mentioned above, the dynamics, or data-generating process,
for the stock in the BSM model are taken as given by a GBM. This
means that in the BSM model we do not intend to ascertain where
the price of the stock comes from, or how it evolves over time. Instead we want a statistical model for how a stock price might evolve.
So it makes sense to look at the time series of stock prices for inspiration.
Figures 2.1 and 2.2 depict time series graphs of the Dow Jones Industrial Average and Amazon’s (AMZN) stock price during selected
times. There seems to be a certain jaggedness to the graphs. Daily
changes seem to be random, and tend to be relatively small most of
the time and occasionally big. If we try to use the tools and functions
of the calculus of real variables or ordinary differential equations to
produce a similar statistical object then it seems we will fail. We need
some new mathematical techniques.
Postpone for a second further discussion of what these new mathematical techniques are and consider the field or domain that we’re
in. We’re trying to model financial asset prices. What does financial
theory have to say about this? One of the central topics and debates
in finance is the efficient markets hypothesis. In brief, the EMH says
that the history of the stock and the associated company is fully re-
18
math 605 introduction to financial mathematics
15000
14000
13000
12000
11000
10000
9000
004n 2005 l 2005n 2006 l 2006n 2007 l 2007
2
l
Ju Ja Ju Ja Ju Ja Ju
Date
12000
11500
11000
10500
10000
9500
9000
99 1999 1999 2000 2000 2000 2000
9
1
Apr Jul Oct Jan Apr Jul Oct
Date
3000
2800
2600
2400
2200
2000
1800
1600
1400
86 1987 1987 1988 1988 1989 1989 1990 1990
9
1
Jul Jan Jul Jan Jul Jan Jul Jan Jul
Date
Figure 2.1: Time series graphs of the
Dow Jones Industrial Average
modeling asset prices
1200
1000
800
600
400
200
0
Figure 2.2: Time series graphs of
Amazon’s (AMZN) stock price
201
3
201
4
5
6
201 201
Date
7
201
90
80
70
60
50
40
30
20
10
002000200020002000200020002000200020002000
0
2
Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec
Date
19
20
math 605 introduction to financial mathematics
flected in its present price, and that its present price does not hold
any further information. Moreover, markets and thus prices respond
immediately to new information about a company that’s relevant to
its valuation. Thus modeling of financial asset price ...

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