MAS362/462/6053/61017 Financial Mathematics
Lecture Notes1
Sam Marsh
Semester 1, 2023
1basedonnotesbyMotyKatzmanandDimitriosRoxanas
Contents
What is this course about? 5
1. What does “correct price” mean? 5
2. Portfolio Theory 6
Chapter 1. Interest, present value and bonds 9
1. Interest, periodic and continuous compounding 9
2. Present value 11
3. Bonds 13
4. Discount and yield curves 14
5. Forward rates 16
6. Exercises 17
Chapter 2. Forward and Futures Contracts 19
1. Short-selling 19
2. The forward price 20
3. Foreign exchange forward contracts 22
4. Futures contracts 22
5. Exercises 23
Chapter 3. Options 25
1. The payoffs of options 25
2. Elementary inequalities satisfied by option prices 26
3. Optimal exercise for American style options 27
4. Parameters affecting the prices of options 28
5. Exercises 28
Chapter 4. Binomial trees and risk neutral valuation 31
1. A motivating example 31
2. Derivatives in a simple, up-down world 32
3. An example 33
4. A two step process 33
5. An example 34
6. n-step trees 35
7. Example: an American option 36
8. Using binomial trees for approximating values of derivatives - Not examinable 37
9. Exercises 38
Chapter 5. The stochastic process followed by stock prices 39
1. Brownian motion 39
3
4 CONTENTS
2. The Ito integral 40
3. Modelling stock prices 41
4. Ito’s Lemma. 43
5. Exercises 45
Chapter 6. The Black-Scholes pricing formulas 47
1. The Black-Scholes differential equation 47
2. Boundary conditions 48
3. The Black-Scholes pricing formulas 49
4. More exotic derivatives 50
5. The Black-Scholes pricing formulas: the risk neutral valuation approach. 51
6. Volatility 52
7. Exercises 52
Chapter 7. Portfolio Theory 55
1. Axioms satisfied by preferences 55
2. Indifference curves 56
3. Portfolios consisting entirely of risky investments 57
4. The feasible set 58
5. Efficient portfolios 58
6. Different choices of portfolios for different appetites for risk 59
7. Portfolios containing risk-free investments 60
8. Exercises 61
Chapter 8. The Capital Asset Pricing Model 63
1. The market portfolio. 63
2. The market price of risk 64
3. Exercises 65
Solutions 67
Chapter 1 67
Chapter 2 69
Index 71
What is this course about?
In this course you will learn some basic facts about an important component of modern life:
finance. This is a mathematics course hence we will be interested in mathematical ideas used in the
context of finance. Our task is to build a mathematical model of financial securities. A crucial first
stageisconcernedwiththepropertiesofthemathematicalobjectsinvolved. Thisisdonebyspecify-
ing a number of assumptions, the purpose of which is to find a compromise between the complexity
of the real world and the limitations and simplifications of a mathematical model imposed to make
ittractable.1 Theassumptionswearegoingtomakereflectourcurrentpositiononthiscompromise
and will be modified in the future. You may find that some situations look simple based on these
assumptions; however I hope you will see they can be sufficiently interesting to convey the flavour
of the theory to be developed later on.
Specifically, in this course we aim to address two questions:
(1) What is the “correct price” of financial assets?
(2) What are optimal investment strategies?
ThefirstquestionwillleadustotheBlack-Scholespricingformula,whichearnedRobertC.Mer-
ton and Myron S. Scholes the 1997 Nobel prize in Economics. William F. Sharpe and Harry M.
Markowitz received the 1990 Nobel prize in Economics for answering the second question.
1. What does “correct price” mean?
Lets first see an example of an incorrect price. Suppose that a barrel of oil trades at ￡100 (i.e.,
one can buy it for ￡100 and one can sell it for ￡100). I want to start trading in oil, too. Would
￡120 be a correct price? No! Other traders will buy oil in large quantities for ￡100, sell it to me for
￡120. Everyone except me will become rich, I end up with a pile of barrels of oil I can’t sell. Would
￡90 be a correct price? Think about it. (The answer is “no”).
Amoresubtleexample: considerthreemerchantswhoarewillingtobuyandsellbagscontaining
apples and oranges (all of identical size and quality) as follows:
Bag content Price
Merchant A 3 apples, 2 oranges ￡5
Merchant B 2 apples, 3 oranges ￡6
Merchant C 4 apples, 3 oranges ￡8
To see that these prices are incorrect, let me show you how to get rich:
(a) borrow ￡36 for a short while (with negligible interest);
(b) buy 6 bags from merchant A and buy 1 bag from merchant B for a total cost of ￡36;
1Thereisawell-knownrelevantquoteforthis: “Allmodelsarewrong,somemodelsareuseful”.
5
6 WHAT IS THIS COURSE ABOUT?
(c) rearrange the fruit in five bags of 4 apples and 3 oranges each;
(d) sell the five bags to merchant C for ￡40;
(e) return the ￡36 loan and pocket a ￡4 profit.
Repeat this process until you are very rich.
Lots and lots of people would be buying from merchants A and B and selling to merchant C.
WhenthishappensthepricesofbagsAandBwillriseandthoseofbagCwillfall, andthisprocess
will continue until there are no more easy profits to be made.
Notice:
(a) we didn’t assume an intrinsic or objective price of apples and oranges,
(b) correct pricing is an equilibrium price,
(c) we expose incorrect pricing by exhibiting profit-making trading strategies which change
prices.
The making of a certain profit with no investment is called arbitrage. Correct pricing means
setting the unique price which does not introduce arbitrage opportunities. The correct price of bag
Cwouldbe￡36/5asthatistheuniquepriceforwhichthestrategyabove(oritsopposite)doesnot
produce a profit. We call this method of pricing no-arbitrage pricing, which is also known colloqui-
ally as “There is no such thing as a free lunch!”
Thegeneralideaforsucharguments, usedtofindthe“correct”priceofsomething, istousethe
assumptionofnoarbitragetoshowthatthecorrectpricecanbeneitherhighernorlowerthanwhat
weclaimtobetherightone. The guiding principle for such arguments is that if something
has a price that is perceived to be “too low”, we buy it in the hopes that we can sell
it for a higher price and make a profit; similarly, if we think that something is priced
“too high”, we seek to sell it.
No-arbitrage pricing is the most fundamental (and mathematically fruitful) assumption we will
make about the market. It is an assumption that’s close enough to reality. As we will witness our-
selves in this course, arguments based on the no-arbitrage-principle are the main tools of financial
mathematics. Arbitrageopportunitiesrarelyexistinpractice. Thisisreasonable: ifarbitrageexists,
arbitragetradingwilloccurandpriceswillquicklychangetoeliminatethesearbitrageopportunities.
Ifandwhentheydo,thegainsaretypicallyextremelysmallcomparedtothevolumeoftransactions,
makingthembeyondthereachofsmallinvestors. Theycanhoweverarisewhenmarketparticipants
make a “mistake”, but they are usually very subtle, short-lived and difficult to spot.
2. Portfolio Theory
Arethereinvestmentstrategieswhicharebetterthanothers? Andwhatcould“better”meanin
thiscontext? Wearegreedy,i.e.,wewanthighreturns. Butaveragehighreturnsarerisky! Suppose
you are offered to make a bet on the outcome of tossing a fair coin: heads – you lose everything
you own, including your clothes, and are forced to live on the street; tails – you double your wealth.
Would you bet? This is a fair game, i.e., the expected gain is zero, but most people would not
bet. People are not only greedy – they also have an aversion towards risk. Even if a win triples or
quadruples their wealth, most people would not place a bet. If we used a coin which produces tails
with 99% probability some people might choose to place a bet. Now the expected gain may tempt
some, but not others.
2. PORTFOLIO THEORY 7
There is an interplay between expected returns and risk investors are willing to take. So a
“good”waytoinvestisastrategythatcompensatesforitsriskwithanappropriatelyhighexpected
return.
The last part of the course will make this statement precise: we will see that these optimal
investments exist and consist of portfolios 2 of risk-free deposits/bonds and market-index trackers.
2Portfolio: acollectionofassets
CHAPTER 1
Interest, present value and bonds
From our daily experience we understand that ￡100 to be received after one year is worth less
than the same amount today. The main reason is that money due in the future (or locked in a
fixed term account) cannot be spent right away. One would therefore expect to be compensated for
postponed consumption. In addition, prices may rise in the meantime, and in that scenario, this
amount will not have the same purchasing power as it would have in the present. Finally, there is
always a risk, even if a negligible one, that the money will never be received. Whenever a future
paymentisuncertaintosomedegree,itsvaluetodaywillbereducedtocompensatefortherisk. For
now, we shall restrict ourselves to risk-free assets, such as a bank account or a bond (more about
bonds later in this chapter.)
Thewayinwhichmoneychangesitsvalueintimeisacomplexissueoffundamentalimportance
in finance. This topic is often referred to as the time value of money. We will be mainly addressing
the following two questions:
What is the future value of an amount invested (or borrowed) today ?
What is the present value of an amount to be paid or received at a certain time in the
future ?
1. Interest, periodic and continuous compounding
Money can be lent and thus earn interest; think of interest as getting paid “rent” for your
money. For what follows, suppose that an amount P is paid into a bank account (you are “lending
thebank”), whereitistoearn(or“accrue”)interest. Thefuture valueofthisinvestmentconsistsof
the original deposit, called the principal, plus all the interest earned since the money was deposited
in the account. We will assume that the principal A is attracting annual interest at a constant rate
r > 0. Interest rates apply for a given period, and interest is often compounded a fixed number
of times in that period. In other words, the interest earned will now be added to the principal
periodically, for example, annually, semi-annually, quarterly, monthly, daily, etc. Subsequently,
interest will be attracted not just by the original deposit, but also by all the interest earned so far.
In such cases, we talk of discrete or periodic compounding.
Example. Let’s consider the case of monthly compounding. The first interest payment of r A
12
willbedueafteronemonth(thisishowmuchinterestouroriginaldepositwillattractinonemonth).
Therefore, now our principal (how much we currently have in the bank) is A+ r A = (1+ r )A.
12 12
This whole amount will now attrack interest in the future. The next interest payment, due after
two months (2 since we started, 1 month since the last payment), will be r (1+ r )A, increasing
12 12
our total capital (what we have in the bank) to (1+ r )2A (check this!). After one year, our capital
12
will have become (1+ r )12A, after m months it will be (1+ r )mA, and after n years it will be
12 12
(1+ r )mnA. The last formula admits n equal to a whole number of months, that is, a multiple of
12
1 .
12
9
10 1. INTEREST, PRESENT VALUE AND BONDS
In general, if m interest payments are made per annum, the time between two consecutive
paymentsmeasuredinyearswillbe 1,thefirstpaymentbeingdueattime 1.Eachinterestpayment
m m
will increase the principal by a factor of 1+ r. Given that the interest rate remains unchanged,
m
after n years the future value of an initial principal A, will become
(cid:16) r (cid:17)mn
(1) V(n)= 1+ A,
m
because there will be nm interest payments during this period. In this formula n nust be a whole
multiple of the period 1. The number (cid:0) 1+ r(cid:1)mn is referred to as the growth factor.
m m
To summarize, an amount A invested for n years at a yearly interest rate r com-
pounded m times a year, yields at the end of this period
A(cid:0)
1+
r(cid:1)mn
.
m
The exact value of the investment may sometimes need to be known at time instants between
interest payments. In particular, this may be so if the account is closed on a day when no inter-
est payment is due. For example, what is the value after 10 days of a deposit of ￡100 subject
to monthly compounding at 10%? One possible answer is ￡100, since the first interest payment
wouldbedueonlyafteronewholemonth. Itishoweverpossibletoextend(1)foralln≥0,notjust
integermultiplesofthecompoundingfrequency,andfromnowthiswillbeastandingassumption.
(Under this assumption, that we allow for any t≥0) formula (1) for the future value at time t
of a principal A attracting interest at a rate r >0 compounded m times a year can be written as
(cid:20)(cid:16)
r
(cid:17)m(cid:21)tr
V(t)= 1+ r A,
m
Now think of a case of periodic compounding with the frequency m is very large (e.g., 1 second or
even milliseconds). In the limit as m→∞, we obtain1
(2) V(t)=etrA,
This is known as continuous compounding (with corresponding growth factor etr). Formula (2) is a
good approximation of the case of periodic compounding when the frequency m is very large. Since
it is a simpler formula, which is easier to manipulate and transform when necessary (as we will see
many times in the course), in this course we want to work with continuously compounded interest
rates (which is also the one most commonly used in practice).
To summarize, an amount A invested for t years accruing a yearly interest of r com-
pounded continuously, yields at the end of this period ￡Aert.
As a closing remark for this section, note that the same payment of interest can be described
usingdifferentinterest-compoundingconventions. Forexample,adepositof￡100yieldingabalance
of ￡110 in one year pays
? 10% annual interest compounded yearly,
√
? 2( 1.1?1)≈9.76% annual interest compounded every six months,
? ln1.1≈9.53% annual interest compounded continuously.
(cid:18) 1(cid:19)n
1Recallthat lim 1+ =e.
n→∞ n
2. PRESENT VALUE 11
(Quick Exercise: verify the claims in the above example.)
Unless stated otherwise, we always assume that interest is compounded continu-
ously.
2. Present value
The notion of interest is linked to another important notion: present value.
Definition. The present value of apayment occurring in the future, isthe present price of the
entitlement to that payment.
Suppose that a deposit of ￡A yields ￡B when we withdraw it in t years (with A < B). We
can now ask how much would we pay now to receive ￡B in t years. The answer is, of course ￡A!
Depositing ￡A now is equivalent to paying that amount now to receive ￡B in t years. We say that
the present value of ￡B paid in t years is ￡A.
To understand better the notion of present value, let’s look at a numerical example:
Example. How much are people willing to pay now in order to receive ￡1 in 10 years? We
make the assumption that 10-year deposits pay 3% per year compounded yearly.
Call the amount we are willing to pay now, X. If you deposit ￡X for 10 years you end up with
￡X ×1.0310 ≈ 1.34×X. To obtain ￡1 in 10 years, you need to deposit ￡1/1.34 ≈ 0.74 now. I
claim that this is the correct present value of ￡1 paid in 10 years. Let’s see why. In what follows, A
denotes the amount of money people are willing to pay now to receive ￡1 in 10 years. We will see
that if A is either higher or lower than 1.03?10, there is a strategy that allows us to take advantage
and get a free lunch!
To make the argument that follows easier to understand, imagine that everybody in the market
can issue certificates that declare “the holder of this certificate is to be paid ￡1, 10 years from this
date”,andhasthesignatureoftheperson/institutionissuingit,thereforemakingitalegalcontract
that people have to uphold. Our original question can be thus rephrased in terms of this certificate:
“what is the correct price for this certificate? How much are people willing to pay now to get ￡1 in
10 years? (by showing this certificate to the issuer)”.
Case 1: Assume that people are willing to pay A>1.03?10 for this certificate. We believe that
thispriceis“toohigh”,soaccordingtoourguidingprincipleforsomethingthatispriced“toohigh”,
we issue such a certificate, sign it, and sell it to them for ￡A. Then we deposit A in the bank, and
since the interest rate is 3%, we will collect A×1.0310 > 12 after 10 years (that’s the capital for a
principal deposit of A). At the same time (this is after 10 years), they will come to us and show us
the certificate we issued. We are under the obligation to pay them ￡1, and so we do. But notice:
our capital in the bank after 10 years is A×1.0310, and we have to pay the holder of the certificate
￡1, so we pocket the profit A×1.0310?1>0.
2usingtheassumptionA>1.03?10 andrearranging
12 1. INTEREST, PRESENT VALUE AND BONDS
Case 2: Now assume that people are willing to pay A < 1.03?10. We think this price to be
“too low” and we saw that in such cases we will buy the asset (here, the certificate). So, we will
borrow ￡A from the bank and spend them on buying the certificate. This means that we can now
present the certificate (in 10 years time) and collect ￡1. This is exactly what we do, wait 10 years
and “exchange” the certificate for ￡1. We have to pay the bank back (with interest), that will be
A×1.0310 <13. But then we make a profit of 1?A×1.0310 >0 (collect 1, pay A×1.0310 >0 and
pocket the positive difference.)
In both cases, a guaranteed profit was achieved with no investment whatsoever; a free lunch!
This contradicts our assumption that there are no arbitrage opportunities in the market, which
implies that A=1.03?10 ≈0.74 pounds. Consequently, the present value of ￡1 paid 10 years from
now is 1.03?10.
Example. An annuity is a sequence of finitely many payments of a fixed amount due at equal
time intervals. An example of this can be a mortgage, which is a loan given with some asset held as
a “security” for the loan. If you want to buy a house, you can ask to get a mortgage, which is the
loan to buy the house; if you don’t repay the loan you will lose your house.
Consider a ￡100,000 mortgage paying a fixed annual interest rate of 6% which is compounded
monthly. What are the monthly repayments if the mortgage is to be repaid in 20 years?
The loan will be repaid as a sequence of finitely many payment of a fixed amount every month.
Call this monthly repayment P, which is this number we seek to calculate. We can find the value
of P in two steps.
First, let’s ask ourselves how much is the bank willing to pay you now in order to receive ￡P
in m months? (e.g. ￡P in 3 months; it’s always the same amount P that the bank will receive).
Basically, we are looking for the present value of ￡P in m months. Under monthly periodic com-
pounding at a fixed annual rate of 6%, ￡1 now will be worth ￡(cid:0) 1+ 0.06(cid:1)m in m months4 therefore
12
the present value of ￡P in m months is clearly P ×(cid:0) 1+ 0.06(cid:1)?m .
12
This means in particular, that an amount of ￡P in 1 month has a present value of P ×
(cid:0) 1+ 0.06(cid:1)?1 , an amount of ￡P in 6 months has a present value of P ×(cid:0) 1+ 0.06(cid:1)?6 , etc. In other
12 12
words, the present value is exactly how much the bank is willing to pay now in order to receive ￡P
in m months.
The second step, is to now find the present value of all such monthly payments of ￡P for 20
2 (cid:88)0×12 (cid:18) 0.06(cid:19)?m
years. This is P 1+ and we must have
12
m=1
(cid:88)240 (cid:18) 0.06(cid:19)?m (cid:88)240 (cid:18) 200(cid:19)m
100,000=P 1+ =P
12 201
m=1 m=1
3exactlyasbeforeusetheassumptiononAandrearrange
4thinkofdepositing￡1inthebankandcollectinginterestattheprescribedrate
3. BONDS 13
The latter is a geometric sum that we can explicitly calculate; then solving for P :
100,000 100,000 100,000
P = = = ≈716.43,
(cid:80)240 (cid:0)200(cid:1)m 2001?(200/201)240 200(cid:16) 1?(cid:0)200(cid:1)240(cid:17)
m=1 201 201 1/201 201
which is how much you will be paying the bank every month for the next 20 years.
3. Bonds
The money market consists of risk-free securities. An example is a bond, which is a financial
security promising the holder a sequence of guaranteed future payments. Risk-free here means that
these payments will be delivered with certainty.
Definition. A bond is a contract in which the issuer commits to pay the holder of the bond
payments of certain amounts on certain dates.
Typically, a bond would specify
a maturity date, which is the date when final payments are made,
a face value, which is an amount paid on the maturity date, and
coupons: a sequence of payments by the issuer, their amounts and payment dates.
In effect, the person or institution who buys the bond is lending money to the bond writer.
Bonds are issued mainly by governments and corporations often in auctions, and can be traded
in exchanges (sometimes referred to as secondary markets). There are many kinds of bonds, like
treasurybillsandnotes,mortgageanddebenturebonds,commercialpapers,andotherswithvarious
arrangements concerning the issuing institution, duration, number of payments, embedded rights
and guarantees. 5
Example. Suppose that we have a bond with face value of ￡100, paying 6% annual interest,
with 2 years to maturity and with semi-annual coupons.
This means that the bond pays the holder 6% of the face value (0.06×100=6) in two install-
mentseveryyear(becauseithassemi-annualcoupons),whichmeansthatitpays￡3every6months.
Consequently, this bond entitles its owner to the following payments:
￡3 in 6 months (first coupon payment),
￡3 in 12 months (second coupon payment),
￡3 in 18 months (third coupon payment), and
￡3+100 in 24 months (fourth -and last- coupon payment plus face value since we have reached the
maturity date).
The simplest case of a bond is a zero-coupon bond, which involves just a single payment. (In
practice, this is not very common, but it is useful for theoretical purposes.)
Definition. A zero coupon bond has no coupons, i.e., the issuer makes only one payment: the
face value is paid at the maturity of the bond.
ThepriceP ofa￡1facevaluezerocouponbondwithmaturityintyearsispreciselytheamount
of money you are willing to pay now in order to be receive ￡1 in t years. So P is the present value
of ￡1 paid t years into the future.
5YoucanfinddescriptionsofbondsissuedbytheUKgovernmenthere.
14 1. INTEREST, PRESENT VALUE AND BONDS
Proposition 1. Let P be the price of a zero-coupon bond with face value ￡1 and maturing in
t years. Let r be the (continuously compounded) interest rate for t-year deposits and loans. Then
P =e?rt.
In proving this statement we will make our usual assumption about the world: there are no
arbitrage opportunities. We also assume that everyone can issue bonds, lend and borrow money
under the same terms (these assumptions certainly don’t apply to you and me, but they do apply
approximately to stable governments and large corporations).
Proof. We will consider two cases (higher/lower and argue by contradiction)
If P >e?rt 6,
issue a zero-coupon bond with face value of ￡1 and maturing in t years, sell it, and collect
its price ￡P,
deposit ￡P for t years earning a (continuously compounded) interest rate of r,
wait t years,
collect the balance of your deposit amounting to Pert,
pay the owner the face value of the bond you issued amounting to ￡1,
pocket Pert?1>0.
This strategy, which required no initial investment and produced a certain profit, is an arbitrage
strategy, which we assume cannot exist.
If P
borrow ￡P for t years; this will earn the lender a (continuously compounded) interest rate
of r,
use these ￡P to buy a zero-coupon bond with face value of ￡1 and maturing in t years,
wait t years,
receive the face value of your bond amounting to ￡1,
repay the balance of your loan amounting to Pert,
pocket 1?Pert >0.
This is another arbitrage strategy, which we assume not to exist.
Since both inequalities cannot occur, we are forced to deduce that P =e?rt. □
4. Discount and yield curves
Definition (Discount curves and yield curves). A discount curve is a function of time P(t)
whose value at any t ≥ 0 is the present value of one unit of currency paid in t years. Equivalently,
P(t) is the price of a zero coupon bond with face value 1 maturing in t years.
A yield curve is a function Y(t) whose value at any t > 0 is the interest rate paid on t-year
deposits.
The values of Y(t) are also called spot (interest) rates or yields. 8
6wethinkthisishighsowe“sell”tomakeprofit
7wethinkthisislowsowe“buy”hopingtosellhighertomakeprofit
8YoucancheckwhatisthecurrentUKyieldcurvehere.
4. DISCOUNT AND YIELD CURVES 15
With these definitions, we can rephrase Proposition 1 in the previous section as,
P(t)=e?Y(t)t.
Yield curves are not directly observable – these are constructed by finding interest rates which
fit observed prices of assets whose prices depend on interest rates (e.g., bonds) and obtaining Y(t)
from these interest rates by linear interpolation9. This process is often called the bootstrap method.
So far, we valued bonds under the assumption that their issuers will make the payments they
contracted to make. In real life this is not always the case: e.g., banks go bankrupt 10 and govern-
ments default on their debt. 11 The price of bonds and the rates of deposit reflect this: risky bonds
arecheaper anddepositsinshakybankspayhigher interest. Thesameistrueofsecuredloans,e.g.,
mortgages, compared to non-secured loans.
Henceforth when working with interest rates we will disregard risk. We always assume that for
everycurrencythereexistsarisk-freeinstitutionissuingbondsinthatcurrencyandouryieldcurves
and discount factors will refer to these bonds. We also assume that everyone can both deposit and
borrow any amount of cash for any maturity date at these risk-free rates. (Don’t try to tell that to
your bank manager.)
4.1. Constructing yield curves: the bootstrap method. We now describe a method of
producing yield curves based on the prices of bonds illustrated in the following example.
Consider the following bonds with semi-annual coupons:
Face value (￡) Maturity (years) Annual interest Price
100 0.25 0 98.3
100 0.5 0 96.5
100 1 0 93.7
100 1.5 4 95.5
100 2 6 97.2
The bond in the first line will have no coupon payments (since 0×100/2=0), we can therefore
see it as a zero coupon bond, and so P(t), which is the price of a zero coupon with face value ￡1
maturing in 0.25 years will be P(0.25)=0.983. Since P(t)=e?Y(t)t, we can immediately calculate
the spot rate Y(0.25)=?ln0.983 ≈6.86%.
0.25
Arguing in exactly the same way:
P(0.5)=0.965, Y(0.5)≈7.13%
P(1)=0.937, Y(1)≈6.51%.
After that, it becomes more interesting. Annual interest of 4% and face value ￡100 means that
the owner of the bond is due 0.04 × 100 = 4 every year, and since they are getting 2 coupon
payments per year, every coupon payment produces ￡2, and the owner will get these coupon
payments in 6,12, and 18 months (semi-annual coupons). But a ￡2 in 6 months is worth now
￡2e?r0.5 = 2e?Y(0.5)0.5(present value). This is however nothing but 2P(0.5)12. Similarly, the ￡2-
coupon payment we will get in one year is worth 2P(1) now. At the 18 month mark, which is
9thismeansthatgivenpoints(t1,Y(t1)),(t2,Y(t2))wewillfindthelinethatgoesthroughbothofthemanduse
theequationofthislineastheformulaforY(t)fort∈[t1,t2].
10E.g.,the Icelandbankcrisis.
11E.g., Russiandebtdefault
12formulaconnectingPandY
16 1. INTEREST, PRESENT VALUE AND BONDS
the maturity time for this bond, the owner will get the last coupon payment and the price of
the face value, i.e., ￡102, whose present value is 102e?Y(1.5)1.5 = 102P(1.5). This means that the
price we are willing to pay now in order to get this face value along with the coupon payments is
2P(0.5)+2P(1)+102P(1.5), therefore,
2P(0.5)+2P(1)+102P(1.5)=95.5
and since we have already calculated P(0.5),P(1), solving for P(1.5) we obtain P(1.5) ≈ 0.89898
and Y(1.5)≈7.10%.
We argue in exactly the same way for the bond on the 5th line (now the time to maturity is 2
years and we will get 4 coupon payments):
3P(0.5)+3P(1)+3P(1.5)+103P(2)=97.2 and solving for P(2) we obtain
P(2)≈0.8621073672, Y(2)≈7.42%.
Throughout the previous example we thought of P(t) as both the present value of one unit of
currency paid in t years, and as the price of a zero coupon bond with face value 1 maturing in t
years.
5. Forward rates
Definition. Aforward rate agreement isacontractinwhichonepartyagreestopaytheother
party a pre-specified interest rate on a deposit occurring during a specified period of time in the
future.
The point is that we want to agree now for the rate of a future deposit/loan.
Let’s say that you plan to deposit ￡1 t years from now and to withdraw the balance t years
1 2
from now (t > t ), but you would like to agree now on the interest rate r for this deposit. Of
2 1 12
course,themostnaturalquestionwouldbetoaskwhat this rate r should be. Letthespotinterest
12
rates for t and t year deposits be r and r respectively. One option you have is to deposit your
1 2 1 2
￡1 for t
1
years at r 1; after t
1
years you will have 1×er1t1. You could then deposit your balance at
the pre-agreed rate r
12
for t 2?t
1
years. At the end of t
2
years you will have er1t1er12(t2?t1).
A second option available to you is to just deposit ￡1 for t
2
years; balance =1×er2t2.
Intuitively, we would expect that either way should result in the same balance, i.e.,
So we should expect to receive
r t ?r t
er1t1er12(t2?t1) =er2t2 ?r = 2 2 1 1.
12 t ?t
2 1
Definition. Let 0≤t
1 2 1 2 1 2
r t ?r t
The forward rate for the period from t to t , is r = 2 2 1 1.
1 2 12 t ?t
2 1
This number was found by comparing two different scenaria and by intuitively expecting the
resulttobethesame. So,atthispointyoucanthinkofr astheratethatweexpecttobecorrect.
12
Nowthatwehaveanideaaboutwhattheanswer“ought”tobe,wewilluseno-arbitragearguments
to reach a rigorous proof.
Proposition 2. In a market with no arbitrage opportunities the interest rates of forward rate
agreements are equal to the corresponding forward rates.
6. EXERCISES 17
Proof. Let the forward rate agreement time period start in t years and end in t years, let r
1 2
be the pre-specified interest rate and let
r t ?r t
r = 2 2 1 1
12 t ?t
2 1
be the corresponding forward rate for the period from t to t .
1 2
If r >r ,
12
enter the agreement as a depositor,
borrow e?r1t1 for t
2
years and deposit this amount for t
1
years.
Attimet=t
1
receivethebalanceofyourdepositwhichwillbee?r1t1er1t1 =1,anddeposit
it until t=t earning an interest rate of r.
2
At time t=t
2
obtain the balance of your deposit which will equal er(t2?t1) and repay the
balance of your loan, which will be e?r1t1er2t2 =er2t2?r1t1.
Since
r t ?r t
r > 2 2 1 1 ?er(t2?t1) >er2t2?r1t1
t ?t
2 1
you can now pocket the difference er(t2?t1)?er2t2?r1t1 >0.
If r
12
enter the agreement as a borrower,
borrow e?r1t1 for t
1
years and deposit this amount for t
2
years.
? At time t=t borrow ￡1 until t=t paying an interest rate of r, and use this ￡1 to repay
1 2
your loan.
? At time t = t
2
receive the balance of your deposit which will be e?r1t1er2t2 = er2t2?r1t1,
and use it to repay the balance of your second loan which will be er(t2?t1).
? Now
r t ?r t
r < 2 2 1 1 ?er(t2?t1)
t ?t
2 1
and you can pocket the difference er2t2?r1t1 ?er(t2?t1) >0.
□
6. Exercises
(1) Find the price of a bond with face value ￡100 and ￡5 annual coupons that matures in 4
years, given that the continuous compounding rate is i) 8% or ii)5%. Hint: in this case
they tell us in advance what the coupon payment is.
(2) How long will it take to earn ￡1 in interest if ￡1,000,000 is deposited at 10% compounded
continously?
(3) What is the interest rate if a deposit subject to annual compounding is doubled after 10
years?
(4) Supposethatwehaveadepositof￡100attractinginterestat10%compounded(i)annually,
and (ii) semi-annually. Find the future value of the deposit after 2 years in both cases.
(5) Whatinitialinvestmentsubjecttoannualcompoundingat12%isneededtoproduce￡1,000
after 2 years?
(6) You have ￡1,000 and you are given the choice between depositing your money attracting
interest (i)at 15% compounded daily, or (ii) at 15.5% compounded semi-annually. Which
option will you choose?
(7) Howmuchcanyouborrowiftheinterestrateis18%, youcanaffordtopay￡10,000atthe
end of each year, and you want to clear the loan in 10 years?
18 1. INTEREST, PRESENT VALUE AND BONDS
(8) Consider the following five bonds with face value of ￡100:
Time to maturity Annual interest Bond price
(in years) (paid every 6 months) (in ￡)
0.25 0 99.0
0.5 0 97.8
1.0 0 95.5
1.5 8% 104.5
2.0 12% 113.0
(a) Find the 0.25, 0.5 and 1-year spot interest rates.
(b) Use the bootstrap method to find the 1.5 and 2-year spot interest rates.
(c) Now, suppose that you are offered by a risk-free institution the opportunity to deposit
or borrow ￡1000 in six months for a period of six months earning an interest rate of 5%.
Describe in detail an arbitrage opportunity available to you.
(9) Consider any traded asset whose value v in T years is known with certainty. Prove that
T
the current value of the asset must be v e?rT where r is the continuously-compounded
T
T-years interest rate.
CHAPTER 2
Forward and Futures Contracts
Oneofthemainaimsofthiscourseistofindthecorrect(i.e.,no-arbitrage)pricesofderivatives.
We now define these formally as follows.
Definition. A derivative is an asset whose price depends on the value of other assets.
In this chapter we study two derivatives, namely, forward contracts and futures contracts.
Definition. Aforward contract isanagreementbetweentwopartiesinwhichonepartyagrees
to sell a particular asset at an agreed price (the forward price), on a certain date (maturity date or
delivery date), and the other party agrees to buy that asset at that price and on that date.
There is an asymmetry in the parties to a forward contract: The party receiving the asset has
a long position and the party delivering the asset has a short position. 1
Forward contracts are entered freely by both parties and have no cost. The question we now
raise is what should the forward price which makes a forward contract valueable be. To answer this
questionwewilldistinguishbetweenforwardcontractsonassetsthatprovideincome(e.g., dividend
yielding shares in a company) and those that do not (e.g., bars of gold). To do this we’ll need to
assume short-selling which we describe next
1. Short-selling
Definition. Short-selling is selling something which one does not own.
Thus one may own a positive or a negative amount of assets.
In this course we will assume that any tradable asset can be bought, sold and
short-sold.
Short-selling sounds vaguely criminal, after all, it involves selling something owned by someone
else without their consent– however this is no different from the situation in which a bank takes
your deposit and lends it to someone else. As long as the bank can repay your deposit on request,
all is well and proper.
How does short-selling work? You short sell an asset through a broker, the broker borrows the
asset from another client and sells it, and gives you the proceeds. You can use most of this money
to invest in assets through the broker, but you have to keep some of the proceeds in cash held in
a margin account managed by the broker. The balance on the margin account should be a certain
percentage of the spot value of the short-sold asset, and it is balanced daily: if the spot price of
the short-sold asset goes up, the balance in the margin account needs to be increased and you will
receive a margin call, i.e., a demand to add cash to your margin account. The broker has the right
to sell your assets and to use your balance in the margin account to buy back the short-sold asset
and return it to the other client, if the other client demands it or if you do not respond to a margin
call.
1Infinance,“short”oftenmeans“sell”and“long”oftenstandsfor“buy”.
19
20 2. FORWARD AND FUTURES CONTRACTS
To “close” a short position, you buy back asset and return it to its owner. A profit is made if
the sale price was higher than the purchase price, i.e., if the asset’s price decreased.
We will use repeatedly the following consequence of our ability to short-sell assets:
Theorem 3. (Present-Future Price Comparison) Let p and p be the current prices of two
1 2
assets, and suppose that we know with certainty that their prices q and q at some point in the
1 2
future satisfy q ≤ q . Then p ≤ p . In particular, if we know that two assets will have the same
1 2 1 2
price at some point in the future, then the two assets must have the same price at any prior time.
Proof. Assume, for a contradiction, that p > p . Short sell item 1, and buy item 2 (we sell
1 2
the “expensive” and buy the “cheap”). The balance after this: p ?p > 0 (by our assumption).
1 2
Then wait until we reach the time when the prices are q ,q , respectively. We buy back item 1 (to
1 2
return to the rightful owner) and sell item 2. The balance of these transactions is q ?q ≥ 0 (by
2 1
assumption). To sum it up, our final balance is (q ?q )+(p ?p )>0. This is a contradiction to
2 1 1 2
the no arbitrage assumption since we exhibited how to make a profit with no investment. □
2. The forward price
In what follows, we will make the usual assumption of no arbitrage. We will also assume that
therearenotaxesongoodsbought/sold,thattherearenotransactioncosts(e.g.,thebrokerdoesn’t
ask for a commission) – these assumptions are often referred to as “no friction” assumptions. Last,
wewillalsoassumethattraderscanborrowandlendmoneyatthesame(risk-free)rate, aswehave
done so far. We are now ready to deduce the forward price in a forward contract:
Theorem 4. Consider a forward contract on an asset which provides no income and whose
price at the present (its spot price) is S. The forward price is F = SerT where T is the time to
maturity and r =Y(T), the T-year spot interest rate.
Proof. If F >SerT there is the following arbitrage strategy:
take a short position in the forward contract,
borrow ￡S for T years at spot interest rate of r,
buy the asset for ￡S [remember that we have agreed to sell it at time T],
wait T years,
deliver (sell) the asset and collect ￡F,
use ￡SerT to repay the loan,
pocket the difference F ?SerT >0.
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