Part Deux of the Rates modeling presentation. This one is a little breather before attacking part III. Usually the morning or a couple of days after part I to review the basics. Have tried to put some graphs. More slides than less so that you can pick and choose when using them / giving a talk.
Lots of errors and typos, hope you enjoy them
2. Luc_Faucheux_2020
Couple of notes on those slides
ยจ Those are part II of the the slides on Rates
ยจ Part I was using very little math, and in particular we glanced over the fact that rates could
move
ยจ Every calculation was done in a โdeterministicโ manner or zero volatility
ยจ We revisit this here, and shows why that was justified for Bonds and โregularโ swaps in a
graphical manner, introducing the commonly used graphs for cash flows that you will find in
most textbooks
ยจ Again, trying to limit the math to a minimum here, but refining a little the notations as we go
along in order to make it rigorous when needed, and gaining intuition when we realize that
our current notation is incomplete and needs to be modified
ยจ This deck can be viewed as a โlittle graphical breakโ where we refine the concept of discount
factor before we attack part III.
ยจ This is usually a morning presentation to go easy before lunch or coffee before we attack
part III which will be a little harder
2
4. Luc_Faucheux_2020
Notations and conventions in the rates world
ยจ The Langevin equation is quite commonly used when modeling interest rates.
ยจ Since interest rates are the โspeedโ or โvelocityโ of the Money Market Numeraire, it is quite
natural to have thought about using the Langevin equation which represents the โspeedโ of
a Brownian particle.
ยจ As a result, a number of quantities in Finance are related to the exponential of the integral
over time of the short-term rate (instantaneous spot rate)
ยจ For example (Fabio Mercurio p. 3), the stochastic discount factor ๐ท(๐ก, ๐) between two time
instants ๐ก and ๐ is the amount at time ๐ก that is โequivalentโ to one unit of currency payable
at time ๐, and is equal to
ยจ ๐ท ๐ก, ๐ =
!(#)
!(%)
= exp(โ โซ#
%
๐ ๐ . ๐๐ )
ยจ The Bank account (Money-market account) is such that:
ยจ ๐๐ต ๐ก = ๐ ๐ก . ๐ต ๐ก . ๐๐ก with ๐ต ๐ก = 0 = 1
ยจ ๐ต ๐ก = exp(โซ&
#
๐ ๐ . ๐๐ )
4
5. Luc_Faucheux_2020
Notations and conventions in the rates world - II
ยจ Mostly following Mercurioโs conventions in the this section.
ยจ We can define a very useful quantity: ZCB: Zero Coupon Bond also called pure discount
bond. It is a contract that guarantees the holder the payment on one unit of currency at
maturity, with no intermediate payment.
ยจ ๐ง๐ ๐ก, ๐ is the value of the contract at time ๐ก
ยจ ๐ง๐ ๐, ๐ = 1
ยจ Note that ๐ง๐ ๐ก, ๐ is a known quantity at time ๐ก. It is the value of a contract (like a Call
option is known, it is no longer a stochastic variable)
ยจ On the other hand,
ยจ ๐ท ๐ก, ๐ =
!(#)
!(%)
= exp(โ โซ#
%
๐ ๐ . ๐๐ ) and ๐ต ๐ก = exp(โซ&
#
๐ ๐ . ๐๐ )
ยจ Are just functions of ๐ ๐ . If we place ourselves in a situation where the short-term rate
๐ ๐ is a stochastic process then both the MMN (BAN) noted ๐ต ๐ก (Money market
numeraire, or Bank Account Numeraire), as well as the discount factor ๐ท ๐ก, ๐ , should not
be expected to not be stochastic (unless a very peculiar situation)
5
6. Luc_Faucheux_2020
Notations and conventions in the rates world - III
ยจ In the case of deterministic short-term rate, there is no stochastic component.
ยจ In that case:
ยจ ๐ท ๐ก, ๐ = ๐ง๐ ๐ก, ๐
ยจ When stochastic, ๐ง๐ ๐ก, ๐ is the expectation of ๐ท ๐ก, ๐ , like the Call option price was the
expectation of the call payoff.
ยจ From the Zero Coupon bond we can define a number of quantities:
6
7. Luc_Faucheux_2020
Notations and conventions in the rates world -IV
ยจ Continuously compounded spot interest rate:
ยจ ๐ ๐ก, ๐ = โ
'(()*(#,%))
,(#,%)
ยจ Where ๐(๐ก, ๐) is the year fraction, using whatever convention (ACT/360, ACT/365, 30/360,
30/250,..) and possible holidays calendar we want. In the simplest case:
ยจ ๐ ๐ก, ๐ = ๐ โ ๐ก
ยจ ๐ง๐ ๐ก, ๐ . exp ๐ ๐ก, ๐ . ๐ ๐ก, ๐ = 1
ยจ ๐ง๐ ๐ก, ๐ = exp โ๐ ๐ก, ๐ . ๐ ๐ก, ๐
ยจ In the deterministic case:
ยจ ๐ง๐ ๐ก, ๐ = exp โ๐ ๐ก, ๐ . ๐ ๐ก, ๐ = ๐ท ๐ก, ๐ =
!(#)
!(%)
= exp(โ โซ#
%
๐ ๐ . ๐๐ )
ยจ ๐ ๐ก, ๐ =
-
, #,%
. โซ#
%
๐ ๐ . ๐๐
7
9. Luc_Faucheux_2020
Notations and conventions in the rates world - VI
ยจ Annually compounded spot interest rate
ยจ ๐ฆ ๐ก, ๐ =
-
)*(#,%)!/#(%,') โ 1
ยจ Or alternatively, in the bootstrap form
ยจ (1 + ๐ฆ ๐ก, ๐ ). ๐ง๐ ๐ก, ๐ -/, #,% = 1
ยจ (1 + ๐ฆ ๐ก, ๐ ), #,% . ๐ง๐ ๐ก, ๐ = 1
ยจ ๐ง๐ ๐ก, ๐ =
-
(-/1 #,% )# %,'
ยจ In the deterministic case:
ยจ ๐ง๐ ๐ก, ๐ =
-
(-/3 #,% )# %,' = ๐ท ๐ก, ๐ =
!(#)
!(%)
= exp(โ โซ#
%
๐ ๐ . ๐๐ )
9
10. Luc_Faucheux_2020
Notations and conventions in the rates world - VII
ยจ ๐-times per year compounded spot interest rate
ยจ ๐ฆ4 ๐ก, ๐ =
4
)*(#,%)!/)#(%,') โ ๐
ยจ Or alternatively, in the bootstrap form
ยจ (1 +
-
4
๐ฆ4 ๐ก, ๐ ). ๐ง๐ ๐ก, ๐ -/4, #,% = 1
ยจ (1 +
-
4
๐ฆ4 ๐ก, ๐ )4., #,% . ๐ง๐ ๐ก, ๐ = 1
ยจ ๐ง๐ ๐ก, ๐ =
-
(-/
!
)
.3) #,% )).# %,'
ยจ In the deterministic case:
ยจ ๐ง๐ ๐ก, ๐ =
-
(-/
!
)
.3) #,% )).# %,'
= ๐ท ๐ก, ๐ =
!(#)
!(%)
= exp(โ โซ#
%
๐ ๐ . ๐๐ )
10
11. Luc_Faucheux_2020
Notations and conventions in the rates world - VIII
ยจ In bootstrap form which is the intuitive way:
ยจ Continuously compounded spot: ๐ง๐ ๐ก, ๐ = exp โ๐ ๐ก, ๐ . ๐ ๐ก, ๐
ยจ Simply compounded spot: ๐ง๐ ๐ก, ๐ =
-
-/, #,% .1 #,%
ยจ Annually compounded spot: ๐ง๐ ๐ก, ๐ =
-
(-/3 #,% )# %,'
ยจ ๐-times per year compounded spot ๐ง๐ ๐ก, ๐ =
-
(-/
!
)
.3) #,% )).# %,'
11
12. Luc_Faucheux_2020
Notations and conventions in the rates world - IX
ยจ In the small ๐ ๐ก, ๐ โ 0 limit (also if the rates themselves are such that they are <<1)
ยจ In bootstrap form which is the intuitive way:
ยจ Continuously compounded spot: ๐ง๐ ๐ก, ๐ = 1 โ ๐ ๐ก, ๐ . ๐ ๐ก, ๐ + ๐ช(๐5. ๐5)
ยจ Simply compounded spot: ๐ง๐ ๐ก, ๐ = 1 โ ๐ ๐ก, ๐ . ๐ ๐ก, ๐ + ๐ช(๐5. ๐5)
ยจ Annually compounded spot: ๐ง๐ ๐ก, ๐ = 1 โ ๐ฆ ๐ก, ๐ . ๐ ๐ก, ๐ + ๐ช(๐5. ๐ฆ5)
ยจ ๐-times per year compounded spot ๐ง๐ ๐ก, ๐ = 1 โ ๐ฆ4 ๐ก, ๐ . ๐ ๐ก, ๐ + ๐ช(๐5. ๐ฆ4
5)
ยจ So in the limit of small ๐ ๐ก, ๐ (and also small rates), in particular when ๐ โ ๐ก, all rates
converge to the same limit we call
ยจ ๐ฟ๐๐ ๐ โ ๐ก = lim
%โ#
(
-.)* #,%
, #,%
)
12
13. Luc_Faucheux_2020
Notations and conventions in the rates world - X
ยจ In the deterministic case using the continuously compounded spot rate for example:
ยจ ๐ง๐ ๐ก, ๐ = exp โ๐ ๐ก, ๐ . ๐ ๐ก, ๐ = ๐ท ๐ก, ๐ =
!(#)
!(%)
= exp(โ โซ#
%
๐ ๐ . ๐๐ )
ยจ ๐ ๐ก, ๐ =
-
, #,%
. โซ#
%
๐ ๐ . ๐๐
ยจ When ๐ โ ๐ก, ๐ ๐ก, ๐ โ ๐ (๐ก)
ยจ So: ๐ฟ๐๐ ๐ โ ๐ก = lim
%โ#
(
-.)* #,%
, #,%
) = ๐ (๐ก)
ยจ So ๐ (๐ก) can be seen as the limit of all the different rates defined above.
ยจ You can also do this using any of the rates defined previously
ยจ You find sometimes the adjective โinstantaneousโ to describe the quantity ๐ (๐ก) which is the
limit of small time intervals: ๐ฟ๐๐ ๐ โ ๐ก = ๐ฟ๐๐ ๐ โ 0
13
15. Luc_Faucheux_2020
Some useful graphs
ยจ We will focus on the case of the simply compounded spot rate, where we are at time ๐ก
ยจ Simply compounded spot: ๐ง๐ ๐ก, ๐ =
-
-/, #,% .1 #,%
ยจ What is the value ๐๐ฃ of a SPOT contract that when entered at time ๐ก guarantees the
payment of one unit of currency at time ๐ (remember that we are at time ๐ก)
ยจ To give a quick glance at the numeraire framework, we will say that we choose the Zero
Coupon bond as a numeraire to value claims.
ยจ In that case the ratio of a claim to that numeraire is a martingale, and in particular at
maturity of the contract
ยจ ๐ผ
-
)* %,%
=
78
)*(#,%)
= ๐ผ
-
)* %,%
= ๐ผ
-
-
= 1 since ๐ง๐ ๐, ๐ = 1
ยจ So the value today at time ๐ก of that spot contract is ๐๐ฃ(๐ก) = ๐ง๐ ๐ก, ๐
ยจ Note that we could also take this as the definition of the โmoney curveโ, we could assume
for now that at time ๐ก the quantities ๐ง๐ ๐ก, ๐ are known or observable and give us the value
of a contract that pays $1 at time ๐
15
16. Luc_Faucheux_2020
Some useful graphs - II
ยจ ๐๐ฃ(๐ก) = ๐ง๐ ๐ก, ๐ is the value at time ๐ก of a SPOT contract that will return $1 at time ๐
ยจ Note that once entered into this contract, the value will change over time, but we know that
we will receive $1 at time ๐ (hint: that is why it is called Fied-Income, unlike equities for
example)
ยจ In lots of textbooks you will see the usual โarrowโ graphs.
16
๐ก๐๐๐
๐ก
๐
๐๐ฃ(๐ก) = ๐ง๐ ๐ก, ๐
๐ง๐ ๐, ๐ = $1
17. Luc_Faucheux_2020
Some useful graphs - III
17
๐ก๐๐๐
๐ก
๐
๐๐ฃ(๐ก) = ๐ง๐ ๐ก, ๐
๐ง๐ ๐, ๐ = $1
ยจ You pay ๐ง๐ ๐ก, ๐ today at time ๐ก in order to get $1 at time ๐
ยจ The net value today of those two cashflows is 0
19. Luc_Faucheux_2020
Some useful graphs - IV
ยจ Simply compounded spot: ๐ง๐ ๐ก, ๐ =
-
-/, #,% .1 #,%
ยจ ๐ง๐ ๐ก, ๐ . (1 + ๐ ๐ก, ๐ . ๐ ๐ก, ๐ ) = 1
ยจ ๐ง๐ ๐ก, ๐ + ๐ง๐ ๐ก, ๐ . ๐ ๐ก, ๐ . ๐ ๐ก, ๐ = 1
ยจ ๐ง๐ ๐ก, ๐ is the value ๐๐ฃ of a SPOT contract that when entered at time ๐ก guarantees the
payment of one unit of currency at time ๐ (remember that we are at time ๐ก)
ยจ 1 is 1
ยจ Similarly, ๐ง๐ ๐ก, ๐ . ๐ ๐ก, ๐ . ๐ ๐ก, ๐ is the value of a SPOT contract that when entered at time ๐ก
guarantees the payment of (๐ ๐ก, ๐ . ๐ ๐ก, ๐ ) of currency at time ๐ (remember that we are at
time ๐ก, and so at time ๐ก all quantities are known). This will NOT be the case when we start
looking at forward. Or at least we will have to exercise some caution.
19
21. Luc_Faucheux_2020
Some useful graphs - VI
ยจ In most textbooks the squiggly line is to represent a FLOATING payment.
ยจ In the general case the value of this payment is not known and fixed, but the payment is
usually based on a rate, like BBA LIBOR, which can be published in a manner that is not
completely transparent or obvious, hence the squiggly representation usually to represent
โfloatingโ payments.
ยจ Note that in our case, we have defined this payment as a function of the ZC
ยจ By the way, it is surprisingly hard to make a squiggly line in PowerPoint
21
22. Luc_Faucheux_2020
Some useful graphs โ VI-a
ยจ Note that we know ๐ง๐ ๐ก, ๐
ยจ So we know
-
-/, #,% .1 #,%
ยจ So we know ๐ ๐ก, ๐
ยจ We know that the value of receiving $1 at time ๐ is ๐ง๐ ๐ก, ๐
ยจ So we know that the value of receiving $X at time ๐ is (๐. ๐ง๐ ๐ก, ๐ )
ยจ If we set ๐ ๐ก, ๐ . ๐ ๐ก, ๐ = ๐, then the value of receiving ๐ ๐ก, ๐ . ๐ ๐ก, ๐ time ๐ is :
ยจ ๐ ๐ก, ๐ . ๐ ๐ก, ๐ . ๐ง๐ ๐ก, ๐ = 1 โ ๐ง๐ ๐ก, ๐
ยจ That is the value of receiving $1 at time ๐ก and paying $1 at time ๐
ยจ That is true if the quantity ๐ ๐ก, ๐ is KNOWN, which it is at time ๐ก as we defined it.
ยจ In order to show that this quantity is also known at time before ๐ก, we need to rely on the
replication concept, also called the โlaw of one priceโ or โno-arbitrageโ
22
23. Luc_Faucheux_2020
Some useful graphs - VII
ยจ In most textbooks also a straight line is for a FIXED payment
ยจ So when we will deal with swaps we will have a stream of FIXED payments (with a fixed
coupon like a FIXED coupon bond) versus a stream of FLOAT cashflows that will be a function
of some index that is โfloatingโ (not known at inception). For most swaps that index is the
LIBOR rate, soon to be replaced with SOFR.
23
24. Luc_Faucheux_2020
Some useful graphs - VIII
ยจ Notional exchange:
ยจ A contract will be known to have a Notional exchange when the principal amount (notional(
is ACTUALLY being paid in/out as an actual cash flows.
ยจ In the example we just saw the notional ($1) is paid out at the beginning of the period at
time ๐ก and received back at the end of the period at time ๐
ยจ The value at time ๐ก of this notional exchange will be:
ยจ ๐๐ฃ ๐ก = ๐ง๐ ๐ก, ๐ โ 1
ยจ Since: ๐ง๐ ๐ก, ๐ + ๐ง๐ ๐ก, ๐ . ๐ ๐ก, ๐ . ๐ ๐ก, ๐ = 1
ยจ ๐๐ฃ ๐ก = ๐ง๐ ๐ก, ๐ โ 1 = โ๐ง๐ ๐ก, ๐ . ๐ ๐ก, ๐ . ๐ ๐ก, ๐
ยจ SO, paying out $1 now at time ๐ก and receiving it back at time ๐ has a negative present value
(because usually rates are positive and money has positive time value, but then again rates
are not necessarily positive right?), and that negative present value is EXACTLY equal and
opposite to the present value of receiving at time ๐ a cash flow equal to ๐ ๐ก, ๐ . ๐ ๐ก, ๐
24
28. Luc_Faucheux_2020
Some useful graphs - XI
ยจ Receiving $1 today at time ๐ก and paying it back at time ๐ (doing a notional principal
exchange from period start to period end) has the SAME exact value as not exchanging any
principal (notional) and just receiving at the end of the period a cashflow equal to:
ยจ ๐ ๐ก, ๐ . ๐ ๐ก, ๐
ยจ Where the simply compounded spot rate ๐ ๐ก, ๐ is given by:
ยจ ๐ ๐ก, ๐ =
-
,(#,%)
.
-.)*(#,%)
)*(#,%)
ยจ Or in a more intuitive bootstrap manner:
ยจ ๐ง๐ ๐ก, ๐ =
-
-/, #,% .1 #,%
= ๐ง๐ ๐ก, ๐ก .
-
-/, #,% .1 #,%
since ๐ง๐ ๐ก, ๐ก = 1
28
30. Luc_Faucheux_2020
Notations and conventions (lower case and UPPER CASE)
ยจ I will try to stick to a convention where the the lower case denotes a regular variable, and an
upper case denotes a stochastic variable, as in before:
ยจ
97(:,#)
9#
= โ
9
9:
[๐ ๐ก . ๐ ๐ฅ, ๐ก โ
9
9:
[
-
5
๐ ๐ก 5. ๐ ๐ฅ, ๐ก ]]
ยจ ๐ ๐ก; โ ๐ ๐ก< = โซ#=#<
#=#;
๐๐ ๐ก = โซ#=#<
#=#;
๐(๐ก). ๐๐ก) + โซ#=#<
#=#;
๐(๐ก). ๐๐(๐ก)
ยจ ๐๐ ๐ก = ๐ ๐ก . ๐๐ก + ๐(๐ก). ๐๐
ยจ Where we should really write to be fully precise:
ยจ ๐ ๐ฅ, ๐ก = ๐>(๐ฅ, ๐ก|๐ฅ ๐ก = ๐ก& = ๐&, ๐ก = ๐ก&)
ยจ PDF Probability Density Function: ๐>(๐ฅ, ๐ก)
ยจ Distribution function : ๐>(๐ฅ, ๐ก)
ยจ ๐> ๐ฅ, ๐ก = ๐๐๐๐๐๐๐๐๐๐ก๐ฆ ๐ โค ๐ฅ, ๐ก = โซ3=.?
3=:
๐> ๐ฆ, ๐ก . ๐๐ฆ ๐>(๐ฅ, ๐ก) =
9
9:
๐> ๐ฅ, ๐ก
30
31. Luc_Faucheux_2020
Notations and conventions (Spot and forward)
ยจ So far we have defined quantities depending on 2 variables in time:
ยจ For example, in the case of the continuously compounded spot interest rate:
ยจ ๐ ๐ก, ๐ = โ
'(()*(#,%))
,(#,%)
ยจ It is the constant rate at which an investment of ๐ง๐(๐ก, ๐) at time ๐ก accrues continuously to
yield 1 unit of currency at maturity ๐.
ยจ ๐ง๐ ๐ก, ๐ = exp โ๐ ๐ก, ๐ . ๐ ๐ก, ๐
ยจ It is observed at time ๐ก until maturity, hence the naming convention SPOT
ยจ CAREFUL: Spot sometimes depending on the markets (US treasury) could mean the
settlement of payment (so T+2). A SPOT-starting swap does NOT start today but T+2, subject
to London and NY holidays
ยจ So different currencies will have different definitions of what SPOT means
ยจ ALWAYS CHECK WHAT PEOPLE MEAN BY โSPOTโ
31
32. Luc_Faucheux_2020
Notations and conventions (Spot and forward) - II
ยจ So really we should have expressed:
ยจ ๐ง๐ ๐ก, ๐ = exp โ๐ ๐ก, ๐ . ๐ ๐ก, ๐ as:
ยจ ๐ง๐ ๐ก, ๐ก, ๐ = exp โ๐ ๐ก, ๐ก, ๐ . ๐ ๐ก, ๐
ยจ This is a SPOT contract that when entered at time ๐ก guarantees the payment of one unit of
currency at time ๐
ยจ To give a quick glance at the numeraire framework, we will say that we choose the Zero
Coupon bond as a numeraire to value claims.
ยจ In that case the ratio of a claim to that numeraire is a martingale, and in particular at
maturity of the contract
ยจ ๐ผ
-
)* %,%,%
=
@
)*(#,#,%)
= ๐ผ
-
)* %,%,%
= ๐ผ
-
-
= 1 since ๐ง๐ ๐, ๐, ๐ = 1
ยจ So the value of a contract at time ๐ก that pays 1 at time ๐ is:
ยจ ๐๐ฃ(๐ก) = ๐ง๐(๐ก, ๐ก, ๐)
32
33. Luc_Faucheux_2020
Notations and conventions (Spot and forward) - III
ยจ This can be viewed at obviously simple or very complicated depending how you look at it
ยจ In the โdeterministicโ world of curve building it is quite simple, until you realize that rates do
have volatility.
ยจ In essence, the question is the following: When pricing swaps and bonds, you only need a
yield curve and you do not need to know anything about the dynamics of rates (volatility
structure), even though you know that they do move.
ยจ Why is that ?
ยจ The answer in short, is that you can only do that for products (coincidentally bonds and
swaps that are 99% of the gamut of products out there) that are LINEAR as a function of the
numeraire which we chose to be the Zero Coupon Bonds, or discount factors.
ยจ There is a neat trick that shows that swaps are LINEAR functions of the discount factors
ยจ magic trick,
:
-/:
=
:/-.-
-/:
=
-/:.-
-/:
= 1 โ
-
-/:
or more simply: ๐ฅ = ๐ฅ + 1 โ 1
33
34. Luc_Faucheux_2020
Notations and conventions (Spot and forward) - IV
ยจ For ANY other structure for which the payoff is NOT a LINEAR functions of the discount
factors, the payoff will exhibit some convexity.
ยจ We showed in the options module that the convexity is the origin of the option value
ยจ The option value is essentially the convexity adjustment, which will require some dynamics
(volatility) to be computed.
ยจ So the fact that a swap desk does not need to know about volatility is somewhat of an
anomaly, and also a result of how the business evolved historically
34
35. Luc_Faucheux_2020
Notations and conventions (Spot and forward) - V
ยจ So ๐๐ฃ(๐ก) = ๐ง๐(๐ก, ๐ก, ๐) is the SPOT value at time ๐ก of a contract that will return 1 unit of
currency at maturity ๐
ยจ From that value we can define a number of arbitrary rates.
ยจ Remember there is ONLY one value of the Zero Coupon Bond, but as many different rates as
you have conventions.
ยจ So the only thing that really matters is the Zero Coupon Bond, as a rule of thumb
ยจ When in doubt, always go back to the Zero Coupon Bond
ยจ We would naturally want to extend this to FORWARD contract: what is the value at time ๐ก&
of a contract that will start at time ๐ก and will return one unit of currency at time ๐
ยจ So we are looking if we can extend the SPOT ๐ง๐(๐ก&, ๐ก&, ๐) to something like a forward
๐ง๐(๐ก&, ๐ก, ๐)
ยจ Do not worry, in part III we will revisit the notation again to make it fully explicit, did not
want to start with the full Monthy first, and go through the steps to build it up
35
36. Luc_Faucheux_2020
Notations and conventions (Spot and forward) - VI
ยจ That seems to be doable based on the time value of money argument.
ยจ At time ๐ก&, I can enter into a SPOT contract that returns $1 at time ๐ก
ยจ The value of that contract is ๐ง๐(๐ก&, ๐ก&, ๐ก)
ยจ At time ๐ก&, I can enter into a SPOT contract that returns $1 at time ๐
ยจ The value of that contract is ๐ง๐(๐ก&, ๐ก&, ๐)
ยจ At time ๐ก&, I can enter into a FORWARD contract, an agreement to be delivered at time ๐ก a
โthen SPOTโ contract that will return $1 at time ๐
ยจ The question at hand is, what is the value of that FORWARD contract over time?
ยจ All that might look like conventions for conventions sake, but trust me when we get to
future versus forward rates you will be happy to have safe ground to stand on
36
37. Luc_Faucheux_2020
Notations and conventions (Spot and forward) - V
ยจ Also quite frankly even people in the industry get so much confused between spot, forward,
future, forward price, forward contract, forward rate,โฆ that it pays to spend a little time on
this.
ยจ But no worry, I have met very senior managers who were completely clueless about the
difference between spot and forward
ยจ So goes to show that knowledge is not always a good thing, especially when it comes to your
careerโฆ
37
38. Luc_Faucheux_2020
Notations and conventions (Spot and forward) - VI
ยจ We can always define mathematically a number of quantities that we will call forward rates
based on the knowledge now a time ๐ก of the curve {๐ง๐ ๐ก, ๐ก, ๐ }
ยจ Just to make things more clear, we just use the convention that now is ๐ก = 0
ยจ SO today we know the curve {๐ง๐ 0,0, ๐ }
ยจ We can define what we called spot quantities (as of today ๐ก = 0)
ยจ In the case of the simply compounded convention for those quantities, for example the
โsimply compounded spot rate, as of todayโ was defined as:
ยจ ๐ ๐ก = 0, ๐ =
-
,(#=&,%)
.
-.)*(#=&,%)
)*(#=&,%)
ยจ Or alternatively, in the bootstrap form
ยจ ๐ง๐ ๐ก = 0, ๐ =
-
-/, #=&,% .1 #=&,%
38
39. Luc_Faucheux_2020
Notations and conventions (Spot and forward) - VII
ยจ Using the extra set on convention we defined, observed at time ๐ก = 0, all the sets of โsimply
compounded spot ratesโ are:
ยจ ๐ 0,0, ๐ =
-
,(&,&,%)
.
-.)*(&,&,%)
)*(&,&,%)
ยจ Or alternatively, in the bootstrap form
ยจ ๐ง๐ 0,0, ๐ =
-
-/, &,&,% .1 &,&,%
ยจ TOMORROW at time ๐ก = 1 we will have a new curve {๐ง๐ ๐ก = 1, ๐ก = 1, ๐ } with new spot
rates:
ยจ ๐ 1,1, ๐ =
-
,(-,-,%)
.
-.)*(-,-,%)
)*(-,-,%)
ยจ Note that ๐(๐ก, ๐ก, ๐) is a daycount fraction so should really not depend on the time of
observation, ๐ ๐ก, ๐ก, ๐ = ๐(๐ก, ๐) but to avoid confusion we will keep as is, the first time
variable is always the present time
39
40. Luc_Faucheux_2020
Notations and conventions (Spot and forward) - VIII
ยจ NOW we are absolutely free to compute the following quantities:
ยจ Bear in mind that for now those are โjustโ definitions, we have not said anything linking
those quantities to any kind of expectations from a distribution or dynamics
ยจ From todayโs curve: {๐ง๐ ๐ก = 0, ๐ก = 0, ๐ }
ยจ We can compute:
ยจ ๐ง๐ ๐ก = 0, ๐ก = ๐ก-, ๐ก = ๐ก5 = ๐ง๐ ๐ก = 0, ๐ก = 0, ๐ก = ๐ก5 /๐ง๐ ๐ก = 0, ๐ก = 0, ๐ก = ๐ก-
ยจ Of course we have trivially: ๐ง๐ ๐ก = 0, ๐ก = ๐ก-, ๐ก = ๐ก-
ยจ In particular it is useful to define the daily increments:
ยจ ๐ง๐ ๐ก = 0, ๐ก = ๐ก-, ๐ก = ๐ก- + 1 = ๐ง๐ ๐ก = 0, ๐ก = 0, ๐ก = ๐ก- + 1 /๐ง๐ ๐ก = 0, ๐ก = 0, ๐ก = ๐ก-
ยจ And from those what we will define as the โsimply compounded forward rate observed as of
today ๐ก = 0)
ยจ ๐ 0, ๐ก-, ๐ก5 =
-
,(&,#!,#+)
.
-.)*(&,#!,#+)
)*(&,#!,#+)
40
45. Luc_Faucheux_2020
Notations and conventions (Spot and forward) - XIII
ยจ ๐ง๐ ๐ก = 0, ๐ก = 0, ๐ก = ๐กA = โ
-
-/, &,#-,#) .1 &,#-,#)
ยจ Usually if ๐ก7 < ๐ก4 the daycount fraction is positive (not time traveling yet)
ยจ Usually the rates tend to be positive, ๐ 0, ๐ก7, ๐ก4 > 0
ยจ Note that this is proven to be absolutely wrong recently, but most textbooks still have the
usual graph, showing the decrease over time ๐ of the quantity ๐ง๐ 0,0, ๐
ยจ This is the famous time value of money principle
45
54. Luc_Faucheux_2020
Notations and conventions (Spot and forward) โ XIV-h
ยจ Note that in the previous slide I did not display a โforward rate curveโ
ยจ Because a โspot rate curveโ is well defined since only the 3rd argument is a variable
ยจ So the curve would be plotting ๐ 0,0, ๐ก as a function of the time ๐ก, (of course subject to the
chosen daycount convention)
ยจ In the forward case, there are now 2 variables ๐ 0, ๐กBM, ๐กBN , so ether we plot it as a two-
dimensional surface, or we fix the distance (๐กBNโ๐กBM) to a fixed value say 1-month, 3-month,
6-month, and then we can plot the โ1-month forward rate curveโ, or the โ3-month forward
rate curve). Since that depends also on ๐ 0, ๐กBM, ๐กBN , technically for example you would have
to say โthis is the 3-month forward rate based on ACT/360 daycount convention"
ยจ This illustrates that unlike the ZC, there are multiple choices for the forward rates (also the
spot rate)
ยจ ALWAYS use the daily discount factors as a general rule, as per deck I
54
55. Luc_Faucheux_2020
Notations and conventions (Spot and forward) - XV
ยจ We can then extend the relationship we had in spot space:
55
๐ก๐๐๐
๐ก
๐
๐ง๐ ๐, ๐ = $1
๐ง๐ ๐ก, ๐ก = $1
๐ ๐ก, ๐ . ๐ ๐ก, ๐
56. Luc_Faucheux_2020
Notations and conventions (Spot and forward) - XVI
ยจ To the forward space
56
๐ก๐๐๐
๐ก
๐
๐ง๐ 0, ๐, ๐ = $1
๐ง๐ 0, ๐ก, ๐ก = $1
๐ 0, ๐ก, ๐ . ๐ 0, ๐ก, ๐
๐ก = 0
57. Luc_Faucheux_2020
Notations and conventions (Spot and forward) - XVII
ยจ To the forward space
57
๐ก๐๐๐
๐ก!
(๐ก")
๐ง๐ 0, ๐ก5, ๐ก5 = $1
๐ง๐ 0, ๐ก-, ๐ก- = $1
๐ 0, ๐ก!, ๐ก" . ๐ 0, ๐ก!, ๐ก"
๐ก = 0
58. Luc_Faucheux_2020
Notations and conventions (Spot and forward) - XVII
ยจ ๐ 0, ๐ก-, ๐ก5 =
-
,(&,#!,#+)
.
-.)*(&,#!,#+)
)*(&,#!,#+)
ยจ NOTE THAT when we evolve in time and reach the time ๐ก = ๐ก- we will have then the relation
for the simply compounded spot rate
ยจ ๐ ๐ก-, ๐ก-, ๐ก5 =
-
,(#!,#!,#+)
.
-.)*(#!,#!,#+)
)*(#!,#!,#+)
ยจ THE CRUX OF THE MATTER IN FIXED-INCOME:
ยจ How are ๐ 0, ๐ก-, ๐ก5 and ๐ ๐ก-, ๐ก-, ๐ก5 related ?
ยจ Or more exactly keeping with our notation for variables that are known and the ones that
are random and not fixed yet, How are ๐ 0, ๐ก-, ๐ก5 and ๐ฟ ๐ก, ๐ก-, ๐ก5 related, with ๐ก < ๐ก-?
ยจ And ๐ 0, ๐ก-, ๐ก5 and ๐ ๐ก-, ๐ก-, ๐ก5 when ๐ฟ ๐ก, ๐ก-, ๐ก5 gets fixed to ๐ ๐ก, ๐ก-, ๐ก5 at time ๐ก = ๐ก-?
58
59. Luc_Faucheux_2020
Notations and conventions (Spot and forward) - XVIII
ยจ Put it another way, I was not trying to be pedantic for pedantic sake with those extra
seemingly redundant notation.
ยจ The issue is essentially the following to try to gain some intuition:
ยจ Using TODAY curve {๐ง๐ ๐ก = 0, ๐ก = 0, ๐ } for the time value of money (a known quantity), we
can define a number of other quantities, including forward rates like:
ยจ ๐ 0, ๐ก-, ๐ก5 =
-
,(&,#!,#+)
.
-.)*(&,#!,#+)
)*(&,#!,#+)
ยจ At this point, this is just a definition.
ยจ HOWEVER, it is a definition that has the very nice property that the value of receiving at
time ๐ก5 an amount equal to ๐ 0, ๐ก-, ๐ก5 . ๐ 0, ๐ก-, ๐ก5 is a function of the two quantities
๐ง๐(0, ๐ก-, ๐ก5) and ๐ง๐(0, ๐ก-, ๐ก-)
ยจ IN PARTICULAR, the the value of receiving at time ๐ก5 an amount equal to
๐ 0, ๐ก-, ๐ก5 . ๐ 0, ๐ก-, ๐ก5 is equal to receiving $1 at time ๐ก-and paying it back at time ๐ก5, as
observed today
59
60. Luc_Faucheux_2020
Notations and conventions (Spot and forward) - XIX
ยจ SO, the the value of receiving at time ๐ก5 an amount equal to ๐ 0, ๐ก-, ๐ก5 . ๐ 0, ๐ก-, ๐ก5 is equal
to receiving $1 at time ๐ก-and paying it back at time ๐ก5, as observed today, wich are
deterministic and known quantities.
ยจ It is a non trivial result once you start thinking about it.
ยจ The curve does move every day (actually every millisecond), but barring something not in
our framework (credit consideration, operational risk,..), the value of paying or receiving a
fixed amount of currency say $1 in the future at time ๐ก will be equal to $1 at that time, and
is thus a โknownโ or โfixedโ quantity.
ยจ This is why it is called FIXED-INCOME and not equity for example.
ยจ That is the profound essence of fixed-income, is that there is a time value for money, and
that this time value is given by todayโs discount curve {๐ง๐ ๐ก = 0, ๐ก = 0, ๐ }
ยจ Contracts can be defined, and AS LONG AS those contract can be expressed as LINEAR
functions of the discount factors (discount factors are only the present value for receiving $1
in the future), THEN the value of those contracts are also known
60
61. Luc_Faucheux_2020
Notations and conventions (Spot and forward) - XX
ยจ BEWARE THE CONVEXITY
ยจ If the contract it NOT a linear function of the discount factors (say for example a function
that is obviously convex, like
ยจ ๐๐ด๐(๐ ๐ก-, ๐ก-, ๐ก5 โ ๐พ, 0) paid at time ๐ก5, once ๐ ๐ก-, ๐ก-, ๐ก5 is observed (fixed) at time ๐ก-
ยจ THEN obviously the value of that contract is NOT known and will depend on the dynamics,
or distribution for rates
ยจ BEWARE THE TIMING
ยจ If the contract LOOKS LIKE the same but the timing is different, then it will NO LONGER be a
linear function of discount factors
ยจ ONLY for ๐ 0, ๐ก-, ๐ก5 . ๐ 0, ๐ก-, ๐ก5 paid at time ๐ก5 is that equal to a difference of discount
factors
ยจ IF PAID AT ANY OTHER TIME, it it no longer a simple linear function of discount factors, or
Fixed cash flows
61
62. Luc_Faucheux_2020
Notations and conventions (Spot and forward) - XXI
ยจ This might seem like minutiae, but this is the famous LIBOR IN ARREARS/IN ADVANCE
convexity adjustment
ยจ Remember from the option deck, if not a linear function of fixed cash flows (discount
factors), then it is convex (duh), and thus the expected value of the function will not be the
function of the expected value, and the difference will eb the convexity adjustment.
ยจ The value of that convexity adjustment will depend on the distribution or dynamics that we
assume
ยจ Goldman Sachs made north of 50m in 1995 (that was a lot of money back then, because
remember there is time value to money), by entering with a number of other dealers with
Libor in advance/in arrears agreements, essentially agreeing to pay the same rate
๐ 0, ๐ก-, ๐ก5 . ๐ 0, ๐ก-, ๐ก5 paid at time ๐ก5 on one side, and paid on time ๐ก- on the other.
ยจ The fact that this was possible shows that this is not a trivial subject, and quite frankly I sill
meet an astonishingly large amount of people who do not understand convexity, and the
Libor in arrears/in advance even 25 years later, which is why it is usually mentioned in the
more modern textbooks
62
64. Luc_Faucheux_2020
Pricing a swap on todayโs yield curve
ยจ It turns out that an amazing thing happened (not sure if that was conscious or how it sort of
evolved, but the end result is quite unique, non-trivial and striking, and a lot of people really
do not appreciate it). Also I have met at least 20 people who claim that they were the one
who traded the first swap
ยจ Here is the amazing thing:
ยจ Most of the instruments traded are swaps.
ยจ Swaps are actually contracts that are linear functions of the discount factors (present value
of fixed cash flows)
ยจ CAREFUL: this is only for regular swaps, no funny business about in arrear/in advance
ยจ And so the present value of a swap contract is known, and does NOT depend on any
distributional assumptions about the randomness of rates and the discount curve over time
ยจ And so swap desks only need a yield curve, and do not need any kind of volatility surface or
assumption (to price a swap, go to the swap desk, not the option desk)
ยจ This is actually quite remarkable and somewhat non-trivial, but sort of got taken as a given
for a while.
64
65. Luc_Faucheux_2020
Pricing a swap on todayโs yield curve - II
ยจ As a result, people started getting confused because they fell into the trap of going from:
ยจ The present value of a contract that is a swap (because it is a linear function of fixed cash
flows) is a known value that can be computed from todayโs yield curve
ยจ To the following (that is not necessarily true)
ยจ The expectation of a forward rate (ANY forward rate) paid at ANY time is the value that is
computed from todayโs yield curve
ยจ CAREFUL: I am not saying that the above statement is wrong (actually in the forward
measure, also called the terminal measure, it is actually correct for a specific forward rate), I
am just saying that it is not trivial, and just because we find some very rare cases where
functions of forward rates are such that the expectation happens to be equal to the value
computed from todayโs yield curve, that does not mean that this can be extended to any
other functions.
ยจ Turns out that the very rare case of simple swap is actually 99% of the volume being traded
ยจ So the weird thing that is of very little mathematical probability is actually very common,
hence the confusion that comes out of habit
65
66. Luc_Faucheux_2020
Pricing a swap on todayโs yield curve - III
ยจ The regular textbook illustration for cash flows is the following:
66
๐ก๐๐๐
๐ก
๐
๐ง๐ 0, ๐, ๐ = $1
๐ง๐ 0, ๐ก, ๐ก = $1
๐ 0, ๐ก, ๐ . ๐ 0, ๐ก, ๐
Above the line:
We receive
Below the line:
We pay
67. Luc_Faucheux_2020
Pricing a swap on todayโs yield curve - IV
ยจ ZERO COUPON BOND PAYING $1 PRINCIPAL
67
๐ก๐๐๐
๐
๐ง๐ 0, ๐, ๐ = $1Above the line:
We receive
Below the line:
We pay
๐ก = 0
๐๐ฃ 0 = ๐ง๐ 0,0, ๐
68. Luc_Faucheux_2020
Pricing a swap on todayโs yield curve - V
ยจ FIXED BOND paying $1 PRINCIPAL and X% fixed coupon
68
๐ก๐๐๐
๐
๐ง๐ 0, ๐, ๐ = $1Above the line:
We receive
Below the line:
We pay
๐ก = 0
๐๐ฃ 0 = ๐ง๐ 0,0, ๐ + 2
#
๐. ๐(0, ๐ก#, ๐ก#$!). ๐ง๐(0,0, ๐ก#$!)
๐ก#
๐. ๐(0, ๐กB, ๐กB/-)
ยจ Note that the coupon are paid AT THE END of the period, and so the discounting factor has
to be: ๐ง๐(0,0, ๐กB/-) for the ๐-period defined by [๐กB, ๐กB/-]
69. Luc_Faucheux_2020
Pricing a swap on todayโs yield curve - VI
ยจ SWAP FIXED RECEIVE VERSUS REGULAR FLOAT PAY (pay Float, receive Fixed)
69
๐ก๐๐๐
Above the line:
We receive
Below the line:
We pay
๐ก = 0 ๐ก#
๐. ๐(0, ๐กB, ๐กB/-)
๐(0, ๐กB, ๐กB/-). ๐(0, ๐กB, ๐กB/-)
70. Luc_Faucheux_2020
Pricing a swap on todayโs yield curve - VII
ยจ Be mindful of the conventions
ยจ Previous graph assumed same frequency of payments and same daycount fraction on both
sides.
ยจ USD swaps tend to be:
ยจ FIXED side pays on a semi-annual basis, using a 30/360 daycount fraction
ยจ FIXED side periods are UNADJUSTED to compute the daycount fraction
ยจ FIXED side payment dates are ADJUSTED on a FOLLOWING basis using NY and LN holidays
ยจ Float side pays LIBOR on a quarterly basis, using ACT/360 daycount fraction.
ยจ Float side periods are ADJUSTED to compute the daycount fraction, on a MODIFIED
FOLLOWING basis using NY and LN holidays
ยจ Float side payment dates are ADJUSTED on a FOLLOWING basis using NY and LN holidays
70
71. Luc_Faucheux_2020
Pricing a swap on todayโs yield curve - VIII
ยจ We know how to price the FIXED leg of the swap, it is just a series of fixed cash flows in the
future, and so the present value of the Fixed leg is equal to the present value of a Fixed
bond, minus the last cash flow which is the principal
71
๐ก๐๐๐
Above the line:
We receive
๐ก#
๐. ๐(0, ๐กB, ๐กB/-)
๐๐ฃ_๐๐๐ฅ๐๐ 0 = 2
#
๐. ๐(0, ๐ก#, ๐ก#$!). ๐ง๐(0,0, ๐ก#$!)
72. Luc_Faucheux_2020
Pricing a swap on todayโs yield curve - IX
ยจ The Floating leg seems to be more complicated because the index is a floating rate that has
not been set yet.
ยจ For a REGULAR swap, this rate will set to the then spot rate at the beginning of the period,
and paid at the end. That rate will fix to ๐ ๐กB, ๐กB, ๐กB/- at time ๐กB and the cashflow will be paid
at time ๐กB/- and will be equal to ๐(0, ๐กB, ๐กB/-). ๐(๐กB, ๐กB, ๐กB/-)
ยจ NOW here is the trick again to convert floating cash flows to fixed cash flows:
ยจ We now that:
ยจ ๐ ๐กB, ๐กB, ๐กB/- =
-
,(#,,#,,#,.!)
.
-.)*(#,,#,,#,.!)
)*(#,,#,,#,.!)
ยจ In the bootstrap form:
ยจ ๐ง๐ ๐กB, ๐กB, ๐กB/- =
-
-/, #,,#,,#,.! .1 #,,#,,#,.!
72
73. Luc_Faucheux_2020
Pricing a swap on todayโs yield curve - V
ยจ NOTE that we are NOT saying that:
ยจ ๐ง๐ ๐กB, ๐กB, ๐กB/- = ๐ง๐ 0, ๐กB, ๐กB/-
ยจ Or:
ยจ ๐ ๐กB, ๐กB, ๐กB/- = ๐ 0, ๐กB, ๐กB/-
ยจ In fact, since we know that the curve moves and exhibits some randomness, we know that
the above is actually incorrect (unless in a world with zero volatility, or deterministic)
ยจ With volatility we would like to write something like this:
ยจ ๐ผ{๐ง๐ ๐กB, ๐กB, ๐กB/- |๐ง๐ 0, ๐กB, ๐กB/- } = ๐ง๐ 0, ๐กB, ๐กB/-
ยจ But we do not even need to get that fancy at this point, as we can decompose a regular float
into a linear functions of fixed cash flows, and avoid any kind of discussion regarding
expectations of random variables, we leave that for the next part
73
74. Luc_Faucheux_2020
Pricing a swap on todayโs yield curve - VI
ยจ So the thinking goes like this:
ยจ The floating leg of a regular swap receives a payment AT THE END of the period, that is given
by ๐๐๐๐๐ ๐กB, ๐กB, ๐กB/- , and paid on an ACT/360 basis
ยจ ๐๐๐๐๐ ๐กB, ๐กB, ๐กB/- is fixed (computed) in a very complicated manner before being published
by the BBA (British Bankers Association) at 11am LN time
ยจ LIBOR=London Inter Bank Offer Rate
ยจ This is the UNSECURED rate at which banks are willing to lend to each others for a given
term (1-month LIBOR, 3-month LIBOR, 6-month LIBOR,..)
ยจ There are talks of replacing LIBOR with SOFR
ยจ SOFR is the SECURED Overnight Financing Rate, it is also calculated in a very complicated
manner and suppose to be a measure of the overnight rate at which banks are willing to
lend cash to each others SECURED (collateralized) by US treasuries
74
75. Luc_Faucheux_2020
Pricing a swap on todayโs yield curve - VII
ยจ LIBOR and SOFR are not the same, and it is going to be interesting to see how one can
replace the other, which is something that regulators are keen on
ยจ But anyways back to our regular swap:
ยจ We assume that ๐๐๐๐๐ ๐กB, ๐กB, ๐กB/- = ๐ ๐กB, ๐กB, ๐กB/-
ยจ NOTE that this could be far from being true (in fact the whole reason why regulators want to
get rid of LIBOR is because it was subject to manipulations and we deemed not
representative of the true borrowing cost)
ยจ BUT assuming that ๐๐๐๐๐ ๐กB, ๐กB, ๐กB/- = ๐ ๐กB, ๐กB, ๐กB/- , the payoff of a single period of the float
side of a swap (float-let, or float side of a swap-let), we assume that the payment will be:
ยจ ๐ ๐กB, ๐กB, ๐กB/- . ๐ ๐กB, ๐กB, ๐กB/- = ๐ ๐กB, ๐กB, ๐กB/- .
-
,(#,,#,,#,.!)
.
-.)*(#,,#,,#,.!)
)*(#,,#,,#,.!)
ยจ ๐ ๐กB, ๐กB, ๐กB/- . ๐ ๐กB, ๐กB, ๐กB/- =
-.)*(#,,#,,#,.!)
)*(#,,#,,#,.!)
75
76. Luc_Faucheux_2020
Pricing a swap on todayโs yield curve - VIII
ยจ One more time:
ยจ ๐ ๐กB, ๐กB, ๐กB/- . ๐ ๐กB, ๐กB, ๐กB/- =
-.)*(#,,#,,#,.!)
)*(#,,#,,#,.!)
ยจ At time ๐กB, the discounted value of that payment occurring at time ๐กB/-back to ๐กB (then
present value), will be:
ยจ ๐ง๐ ๐กB, ๐กB, ๐กB/- . ๐ ๐กB, ๐กB, ๐กB/- . ๐ ๐กB, ๐กB, ๐กB/- = 1 โ ๐ง๐(๐กB, ๐กB, ๐กB/-)
ยจ This is exactly equal to receiving $1 at time ๐กB and paying $1 at time ๐กB/-
ยจ It is a linear sum of fixed cash flows
ยจ So it can be hedged (replicated) by a portfolio equal to paying $1 at time ๐กB and receiving $1
at time ๐กB/-
ยจ The price at any point in time of this contract should then ALSO be equal to the price of the
replicating portfolio (otherwise there would be arbitrage)
76
77. Luc_Faucheux_2020
Pricing a swap on todayโs yield curve โ VIII-a
ยจ This concept is crucial to the field of Fixed-Income, and goes by many names
ยจ Replication
ยจ No arbitrage
ยจ Law of one price
ยจ (I like this one the best, because it puts the emphasis on โpriceโ, so it tells you that you
cannot look at forward rates, or payoff, only if you convert them to a price)
ยจ There is also some theory behind it, in particular how a replicating portfolio โholdsโ in time,
meaning if it is replicating for a given time, it is also for all times and under which conditions
77
78. Luc_Faucheux_2020
Pricing a swap on todayโs yield curve - IX
ยจ And SO we would like to write something like this: at any point in time the value of the
replicating portfolio is:
ยจ ๐๐ฃ ๐กB = 1 โ ๐ง๐(๐กB, ๐กB, ๐กB/-)
ยจ ๐๐ฃ ๐ก < ๐กB = ๐ง๐ ๐ก, ๐ก, ๐กB . (1 โ ๐ง๐ ๐กB, ๐กB, ๐กB/- )
ยจ ๐๐ฃ ๐ก < ๐กB = ๐ง๐ ๐ก, ๐ก, ๐กB โ ๐ง๐ ๐ก, ๐ก, ๐กB . ๐ง๐ ๐กB, ๐กB, ๐กB/-
ยจ At time ๐ก, the value of ๐ง๐ ๐ก, ๐ก, ๐กB is receiving $1 at time ๐กB
ยจ NOW comes the question: What is ๐ง๐ ๐ก, ๐ก, ๐กB . ๐ง๐ ๐กB, ๐กB, ๐กB/- ?
ยจ More crucially, at time ๐ก we DO NOT KNOW what will be ๐ง๐ ๐กB, ๐กB, ๐กB/-
ยจ SO we cannot really write something like we did above
ยจ BUT We also know that this portfolio is also just receiving $1 at time ๐กB and paying $1 at time
๐กB/-, and so the present value at time ๐ก of this portfolio is:
ยจ ๐๐ฃ ๐ก < ๐กB = ๐ง๐ ๐ก, ๐ก, ๐กB โ ๐ง๐ ๐ก, ๐ก, ๐กB/-
78
79. Luc_Faucheux_2020
Pricing a swap on todayโs yield curve - X
ยจ IN PARTICULAR the above holds for todayโs yield curve
ยจ To summarize:
ยจ The fixed leg of a swap is easy to price using todayโs yield curve, it is a series of fixed and
known cash flows
ยจ The float leg of a swap is also easy to price as it turns out that for a REGULAR swap (libor
rate set at the beginning of the period, paid at the end) the floating cash flow is exactly
equal to a replicating portfolio of receiving $1 at the beginning of the period and receiving
$1 back at the end of the period
ยจ So in most textbooks you might find any of the following graphs (apologies for the poor
drawing skills).
79
80. Luc_Faucheux_2020
Pricing a swap on todayโs yield curve - XI
ยจ SWAP FIXED RECEIVE VERSUS REGULAR FLOAT PAY (pay Float, receive Fixed)
80
๐ก๐๐๐
Above the line:
We receive
Below the line:
We pay
๐ก = 0 ๐ก#
๐. ๐(0, ๐กB, ๐กB/-)
๐(0, ๐กB, ๐กB/-). ๐(0, ๐กB, ๐กB/-)
87. Luc_Faucheux_2020
Pricing a swap on todayโs yield curve - XVI
ยจ Note that we can cancel out all the inter=period $1 payments
87
๐ก๐๐๐
$1
๐. ๐ 0, ๐กB, ๐กB/- = ๐. ๐ ๐. ๐ ๐. ๐ ๐. ๐
$1
88. Luc_Faucheux_2020
Pricing a swap on todayโs yield curve - XVII
ยจ So the entire floating leg just is equivalent to receiving $1 at the beginning of the swap and
paying it back at the end.
ยจ IN PARTICULAR, the theoretical value of the floating leg of a swap is independent of the
frequency (meaning that it is the same value to receive LIBOR3M paid on a quarterly basis
that to receive LIBOR6M on a semi-annual basis).
ยจ Note that obviously this would not be the case if say we were to receive LIBOR3M on a
semi-annual basis
ยจ Note that this also means that the theoretical value for the 3s6s basis (observed market
value of receiving LIBOR3M quarterly versus paying LIBOR6M semi-annually) is 0.
ยจ It is absolutely NOT the case in real life, for a number of reasons having to do with structural
flows, but more importantly a risk premium over the life of the swap to experience a large
move in the second LIBOR3M leg as the first one is already set and the LIBOR6M is also set.
ยจ Entire books can be written on the 3s6s basis, just ask Sandeep Shukla, the super
stupendous Sensei of the 3s6s simple swap sneaky basis
88
89. Luc_Faucheux_2020
Pricing a swap on todayโs yield curve - XVII
ยจ This is why a regular fixed-float swap is still priced on a swap desk with NO consideration to
volatility or distribution on how the rates move over time:
89
๐ก๐๐๐
$1
๐. ๐ 0, ๐กB, ๐กB/- = ๐. ๐ ๐. ๐ ๐. ๐ ๐. ๐
$1
90. Luc_Faucheux_2020
Pricing a swap on todayโs yield curve - XVIII
ยจ Gotta quote Blyth (p. 29) (changing the notations a little)
ยจ The result is non-trivial, in that we have shown that the value at time ๐ก of agreeing to
receive the random quantity ๐ ๐กB, ๐กB, ๐กB/- at time ๐กB/- is a deterministic (and LINEAR)
function of the known quantities ๐ง๐ ๐ก, ๐ก, ๐กB and ๐ง๐ ๐ก, ๐ก, ๐กB/- .
ยจ The value does NOT depend on any distributional assumptions about the random variable
๐ ๐กB, ๐กB, ๐กB/- .
ยจ This is a key characteristic of forward contracts
90
91. Luc_Faucheux_2020
Pricing a swap on todayโs yield curve - XIX
ยจ So letโs go back to the swap pv for the Fixed leg, X% fixed coupon
91
๐ก๐๐๐
๐
๐ก = 0
๐๐ฃ_๐๐๐ฅ๐๐ 0 = 2
#
๐. ๐(0, ๐ก#, ๐ก#$!). ๐ง๐(0,0, ๐ก#$!)
๐ก#
๐. ๐(0, ๐กB, ๐กB/-)
92. Luc_Faucheux_2020
Pricing a swap on todayโs yield curve - XX
ยจ And the Floating leg:
92
๐ก๐๐๐
๐ก = 0 ๐ก#
๐(0, ๐กB, ๐กB/-). ๐(0, ๐กB, ๐กB/-)
๐๐ฃ_๐๐๐๐๐ก 0 = 2
#
๐(0, ๐ก#, ๐ก#$!). ๐(0, ๐ก#, ๐ก#$!). ๐ง๐(0,0, ๐ก#$!)
93. Luc_Faucheux_2020
Pricing a swap on todayโs yield curve - XXI
ยจ Floating leg is also equal to:
93
๐ก๐๐๐
$1
$1
$1
$1
$1
$1
$1
$1
๐๐ฃ_๐๐๐๐๐ก 0 = 2
#
{โ๐ง๐ 0,0, ๐ก#$! + ๐ง๐(0,0, ๐ก#)}
94. Luc_Faucheux_2020
Pricing a swap on todayโs yield curve - XXII
ยจ ๐๐ฃ_๐๐๐๐๐ก 0 = โB ๐(0, ๐กB, ๐กB/-). ๐(0, ๐กB, ๐กB/-). ๐ง๐(0,0, ๐กB/-)
ยจ ๐๐ฃ_๐๐๐๐๐ก 0 = โB{โ๐ง๐ 0,0, ๐กB/- + ๐ง๐(0,0, ๐กB)}
ยจ ๐๐ฃ_๐๐๐ฅ๐๐ 0 = โB ๐. ๐(0, ๐กB, ๐กB/-). ๐ง๐(0,0, ๐กB/-)
ยจ Note that we assumed that both fixed and float side has same frequency and daycount
convention for sake of simplicity. Having different frequency and daycount convention,
which is the usual case, does not change anything, only add some more notation (see the
deck on the curve)
ยจ Note that this is also BEFORE the swap โstartsโ. Once time passes by, the Floating leg gets
set to a fixed amount (BBA LIBOR fixing), and that float swaplet just becomes a simple fixed
period
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95. Luc_Faucheux_2020
Pricing a swap on todayโs yield curve - XXIII
ยจ The Swap Rate is the value of the coupon on the Fixed side such that the present value of
the swap is 0 (swap is on market)
ยจ ๐๐ฃ_๐๐๐๐๐ก 0 = โB ๐(0, ๐กB, ๐กB/-). ๐(0, ๐กB, ๐กB/-). ๐ง๐(0,0, ๐กB/-)
ยจ ๐๐ฃ_๐๐๐๐๐ก 0 = โB{โ๐ง๐ 0,0, ๐กB/- + ๐ง๐(0,0, ๐กB)}
ยจ ๐๐ฃ_๐๐๐ฅ๐๐ 0 = โB ๐. ๐(0, ๐กB, ๐กB/-). ๐ง๐(0,0, ๐กB/-)
ยจ ๐๐ฃ_๐๐๐๐๐ก 0 = ๐๐ฃ_๐๐๐ฅ๐๐ 0 = โB ๐๐ . ๐(0, ๐กB, ๐กB/-). ๐ง๐(0,0, ๐กB/-)
ยจ ๐๐ (0, ๐R, ๐S) =
โ, 1(&,#,,#,.!).,(&,#,,#,.!).)*(&,&,#,.!)
โ, ,(&,#,,#,.!).)*(&,&,#,.!)
ยจ The Swap Rate is a weighted average of the forward rates ๐(0, ๐กB, ๐กB/-) for a given start of the
swap ๐R and maturity ๐S
95
96. Luc_Faucheux_2020
Pricing a swap on todayโs yield curve โ XXIII-a
ยจ An โat-the-money Swapโ is also said to be โat parโ following the terminology of bonds.
ยจ This is because the cash flows of a swap are the following:
96
๐ก๐๐๐
$1
๐. ๐ 0, ๐กB, ๐กB/- = ๐. ๐ ๐. ๐ ๐. ๐ ๐. ๐
$1
97. Luc_Faucheux_2020
Pricing a swap on todayโs yield curve โ XXIII-b
ยจ For an โat-the-moneyโ swap the pv of all those cash flows is 0
97
๐ก๐๐๐
$1
๐. ๐ 0, ๐กB, ๐กB/- = ๐. ๐ ๐. ๐ ๐. ๐ ๐. ๐
$1
100. Luc_Faucheux_2020
Pricing a swap on todayโs yield curve โ XXIII-e
ยจ And so the Bond that will pay a coupon equal to the Swap Rate plus the principal amount at
the end of the last period will have a present value equal to $1, and is called โat-parโ.
ยจ So naturally the swap that is โat-the-moneyโ is also sometimes called a โpar-swapโ, because
the swap rate is equal in value to the fixed coupon of a par bond (assuming same daycount
convention, periods, holiday,โฆall other things being equalโฆ)
100
101. Luc_Faucheux_2020
Pricing a swap on todayโs yield curve - XXIV
ยจ The very powerful concept of the flat curve
ยจ Yield curve are not that steep (especially now)
ยจ Also yield curves tend to move in parallel fashion (usually 1st factor in a PCA analysis has an
eigenvalue around 95% and is a parallel move)
ยจ And so it is always useful as a first model to understand and gain intuition to assume a flat
yield curve evolving in a parallel fashion, so that the yield curve is always flat
ยจ IF for all forward rates ๐(0, ๐กB, ๐กB/-), ๐ 0, ๐กB, ๐กB/- = ๐(0)
ยจ THEN for all Swap rates ๐๐ 0, ๐R, ๐S = ๐(0) when identical swap periods and daycount
fractions
ยจ ๐๐ 0, ๐R, ๐S =
โ, 1 &,#,,#,.! ., &,#,,#,.! .)* &,&,#,.!
โ, , &,#,,#,.! .)* &,&,#,.!
= ๐ 0 .
โ, , &,#,,#,.! .)* &,&,#,.!
โ, , &,#,,#,.! .)* &,&,#,.!
= ๐(0)
101
103. Luc_Faucheux_2020
Pricing a swap on todayโs yield curve - XXV
ยจ Just because the Powerpoint file size is already at 80MB (lots of graphs), will stop here for
this deck, and just include the first memo I wrote the week I started in Finance in May 1995.
ยจ That week was Golden week in Asia, so desks of APAC based banks were thinly staffed
ยจ Volatilities were quite high
ยจ Rates were also quite high by todayโs standards of โnew normalโ
ยจ Goldman Sachs had just done a relatively large โIn Arrears โ In Advanceโ trade on
LIBOR12M, exploiting the fact that most dealers were pricing swaps on the swap desk based
on todayโs yield curve without checking that they could do it (meaning that ALL cash flows
are LINEAR functions of the discount factors).
ยจ Some dealers thought they had made money facing GS and actually some of those dealers (a
Canadian one that should remain nameless) threw a party that evening because of all the
money they thought they had made
103
104. Luc_Faucheux_2020
Pricing a swap on todayโs yield curve - XXVI
ยจ Luckily for us, the head of trading at DKB Financial Products was one of those Chicago
traders, realized that there was convexity in the trade, and we priced it on an option model
instead of a swap model
ยจ His name is Richard Robb and he has a great book that just came out.
104
105. Luc_Faucheux_2020
Pricing a swap on todayโs yield curve - XXVII
ยจ He also alerted his friend Andy Morton at Lehman Brothers (the M in the HJM model, as a
general rule for some reason you always want to hire the one who is the last letter in the
model name).
ยจ Lehman ran the swap on their HJM model and also realized that there was convexity and
avoided being โpicked offโ
ยจ GS made around 50m that week, which at the time was quite the talk of the town.
ยจ I have included below (all I could find was a pdf version so I could only copy/paste as picture)
the memo I wrote that week explaining the arrears/advance, and the connection with the
future convexity adjustment. Blythe also has a great chapter in his book, essentially without
giving too much details that was the 1995 GS trade
ยจ This is a very simple and crude ersatz (flat curve, so only one variable, moving in a parallel
fashion so always only one variable, and full correlation between the move in the future
price and the discounting factors), but this kind of simple model is quite useful to gain
intuition.
ยจ It is also not too far from reality given the large eigenvalue of the 1st factor in PCA analysis of
the yield curve
105