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

1. Luc_Faucheux_2020
THE RATES WORLD – Part II
Second pass, this time with graphs
1

2. Luc_Faucheux_2020
Couple of notes on those slides
¨ Those are part II of the the slides on Rates
¨ Part I was using very little math, and in particular we glanced over the fact that rates could
move
¨ Every calculation was done in a “deterministic” manner or zero volatility
¨ We revisit this here, and shows why that was justified for Bonds and “regular” swaps in a
graphical manner, introducing the commonly used graphs for cash flows that you will find in
most textbooks
¨ 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
¨ This deck can be viewed as a “little graphical break” where we refine the concept of discount
factor before we attack part III.
¨ 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

3. Luc_Faucheux_2020
Notations in the Rates world
Spot section
Deterministic view
3

4. Luc_Faucheux_2020
Notations and conventions in the rates world
¨ The Langevin equation is quite commonly used when modeling interest rates.
¨ Since interest rates are the “speed” or “velocity” of the Money Market Numeraire, it is quite
natural to have thought about using the Langevin equation which represents the “speed” of
a Brownian particle.
¨ 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)
¨ For example (Fabio Mercurio p. 3), the stochastic discount factor 𝐷(𝑡, 𝑇) between two time
instants 𝑡 and 𝑇 is the amount at time 𝑡 that is “equivalent” to one unit of currency payable
at time 𝑇, and is equal to
¨ 𝐷 𝑡, 𝑇 =
!(#)
!(%)
= exp(− ∫#
%
𝑅 𝑠 . 𝑑𝑠)
¨ The Bank account (Money-market account) is such that:
¨ 𝑑𝐵 𝑡 = 𝑅 𝑡 . 𝐵 𝑡 . 𝑑𝑡 with 𝐵 𝑡 = 0 = 1
¨ 𝐵 𝑡 = exp(∫&
#
𝑅 𝑠 . 𝑑𝑠)
4

5. Luc_Faucheux_2020
Notations and conventions in the rates world - II
¨ Mostly following Mercurio’s conventions in the this section.
¨ 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.
¨ 𝑧𝑐 𝑡, 𝑇 is the value of the contract at time 𝑡
¨ 𝑧𝑐 𝑇, 𝑇 = 1
¨ Note that 𝑧𝑐 𝑡, 𝑇 is a known quantity at time 𝑡. It is the value of a contract (like a Call
option is known, it is no longer a stochastic variable)
¨ On the other hand,
¨ 𝐷 𝑡, 𝑇 =
!(#)
!(%)
= exp(− ∫#
%
𝑅 𝑠 . 𝑑𝑠) and 𝐵 𝑡 = exp(∫&
#
𝑅 𝑠 . 𝑑𝑠)
¨ Are just functions of 𝑅 𝑠 . If we place ourselves in a situation where the short-term rate
𝑅 𝑠 is a stochastic process then both the MMN (BAN) noted 𝐵 𝑡 (Money market
numeraire, or Bank Account Numeraire), as well as the discount factor 𝐷 𝑡, 𝑇 , 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
¨ In the case of deterministic short-term rate, there is no stochastic component.
¨ In that case:
¨ 𝐷 𝑡, 𝑇 = 𝑧𝑐 𝑡, 𝑇
¨ When stochastic, 𝑧𝑐 𝑡, 𝑇 is the expectation of 𝐷 𝑡, 𝑇 , like the Call option price was the
expectation of the call payoff.
¨ 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
¨ Continuously compounded spot interest rate:
¨ 𝑟 𝑡, 𝑇 = −
'(()*(#,%))
,(#,%)
¨ Where 𝜏(𝑡, 𝑇) 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:
¨ 𝜏 𝑡, 𝑇 = 𝑇 − 𝑡
¨ 𝑧𝑐 𝑡, 𝑇 . exp 𝑟 𝑡, 𝑇 . 𝜏 𝑡, 𝑇 = 1
¨ 𝑧𝑐 𝑡, 𝑇 = exp −𝑟 𝑡, 𝑇 . 𝜏 𝑡, 𝑇
¨ In the deterministic case:
¨ 𝑧𝑐 𝑡, 𝑇 = exp −𝑟 𝑡, 𝑇 . 𝜏 𝑡, 𝑇 = 𝐷 𝑡, 𝑇 =
!(#)
!(%)
= exp(− ∫#
%
𝑅 𝑠 . 𝑑𝑠)
¨ 𝑟 𝑡, 𝑇 =
-
, #,%
. ∫#
%
𝑅 𝑠 . 𝑑𝑠
7

8. Luc_Faucheux_2020
Notations and conventions in the rates world - V
¨ Simply compounded spot interest rate
¨ 𝑙 𝑡, 𝑇 =
-
,(#,%)
.
-.)*(#,%)
)*(#,%)
¨ Or alternatively, in the bootstrap form
¨ 𝜏 𝑡, 𝑇 . 𝑙 𝑡, 𝑇 . 𝑧𝑐 𝑡, 𝑇 = 1 − 𝑧𝑐(𝑡, 𝑇)
¨ 1 + 𝜏 𝑡, 𝑇 . 𝑙 𝑡, 𝑇 . 𝑧𝑐 𝑡, 𝑇 = 1
¨ 𝑧𝑐 𝑡, 𝑇 =
-
-/, #,% .1 #,%
¨ In the deterministic case:
¨ 𝑧𝑐 𝑡, 𝑇 =
-
-/, #,% .1 #,%
= 𝐷 𝑡, 𝑇 =
!(#)
!(%)
= exp(− ∫#
%
𝑅 𝑠 . 𝑑𝑠)
¨ 𝑙 𝑡, 𝑇 =
-
, #,%
. [1 − exp − ∫#
%
𝑅 𝑠 . 𝑑𝑠 ]
8

9. Luc_Faucheux_2020
Notations and conventions in the rates world - VI
¨ Annually compounded spot interest rate
¨ 𝑦 𝑡, 𝑇 =
-
)*(#,%)!/#(%,') − 1
¨ Or alternatively, in the bootstrap form
¨ (1 + 𝑦 𝑡, 𝑇 ). 𝑧𝑐 𝑡, 𝑇 -/, #,% = 1
¨ (1 + 𝑦 𝑡, 𝑇 ), #,% . 𝑧𝑐 𝑡, 𝑇 = 1
¨ 𝑧𝑐 𝑡, 𝑇 =
-
(-/1 #,% )# %,'
¨ In the deterministic case:
¨ 𝑧𝑐 𝑡, 𝑇 =
-
(-/3 #,% )# %,' = 𝐷 𝑡, 𝑇 =
!(#)
!(%)
= exp(− ∫#
%
𝑅 𝑠 . 𝑑𝑠)
9

10. Luc_Faucheux_2020
Notations and conventions in the rates world - VII
¨ 𝑞-times per year compounded spot interest rate
¨ 𝑦4 𝑡, 𝑇 =
4
)*(#,%)!/)#(%,') − 𝑞
¨ Or alternatively, in the bootstrap form
¨ (1 +
-
4
𝑦4 𝑡, 𝑇 ). 𝑧𝑐 𝑡, 𝑇 -/4, #,% = 1
¨ (1 +
-
4
𝑦4 𝑡, 𝑇 )4., #,% . 𝑧𝑐 𝑡, 𝑇 = 1
¨ 𝑧𝑐 𝑡, 𝑇 =
-
(-/
!
)
.3) #,% )).# %,'
¨ In the deterministic case:
¨ 𝑧𝑐 𝑡, 𝑇 =
-
(-/
!
)
.3) #,% )).# %,'
= 𝐷 𝑡, 𝑇 =
!(#)
!(%)
= exp(− ∫#
%
𝑅 𝑠 . 𝑑𝑠)
10

11. Luc_Faucheux_2020
Notations and conventions in the rates world - VIII
¨ In bootstrap form which is the intuitive way:
¨ Continuously compounded spot: 𝑧𝑐 𝑡, 𝑇 = exp −𝑟 𝑡, 𝑇 . 𝜏 𝑡, 𝑇
¨ Simply compounded spot: 𝑧𝑐 𝑡, 𝑇 =
-
-/, #,% .1 #,%
¨ Annually compounded spot: 𝑧𝑐 𝑡, 𝑇 =
-
(-/3 #,% )# %,'
¨ 𝑞-times per year compounded spot 𝑧𝑐 𝑡, 𝑇 =
-
(-/
!
)
.3) #,% )).# %,'
11

12. Luc_Faucheux_2020
Notations and conventions in the rates world - IX
¨ In the small 𝜏 𝑡, 𝑇 → 0 limit (also if the rates themselves are such that they are <<1)
¨ In bootstrap form which is the intuitive way:
¨ Continuously compounded spot: 𝑧𝑐 𝑡, 𝑇 = 1 − 𝑟 𝑡, 𝑇 . 𝜏 𝑡, 𝑇 + 𝒪(𝜏5. 𝑟5)
¨ Simply compounded spot: 𝑧𝑐 𝑡, 𝑇 = 1 − 𝑙 𝑡, 𝑇 . 𝜏 𝑡, 𝑇 + 𝒪(𝜏5. 𝑙5)
¨ Annually compounded spot: 𝑧𝑐 𝑡, 𝑇 = 1 − 𝑦 𝑡, 𝑇 . 𝜏 𝑡, 𝑇 + 𝒪(𝜏5. 𝑦5)
¨ 𝑞-times per year compounded spot 𝑧𝑐 𝑡, 𝑇 = 1 − 𝑦4 𝑡, 𝑇 . 𝜏 𝑡, 𝑇 + 𝒪(𝜏5. 𝑦4
5)
¨ So in the limit of small 𝜏 𝑡, 𝑇 (and also small rates), in particular when 𝑇 → 𝑡, all rates
converge to the same limit we call
¨ 𝐿𝑖𝑚 𝑇 → 𝑡 = lim
%→#
(
-.)* #,%
, #,%
)
12

13. Luc_Faucheux_2020
Notations and conventions in the rates world - X
¨ In the deterministic case using the continuously compounded spot rate for example:
¨ 𝑧𝑐 𝑡, 𝑇 = exp −𝑟 𝑡, 𝑇 . 𝜏 𝑡, 𝑇 = 𝐷 𝑡, 𝑇 =
!(#)
!(%)
= exp(− ∫#
%
𝑅 𝑠 . 𝑑𝑠)
¨ 𝑟 𝑡, 𝑇 =
-
, #,%
. ∫#
%
𝑅 𝑠 . 𝑑𝑠
¨ When 𝑇 → 𝑡, 𝑟 𝑡, 𝑇 → 𝑅(𝑡)
¨ So: 𝐿𝑖𝑚 𝑇 → 𝑡 = lim
%→#
(
-.)* #,%
, #,%
) = 𝑅(𝑡)
¨ So 𝑅(𝑡) can be seen as the limit of all the different rates defined above.
¨ You can also do this using any of the rates defined previously
¨ You find sometimes the adjective “instantaneous” to describe the quantity 𝑅(𝑡) which is the
limit of small time intervals: 𝐿𝑖𝑚 𝑇 → 𝑡 = 𝐿𝑖𝑚 𝜏 → 0
13

14. Luc_Faucheux_2020
Usual graphs for cash flows
14

15. Luc_Faucheux_2020
Some useful graphs
¨ We will focus on the case of the simply compounded spot rate, where we are at time 𝑡
¨ Simply compounded spot: 𝑧𝑐 𝑡, 𝑇 =
-
-/, #,% .1 #,%
¨ What is the value 𝑝𝑣 of a SPOT contract that when entered at time 𝑡 guarantees the
payment of one unit of currency at time 𝑇 (remember that we are at time 𝑡)
¨ 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.
¨ In that case the ratio of a claim to that numeraire is a martingale, and in particular at
maturity of the contract
¨ 𝔼
-
)* %,%
=
78
)*(#,%)
= 𝔼
-
)* %,%
= 𝔼
-
-
= 1 since 𝑧𝑐 𝑇, 𝑇 = 1
¨ So the value today at time 𝑡 of that spot contract is 𝑝𝑣(𝑡) = 𝑧𝑐 𝑡, 𝑇
¨ Note that we could also take this as the definition of the “money curve”, we could assume
for now that at time 𝑡 the quantities 𝑧𝑐 𝑡, 𝑇 are known or observable and give us the value
of a contract that pays $1 at time 𝑇
15

16. Luc_Faucheux_2020
Some useful graphs - II
¨ 𝑝𝑣(𝑡) = 𝑧𝑐 𝑡, 𝑇 is the value at time 𝑡 of a SPOT contract that will return $1 at time 𝑇
¨ Note that once entered into this contract, the value will change over time, but we know that
we will receive $1 at time 𝑇 (hint: that is why it is called Fied-Income, unlike equities for
example)
¨ In lots of textbooks you will see the usual “arrow” graphs.
16
𝑡𝑖𝑚𝑒
𝑡
𝑇
𝑝𝑣(𝑡) = 𝑧𝑐 𝑡, 𝑇
𝑧𝑐 𝑇, 𝑇 = $1

17. Luc_Faucheux_2020
Some useful graphs - III
17
𝑡𝑖𝑚𝑒
𝑡
𝑇
𝑝𝑣(𝑡) = 𝑧𝑐 𝑡, 𝑇
𝑧𝑐 𝑇, 𝑇 = $1
¨ You pay 𝑧𝑐 𝑡, 𝑇 today at time 𝑡 in order to get $1 at time 𝑇
¨ The net value today of those two cashflows is 0

18. Luc_Faucheux_2020
Some useful graphs – III-a
18
𝑡𝑖𝑚𝑒
𝑡
𝑇
𝑝𝑣(𝑡) = 𝑧𝑐 𝑡, 𝑇
𝑧𝑐 𝑇, 𝑇 = $1
= 0

19. Luc_Faucheux_2020
Some useful graphs - IV
¨ Simply compounded spot: 𝑧𝑐 𝑡, 𝑇 =
-
-/, #,% .1 #,%
¨ 𝑧𝑐 𝑡, 𝑇 . (1 + 𝜏 𝑡, 𝑇 . 𝑙 𝑡, 𝑇 ) = 1
¨ 𝑧𝑐 𝑡, 𝑇 + 𝑧𝑐 𝑡, 𝑇 . 𝜏 𝑡, 𝑇 . 𝑙 𝑡, 𝑇 = 1
¨ 𝑧𝑐 𝑡, 𝑇 is the value 𝑝𝑣 of a SPOT contract that when entered at time 𝑡 guarantees the
payment of one unit of currency at time 𝑇 (remember that we are at time 𝑡)
¨ 1 is 1
¨ Similarly, 𝑧𝑐 𝑡, 𝑇 . 𝜏 𝑡, 𝑇 . 𝑙 𝑡, 𝑇 is the value of a SPOT contract that when entered at time 𝑡
guarantees the payment of (𝜏 𝑡, 𝑇 . 𝑙 𝑡, 𝑇 ) of currency at time 𝑇 (remember that we are at
time 𝑡, and so at time 𝑡 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

20. Luc_Faucheux_2020
Some useful graphs - V
20
𝑡𝑖𝑚𝑒
𝑡
𝑇
𝑧𝑐 𝑇, 𝑇 = $1
𝑧𝑐 𝑡, 𝑡 = $1
𝜏 𝑡, 𝑇 . 𝑙 𝑡, 𝑇

21. Luc_Faucheux_2020
Some useful graphs - VI
¨ In most textbooks the squiggly line is to represent a FLOATING payment.
¨ 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
“floating” payments.
¨ Note that in our case, we have defined this payment as a function of the ZC
¨ By the way, it is surprisingly hard to make a squiggly line in PowerPoint
21

22. Luc_Faucheux_2020
Some useful graphs – VI-a
¨ Note that we know 𝑧𝑐 𝑡, 𝑇
¨ So we know
-
-/, #,% .1 #,%
¨ So we know 𝑙 𝑡, 𝑇
¨ We know that the value of receiving $1 at time 𝑇 is 𝑧𝑐 𝑡, 𝑇
¨ So we know that the value of receiving $X at time 𝑇 is (𝑋. 𝑧𝑐 𝑡, 𝑇 )
¨ If we set 𝜏 𝑡, 𝑇 . 𝑙 𝑡, 𝑇 = 𝑋, then the value of receiving 𝜏 𝑡, 𝑇 . 𝑙 𝑡, 𝑇 time 𝑇 is :
¨ 𝜏 𝑡, 𝑇 . 𝑙 𝑡, 𝑇 . 𝑧𝑐 𝑡, 𝑇 = 1 − 𝑧𝑐 𝑡, 𝑇
¨ That is the value of receiving $1 at time 𝑡 and paying $1 at time 𝑇
¨ That is true if the quantity 𝑙 𝑡, 𝑇 is KNOWN, which it is at time 𝑡 as we defined it.
¨ In order to show that this quantity is also known at time before 𝑡, we need to rely on the
replication concept, also called the “law of one price” or “no-arbitrage”
22

23. Luc_Faucheux_2020
Some useful graphs - VII
¨ In most textbooks also a straight line is for a FIXED payment
¨ 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 “floating” (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
¨ Notional exchange:
¨ 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.
¨ In the example we just saw the notional ($1) is paid out at the beginning of the period at
time 𝑡 and received back at the end of the period at time 𝑇
¨ The value at time 𝑡 of this notional exchange will be:
¨ 𝑝𝑣 𝑡 = 𝑧𝑐 𝑡, 𝑇 − 1
¨ Since: 𝑧𝑐 𝑡, 𝑇 + 𝑧𝑐 𝑡, 𝑇 . 𝜏 𝑡, 𝑇 . 𝑙 𝑡, 𝑇 = 1
¨ 𝑝𝑣 𝑡 = 𝑧𝑐 𝑡, 𝑇 − 1 = −𝑧𝑐 𝑡, 𝑇 . 𝜏 𝑡, 𝑇 . 𝑙 𝑡, 𝑇
¨ SO, paying out $1 now at time 𝑡 and receiving it back at time 𝑇 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 𝑇 a cash flow equal to 𝜏 𝑡, 𝑇 . 𝑙 𝑡, 𝑇
24

25. Luc_Faucheux_2020
Some useful graphs - IX
25
𝑡𝑖𝑚𝑒
𝑡
𝑇
𝑧𝑐 𝑇, 𝑇 = $1
𝑧𝑐 𝑡, 𝑡 = $1
𝑡𝑖𝑚𝑒𝑡
𝑇
𝜏 𝑡, 𝑇 . 𝑙 𝑡, 𝑇
+ = 0

26. Luc_Faucheux_2020
Some useful graphs – IX-a (obviously)
26
𝑡𝑖𝑚𝑒
𝑡
𝑇
𝑧𝑐 𝑇, 𝑇 = $1
𝑧𝑐 𝑡, 𝑡 = $1
+ = 0
𝑡𝑖𝑚𝑒
𝑡
𝑇
𝑧𝑐 𝑇, 𝑇 = $1
𝑧𝑐 𝑡, 𝑡 = $1

27. Luc_Faucheux_2020
Some useful graphs - X
27
𝑡𝑖𝑚𝑒
𝑡
𝑇
𝑧𝑐 𝑇, 𝑇 = $1
𝑧𝑐 𝑡, 𝑡 = $1
𝑡𝑖𝑚𝑒𝑡
𝑇
𝜏 𝑡, 𝑇 . 𝑙 𝑡, 𝑇
=

28. Luc_Faucheux_2020
Some useful graphs - XI
¨ Receiving $1 today at time 𝑡 and paying it back at time 𝑇 (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:
¨ 𝜏 𝑡, 𝑇 . 𝑙 𝑡, 𝑇
¨ Where the simply compounded spot rate 𝑙 𝑡, 𝑇 is given by:
¨ 𝑙 𝑡, 𝑇 =
-
,(#,%)
.
-.)*(#,%)
)*(#,%)
¨ Or in a more intuitive bootstrap manner:
¨ 𝑧𝑐 𝑡, 𝑇 =
-
-/, #,% .1 #,%
= 𝑧𝑐 𝑡, 𝑡 .
-
-/, #,% .1 #,%
since 𝑧𝑐 𝑡, 𝑡 = 1
28

29. Luc_Faucheux_2020
Notations in the Rates world
Forward section
Deterministic view
29

30. Luc_Faucheux_2020
Notations and conventions (lower case and UPPER CASE)
¨ 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:
¨
97(:,#)
9#
= −
9
9:
[𝑎 𝑡 . 𝑝 𝑥, 𝑡 −
9
9:
[
-
5
𝑏 𝑡 5. 𝑝 𝑥, 𝑡 ]]
¨ 𝑋 𝑡; − 𝑋 𝑡< = ∫#=#<
#=#;
𝑑𝑋 𝑡 = ∫#=#<
#=#;
𝑎(𝑡). 𝑑𝑡) + ∫#=#<
#=#;
𝑏(𝑡). 𝑑𝑊(𝑡)
¨ 𝑑𝑋 𝑡 = 𝑎 𝑡 . 𝑑𝑡 + 𝑏(𝑡). 𝑑𝑊
¨ Where we should really write to be fully precise:
¨ 𝑝 𝑥, 𝑡 = 𝑝>(𝑥, 𝑡|𝑥 𝑡 = 𝑡& = 𝑋&, 𝑡 = 𝑡&)
¨ PDF Probability Density Function: 𝑝>(𝑥, 𝑡)
¨ Distribution function : 𝑃>(𝑥, 𝑡)
¨ 𝑃> 𝑥, 𝑡 = 𝑃𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 𝑋 ≤ 𝑥, 𝑡 = ∫3=.?
3=:
𝑝> 𝑦, 𝑡 . 𝑑𝑦 𝑝>(𝑥, 𝑡) =
9
9:
𝑃> 𝑥, 𝑡
30

31. Luc_Faucheux_2020
Notations and conventions (Spot and forward)
¨ So far we have defined quantities depending on 2 variables in time:
¨ For example, in the case of the continuously compounded spot interest rate:
¨ 𝑟 𝑡, 𝑇 = −
'(()*(#,%))
,(#,%)
¨ It is the constant rate at which an investment of 𝑧𝑐(𝑡, 𝑇) at time 𝑡 accrues continuously to
yield 1 unit of currency at maturity 𝑇.
¨ 𝑧𝑐 𝑡, 𝑇 = exp −𝑟 𝑡, 𝑇 . 𝜏 𝑡, 𝑇
¨ It is observed at time 𝑡 until maturity, hence the naming convention SPOT
¨ 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
¨ So different currencies will have different definitions of what SPOT means
¨ ALWAYS CHECK WHAT PEOPLE MEAN BY “SPOT”
31

32. Luc_Faucheux_2020
Notations and conventions (Spot and forward) - II
¨ So really we should have expressed:
¨ 𝑧𝑐 𝑡, 𝑇 = exp −𝑟 𝑡, 𝑇 . 𝜏 𝑡, 𝑇 as:
¨ 𝑧𝑐 𝑡, 𝑡, 𝑇 = exp −𝑟 𝑡, 𝑡, 𝑇 . 𝜏 𝑡, 𝑇
¨ This is a SPOT contract that when entered at time 𝑡 guarantees the payment of one unit of
currency at time 𝑇
¨ 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.
¨ In that case the ratio of a claim to that numeraire is a martingale, and in particular at
maturity of the contract
¨ 𝔼
-
)* %,%,%
=
@
)*(#,#,%)
= 𝔼
-
)* %,%,%
= 𝔼
-
-
= 1 since 𝑧𝑐 𝑇, 𝑇, 𝑇 = 1
¨ So the value of a contract at time 𝑡 that pays 1 at time 𝑇 is:
¨ 𝑝𝑣(𝑡) = 𝑧𝑐(𝑡, 𝑡, 𝑇)
32

33. Luc_Faucheux_2020
Notations and conventions (Spot and forward) - III
¨ This can be viewed at obviously simple or very complicated depending how you look at it
¨ In the “deterministic” world of curve building it is quite simple, until you realize that rates do
have volatility.
¨ 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.
¨ Why is that ?
¨ 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.
¨ There is a neat trick that shows that swaps are LINEAR functions of the discount factors
¨ magic trick,
:
-/:
=
:/-.-
-/:
=
-/:.-
-/:
= 1 −
-
-/:
or more simply: 𝑥 = 𝑥 + 1 − 1
33

34. Luc_Faucheux_2020
Notations and conventions (Spot and forward) - IV
¨ For ANY other structure for which the payoff is NOT a LINEAR functions of the discount
factors, the payoff will exhibit some convexity.
¨ We showed in the options module that the convexity is the origin of the option value
¨ The option value is essentially the convexity adjustment, which will require some dynamics
(volatility) to be computed.
¨ 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
¨ So 𝑝𝑣(𝑡) = 𝑧𝑐(𝑡, 𝑡, 𝑇) is the SPOT value at time 𝑡 of a contract that will return 1 unit of
currency at maturity 𝑇
¨ From that value we can define a number of arbitrary rates.
¨ Remember there is ONLY one value of the Zero Coupon Bond, but as many different rates as
you have conventions.
¨ So the only thing that really matters is the Zero Coupon Bond, as a rule of thumb
¨ When in doubt, always go back to the Zero Coupon Bond
¨ We would naturally want to extend this to FORWARD contract: what is the value at time 𝑡&
of a contract that will start at time 𝑡 and will return one unit of currency at time 𝑇
¨ So we are looking if we can extend the SPOT 𝑧𝑐(𝑡&, 𝑡&, 𝑇) to something like a forward
𝑧𝑐(𝑡&, 𝑡, 𝑇)
¨ 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
¨ That seems to be doable based on the time value of money argument.
¨ At time 𝑡&, I can enter into a SPOT contract that returns $1 at time 𝑡
¨ The value of that contract is 𝑧𝑐(𝑡&, 𝑡&, 𝑡)
¨ At time 𝑡&, I can enter into a SPOT contract that returns $1 at time 𝑇
¨ The value of that contract is 𝑧𝑐(𝑡&, 𝑡&, 𝑇)
¨ At time 𝑡&, I can enter into a FORWARD contract, an agreement to be delivered at time 𝑡 a
“then SPOT” contract that will return $1 at time 𝑇
¨ The question at hand is, what is the value of that FORWARD contract over time?
¨ 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
¨ Also quite frankly even people in the industry get so much confused between spot, forward,
future, forward price, forward contract, forward rate,… that it pays to spend a little time on
this.
¨ But no worry, I have met very senior managers who were completely clueless about the
difference between spot and forward
¨ So goes to show that knowledge is not always a good thing, especially when it comes to your
career…
37

38. Luc_Faucheux_2020
Notations and conventions (Spot and forward) - VI
¨ We can always define mathematically a number of quantities that we will call forward rates
based on the knowledge now a time 𝑡 of the curve {𝑧𝑐 𝑡, 𝑡, 𝑇 }
¨ Just to make things more clear, we just use the convention that now is 𝑡 = 0
¨ SO today we know the curve {𝑧𝑐 0,0, 𝑇 }
¨ We can define what we called spot quantities (as of today 𝑡 = 0)
¨ In the case of the simply compounded convention for those quantities, for example the
“simply compounded spot rate, as of today” was defined as:
¨ 𝑙 𝑡 = 0, 𝑇 =
-
,(#=&,%)
.
-.)*(#=&,%)
)*(#=&,%)
¨ Or alternatively, in the bootstrap form
¨ 𝑧𝑐 𝑡 = 0, 𝑇 =
-
-/, #=&,% .1 #=&,%
38

39. Luc_Faucheux_2020
Notations and conventions (Spot and forward) - VII
¨ Using the extra set on convention we defined, observed at time 𝑡 = 0, all the sets of “simply
compounded spot rates” are:
¨ 𝑙 0,0, 𝑇 =
-
,(&,&,%)
.
-.)*(&,&,%)
)*(&,&,%)
¨ Or alternatively, in the bootstrap form
¨ 𝑧𝑐 0,0, 𝑇 =
-
-/, &,&,% .1 &,&,%
¨ TOMORROW at time 𝑡 = 1 we will have a new curve {𝑧𝑐 𝑡 = 1, 𝑡 = 1, 𝑇 } with new spot
rates:
¨ 𝑙 1,1, 𝑇 =
-
,(-,-,%)
.
-.)*(-,-,%)
)*(-,-,%)
¨ Note that 𝜏(𝑡, 𝑡, 𝑇) is a daycount fraction so should really not depend on the time of
observation, 𝜏 𝑡, 𝑡, 𝑇 = 𝜏(𝑡, 𝑇) 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
¨ NOW we are absolutely free to compute the following quantities:
¨ Bear in mind that for now those are “just” definitions, we have not said anything linking
those quantities to any kind of expectations from a distribution or dynamics
¨ From today’s curve: {𝑧𝑐 𝑡 = 0, 𝑡 = 0, 𝑇 }
¨ We can compute:
¨ 𝑧𝑐 𝑡 = 0, 𝑡 = 𝑡-, 𝑡 = 𝑡5 = 𝑧𝑐 𝑡 = 0, 𝑡 = 0, 𝑡 = 𝑡5 /𝑧𝑐 𝑡 = 0, 𝑡 = 0, 𝑡 = 𝑡-
¨ Of course we have trivially: 𝑧𝑐 𝑡 = 0, 𝑡 = 𝑡-, 𝑡 = 𝑡-
¨ In particular it is useful to define the daily increments:
¨ 𝑧𝑐 𝑡 = 0, 𝑡 = 𝑡-, 𝑡 = 𝑡- + 1 = 𝑧𝑐 𝑡 = 0, 𝑡 = 0, 𝑡 = 𝑡- + 1 /𝑧𝑐 𝑡 = 0, 𝑡 = 0, 𝑡 = 𝑡-
¨ And from those what we will define as the “simply compounded forward rate observed as of
today 𝑡 = 0)
¨ 𝑙 0, 𝑡-, 𝑡5 =
-
,(&,#!,#+)
.
-.)*(&,#!,#+)
)*(&,#!,#+)
40

41. Luc_Faucheux_2020
Notations and conventions (Spot and forward) - IX
¨ 𝑙 0, 𝑡-, 𝑡5 =
-
,(&,#!,#+)
.
-.)*(&,#!,#+)
)*(&,#!,#+)
¨ In the bootstrap form:
¨ 𝑧𝑐 0, 𝑡-, 𝑡5 =
-
-/, &,#!,#+ .1 &,#!,#+
¨ Using the daily:
¨ 𝑧𝑐 0, 𝑡-, 𝑡- + 1 =
-
-/, &,#!,#!/- .1 &,#!,#!/-
¨ And since:
¨ 𝑧𝑐 𝑡 = 0, 𝑡 = 𝑡-, 𝑡 = 𝑡- + 1 = 𝑧𝑐 𝑡 = 0, 𝑡 = 0, 𝑡 = 𝑡- + 1 /𝑧𝑐 𝑡 = 0, 𝑡 = 0, 𝑡 = 𝑡-
¨ 𝑧𝑐 𝑡 = 0, 𝑡 = 0, 𝑡 = 𝑡- + 1 = 𝑧𝑐 𝑡 = 0, 𝑡 = 𝑡-, 𝑡 = 𝑡- + 1 ∗ 𝑧𝑐 𝑡 = 0, 𝑡 = 0, 𝑡 = 𝑡-
41

42. Luc_Faucheux_2020
Notations and conventions (Spot and forward) - X
¨ 𝑧𝑐 𝑡 = 0, 𝑡 = 0, 𝑡 = 𝑡- + 1 = 𝑧𝑐 𝑡 = 0, 𝑡 = 𝑡-, 𝑡 = 𝑡- + 1 ∗ 𝑧𝑐 𝑡 = 0, 𝑡 = 0, 𝑡 = 𝑡-
¨ So we also have:
¨ 𝑧𝑐 𝑡 = 0, 𝑡 = 0, 𝑡 = 𝑡A = 𝑧𝑐 𝑡 = 0, 𝑡 = 0, 𝑡 = 0 ∗ ∏B=&
B=A
𝑧𝑐 𝑡 = 0, 𝑡 = 𝑡B, 𝑡 = 𝑡B + 1
¨ Since: 𝑧𝑐 𝑡 = 0, 𝑡 = 0, 𝑡 = 0 = 1
¨ Note that we also have: 𝑧𝑐 𝑡 = 0, 𝑡 = 𝑇, 𝑡 = 𝑇 = 1
¨ 𝑧𝑐 𝑡 = 0, 𝑡 = 0, 𝑡 = 𝑡A = ∏B=&
B=A
𝑧𝑐 𝑡 = 0, 𝑡 = 𝑡B, 𝑡 = 𝑡B + 1
¨ 𝑧𝑐 0, 𝑡-, 𝑡- + 1 =
-
-/, &,#!,#!/- .1 &,#!,#!/-
¨ 𝑧𝑐 𝑡 = 0, 𝑡 = 0, 𝑡 = 𝑡A = ∏B=&
B=A -
-/, &,#,,#,/- .1 &,#,,#,/-
42

43. Luc_Faucheux_2020
Notations and conventions (Spot and forward) - XI
¨ We could also define any kind of buckets:
¨ 𝑧𝑐 𝑡 = 0, 𝑡 = 𝑡7, 𝑡 = 𝑡4 = ∏B=7
B=4
𝑧𝑐 𝑡 = 0, 𝑡 = 𝑡B, 𝑡 = 𝑡B + 1
¨ 𝑧𝑐 𝑡 = 0, 𝑡7, 𝑡4 =
-
-/, &,#-,#) .1 &,#-,#)
¨ And so for any joint sequence of buckets, we have the usual bootstrap equation
¨ 𝑧𝑐 𝑡 = 0, 𝑡 = 0, 𝑡 = 𝑡A = ∏ 𝑧𝑐 𝑡 = 0, 𝑡 = 𝑡7, 𝑡 = 𝑡4
¨ 𝑧𝑐 𝑡 = 0, 𝑡 = 0, 𝑡 = 𝑡A = ∏
-
-/, &,#-,#) .1 &,#-,#)
¨ Where the successive buckets [𝑡7, 𝑡4] covers the [0, 𝑡A] interval
¨ A picture usually being worth a thousand words:
43

44. Luc_Faucheux_2020
Notations and conventions (Spot and forward) - XII
44
𝑧𝑐 0,0,0 = 1
𝑧𝑐 0,0, 𝑡!" = 𝑧𝑐 0,0,0 ∗
1
1 + 𝜏 0, 𝑡!#, 𝑡!" . 𝑙 0, 𝑡!#, 𝑡!"
𝑡𝑖𝑚𝑒
𝑡 = 0 = 𝑡B&
𝑡B- 𝑡B5 𝑡BM 𝑡BN
𝑧𝑐 0,0, 𝑡!$ = 𝑧𝑐 0,0, 𝑡!" ∗
1
1 + 𝜏 0, 𝑡!", 𝑡!$ . 𝑙 0, 𝑡!", 𝑡!$
𝑧𝑐 0,0, 𝑡!% = 𝑧𝑐 0,0, 𝑡!$ ∗
1
1 + 𝜏 0, 𝑡!$, 𝑡!% . 𝑙 0, 𝑡!$, 𝑡!%
𝑧𝑐 0,0, 𝑡!& = 𝑧𝑐 0,0, 𝑡!% ∗
1
1 + 𝜏 0, 𝑡!%, 𝑡!& . 𝑙 0, 𝑡!%, 𝑡!&

45. Luc_Faucheux_2020
Notations and conventions (Spot and forward) - XIII
¨ 𝑧𝑐 𝑡 = 0, 𝑡 = 0, 𝑡 = 𝑡A = ∏
-
-/, &,#-,#) .1 &,#-,#)
¨ Usually if 𝑡7 < 𝑡4 the daycount fraction is positive (not time traveling yet)
¨ Usually the rates tend to be positive, 𝑙 0, 𝑡7, 𝑡4 > 0
¨ Note that this is proven to be absolutely wrong recently, but most textbooks still have the
usual graph, showing the decrease over time 𝑇 of the quantity 𝑧𝑐 0,0, 𝑇
¨ This is the famous time value of money principle
45

46. Luc_Faucheux_2020
Notations and conventions (Spot and forward) - XIV
46
𝑧𝑐 0,0,0 = 1
𝑧𝑐 0,0, 𝑡!" = 𝑧𝑐 0,0,0 ∗
1
1 + 𝜏 0, 𝑡!#, 𝑡!" . 𝑙 0, 𝑡!#, 𝑡!"
𝑡𝑖𝑚𝑒 𝑇𝑡B- 𝑡B5 𝑡BM 𝑡BN
𝑧𝑐 0,0, 𝑡!$ = 𝑧𝑐 0,0, 𝑡!" ∗
1
1 + 𝜏 0, 𝑡!", 𝑡!$ . 𝑙 0, 𝑡!", 𝑡!$
𝑧𝑐 0,0, 𝑡!% = 𝑧𝑐 0,0, 𝑡!$ ∗
1
1 + 𝜏 0, 𝑡!$, 𝑡!% . 𝑙 0, 𝑡!$, 𝑡!%
𝑧𝑐 0,0, 𝑡!& = 𝑧𝑐 0,0, 𝑡!% ∗
1
1 + 𝜏 0, 𝑡!%, 𝑡!& . 𝑙 0, 𝑡!%, 𝑡!&
𝑧𝑐 0,0,0 = 1
𝑧𝑐 0,0, 𝑇
𝑧𝑐 0,0, 𝑇 → ∞ = 0
USING
FORWARD
RATES
𝑡 = 𝑡B& = 0

47. Luc_Faucheux_2020
Notations and conventions (Spot and forward) – XIV-a
47
𝑧𝑐 0,0,0 = 1
𝑧𝑐 0,0, 𝑡!" = 𝑧𝑐 0,0,0 ∗
1
1 + 𝜏 0, 𝑡!#, 𝑡!" . 𝑙 0, 𝑡!#, 𝑡!"
𝑡𝑖𝑚𝑒 𝑇𝑡B- 𝑡B5 𝑡BM 𝑡BN
𝑧𝑐 0,0, 𝑡!$ = 𝑧𝑐 0,0,0 ∗
1
1 + 𝜏 0, 𝑡!#, 𝑡!$ . 𝑙 0, 𝑡!#, 𝑡!$
𝑧𝑐 0,0, 𝑡!% = 𝑧𝑐 0,0,0 ∗
1
1 + 𝜏 0, 𝑡!#, 𝑡!% . 𝑙 0, 𝑡!#, 𝑡!%
𝑧𝑐 0,0, 𝑡!& = 𝑧𝑐 0,0,0 ∗
1
1 + 𝜏 0, 𝑡!#, 𝑡!& . 𝑙 0, 𝑡!#, 𝑡!&
𝑧𝑐 0,0,0 = 1
𝑧𝑐 0,0, 𝑇
𝑧𝑐 0,0, 𝑇 → ∞ = 0
USING
SPOT
RATES
𝑡 = 𝑡B& = 0

48. Luc_Faucheux_2020
Notations and conventions (Spot and forward) – XIV-b
48
𝑧𝑐 0,0,0 = 1
𝑧𝑐 0, 𝑡!#, 𝑡!" = 1 ∗
1
1 + 𝜏 0, 𝑡!#, 𝑡!" . 𝑙 0, 𝑡!#, 𝑡!"
𝑡𝑖𝑚𝑒 𝑇𝑡B- 𝑡B5 𝑡BM 𝑡BN
𝑧𝑐 0, 𝑡!", 𝑡!$ = 1 ∗
1
1 + 𝜏 0, 𝑡!", 𝑡!$ . 𝑙 0, 𝑡!", 𝑡!$
𝑧𝑐 0, 𝑡!$, 𝑡!% = 1 ∗
1
1 + 𝜏 0, 𝑡!$, 𝑡!% . 𝑙 0, 𝑡!$, 𝑡!%
𝑧𝑐 0, 𝑡!%, 𝑡!& = 1 ∗
1
1 + 𝜏 0, 𝑡!%, 𝑡!& . 𝑙 0, 𝑡!%, 𝑡!&
𝑧𝑐 0,0,0 = 1
𝑧𝑐 0,0, 𝑇
𝑧𝑐 0,0, 𝑇 → ∞ = 0
USING
FORWARD
RATES
𝑡 = 𝑡B& = 0

49. Luc_Faucheux_2020
Notations and conventions (Spot and forward) – XIV-c
49
𝑧𝑐 0,0,0 = 1
𝑧𝑐 0, 𝑡!#, 𝑡!" = 𝑧𝑐 0,0,0 ∗
1
1 + 𝜏 0, 𝑡!#, 𝑡!" . 𝑙 0, 𝑡!#, 𝑡!"
𝑡𝑖𝑚𝑒 𝑇𝑡B- 𝑡B5 𝑡BM 𝑡BN
𝑧𝑐 0,0, 𝑡!$ = 𝑧𝑐 0, 𝑡!#, 𝑡!" ∗ 𝑧𝑐 0, 𝑡!", 𝑡!$
𝑧𝑐 0,0, 𝑡!% = 𝑧𝑐 0,0, 𝑡!$ ∗ 𝑧𝑐 0, 𝑡!$, 𝑡!%
𝑧𝑐 0,0, 𝑡!& = 𝑧𝑐 0,0, 𝑡!% ∗ 𝑧𝑐 0, 𝑡!%, 𝑡!&
𝑧𝑐 0,0,0 = 1
𝑧𝑐 0,0, 𝑇
𝑧𝑐 0,0, 𝑇 → ∞ = 0
USING ZC
𝑡 = 𝑡B& = 0

50. Luc_Faucheux_2020
Notations and conventions (Spot and forward) – XIV-d
50
𝑧𝑐 0,0,0 = 1
𝑧𝑐 0,0, 𝑡!" =
1
1 + 𝜏 0, 𝑡!#, 𝑡!" . 𝑙 0, 𝑡!#, 𝑡!"
𝑡𝑖𝑚𝑒 𝑇𝑡B- 𝑡B5 𝑡BM 𝑡BN
𝑧𝑐 0,0, 𝑡!$ =
1
1 + 𝜏 0, 𝑡!#, 𝑡!$ . 𝑙 0, 𝑡!#, 𝑡!$
𝑧𝑐 0,0, 𝑡!% =
1
1 + 𝜏 0, 𝑡!#, 𝑡!% . 𝑙 0, 𝑡!#, 𝑡!%
𝑧𝑐 0,0, 𝑡!& =
1
1 + 𝜏 0, 𝑡!#, 𝑡!& . 𝑙 0, 𝑡!#, 𝑡!&
𝑧𝑐 0,0,0 = 1
𝑧𝑐 0,0, 𝑇
𝑧𝑐 0,0, 𝑇 → ∞ = 0
USING
SPOT
RATES
𝑡 = 𝑡B& = 0

51. Luc_Faucheux_2020
Notations and conventions (Spot and forward) – XIV-e
51
𝑧𝑐 0,0,0 = 1
𝑙 0, 𝑡!#, 𝑡!" =
1
𝜏 0, 𝑡!#, 𝑡!"
. [
1
𝑧𝑐 0,0, 𝑡!"
− 1]
𝑡𝑖𝑚𝑒 𝑇𝑡B- 𝑡B5 𝑡BM 𝑡BN
𝑙 0, 𝑡!#, 𝑡!$ =
1
𝜏 0, 𝑡!#, 𝑡!$
. [
1
𝑧𝑐 0,0, 𝑡!$
− 1]
𝑙 0, 𝑡!#, 𝑡!% =
1
𝜏 0, 𝑡!#, 𝑡!%
. [
1
𝑧𝑐 0,0, 𝑡!%
− 1]
𝑙 0, 𝑡!#, 𝑡!& =
1
𝜏 0, 𝑡!#, 𝑡!&
. [
1
𝑧𝑐 0,0, 𝑡!&
− 1]
𝑧𝑐 0,0,0 = 1
𝑧𝑐 0,0, 𝑇
𝑧𝑐 0,0, 𝑇 → ∞ = 0
USING
SPOT
RATES
𝑡 = 𝑡B& = 0

52. Luc_Faucheux_2020
Notations and conventions (Spot and forward) – XIV-f
52
𝑙 0, 𝑡!#, 𝑡!" =
1
𝜏 0, 𝑡!#, 𝑡!"
. [
1
𝑧𝑐 0,0, 𝑡!"
− 1]
𝑡𝑖𝑚𝑒 𝑇𝑡 = 𝑡B& = 0 𝑡B- 𝑡B5 𝑡BM 𝑡BN
𝑙 0, 𝑡!#, 𝑡!$ =
1
𝜏 0, 𝑡!#, 𝑡!$
. [
1
𝑧𝑐 0,0, 𝑡!$
− 1]
𝑙 0, 𝑡!#, 𝑡!% =
1
𝜏 0, 𝑡!#, 𝑡!%
. [
1
𝑧𝑐 0,0, 𝑡!%
− 1]
𝑙 0, 𝑡!#, 𝑡!& =
1
𝜏 0, 𝑡!#, 𝑡!&
. [
1
𝑧𝑐 0,0, 𝑡!&
− 1]
𝑆𝑝𝑜𝑡 𝑅𝑎𝑡𝑒 𝐶𝑢𝑟𝑣𝑒
USING
SPOT
RATES

53. Luc_Faucheux_2020
Notations and conventions (Spot and forward) – XIV-g
53
𝑙 0, 𝑡!#, 𝑡!" =
1
𝜏 0, 𝑡!#, 𝑡!"
. [
1
𝑧𝑐 0,0, 𝑡!"
− 1]
𝑡𝑖𝑚𝑒 𝑇𝑡B- 𝑡B5 𝑡BM 𝑡BN
𝑙 0, 𝑡!", 𝑡!$ =
1
𝜏 0, 𝑡!", 𝑡!$
. [
1
𝑧𝑐 0, 𝑡!", 𝑡!$
− 1]
𝑙 0, 𝑡!$, 𝑡!% =
1
𝜏 0, 𝑡!$, 𝑡!%
. [
1
𝑧𝑐 0, 𝑡!$, 𝑡!%
− 1]
𝑙 0, 𝑡!%, 𝑡!& =
1
𝜏 0, 𝑡!%, 𝑡!&
. [
1
𝑧𝑐 0, 𝑡!%, 𝑡!&
− 1]
USING
FORWARD
RATES
𝑡 = 𝑡B& = 0

54. Luc_Faucheux_2020
Notations and conventions (Spot and forward) – XIV-h
¨ Note that in the previous slide I did not display a “forward rate curve”
¨ Because a “spot rate curve” is well defined since only the 3rd argument is a variable
¨ So the curve would be plotting 𝑙 0,0, 𝑡 as a function of the time 𝑡, (of course subject to the
chosen daycount convention)
¨ In the forward case, there are now 2 variables 𝑙 0, 𝑡BM, 𝑡BN , so ether we plot it as a two-
dimensional surface, or we fix the distance (𝑡BN−𝑡BM) to a fixed value say 1-month, 3-month,
6-month, and then we can plot the “1-month forward rate curve”, or the “3-month forward
rate curve). Since that depends also on 𝜏 0, 𝑡BM, 𝑡BN , technically for example you would have
to say “this is the 3-month forward rate based on ACT/360 daycount convention"
¨ This illustrates that unlike the ZC, there are multiple choices for the forward rates (also the
spot rate)
¨ 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
¨ We can then extend the relationship we had in spot space:
55
𝑡𝑖𝑚𝑒
𝑡
𝑇
𝑧𝑐 𝑇, 𝑇 = $1
𝑧𝑐 𝑡, 𝑡 = $1
𝜏 𝑡, 𝑇 . 𝑙 𝑡, 𝑇

56. Luc_Faucheux_2020
Notations and conventions (Spot and forward) - XVI
¨ To the forward space
56
𝑡𝑖𝑚𝑒
𝑡
𝑇
𝑧𝑐 0, 𝑇, 𝑇 = $1
𝑧𝑐 0, 𝑡, 𝑡 = $1
𝜏 0, 𝑡, 𝑇 . 𝑙 0, 𝑡, 𝑇
𝑡 = 0

57. Luc_Faucheux_2020
Notations and conventions (Spot and forward) - XVII
¨ To the forward space
57
𝑡𝑖𝑚𝑒
𝑡!
(𝑡")
𝑧𝑐 0, 𝑡5, 𝑡5 = $1
𝑧𝑐 0, 𝑡-, 𝑡- = $1
𝜏 0, 𝑡!, 𝑡" . 𝑙 0, 𝑡!, 𝑡"
𝑡 = 0

58. Luc_Faucheux_2020
Notations and conventions (Spot and forward) - XVII
¨ 𝑙 0, 𝑡-, 𝑡5 =
-
,(&,#!,#+)
.
-.)*(&,#!,#+)
)*(&,#!,#+)
¨ NOTE THAT when we evolve in time and reach the time 𝑡 = 𝑡- we will have then the relation
for the simply compounded spot rate
¨ 𝑙 𝑡-, 𝑡-, 𝑡5 =
-
,(#!,#!,#+)
.
-.)*(#!,#!,#+)
)*(#!,#!,#+)
¨ THE CRUX OF THE MATTER IN FIXED-INCOME:
¨ How are 𝑙 0, 𝑡-, 𝑡5 and 𝑙 𝑡-, 𝑡-, 𝑡5 related ?
¨ Or more exactly keeping with our notation for variables that are known and the ones that
are random and not fixed yet, How are 𝑙 0, 𝑡-, 𝑡5 and 𝐿 𝑡, 𝑡-, 𝑡5 related, with 𝑡 < 𝑡-?
¨ And 𝑙 0, 𝑡-, 𝑡5 and 𝑙 𝑡-, 𝑡-, 𝑡5 when 𝐿 𝑡, 𝑡-, 𝑡5 gets fixed to 𝑙 𝑡, 𝑡-, 𝑡5 at time 𝑡 = 𝑡-?
58

59. Luc_Faucheux_2020
Notations and conventions (Spot and forward) - XVIII
¨ Put it another way, I was not trying to be pedantic for pedantic sake with those extra
seemingly redundant notation.
¨ The issue is essentially the following to try to gain some intuition:
¨ Using TODAY curve {𝑧𝑐 𝑡 = 0, 𝑡 = 0, 𝑇 } for the time value of money (a known quantity), we
can define a number of other quantities, including forward rates like:
¨ 𝑙 0, 𝑡-, 𝑡5 =
-
,(&,#!,#+)
.
-.)*(&,#!,#+)
)*(&,#!,#+)
¨ At this point, this is just a definition.
¨ HOWEVER, it is a definition that has the very nice property that the value of receiving at
time 𝑡5 an amount equal to 𝜏 0, 𝑡-, 𝑡5 . 𝑙 0, 𝑡-, 𝑡5 is a function of the two quantities
𝑧𝑐(0, 𝑡-, 𝑡5) and 𝑧𝑐(0, 𝑡-, 𝑡-)
¨ IN PARTICULAR, the the value of receiving at time 𝑡5 an amount equal to
𝜏 0, 𝑡-, 𝑡5 . 𝑙 0, 𝑡-, 𝑡5 is equal to receiving $1 at time 𝑡-and paying it back at time 𝑡5, as
observed today
59

60. Luc_Faucheux_2020
Notations and conventions (Spot and forward) - XIX
¨ SO, the the value of receiving at time 𝑡5 an amount equal to 𝜏 0, 𝑡-, 𝑡5 . 𝑙 0, 𝑡-, 𝑡5 is equal
to receiving $1 at time 𝑡-and paying it back at time 𝑡5, as observed today, wich are
deterministic and known quantities.
¨ It is a non trivial result once you start thinking about it.
¨ 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 𝑡 will be equal to $1 at that time, and
is thus a “known” or “fixed” quantity.
¨ This is why it is called FIXED-INCOME and not equity for example.
¨ 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 today’s discount curve {𝑧𝑐 𝑡 = 0, 𝑡 = 0, 𝑇 }
¨ 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
¨ BEWARE THE CONVEXITY
¨ If the contract it NOT a linear function of the discount factors (say for example a function
that is obviously convex, like
¨ 𝑀𝐴𝑋(𝑙 𝑡-, 𝑡-, 𝑡5 − 𝐾, 0) paid at time 𝑡5, once 𝑙 𝑡-, 𝑡-, 𝑡5 is observed (fixed) at time 𝑡-
¨ THEN obviously the value of that contract is NOT known and will depend on the dynamics,
or distribution for rates
¨ BEWARE THE TIMING
¨ If the contract LOOKS LIKE the same but the timing is different, then it will NO LONGER be a
linear function of discount factors
¨ ONLY for 𝜏 0, 𝑡-, 𝑡5 . 𝑙 0, 𝑡-, 𝑡5 paid at time 𝑡5 is that equal to a difference of discount
factors
¨ 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
¨ This might seem like minutiae, but this is the famous LIBOR IN ARREARS/IN ADVANCE
convexity adjustment
¨ 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.
¨ The value of that convexity adjustment will depend on the distribution or dynamics that we
assume
¨ 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
𝜏 0, 𝑡-, 𝑡5 . 𝑙 0, 𝑡-, 𝑡5 paid at time 𝑡5 on one side, and paid on time 𝑡- on the other.
¨ 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

63. Luc_Faucheux_2020
Swap Pricing
Deterministic view
63

64. Luc_Faucheux_2020
Pricing a swap on today’s yield curve
¨ 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
¨ Here is the amazing thing:
¨ Most of the instruments traded are swaps.
¨ Swaps are actually contracts that are linear functions of the discount factors (present value
of fixed cash flows)
¨ CAREFUL: this is only for regular swaps, no funny business about in arrear/in advance
¨ 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
¨ 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)
¨ 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 today’s yield curve - II
¨ As a result, people started getting confused because they fell into the trap of going from:
¨ 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 today’s yield curve
¨ To the following (that is not necessarily true)
¨ The expectation of a forward rate (ANY forward rate) paid at ANY time is the value that is
computed from today’s yield curve
¨ 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 today’s yield curve, that does not mean that this can be extended to any
other functions.
¨ Turns out that the very rare case of simple swap is actually 99% of the volume being traded
¨ 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 today’s yield curve - III
¨ The regular textbook illustration for cash flows is the following:
66
𝑡𝑖𝑚𝑒
𝑡
𝑇
𝑧𝑐 0, 𝑇, 𝑇 = $1
𝑧𝑐 0, 𝑡, 𝑡 = $1
𝜏 0, 𝑡, 𝑇 . 𝑙 0, 𝑡, 𝑇
Above the line:
We receive
Below the line:
We pay

67. Luc_Faucheux_2020
Pricing a swap on today’s yield curve - IV
¨ ZERO COUPON BOND PAYING $1 PRINCIPAL
67
𝑡𝑖𝑚𝑒
𝑇
𝑧𝑐 0, 𝑇, 𝑇 = $1Above the line:
We receive
Below the line:
We pay
𝑡 = 0
𝑝𝑣 0 = 𝑧𝑐 0,0, 𝑇

68. Luc_Faucheux_2020
Pricing a swap on today’s yield curve - V
¨ FIXED BOND paying $1 PRINCIPAL and X% fixed coupon
68
𝑡𝑖𝑚𝑒
𝑇
𝑧𝑐 0, 𝑇, 𝑇 = $1Above the line:
We receive
Below the line:
We pay
𝑡 = 0
𝑝𝑣 0 = 𝑧𝑐 0,0, 𝑇 + 2
#
𝑋. 𝜏(0, 𝑡#, 𝑡#$!). 𝑧𝑐(0,0, 𝑡#$!)
𝑡#
𝑋. 𝜏(0, 𝑡B, 𝑡B/-)
¨ Note that the coupon are paid AT THE END of the period, and so the discounting factor has
to be: 𝑧𝑐(0,0, 𝑡B/-) for the 𝑖-period defined by [𝑡B, 𝑡B/-]

69. Luc_Faucheux_2020
Pricing a swap on today’s yield curve - VI
¨ SWAP FIXED RECEIVE VERSUS REGULAR FLOAT PAY (pay Float, receive Fixed)
69
𝑡𝑖𝑚𝑒
Above the line:
We receive
Below the line:
We pay
𝑡 = 0 𝑡#
𝑋. 𝜏(0, 𝑡B, 𝑡B/-)
𝜏(0, 𝑡B, 𝑡B/-). 𝑙(0, 𝑡B, 𝑡B/-)

70. Luc_Faucheux_2020
Pricing a swap on today’s yield curve - VII
¨ Be mindful of the conventions
¨ Previous graph assumed same frequency of payments and same daycount fraction on both
sides.
¨ USD swaps tend to be:
¨ FIXED side pays on a semi-annual basis, using a 30/360 daycount fraction
¨ FIXED side periods are UNADJUSTED to compute the daycount fraction
¨ FIXED side payment dates are ADJUSTED on a FOLLOWING basis using NY and LN holidays
¨ Float side pays LIBOR on a quarterly basis, using ACT/360 daycount fraction.
¨ Float side periods are ADJUSTED to compute the daycount fraction, on a MODIFIED
FOLLOWING basis using NY and LN holidays
¨ Float side payment dates are ADJUSTED on a FOLLOWING basis using NY and LN holidays
70

71. Luc_Faucheux_2020
Pricing a swap on today’s yield curve - VIII
¨ 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
𝑡𝑖𝑚𝑒
Above the line:
We receive
𝑡#
𝑋. 𝜏(0, 𝑡B, 𝑡B/-)
𝑝𝑣_𝑓𝑖𝑥𝑒𝑑 0 = 2
#
𝑋. 𝜏(0, 𝑡#, 𝑡#$!). 𝑧𝑐(0,0, 𝑡#$!)

72. Luc_Faucheux_2020
Pricing a swap on today’s yield curve - IX
¨ The Floating leg seems to be more complicated because the index is a floating rate that has
not been set yet.
¨ 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 𝑙 𝑡B, 𝑡B, 𝑡B/- at time 𝑡B and the cashflow will be paid
at time 𝑡B/- and will be equal to 𝜏(0, 𝑡B, 𝑡B/-). 𝑙(𝑡B, 𝑡B, 𝑡B/-)
¨ NOW here is the trick again to convert floating cash flows to fixed cash flows:
¨ We now that:
¨ 𝑙 𝑡B, 𝑡B, 𝑡B/- =
-
,(#,,#,,#,.!)
.
-.)*(#,,#,,#,.!)
)*(#,,#,,#,.!)
¨ In the bootstrap form:
¨ 𝑧𝑐 𝑡B, 𝑡B, 𝑡B/- =
-
-/, #,,#,,#,.! .1 #,,#,,#,.!
72

73. Luc_Faucheux_2020
Pricing a swap on today’s yield curve - V
¨ NOTE that we are NOT saying that:
¨ 𝑧𝑐 𝑡B, 𝑡B, 𝑡B/- = 𝑧𝑐 0, 𝑡B, 𝑡B/-
¨ Or:
¨ 𝑙 𝑡B, 𝑡B, 𝑡B/- = 𝑙 0, 𝑡B, 𝑡B/-
¨ 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)
¨ With volatility we would like to write something like this:
¨ 𝔼{𝑧𝑐 𝑡B, 𝑡B, 𝑡B/- |𝑧𝑐 0, 𝑡B, 𝑡B/- } = 𝑧𝑐 0, 𝑡B, 𝑡B/-
¨ 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 today’s yield curve - VI
¨ So the thinking goes like this:
¨ The floating leg of a regular swap receives a payment AT THE END of the period, that is given
by 𝑙𝑖𝑏𝑜𝑟 𝑡B, 𝑡B, 𝑡B/- , and paid on an ACT/360 basis
¨ 𝑙𝑖𝑏𝑜𝑟 𝑡B, 𝑡B, 𝑡B/- is fixed (computed) in a very complicated manner before being published
by the BBA (British Bankers Association) at 11am LN time
¨ LIBOR=London Inter Bank Offer Rate
¨ 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,..)
¨ There are talks of replacing LIBOR with SOFR
¨ 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 today’s yield curve - VII
¨ 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
¨ But anyways back to our regular swap:
¨ We assume that 𝑙𝑖𝑏𝑜𝑟 𝑡B, 𝑡B, 𝑡B/- = 𝑙 𝑡B, 𝑡B, 𝑡B/-
¨ 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)
¨ BUT assuming that 𝑙𝑖𝑏𝑜𝑟 𝑡B, 𝑡B, 𝑡B/- = 𝑙 𝑡B, 𝑡B, 𝑡B/- , 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:
¨ 𝜏 𝑡B, 𝑡B, 𝑡B/- . 𝑙 𝑡B, 𝑡B, 𝑡B/- = 𝜏 𝑡B, 𝑡B, 𝑡B/- .
-
,(#,,#,,#,.!)
.
-.)*(#,,#,,#,.!)
)*(#,,#,,#,.!)
¨ 𝜏 𝑡B, 𝑡B, 𝑡B/- . 𝑙 𝑡B, 𝑡B, 𝑡B/- =
-.)*(#,,#,,#,.!)
)*(#,,#,,#,.!)
75

76. Luc_Faucheux_2020
Pricing a swap on today’s yield curve - VIII
¨ One more time:
¨ 𝜏 𝑡B, 𝑡B, 𝑡B/- . 𝑙 𝑡B, 𝑡B, 𝑡B/- =
-.)*(#,,#,,#,.!)
)*(#,,#,,#,.!)
¨ At time 𝑡B, the discounted value of that payment occurring at time 𝑡B/-back to 𝑡B (then
present value), will be:
¨ 𝑧𝑐 𝑡B, 𝑡B, 𝑡B/- . 𝜏 𝑡B, 𝑡B, 𝑡B/- . 𝑙 𝑡B, 𝑡B, 𝑡B/- = 1 − 𝑧𝑐(𝑡B, 𝑡B, 𝑡B/-)
¨ This is exactly equal to receiving $1 at time 𝑡B and paying $1 at time 𝑡B/-
¨ It is a linear sum of fixed cash flows
¨ So it can be hedged (replicated) by a portfolio equal to paying $1 at time 𝑡B and receiving $1
at time 𝑡B/-
¨ 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 today’s yield curve – VIII-a
¨ This concept is crucial to the field of Fixed-Income, and goes by many names
¨ Replication
¨ No arbitrage
¨ Law of one price
¨ (I like this one the best, because it puts the emphasis on “price”, so it tells you that you
cannot look at forward rates, or payoff, only if you convert them to a price)
¨ There is also some theory behind it, in particular how a replicating portfolio ”holds” 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 today’s yield curve - IX
¨ And SO we would like to write something like this: at any point in time the value of the
replicating portfolio is:
¨ 𝑝𝑣 𝑡B = 1 − 𝑧𝑐(𝑡B, 𝑡B, 𝑡B/-)
¨ 𝑝𝑣 𝑡 < 𝑡B = 𝑧𝑐 𝑡, 𝑡, 𝑡B . (1 − 𝑧𝑐 𝑡B, 𝑡B, 𝑡B/- )
¨ 𝑝𝑣 𝑡 < 𝑡B = 𝑧𝑐 𝑡, 𝑡, 𝑡B − 𝑧𝑐 𝑡, 𝑡, 𝑡B . 𝑧𝑐 𝑡B, 𝑡B, 𝑡B/-
¨ At time 𝑡, the value of 𝑧𝑐 𝑡, 𝑡, 𝑡B is receiving $1 at time 𝑡B
¨ NOW comes the question: What is 𝑧𝑐 𝑡, 𝑡, 𝑡B . 𝑧𝑐 𝑡B, 𝑡B, 𝑡B/- ?
¨ More crucially, at time 𝑡 we DO NOT KNOW what will be 𝑧𝑐 𝑡B, 𝑡B, 𝑡B/-
¨ SO we cannot really write something like we did above
¨ BUT We also know that this portfolio is also just receiving $1 at time 𝑡B and paying $1 at time
𝑡B/-, and so the present value at time 𝑡 of this portfolio is:
¨ 𝑝𝑣 𝑡 < 𝑡B = 𝑧𝑐 𝑡, 𝑡, 𝑡B − 𝑧𝑐 𝑡, 𝑡, 𝑡B/-
78

79. Luc_Faucheux_2020
Pricing a swap on today’s yield curve - X
¨ IN PARTICULAR the above holds for today’s yield curve
¨ To summarize:
¨ The fixed leg of a swap is easy to price using today’s yield curve, it is a series of fixed and
known cash flows
¨ 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
¨ 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 today’s yield curve - XI
¨ SWAP FIXED RECEIVE VERSUS REGULAR FLOAT PAY (pay Float, receive Fixed)
80
𝑡𝑖𝑚𝑒
Above the line:
We receive
Below the line:
We pay
𝑡 = 0 𝑡#
𝑋. 𝜏(0, 𝑡B, 𝑡B/-)
𝜏(0, 𝑡B, 𝑡B/-). 𝑙(0, 𝑡B, 𝑡B/-)

81. Luc_Faucheux_2020
Pricing a swap on today’s yield curve - XII
81
𝑡𝑖𝑚𝑒
𝑡
𝑇
𝑧𝑐 0, 𝑇, 𝑇 = $1
𝑧𝑐 0, 𝑡, 𝑡 = $1
𝑡𝑖𝑚𝑒𝑡
𝑇
𝜏 0, 𝑡, 𝑇 . 𝑙 0, 𝑡, 𝑇
=

82. Luc_Faucheux_2020
Pricing a swap on today’s yield curve – XII-a
¨ SWAP FIXED RECEIVE VERSUS REGULAR FLOAT PAY (pay Float, receive Fixed)
82
𝑡𝑖𝑚𝑒
Above the line:
We receive
Below the line:
We pay
𝑡 = 0 𝑡#
𝑋. 𝜏(0, 𝑡B, 𝑡B/-)
𝜏(0, 𝑡B, 𝑡B/-). 𝑙(0, 𝑡B, 𝑡B/-)

83. Luc_Faucheux_2020
Pricing a swap on today’s yield curve - XIII
¨ SWAP FIXED RECEIVE VERSUS REGULAR FLOAT PAY (pay Float, receive Fixed)
83
𝑡𝑖𝑚𝑒
Above the line:
We receive
𝑡 = 0 𝑡#
𝜏(0, 𝑡B, 𝑡B/-). 𝑙(0, 𝑡B, 𝑡B/-)
$1
$1
𝑋. 𝜏 0, 𝑡B, 𝑡B/- = 𝑋. 𝜏 𝑋. 𝜏 𝑋. 𝜏 𝑋. 𝜏

84. Luc_Faucheux_2020
Pricing a swap on today’s yield curve - XIV
¨ SWAP FIXED RECEIVE VERSUS REGULAR FLOAT PAY (pay Float, receive Fixed)
84
𝑡𝑖𝑚𝑒
𝑡#
𝜏(0, 𝑡B, 𝑡B/-). 𝑙(0, 𝑡B, 𝑡B/-)
$1
$1
𝑋. 𝜏 0, 𝑡B, 𝑡B/- = 𝑋. 𝜏 𝑋. 𝜏 𝑋. 𝜏 𝑋. 𝜏
$1
$1

85. Luc_Faucheux_2020
Pricing a swap on today’s yield curve - XV
¨ SWAP FIXED RECEIVE VERSUS REGULAR FLOAT PAY (pay Float, receive Fixed)
85
𝑡𝑖𝑚𝑒
𝑡#
𝜏(0, 𝑡B, 𝑡B/-). 𝑙(0, 𝑡B, 𝑡B/-)
$1
$1
𝑋. 𝜏 0, 𝑡B, 𝑡B/- = 𝑋. 𝜏 𝑋. 𝜏 𝑋. 𝜏 𝑋. 𝜏
$1
$1
$1
$1

86. Luc_Faucheux_2020
Pricing a swap on today’s yield curve - XV
¨ SWAP FIXED RECEIVE VERSUS REGULAR FLOAT PAY (pay Float, receive Fixed)
86
𝑡𝑖𝑚𝑒
$1
$1
𝑋. 𝜏 0, 𝑡B, 𝑡B/- = 𝑋. 𝜏 𝑋. 𝜏 𝑋. 𝜏 𝑋. 𝜏
$1
$1
$1
$1
$1
$1

87. Luc_Faucheux_2020
Pricing a swap on today’s yield curve - XVI
¨ Note that we can cancel out all the inter=period $1 payments
87
𝑡𝑖𝑚𝑒
$1
𝑋. 𝜏 0, 𝑡B, 𝑡B/- = 𝑋. 𝜏 𝑋. 𝜏 𝑋. 𝜏 𝑋. 𝜏
$1

88. Luc_Faucheux_2020
Pricing a swap on today’s yield curve - XVII
¨ So the entire floating leg just is equivalent to receiving $1 at the beginning of the swap and
paying it back at the end.
¨ 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).
¨ Note that obviously this would not be the case if say we were to receive LIBOR3M on a
semi-annual basis
¨ 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.
¨ 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.
¨ 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 today’s yield curve - XVII
¨ 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
𝑡𝑖𝑚𝑒
$1
𝑋. 𝜏 0, 𝑡B, 𝑡B/- = 𝑋. 𝜏 𝑋. 𝜏 𝑋. 𝜏 𝑋. 𝜏
$1

90. Luc_Faucheux_2020
Pricing a swap on today’s yield curve - XVIII
¨ Gotta quote Blyth (p. 29) (changing the notations a little)
¨ The result is non-trivial, in that we have shown that the value at time 𝑡 of agreeing to
receive the random quantity 𝑙 𝑡B, 𝑡B, 𝑡B/- at time 𝑡B/- is a deterministic (and LINEAR)
function of the known quantities 𝑧𝑐 𝑡, 𝑡, 𝑡B and 𝑧𝑐 𝑡, 𝑡, 𝑡B/- .
¨ The value does NOT depend on any distributional assumptions about the random variable
𝑙 𝑡B, 𝑡B, 𝑡B/- .
¨ This is a key characteristic of forward contracts
90

91. Luc_Faucheux_2020
Pricing a swap on today’s yield curve - XIX
¨ So let’s go back to the swap pv for the Fixed leg, X% fixed coupon
91
𝑡𝑖𝑚𝑒
𝑇
𝑡 = 0
𝑝𝑣_𝑓𝑖𝑥𝑒𝑑 0 = 2
#
𝑋. 𝜏(0, 𝑡#, 𝑡#$!). 𝑧𝑐(0,0, 𝑡#$!)
𝑡#
𝑋. 𝜏(0, 𝑡B, 𝑡B/-)

92. Luc_Faucheux_2020
Pricing a swap on today’s yield curve - XX
¨ And the Floating leg:
92
𝑡𝑖𝑚𝑒
𝑡 = 0 𝑡#
𝜏(0, 𝑡B, 𝑡B/-). 𝑙(0, 𝑡B, 𝑡B/-)
𝑝𝑣_𝑓𝑙𝑜𝑎𝑡 0 = 2
#
𝑙(0, 𝑡#, 𝑡#$!). 𝜏(0, 𝑡#, 𝑡#$!). 𝑧𝑐(0,0, 𝑡#$!)

93. Luc_Faucheux_2020
Pricing a swap on today’s yield curve - XXI
¨ Floating leg is also equal to:
93
𝑡𝑖𝑚𝑒
$1
$1
$1
$1
$1
$1
$1
$1
𝑝𝑣_𝑓𝑙𝑜𝑎𝑡 0 = 2
#
{−𝑧𝑐 0,0, 𝑡#$! + 𝑧𝑐(0,0, 𝑡#)}

94. Luc_Faucheux_2020
Pricing a swap on today’s yield curve - XXII
¨ 𝑝𝑣_𝑓𝑙𝑜𝑎𝑡 0 = ∑B 𝑙(0, 𝑡B, 𝑡B/-). 𝜏(0, 𝑡B, 𝑡B/-). 𝑧𝑐(0,0, 𝑡B/-)
¨ 𝑝𝑣_𝑓𝑙𝑜𝑎𝑡 0 = ∑B{−𝑧𝑐 0,0, 𝑡B/- + 𝑧𝑐(0,0, 𝑡B)}
¨ 𝑝𝑣_𝑓𝑖𝑥𝑒𝑑 0 = ∑B 𝑋. 𝜏(0, 𝑡B, 𝑡B/-). 𝑧𝑐(0,0, 𝑡B/-)
¨ 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)
¨ Note that this is also BEFORE the swap “starts”. 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
94

95. Luc_Faucheux_2020
Pricing a swap on today’s yield curve - XXIII
¨ 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)
¨ 𝑝𝑣_𝑓𝑙𝑜𝑎𝑡 0 = ∑B 𝑙(0, 𝑡B, 𝑡B/-). 𝜏(0, 𝑡B, 𝑡B/-). 𝑧𝑐(0,0, 𝑡B/-)
¨ 𝑝𝑣_𝑓𝑙𝑜𝑎𝑡 0 = ∑B{−𝑧𝑐 0,0, 𝑡B/- + 𝑧𝑐(0,0, 𝑡B)}
¨ 𝑝𝑣_𝑓𝑖𝑥𝑒𝑑 0 = ∑B 𝑋. 𝜏(0, 𝑡B, 𝑡B/-). 𝑧𝑐(0,0, 𝑡B/-)
¨ 𝑝𝑣_𝑓𝑙𝑜𝑎𝑡 0 = 𝑝𝑣_𝑓𝑖𝑥𝑒𝑑 0 = ∑B 𝑆𝑅. 𝜏(0, 𝑡B, 𝑡B/-). 𝑧𝑐(0,0, 𝑡B/-)
¨ 𝑆𝑅(0, 𝑇R, 𝑇S) =
∑, 1(&,#,,#,.!).,(&,#,,#,.!).)*(&,&,#,.!)
∑, ,(&,#,,#,.!).)*(&,&,#,.!)
¨ The Swap Rate is a weighted average of the forward rates 𝑙(0, 𝑡B, 𝑡B/-) for a given start of the
swap 𝑇R and maturity 𝑇S
95

96. Luc_Faucheux_2020
Pricing a swap on today’s yield curve – XXIII-a
¨ An “at-the-money Swap” is also said to be “at par” following the terminology of bonds.
¨ This is because the cash flows of a swap are the following:
96
𝑡𝑖𝑚𝑒
$1
𝑋. 𝜏 0, 𝑡B, 𝑡B/- = 𝑋. 𝜏 𝑋. 𝜏 𝑋. 𝜏 𝑋. 𝜏
$1

97. Luc_Faucheux_2020
Pricing a swap on today’s yield curve – XXIII-b
¨ For an “at-the-money” swap the pv of all those cash flows is 0
97
𝑡𝑖𝑚𝑒
$1
𝑋. 𝜏 0, 𝑡B, 𝑡B/- = 𝑋. 𝜏 𝑋. 𝜏 𝑋. 𝜏 𝑋. 𝜏
$1

98. Luc_Faucheux_2020
Pricing a swap on today’s yield curve – XXIII-c
98
= 0
$1
𝑆𝑅. 𝜏 𝑆𝑅. 𝜏 𝑆𝑅. 𝜏 𝑆𝑅. 𝜏
$1

99. Luc_Faucheux_2020
Pricing a swap on today’s yield curve – XXIII-d
99
𝑆𝑅. 𝜏 $1
=
$1

100. Luc_Faucheux_2020
Pricing a swap on today’s yield curve – XXIII-e
¨ 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 “at-par”.
¨ So naturally the swap that is “at-the-money” is also sometimes called a “par-swap”, because
the swap rate is equal in value to the fixed coupon of a par bond (assuming same daycount
convention, periods, holiday,…all other things being equal…)
100

101. Luc_Faucheux_2020
Pricing a swap on today’s yield curve - XXIV
¨ The very powerful concept of the flat curve
¨ Yield curve are not that steep (especially now)
¨ 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)
¨ 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
¨ IF for all forward rates 𝑙(0, 𝑡B, 𝑡B/-), 𝑙 0, 𝑡B, 𝑡B/- = 𝑙(0)
¨ THEN for all Swap rates 𝑆𝑅 0, 𝑇R, 𝑇S = 𝑙(0) when identical swap periods and daycount
fractions
¨ 𝑆𝑅 0, 𝑇R, 𝑇S =
∑, 1 &,#,,#,.! ., &,#,,#,.! .)* &,&,#,.!
∑, , &,#,,#,.! .)* &,&,#,.!
= 𝑙 0 .
∑, , &,#,,#,.! .)* &,&,#,.!
∑, , &,#,,#,.! .)* &,&,#,.!
= 𝑙(0)
101

102. Luc_Faucheux_2020
Pricing a swap on today’s yield curve – XXIV-a
¨ A flat curve is not a bad approximation these days
102

103. Luc_Faucheux_2020
Pricing a swap on today’s yield curve - XXV
¨ 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.
¨ That week was Golden week in Asia, so desks of APAC based banks were thinly staffed
¨ Volatilities were quite high
¨ Rates were also quite high by today’s standards of “new normal”
¨ Goldman Sachs had just done a relatively large “In Arrears – In Advance” trade on
LIBOR12M, exploiting the fact that most dealers were pricing swaps on the swap desk based
on today’s yield curve without checking that they could do it (meaning that ALL cash flows
are LINEAR functions of the discount factors).
¨ 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 today’s yield curve - XXVI
¨ 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
¨ His name is Richard Robb and he has a great book that just came out.
104

105. Luc_Faucheux_2020
Pricing a swap on today’s yield curve - XXVII
¨ 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).
¨ Lehman ran the swap on their HJM model and also realized that there was convexity and
avoided being “picked off”
¨ GS made around 50m that week, which at the time was quite the talk of the town.
¨ 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
¨ 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.
¨ It is also not too far from reality given the large eigenvalue of the 1st factor in PCA analysis of
the yield curve
105

106. Luc_Faucheux_2020
May 1995
Libor in arrears
The Goldman Sachs trade
Future/Forward convexity
106

107. Luc_Faucheux_2020
May 1995
107

108. Luc_Faucheux_2020
May 1995 - II
108

109. Luc_Faucheux_2020
May 1995 - III
109

110. Luc_Faucheux_2020
May 1995 - IV
110

111. Luc_Faucheux_2020
May 1995 - V
111

112. Luc_Faucheux_2020
May 1995 - VI
112

113. Luc_Faucheux_2020
May 1995 - VII
113

114. Luc_Faucheux_2020
May 1995 - VIII
114

115. Luc_Faucheux_2020
So at least for now…..
115