Lf 2020 rates_v_a

Luc_Faucheux_2020
THE RATES WORLD – Part V_a
Summary of part IV, some notes on Ho-Lee
model
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Luc_Faucheux_2020
That deck
2
¨ Could have been named “Everything that you ever wanted to know about Ho-Lee but were
too afraid to ask”
¨ Using Ho Lee as a working example to introduce a lot of concepts, mostly within the HJM
framework, but also illustrating some of the properties of the affine models
¨ Reached 280 slides and 82M, so I had to split it into two sections
¨ Apologies for that.
¨ Should be able to finish up part b shortly, and then, this will truly be everything and anything
that you wanted to know about Ho-Lee

Luc_Faucheux_2020
SUMMARY OF PART IV
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Luc_Faucheux_2020
Summary - I
¨ When looking at payoffs, we should ALWAYS specify the following: What is the payoff
function, when is it fixed, when is it paid, at what time are we trying to compute its value
¨ 𝑉(𝑡) = 𝑉 𝑡, $𝐻(𝑡), 𝑡!, 𝑡"
4
𝑃𝑎𝑖𝑑 𝑎𝑡 𝑡𝑖𝑚𝑒 𝑡"
𝐹𝑖𝑥𝑒𝑑 𝑜𝑟 𝑠𝑒𝑡 𝑎𝑡 𝑡𝑖𝑚𝑒 𝑡!
𝐺𝑒𝑛𝑒𝑟𝑎𝑙 𝑃𝑎𝑦𝑜𝑓𝑓 𝐻 𝑡 𝑖𝑛 𝑐𝑢𝑟𝑟𝑒𝑛𝑐𝑦 $
𝑉𝑎𝑙𝑢𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑝𝑎𝑦𝑜𝑓𝑓 𝑜𝑏𝑠𝑒𝑟𝑣𝑒𝑑 𝑎𝑡 𝑡𝑖𝑚𝑒 𝑡

Luc_Faucheux_2020
Summary – I -a
¨ 𝑉(𝑡) = 𝑉 𝑡, $𝐻(𝑡), 𝑡!, 𝑡"
¨ Most simple payoffs $𝐻(𝑡) are a function of random variables that gets fixed at the same
time 𝑡!, hence why I isolated 𝑡!
¨ However (say SOFR or OIS), the function $𝐻(𝑡) could be as complicated as it can be, and in
the case of averaging indices, could be an integral or a discrete sum over a number of
observations point.
¨ It could also be the MAX or MIN over a given period, or a range accrual
¨ So the possibilities are endless in order to customize this function, making the observation
time 𝑡! meaningless in the very general case
¨ Again, a lot of the simple payoffs have a single discrete time 𝑡! for “fixing”, which is generally
different from the payment time 𝑡", hence the reason why I explicitly kept it as a variable on
its own
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Luc_Faucheux_2020
Summary – I -b
¨ In some ways, this is why quantitative finance can be so tricky for people used to simple
stochastic processes.
¨ Usually we deal with random variables 𝑋(𝑡), which are observed at time 𝑡
¨ HOWEVER in finance, we are looking at random payoff that are observed at time 𝑡! and PAID
at time 𝑡!, where those two points in time usually do not align
¨ This is what usually creates most of the confusion because the deferred payment is actually
a big deal as soon as we introduce volatility (non-deterministic) and correlation between the
payoffs and the Zero discount factors
¨ So ALWAYS explicitly describe the actual payoff and especially WHEN it is paid out
¨ A perfect example of the consequence of this timing difference is the Libor in arrears / in
advance trade or the CMS versus swap rate
¨ BTW, those trades are not that common, but you see in most textbooks, because they were
famous at the time, but also they are a great way to check our understanding and
knowledge, to make sure that we do not get tricked.
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Luc_Faucheux_2020
Summary - II
¨ At each point in time 𝑡, we observe the discount curve 𝑧𝑐 𝑡, 𝑡, 𝑡"
7
𝑂𝑏𝑠𝑒𝑟𝑣𝑒𝑑 𝑎𝑡 𝑡𝑖𝑚𝑒 𝑡
𝑆𝑡𝑎𝑟𝑡 𝑜𝑓 𝑡ℎ𝑒 𝑝𝑒𝑟𝑖𝑜𝑑
𝐸𝑛𝑑 𝑜𝑓 𝑡ℎ𝑒 𝑝𝑒𝑟𝑖𝑜𝑑

Luc_Faucheux_2020
Summary - III
¨ 𝑧𝑐 𝑡, 𝑡, 𝑡" is the price at time 𝑡 of a contract that will pay $1 at time 𝑡"
¨ At that point in time 𝑡 one can define the “then-spot simply compounded rate” as:
¨ 𝑧𝑐 𝑡, 𝑡, 𝑡" =
#
#$% &,&,&! .) &,&,&!
¨ For any point 𝑡! such that 𝑡 < 𝑡! < 𝑡" we can bootstrap the following discount factors:
¨ 𝑧𝑐 𝑡, 𝑡, 𝑡" = 𝑧𝑐 𝑡, 𝑡, 𝑡! ∗ 𝑧𝑐 𝑡, 𝑡!, 𝑡"
¨ We can then also define the “then-forward simply compounded rate” as:
¨ 𝑧𝑐 𝑡, 𝑡!, 𝑡" =
#
#$% &,&",&! .) &,&",&!
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Luc_Faucheux_2020
Summary - IV
¨ Lower case means that the value is known, or fixed or observed
¨ Upper case means the random variable
¨ At each point in time 𝑡, we observe the bootstrapped discount curve 𝑧𝑐 𝑡, 𝑡!, 𝑡"
¨ The discount factors 𝑍𝐶 𝑡, 𝑡!, 𝑡" evolve randomly in time 𝑡 for a given period [𝑡!, 𝑡"]
¨ The corresponding rates we defined as:
¨ 𝐿 𝑡, 𝑡!, 𝑡" =
#
) &,&",&!
. [
#
*+ &,&",&!
− 1]
¨ Also evolves randomly in time 𝑡 for a given period [𝑡!, 𝑡"]
¨ Note that we have not yet defined any dynamics (normal, lognormal,..) of those variables
yet
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Luc_Faucheux_2020
Summary - V
¨ 𝐿 𝑡, 𝑡!, 𝑡" =
#
) &,&",&!
. [
#
*+ &,&",&!
− 1]
¨ 𝐿 𝑡, 𝑡!, 𝑡" . 𝜏 𝑡, 𝑡!, 𝑡" =
#
*+ &,&",&!
. [1 − 𝑍𝐶 𝑡, 𝑡!, 𝑡" ]
¨ When 𝑡 reaches 𝑡!, the random rate 𝐿 𝑡, 𝑡!, 𝑡" gets fixed to 𝑙 𝑡 = 𝑡!, 𝑡!, 𝑡"
¨ (The forward rate becomes fixed to the spot rate)
¨ When 𝑡 reaches 𝑡!, the random discount 𝑍𝐶 𝑡, 𝑡!, 𝑡" gets fixed to 𝑧𝑐 𝑡 = 𝑡!, 𝑡!, 𝑡"
¨ Random variables are observed at a given point in time
¨ HOWEVER what matters in Finance is not only the observation (“fixing”) time, but WHEN a
particular payoff function of those random variables is paid.
¨ The fixing time and the payment time do not have to be the same
¨ In fact most of the time they are not
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Luc_Faucheux_2020
Summary - VI
¨ A very common and useful numeraire is the Zero Discount factor whose period end is the
payment date for the payoff.
¨ The value of a claim that pays on the payment date, normalized by the Zeros, is a martingale
(under the terminal measure, ALWAYS specify which measure you work with)
¨ The measure under which we compute expectations, that is associated to the Zeros whose
period end is the payment date is often referred to as the Terminal measure of Forward
measure
¨ You are free to choose another numeraire or another measure of course (see the deck on
Numeraire), it is a matter of what makes the computation convenient without obscuring the
intuition.
¨ In particular if the claim always pays $1 at time 𝑡"
¨
, &,$#,&",&!
./ &,&,&!
= 𝔼&!
*+ , &!,$#,&",&!
*+ &!,&!,&!
|𝔉(𝑡) = 𝔼&!
*+ , &!,$#,&",&!
#
|𝔉(𝑡) = 𝔼&!
*+ #
#
|𝔉(𝑡) = 1
¨ 𝑉 𝑡, $1, 𝑡!, 𝑡" = 𝑧𝑐 𝑡, 𝑡, 𝑡"
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Luc_Faucheux_2020
Summary - VII
¨ We have derived a couple of useful formulas in part III
¨ Zero coupons:
¨ 𝔼&"
*+ 𝑉 𝑡!, $1 𝑡 , 𝑡!, 𝑡! |𝔉(𝑡) = 1
¨
, &,$#,&",&"
./ &,&,&"
= 𝔼&"
*+ , &",$# & ,&",&"
*+ &",&",&"
|𝔉(𝑡) = 1
¨ 𝑉 𝑡, $1, 𝑡!, 𝑡! = 𝑧𝑐 𝑡, 𝑡, 𝑡!
¨
, &,$#,&",&!
./ &,&,&!
= 𝔼&!
*+ , &",$# & ,&",&!
*+ &!,&!,&!
|𝔉(𝑡) = 1
¨ 𝑉 𝑡, $1, 𝑡!, 𝑡" = 𝑧𝑐 𝑡, 𝑡, 𝑡"
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Luc_Faucheux_2020
Summary - VIII
¨ Deferred premium
¨ 𝔼&"
*+ 𝑉 𝑡!, $1 𝑡 , 𝑡!, 𝑡" |𝔉(𝑡) = 𝔼&"
*+ 𝑉 𝑡!, $𝑍𝐶(𝑡, 𝑡!, 𝑡"), 𝑡!, 𝑡! |𝔉(𝑡) = 𝑧𝑐(𝑡, 𝑡!, 𝑡")
¨
, &,$#,&",&"
./ &,&,&"
= 𝔼&"
*+ , &",$# & ,&",&"
*+ &",&",&"
|𝔉(𝑡) = 1
¨
, &,$#,&",&!
./ &,&,&!
= 𝔼&!
*+ , &",$# & ,&",&!
*+ &!,&!,&!
|𝔉(𝑡) = 1
¨
, &,$#,&",&!
./ &,&,&"
= 𝔼&"
*+ , &",$# & ,&",&!
*+ &",&",&"
|𝔉(𝑡) = 𝔼&"
*+ 𝑉 𝑡!, $𝑍𝐶(𝑡, 𝑡!, 𝑡"), 𝑡!, 𝑡! |𝔉(𝑡) = 𝑧𝑐(𝑡, 𝑡!, 𝑡")
¨ 𝑉 𝑡, $1, 𝑡!, 𝑡" = 𝑧𝑐 𝑡, 𝑡, 𝑡! . 𝑧𝑐 𝑡, 𝑡!, 𝑡" = 𝑧𝑐 𝑡, 𝑡, 𝑡"
¨ If the general claim $𝐻 𝑡 is fixed at time 𝑡!
¨ 𝔼&"
*+
𝑉 𝑡!, $𝐻 𝑡 , 𝑡!, 𝑡" |𝔉(𝑡) = 𝔼&"
*+
𝑉 𝑡!, $𝐻 𝑡 . 𝑍𝐶(𝑡, 𝑡!, 𝑡"), 𝑡!, 𝑡! |𝔉(𝑡)
13

Luc_Faucheux_2020
Summary - IX
¨ 𝔼&"
*+
𝑉 𝑡!, $1 𝑡 , 𝑡!, 𝑡" |𝔉(𝑡) = 𝔼&"
*+
𝑉 𝑡!, $𝑍𝐶(𝑡, 𝑡!, 𝑡"), 𝑡!, 𝑡! |𝔉(𝑡) = 𝑧𝑐(𝑡, 𝑡!, 𝑡")
¨ 𝔼&"
*+ 𝑉 𝑡!, $𝐻 𝑡 , 𝑡!, 𝑡" |𝔉(𝑡) = 𝔼&"
*+ 𝑉 𝑡!, $𝐻 𝑡 . 𝑍𝐶(𝑡, 𝑡!, 𝑡"), 𝑡!, 𝑡! |𝔉(𝑡)
¨ Note that in the case of a general claim that could be a function of the 𝑍𝐶(𝑡, 𝑡!, 𝑡"), we cannot
split the expectation of the products into a product of expectation
¨ But we can use the covariance formula, which is a useful trick used in Tuckmann book, especially
when computing the forward-future convexity adjustment
¨ 𝐶𝑜𝑣𝑎𝑟 𝑋, 𝑌 = 𝔼{𝑋 − 𝔼 𝑋 }. 𝔼{𝑌 − 𝔼[𝑌]}
¨ 𝐶𝑜𝑣𝑎𝑟 𝑋, 𝑌 = 𝔼[𝑋. 𝑌] − 𝔼 𝑋 . 𝔼 𝑌
¨ So in the above, something we should start getting used to:
¨ 𝔼&"
*+ 𝑉 𝑡!, $𝐻 𝑡 . 𝑍𝐶(𝑡, 𝑡!, 𝑡"), 𝑡!, 𝑡! |𝔉(𝑡) =
𝔼&"
*+ 𝑉 𝑡!, $𝐻 𝑡 , 𝑡!, 𝑡! |𝔉(𝑡) . 𝔼&"
*+ 𝑉 𝑡!, $𝑍𝐶(𝑡, 𝑡!, 𝑡"), 𝑡!, 𝑡! |𝔉(𝑡) +
𝐶𝑂𝑉𝐴𝑅{𝑉 𝑡!, $𝐻 𝑡 , 𝑍𝐶(𝑡, 𝑡!, 𝑡"), 𝑡!, 𝑡! |𝔉(𝑡)}
14

Luc_Faucheux_2020
Summary - X
¨ 𝔼&"
*+
𝑉 𝑡!, $𝐻 𝑡 . 𝑍𝐶(𝑡, 𝑡!, 𝑡"), 𝑡!, 𝑡! |𝔉(𝑡) =
𝔼&"
*+ 𝑉 𝑡!, $𝐻 𝑡 , 𝑡!, 𝑡! |𝔉(𝑡) . 𝔼&"
*+ 𝑉 𝑡!, $𝑍𝐶(𝑡, 𝑡!, 𝑡"), 𝑡!, 𝑡! |𝔉(𝑡) +
¨ 𝔼&"
*+
𝑉 𝑡!, $𝐻 𝑡 . 𝑍𝐶(𝑡, 𝑡!, 𝑡"), 𝑡!, 𝑡! |𝔉(𝑡) = 𝔼&"
*+
𝑉 𝑡!, $𝐻 𝑡 , 𝑡!, 𝑡! |𝔉(𝑡) . 𝑧𝑐 𝑡, 𝑡!, 𝑡" +
¨ This looks like we just replaced something by something more complicated, but it highlights
the fact that if the claim is NOT correlated with the discount 𝑍𝐶(𝑡, 𝑡!, 𝑡")
¨ Then:
¨ 𝐶𝑂𝑉𝐴𝑅 𝑉 𝑡!, $𝐻 𝑡 , 𝑍𝐶 𝑡, 𝑡!, 𝑡" , 𝑡!, 𝑡! 𝔉 𝑡 = 0
¨ And:
¨ 𝔼&"
*+ 𝑉 𝑡!, $𝐻 𝑡 . 𝑍𝐶(𝑡, 𝑡!, 𝑡"), 𝑡!, 𝑡! |𝔉(𝑡) = 𝑧𝑐 𝑡, 𝑡!, 𝑡" . 𝔼&"
*+ 𝑉 𝑡!, $𝐻 𝑡 , 𝑡!, 𝑡! |𝔉(𝑡)
15

Luc_Faucheux_2020
Summary - XI
¨ When there is NO correlation between the claim and the Zeros
¨ 𝔼&"
*+ 𝑉 𝑡!, $𝐻 𝑡 . 𝑍𝐶(𝑡, 𝑡!, 𝑡"), 𝑡!, 𝑡! |𝔉(𝑡) = 𝑧𝑐 𝑡, 𝑡!, 𝑡" . 𝔼&"
*+ 𝑉 𝑡!, $𝐻 𝑡 , 𝑡!, 𝑡! |𝔉(𝑡)
¨ If the general claim $𝐻 𝑡 is fixed at time 𝑡!
¨ 𝔼&"
*+
𝑉 𝑡!, $𝐻 𝑡 , 𝑡!, 𝑡" |𝔉(𝑡) = 𝔼&"
*+
𝑉 𝑡!, $𝐻 𝑡 . 𝑍𝐶(𝑡, 𝑡!, 𝑡"), 𝑡!, 𝑡! |𝔉(𝑡)
¨ 𝔼&"
*+
𝑉 𝑡!, $𝐻 𝑡 , 𝑡!, 𝑡" |𝔉(𝑡) = 𝑧𝑐 𝑡, 𝑡!, 𝑡" . 𝔼&"
*+
𝑉 𝑡!, $𝐻 𝑡 , 𝑡!, 𝑡! |𝔉(𝑡)
¨
, &,$0(&),&",&!
./ &,&,&"
= 𝔼&"
*+ , &",$0 & ,&",&!
*+ &",&",&"
|𝔉(𝑡) = 𝔼&"
*+ 𝑉 𝑡!, $𝐻 𝑡 , 𝑡!, 𝑡" |𝔉(𝑡)
¨
, &,$0(&),&",&!
./ &,&,&"
= 𝑧𝑐 𝑡, 𝑡!, 𝑡" . 𝔼&"
*+
𝑉 𝑡!, $𝐻 𝑡 , 𝑡!, 𝑡! |𝔉(𝑡)
¨ 𝑉 𝑡, $𝐻(𝑡), 𝑡!, 𝑡" = 𝑧𝑐 𝑡, 𝑡, 𝑡! . 𝑧𝑐 𝑡, 𝑡!, 𝑡" . 𝔼&"
*+
𝑉 𝑡!, $𝐻 𝑡 , 𝑡!, 𝑡! |𝔉(𝑡)
¨ 𝑉 𝑡, $𝐻(𝑡), 𝑡!, 𝑡" = 𝑧𝑐 𝑡, 𝑡, 𝑡" . 𝔼&"
*+ 𝑉 𝑡!, $𝐻 𝑡 , 𝑡!, 𝑡! |𝔉(𝑡)
16

Luc_Faucheux_2020
Summary - XI
¨ 𝑉 𝑡, $𝐻(𝑡), 𝑡!, 𝑡" = 𝑧𝑐 𝑡, 𝑡, 𝑡" . 𝔼&"
*+ 𝑉 𝑡!, $𝐻 𝑡 , 𝑡!, 𝑡! |𝔉(𝑡)
¨ Note again that the above is ONLY true if there is no correlation between the claim and the
discount
¨ If there is, the Covariance term will appear, (this will be the famed convexity adjustment)
¨ Expressing the convexity adjustment as a covariance term sometimes makes it easier to
compute (Tuckmann book) but also put front and center the fact that if you value a claim
that is a function of the Zeros, and the timing is not the regular timing for the payment
(value a LIBOR in ARREARS trade for example), or that function is not a linear combination of
the Zeros (value a LIBOR square trade for example) YOU WILL HAVE a convexity adjustment
to take into account
¨ IF CORRELATION
¨ 𝑉 𝑡, $𝐻(𝑡), 𝑡!, 𝑡" = 𝑧𝑐 𝑡, 𝑡, 𝑡" . 𝔼&"
*+
𝑉 𝑡!, $𝐻 𝑡 , 𝑡!, 𝑡! |𝔉(𝑡) +
𝑧𝑐 𝑡, 𝑡, 𝑡! . 𝐶𝑂𝑉𝐴𝑅&"
*+
𝑉 𝑡!, $𝐻 𝑡 , 𝑍𝐶 𝑡, 𝑡!, 𝑡" , 𝑡!, 𝑡! 𝔉 𝑡
17

Luc_Faucheux_2020
Summary - XII
¨ If the payoff has no correlation, you can “move” the payment up and down the curve as per
the deterministic zeros (lower case), like you would on a swap desk
¨ If the payoff has ANY correlation with the zeros, go talk to the option desk because there is
some convexity
¨ There are however some special payoffs that ARE function of the zeros but for which the
convexity magically disappear, and you can price them in the deterministic world of lower
case, and go talk to the swap trader (hint: those payoffs are the regular swaps).
¨ Those are in the next slide
¨ The magic trick is usually (1 = 1), or (𝑋 = 𝑋), or (𝑋 − 𝑋 = 0) or (
3
3
= 1) or (1 − 1 = 0)
18

Luc_Faucheux_2020
¨ 𝑍𝐶 𝑡, 𝑡!, 𝑡" =
#
#$4 &,&",&! .)
and 𝑧𝑐 𝑡, 𝑡!, 𝑡" =
#
#$% &,&",&! .)
¨ $𝐻 𝑡 = $𝐿 𝑡, 𝑡!, 𝑡" = $
#
)
(
#
*+ &,&",&!
− 1)
¨ 𝔼&!
*+ 𝑉 𝑡", $𝐿 𝑡, 𝑡!, 𝑡" , 𝑡!, 𝑡" |𝔉(𝑡) = 𝑙 𝑡, 𝑡!, 𝑡" =
#
)
(
#
./ &,&",&!
− 1)
¨ 𝔼&"
*+
𝑉 𝑡!, $
4 &,&",&! .)
#$4 &,&",&! .)
, 𝑡!, 𝑡! |𝔉(𝑡) =
% &,&",&! .)
#$% &,&",&! .)
¨ 𝔼&"
*+
𝑉 𝑡!, $
#
#$4 &,&",&! .)
, 𝑡!, 𝑡! |𝔉(𝑡) =
#
#$% &,&",&! .)
= 𝑧𝑐 𝑡, 𝑡!, 𝑡" =
./ &,&,&!
./ &,&,&"
¨ 𝔼&"
*+ 𝑉 𝑡!, $𝑍𝐶 𝑡, 𝑡!, 𝑡" , 𝑡!, 𝑡! |𝔉(𝑡) = 𝑧𝑐 𝑡, 𝑡!, 𝑡" = 𝔼&"
*+ 𝑉 𝑡!, $1 𝑡 , 𝑡!, 𝑡" |𝔉(𝑡)
¨ THIS is why you can value a swap in the deterministic world (lower case, no volatility, no
convexity, no dynamics, no option trader involved, just a swap trader and one discount
curve)
¨ All right that was a good summary
Summary - XIII
19

Luc_Faucheux_2020
Another nice little summary
20

Luc_Faucheux_2020
Another summary - I
¨ Everything is based of the zeros 𝑍𝐶 𝑡, 𝑡, 𝑡"
¨ For a fixed value of time 𝑡, the observed zeros 𝑍𝐶 𝑡, 𝑡, 𝑡" is a regular function of the
maturity 𝑡". The graph of that function is called the “ZCB, Zero Coupon Bond, price curve at
time 𝑡”, or the “term structure curve at time 𝑡”.
¨ For a fixed value of time 𝑡", 𝑍𝐶 𝑡, 𝑡, 𝑡" will be a STOCHASTIC process of the first variable 𝑡.
¨ The term structure is further defined through the bootstrap equations:
¨ 𝑍𝐶 𝑡, 𝑡!, 𝑡" = 𝑍𝐶 𝑡, 𝑡, 𝑡" /𝑍𝐶 𝑡, 𝑡, 𝑡!
¨ 𝑍𝐶 𝑡, 𝑡, 𝑡" = 𝑍𝐶 𝑡, 𝑡, 𝑡! ∗ 𝑍𝐶 𝑡, 𝑡!, 𝑡"
¨ Only the first variable is the “real” time 𝑡 over which the stochastic process evolve. The
others are just parametrizations on the term structure curve, and you can apply all the rules
and tricks of regular calculus on those. On the first one though, be super duper extra careful
that you need to deal with all the nastiness of full blown stochastic calculus.
21

Luc_Faucheux_2020
Another summary - II
¨ Because again remember that not all 3 time variables are equal, the last two are just indices
on the curve and we can perform all the usual calculus that we want on those two. The first
one is the “real” time, and some care needs to be taken when trying to write differential
equations in the case of stochastic variables
22
𝑂𝑏𝑠𝑒𝑟𝑣𝑒𝑑 𝑎𝑡 𝑡𝑖𝑚𝑒 𝑡,
STOCHASTIC CALCULUS RULES APPLY
𝑆𝑡𝑎𝑟𝑡 𝑜𝑓 𝑡ℎ𝑒 𝑝𝑒𝑟𝑖𝑜𝑑 𝑜𝑛 𝑡ℎ𝑒 𝑐𝑢𝑟𝑣𝑒,
REGULAR CALCULUS RULES APPLY
𝑍𝐶 𝑡, 𝑡!, 𝑡"
𝐸𝑛𝑑 𝑜𝑓 𝑡ℎ𝑒 𝑝𝑒𝑟𝑖𝑜𝑑 𝑜𝑛 𝑡ℎ𝑒 𝑐𝑢𝑟𝑣𝑒,
REGULAR CALCULUS RULES APPLY

Luc_Faucheux_2020
Another summary - II
¨ Because again remember that not all 3 time variables are equal, the last two are just indices
on the curve and we can perform all the usual calculus that we want on those two. The first
one is the “real” time, and some care needs to be taken when trying to write differential
equations in the case of stochastic variables
23
𝑂𝑏𝑠𝑒𝑟𝑣𝑒𝑑 𝑎𝑡 𝑡𝑖𝑚𝑒 𝑡,
BROWNIAN CALCULUS RULES APPLY
𝑆𝑡𝑎𝑟𝑡 𝑜𝑓 𝑡ℎ𝑒 𝑝𝑒𝑟𝑖𝑜𝑑 𝑜𝑛 𝑡ℎ𝑒 𝑐𝑢𝑟𝑣𝑒,
NEWTONIAN CALCULUS RULES APPLY
𝑍𝐶 𝑡, 𝑡!, 𝑡"
𝐸𝑛𝑑 𝑜𝑓 𝑡ℎ𝑒 𝑝𝑒𝑟𝑖𝑜𝑑 𝑜𝑛 𝑡ℎ𝑒 𝑐𝑢𝑟𝑣𝑒,
NEWTONIAN CALCULUS RULES APPLY

Luc_Faucheux_2020
Another summary – II - a
¨ Another way to think about it is that the first variable in time 𝑡 is “truly” continuous.
¨ The price of anything moves all the time, every milliseconds.
¨ On the other hand the two other variables 𝑡!and 𝑡" describes dates, like for example the
period [𝑡!, 𝑡"] = [“12 December 2022”, ”12 June 2023”]
¨ The two other dates are really about actual cashflows, like for example wiring money, using
the wire transfer, or sweeping money in and out of a margin account. Those usually are
done once a day at a precise time.
¨ Except for margin call, there the amount and the timing of the cashflow depend on the
actual risk in the portfolio and the market move, try modeling this one… (although even in
the case of a margin call, you might get the call say at 11:30am, and if you wire funds, that
wire will still not be instantaneous, if you cannot wire funds then the closing of the positions
will be instantaneous)
¨ So the two other variables 𝑡! and 𝑡" really have a minimal discrete daily increment, but it is
convenient to treat them as continuous for a number of reasons, knowing that we will
mostly use them in a discrete daily manner
24

Luc_Faucheux_2020
Another summary - III
¨ From the variable 𝑍𝐶 𝑡, 𝑡!, 𝑡" , we are absolutely free to define a bunch of other variables,
and we certainly did not deprive ourselves of doing so:
¨ Continuously compounded FORWARD : 𝑍𝐶 𝑡, 𝑡!, 𝑡" = exp −𝑅 𝑡, 𝑡!, 𝑡" . 𝜏 𝑡, 𝑡!, 𝑡"
¨ Simply compounded FORWARD: 𝑍𝐶 𝑡, 𝑡!, 𝑡" =
#
#$) &,&",&! .4 &,&",&!
¨ Annually compounded FORWARD : 𝑍𝐶 𝑡, 𝑡!, 𝑡" =
#
(#$5 &,&",&! )
# $,$",$!
¨ 𝑞-times per year compounded FOWARD: 𝑍𝐶 𝑡, 𝑡!, 𝑡" =
#
(#$
&
'
.5' &,&",&! )
'.# $,$",$!
¨ The function 𝜏 𝑡, 𝑡!, 𝑡" is the daycount fraction, will usually depends on what convention
(ACT/ACT, ACT/360, 30/360,…) you will choose, and potentially adjustment for holidays and
what holiday center
25

Luc_Faucheux_2020
Another summary - IV
¨ The SPOT are obtained from the FORWARD by setting 𝑡! = 𝑡
¨ Continuously compounded SPOT : 𝑍𝐶 𝑡, 𝑡, 𝑡" = exp −𝑅 𝑡, 𝑡, 𝑡" . 𝜏 𝑡, 𝑡, 𝑡"
¨ Simply compounded SPOT: 𝑍𝐶 𝑡, 𝑡, 𝑡" =
#
#$) &,&,&! .4 &,&,&!
¨ Annually compounded SPOT : 𝑍𝐶 𝑡, 𝑡, 𝑡" =
#
(#$5 &,&,&! )
# $,$,$!
¨ 𝑞-times per year compounded SPOT: 𝑍𝐶 𝑡, 𝑡, 𝑡" =
#
(#$
&
'
.5' &,&,&! )
'.# $,$,$!
¨ CAREFUL: I like to call those “then-SPOT” and “then-FORWARD”, because a lot of people use
“SPOT” for saying (𝑡 = 0), or even depending on the location (𝑡 = 1) or (𝑡 = 2).
¨ When you hear someone say SPOT, be always careful to ask for an exact definition !
26

Luc_Faucheux_2020
Another summary - V
¨ And that is really it.
¨ Any kind of modeling is usually done on the 𝐿 𝑡, 𝑡!, 𝑡" , or sometimes on the 𝑅 𝑡, 𝑡!, 𝑡"
¨ You end up writing something like:
¨ 𝑑𝐿 𝑡, 𝑡!, 𝑡" = 𝐴 𝑡, 𝑡!, 𝑡" . 𝑑𝑡 + 𝐵 𝑡, 𝑡!, 𝑡" . ([). 𝑑𝑊 𝑡, 𝑡!, 𝑡" or more exactly in SIE form:
¨ 𝐿 𝑡 + 𝛿𝑡, 𝑡!, 𝑡" − 𝐿 𝑡, 𝑡!, 𝑡" = ∫&
&$6&
𝐴 𝑠, 𝑡!, 𝑡" . 𝑑𝑠 + ∫&
&$6&
𝐵 𝑠, 𝑡!, 𝑡" . ([). 𝑑𝑊 𝑠, 𝑡!, 𝑡"
¨ Note that we chose only one-factor, meaning one driver of stochastic process, for sake of
notation, you could even say the above can be interpreted as a vector notation), but you
could write a multi factor as:
¨ 𝑑𝐿 𝑡, 𝑡!, 𝑡" = 𝐴 𝑡, 𝑡!, 𝑡" . 𝑑𝑡 + ∑7 𝐵7 𝑡, 𝑡!, 𝑡" . ([). 𝑑𝑊7 𝑡, 𝑡!, 𝑡"
27

Luc_Faucheux_2020
Another summary - VI
¨ Next time as an industry we rebuild the whole thing all over again (which should be any time
now), we will make sure to build all the modeling off the Zeros 𝑍𝐶 𝑡, 𝑡!, 𝑡" and not the
rates.
¨ This will be a lot more transparent and less confusing.
¨ Because Bonds and regular swaps are LINEAR functions of the Zeros 𝑍𝐶 𝑡, 𝑡!, 𝑡" but they
are NON-LINEAR (CONVEX) functions of the rates, say 𝐿 𝑡, 𝑡!, 𝑡" .
¨ AND SO you can calculate the convexity of a swap or a bond against the yield
¨ AND THEN everyone gets confused about convexity, because if there is convexity, there
should be a convexity adjustment, and so the value of a bond should depend on the
volatility right ?
¨ AND SO to price a swap I cannot just go the swap desk, because they only have a yield curve
and they do not have any volatility as an input.
28

Luc_Faucheux_2020
Another summary - VII
¨ So again, when someone start telling you about convexity, make sure to ask against which
variable?
¨ Plotting a swap value (y-axis) as a function of the yield (x-axis) is the “wrong” way to plot it
because it is a convex function.
¨ 𝑝𝑣_𝑓𝑙𝑜𝑎𝑡 0 = ∑! 𝑙(0, 𝑡!, 𝑡!$#). 𝜏(0, 𝑡!, 𝑡!$#). 𝑧𝑐(0,0, 𝑡!$#)
¨ 𝑝𝑣_𝑓𝑙𝑜𝑎𝑡 0 = ∑!{−𝑧𝑐 0,0, 𝑡!$# + 𝑧𝑐(0,0, 𝑡!)}
¨ 𝑝𝑣_𝑓𝑖𝑥𝑒𝑑 0 = ∑! 𝑋. 𝜏(0, 𝑡!, 𝑡!$#). 𝑧𝑐(0,0, 𝑡!$#)
¨ 𝑃𝑉 0 = 𝑝𝑣𝐹𝑖𝑥𝑒𝑑 0 − 𝑝𝑣𝐹𝑖𝑥𝑒𝑑(0)
¨ For a swap where we pay float and receive Fixed
¨ 𝑃𝑉 0 = ∑! 𝑋. 𝜏(0, 𝑡!, 𝑡!$#). 𝑧𝑐(0,0, 𝑡!$#) − ∑!{−𝑧𝑐 0,0, 𝑡!$# + 𝑧𝑐(0,0, 𝑡!)}
¨ That is a linear function indeed of the 𝑧𝑐(0,0, 𝑡!$#)
¨ Note that I chose the lower case notation here because we are looking those zeros up on
today’s yield curve
29

Luc_Faucheux_2020
Another summary - VIII
¨ SWAP FIXED RECEIVE VERSUS REGULAR FLOAT PAY (pay Float, receive Fixed)
30
𝑡𝑖𝑚𝑒
Above the line:
We receive
Below the line:
We pay
𝑡 = 0 𝑡!
𝑋. 𝜏(0, 𝑡!, 𝑡!$#)
𝜏(0, 𝑡!, 𝑡!$#). 𝑙(0, 𝑡!, 𝑡!$#)

Luc_Faucheux_2020
Another summary - IX
¨ SWAP FIXED RECEIVE VERSUS REGULAR FLOAT PAY (pay Float, receive Fixed)
31
𝑡𝑖𝑚𝑒
$1
$1
𝑋. 𝜏 0, 𝑡!, 𝑡!$# = 𝑋. 𝜏 𝑋. 𝜏 𝑋. 𝜏 𝑋. 𝜏
$1
$1
$1
$1
$1
$1

Luc_Faucheux_2020
Another summary - X
¨ But if we were to plot the swap pv as a function of the rates (say in the simplest case of a flat
yield curve) we would get a graph like below (blue line is swap PV, orange is straight line)
32

Luc_Faucheux_2020
Another summary - XI
¨ So a swap PV or a Bond PV will be usually a CONVEX function of the yield.
¨ You already know that because you have formula for the duration (first derivative with
respect to rate) and the convexity (second derivative) for Bonds and Swaps.
¨ HOWEVER, again, and sorry to stress that one more time, the rate is NOT the natural x-axis
that we should choose. We should pick the Zeros.
¨ Bonds and regular swaps (no funny Libor in arrears for example) are linear functions of the
Zeros, hence there is NO convexity, hence you can compute the present value using only
today’s yield curve, and only go to your swap desk, and do not involve your option trader
with it.
33

Luc_Faucheux_2020
Another summary - XII
¨ But nonetheless plenty of people get confused about it, I can bet a lot of money that you
can ask any “senior” manager, let’s call him Mark, and let’s say your name is Jeremy
¨ Jeremy: “hey swaps have convexity right? Just like bonds because the duration is a function
of the swap rate (or implied bond yield in the case of a bond) ?”
¨ Mark: “of course you imbecile”
¨ Jeremy: “ cool, cool cool, cool cool cool, so…yeah…so swaps have convexity adjustments?”
¨ Mark:”well of course they do.”
¨ Jeremy:” ok ok, right, all right all right all right, so what volatility do I plug in to compute the
convexity adjustment? The swap trader tells me that she is not using any volatility surface
to price a swap, just the yield curve?”
¨ Mark:”……I am very busy, figure it out yourself, I am way too senior to answer those
questions”
34

Luc_Faucheux_2020
Another summary - XIII
¨ Jeremy: “Hey Mark, sooo…yeah…what volatility do I use to
compute the present value of a swap, since you just told
me that a swap is convex, and so there has to be a
convexity adjustment?”
35
¨ Mark: ”Leave me alone Jeremy, I am way too senior and I
make way too much money to use my brain on things like
your stupid questions”

Luc_Faucheux_2020
Another summary - XIV
¨ ALSO, there is another twist to the story. Most Risk Management is built on bumping the
yield curve, NOT the Zero curve, by usually what are pre-defined but equally spaced and
symmetrical
¨ For example, most risk report you will see are “ bump the yield curve up 1 basis point,
recalculate the PV of the portfolio PV(+1), bump down by 1 basis point, recalculate the PV
again PV(-1), and then you get:
¨ Duration_up = Delta_up =PV(+1)-PV(0)
¨ Duration_down = Delta_down=PV(0)-PV(-1)
¨ Gamma = Convexity = Delta_up – Delta_down
¨ So every risk report out there will show Gamma and convexity for a Bond or swap portfolio.
¨ Note that this is NOT the same Gamma as an option Gamma for example, there is no THETA
associated to that Gamma
36

Luc_Faucheux_2020
Another summary - XV
¨ You will never see (not yet at least) a risk report that says:
¨ “We are bumping the yield curve up by X, and then down by Y, so that X and Y are such that
the associated differences in the Zeros from the base curve are symmetrical and equal in
value, because we know that a bond and swap portfolio has 0 convexity against the Zeros,
and we want to express our risk in the right choice of x-axis”
¨ If you ever see a risk report like this, call me.
37

Luc_Faucheux_2020
Another summary – XV - a
¨ So on swap desks and around trading floors you will usually hear people saying that
“receiving on swaps is being long convexity” (usually the convention is that the “fixed” is
omitted, so that meant “receiving Fixed and paying Float on a regular swap result in a
position that is long convexity, that has positive convexity, that has a positive second order
derivative of the present value of that position WITH RESPECT TO rates”
¨ The “WITH RESPECT TO rates” is the important part here.
¨ It is true that a receiver swap will be positively convex with respect to the level of rates.
¨ That will not mean that when valuing such a swap you will need to value a convexity
adjustment and input some sort of volatility, because a swap is a linear function of Zeros,
meaning a linear function of fixed cashflows paid in the future, and as such can be valued
without consideration to the dynamics of rates or Zeros, because the ratio of the $1 payoff
to the Zero at maturity is a martingale under the terminal measure
¨ This is of course true in the Terminal (or Forward) measure, but also in the Early (discount)
measure. This is the measure we have essentially used up to now (except when dealing with
HJM, where we used the ℚ risk-free measure). I will add a summary on all the different
measures and how to go from one to the other, either in part VI or part VII
38

Luc_Faucheux_2020
Another summary – XV - b
¨ Zero coupons:
¨ 𝔼&"
*+
𝑉 𝑡!, $1 𝑡 , 𝑡!, 𝑡! |𝔉(𝑡) = 1
¨
, &,$#,&",&"
./ &,&,&"
= 𝔼&"
*+ , &",$# & ,&",&"
*+ &",&",&"
|𝔉(𝑡) = 1
¨ 𝑉 𝑡, $1, 𝑡!, 𝑡! = 𝑧𝑐 𝑡, 𝑡, 𝑡!
39

Luc_Faucheux_2020
Another summary – XV - c
¨ So a swap receiver position will be said to be positively convex in rates
40

Luc_Faucheux_2020
Another summary – XV - d
¨ Similarly a paying fixed position in swaps will be said to be negatively convex in rates
41

Luc_Faucheux_2020
Another seemingly random rambling about
convexity
42

Luc_Faucheux_2020
Another summary – XV - e
¨ Please note that there is usually another twist to the story, which makes it a little more
confusing, so worth to go over.
¨ It has to do with what you have on the other side of your swap position to hedge your
duration.
¨ Suppose that you pay fixed on a swap, you are short duration (you are short the market, if
rates go up you will be up more on the leg where you receive floating payments, your PL will
be positive, but the convention is that the rates market is quoted in terms of bonds, as you
know from the Wall Street Journal, Bond prices and Bond yields move in opposite direction,
so if rates go up, bond prices will go down, the market goes down, so if your PL is positive
when the market goes down, you are “short the market”, in terms of rates, you are “short
duration”
¨ If the maturity of the swap is not too long, a good hedge against being short duration is to
buy a strip of Eurodollar futures along the curve (see Part I).
¨ So to be duration neutral on a swap where you pay fixed, you will buy Eurodollar futures.
43

Luc_Faucheux_2020
Another summary – XV - f
¨ As we saw in part I (but still have not formally computed in any model, hopefully we can do
this in this part), if you are long a Eurodollar future contract, you are short convexity, and
short volatility (this time this is real, you are short convexity and short volatility), if the rate
at which you lend/borrow overnight to offset the daily cashflow on the future contract, is
positively correlated with the rate of the underlying future contract (let’s face it, this is
usually a pretty good assumption, but in those weird times that we live in, especially on the
front end of the curve, do not take for granted that being long a Eurodollar future makes you
automatically short volatility, always good to check yourself before you wreck yourself)
¨ But the standard argument is the one we went over in part I.
44

Luc_Faucheux_2020
Another summary – XV - g
¨ Suppose that you are long a Eurodollar future contract on a specific bucket on the curve, say
EDZ2 (December 3-month covering the period dec 2022 – mar 2023)
¨ More exactly if you remember part I, the dates are the IMM dates (the IMM Wednesday is
the 3rd Wednesday of the month, the IMM Monday is the Monday preceding the IMM
Wednesday)
45

Luc_Faucheux_2020
Another summary – XV - h
¨ So you are long the EDZ2 future contract
¨ The price of the contract is 𝑃𝑟𝑖𝑐𝑒 = 1 − 𝑅𝑎𝑡𝑒
¨ It was constructed like that so that Price and Rate would go in opposite direction when the
market move (so that we are in the Bond familiar world where prices go up when rates go
down, and inversely)
¨ At the time they kept it simple, and just chose a linear relationship, as opposed to more of a
Bond-like formula like for example:
¨ 𝑃𝑟𝑖𝑐𝑒 =
#
#$8.9:&;
¨ Where the daycount fraction 𝑞 would be 3-months or (1/4)
¨ Note that some currencies (like Australia for example have defined their contracts this way)
46

Luc_Faucheux_2020
Another summary – XV - i
¨ Let’s go back to some of our favorite futures in the world from Part 1
47

Luc_Faucheux_2020
Another summary – XV - j
¨ 𝑃𝑟𝑖𝑐𝑒 = 1 − 𝑅𝑎𝑡𝑒 (to look like a bond, so that price and yield/rate go in opposite direction)
¨ Contract size = 1mm USD…ohh that is a big size, essentially it is built so that if you want to
hedge 1mm of a swap you will buy a strip of 1 contract in each bucket
¨ Value of 1pt = $2,500
¨ That is to essentially ‘mock” the fact that it is a 3-month period, hence the daycount farction
will be close to (1/4)
¨ So the value of 1 basis point will be (2,500/100)=25
¨ The contract change value by $25 per basis point.
¨ The convention to quote it is in basis point
¨ 𝑃𝑟𝑖𝑐𝑒(𝑖𝑛 𝑝𝑜𝑖𝑛𝑡) = 1 − 𝑅𝑎𝑡𝑒
¨ 𝑃𝑟𝑖𝑐𝑒 𝑖𝑛 𝑏𝑎𝑠𝑖𝑠 𝑝𝑜𝑖𝑛𝑡 = 100 − 100 ∗ 𝑅𝑎𝑡𝑒
¨ Remember rate are also usually quoted in %, say for example 2.3%, which is really 0.023, so
the price in basis point will be:
¨ 𝑃𝑟𝑖𝑐𝑒 𝑖𝑛 𝑏𝑎𝑠𝑖𝑠 𝑝𝑜𝑖𝑛𝑡 = 100 − 100 ∗ 0.023 = 100 − 2.3 = 97.7
48

Luc_Faucheux_2020
Another summary – XV - k
¨ Tick size and tick value: this is somewhat arbitrary
¨ Most recent is:
¨ Tick size = 0.005 = 0.5 of a basis point
¨ The tick value would then be : $25*.5=$12.5
¨ That did not use to be always the case, and also it depends on the actual contract.
¨ The daily settlement is usually on the tick size.
¨ Only exception (that changed in 1999 I think) is for the contract that will expire and set to
the LIBOR that will fix on Monday IMM 10am London time.
¨ The contract last trading day is Friday before, it will settle on the tick size and then on
Monday it will re-settle on the actual LIBOR fixing (with all the decimals)
¨ Before that, swap desks used to have huge PL on IMM because the ED future contract would
not settle on the same rate as the LIBOR rate…good times….am dating myself, that was a
generation ago
49

Luc_Faucheux_2020
Another summary – XV - l
¨ Contract size = 1mm
¨ If rate moves by 1bp, then future price moves by $25
¨ If you think about say a bond with only a 3 month period, this makes sense (this is why they
designed the contract that way, the Chicago boys are not stupid)
¨ 𝑃𝑟𝑖𝑐𝑒 =
#
#$8.9:&;
~
#
#$(#/=).9:&;
~1 −
#
=
. 𝑟𝑎𝑡𝑒
¨ So:
>?@!/;
>9:&;
~(
#
=
)
¨
>AB&B@;
>9:&;
~
#
=
. 𝐶𝑜𝑛𝑡𝑟𝑎𝑐𝑡𝑆𝑖𝑧𝑒 =
#
=
. 1𝑚𝑚 = 250,000.
¨ So $250,000 per unit (actual number) is $2,500 per point, or $25 per basis point
¨ This is really math 101, but at times can be confusing, so worth always getting your units
right. The good news is that in Rates, basis points, points, percentage and numbers are
usually related to each other by factors 100, so if you messed up, it will be obvious relatively
quickly, it’s not like you are off by a factor 𝜋 or something that small which will make it
hard to see that you are wrong
50

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Another summary – XV - m
¨ One small note that might illuminate any confusion you might have.
¨ The “quoted price” does not have to be the price of an equivalent bond, we can decide to
quote however we want, in our case
¨ 𝑄𝑢𝑜𝑡𝑒𝑑𝑃𝑟𝑖𝑐𝑒 = 100 ∗ (1 − 𝑅𝑎𝑡𝑒), with the Rate quoted in number
¨ 𝐴𝑐𝑡𝑢𝑎𝑙𝑃𝑟𝑖𝑐𝑒 = 1,000,000 ∗ {1 −
#
=
. 𝑅𝑎𝑡𝑒}, so you get $25/bp in rate
¨ For example you can look at the Australian contract, where they actually built it like a real
bond, you do not have a constant $25 per basis point, because the relationship between the
price and the rate is NOT linear
¨ This is not too surprising they never do anything like everyone else down there in the land of
plenty, eating vegemite sandwiches in fried out combies
¨ That makes for some fun workaround when you pull Bloomberg fields into a spreadsheet, or
try to incorporate those contracts into a Macro portfolio and try to fit it into the mean
variance formula (MPT), just ask Felix Turton
51

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Another summary – XV - n
¨ The Australian contract, pay attention to the variable value per basis point, look at the field
“Value of 1.0 pt”, result is “Varies” when you pull that into an Excel spreadsheet. Needless
to say, any formula that expects a number and gets “Varies” will most certainly puke on you
52

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Another summary – XV - o
¨ Also, thanks Bloomberg! “Varies”…super helpful and useful.. How does it vary? How do I
compute duration delta, how do I hedge a swap?
¨ As a little exercise for you (we did it in the spreadsheet when looking at the Macro portfolio
under MPT in the Risk Management module)
¨ https://www.asx.com.au/documents/products/ird-pricing-guide.pdf
¨ And then it gets even more confusing because the quoted price will still be (100-yield), even
though you have a completely different formula for the actual value as a function of yield
(next page)
¨ So the quoted price times the point value will be the contract value, but in the case of IRZ2,
the point value “varies”….good luck….
¨ You will notice that in the case of EDZ2, Bloomberg does not bother indicating “Contract
Value”, they just say “Pt Value x Price”
53

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Another summary – XV - p
54

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Another summary – XV – p -2
¨ So for EDZ2:
¨ 𝐴𝑐𝑡𝑢𝑎𝑙𝑃𝑟𝑖𝑐𝑒 = 1,000,000 ∗ {1 −
#
=
. 𝑅𝑎𝑡𝑒}, so you get $25/bp in rate
¨ And for IRZ2:
¨ 𝐴𝑐𝑡𝑢𝑎𝑙𝑃𝑟𝑖𝑐𝑒 = 1,000,000 ∗
#
#$
)*
+,-
.9:&;
so you get $XX/bp in rate
¨ The $XX/bp varies with the actual level of rate
55

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Another summary – XV - q
¨ OK so back to our EDZ2 contract, and the hand waving argument why you are short volatility
if you are long the contract (we did that in Part I)
¨ If Rates go up, the value of the future goes down (by $25 per basis point).
¨ If you are long that future, your PL is negative
¨ BECAUSE it is a FUTURE contract, and NOT a FORWARD contract (see section in Future
contract later), an ACTUAL cash flow has to occur after settlement to offset the PL (make the
exchange whole if you want)
¨ So you need to pay
¨ So you need to borrow to pay
¨ But rates just went higher (again assuming that the rate at which you have to lend and
borrow is somewhat positively correlated with the rate of the underlying future contract,
not always true but that gives you an idea of what assumptions we will have to build in the
modeling), so you will have to borrow at a higher rate, borrowing is more expensive
56

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Another summary – XV - r
¨ So now assume that rates do exhibit some volatility (do move around) over the life of the
future contract.
¨ When rates go up, future price goes down, you have negative PL, you need to borrow at a
higher rate
¨ When rates go down, future price goes up, you have positive PL, you can now lend at a
lower rate
¨ Over time this borrowing at a high rate and lending at a low rate will accumulate into a loss
¨ The more violent the moves the higher the accumulated loss
¨ The more frequent the moves the higher the accumulated loss
¨ The higher the correlation between the future rate and the borrowing/lending/financing
rate the higher the accumulated loss
¨ The higher the volatility, the higher the correlation, the higher the accumulated loss
¨ The higher the Covar, the higher the accumulated loss
57

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Another summary – XV - s
¨ This should give us a hint that when we will be looking at calculating that convexity
adjustment, we will be looking at something like:
¨ 𝔼.!
/0
𝑉 𝑡1, $𝐻 𝑡 . 𝑍𝐶(𝑡, 𝑡1, 𝑡2), 𝑡1, 𝑡1 |𝔉(𝑡) = 𝔼.!
/0
𝑉 𝑡1, $𝐻 𝑡 , 𝑡1, 𝑡1 |𝔉(𝑡) . 𝑧𝑐 𝑡, 𝑡1, 𝑡2 +
𝐶𝑂𝑉𝐴𝑅{𝑉 𝑡1, $𝐻 𝑡 , 𝑍𝐶(𝑡, 𝑡1, 𝑡2), 𝑡1, 𝑡1 |𝔉(𝑡)}
¨ As a general rule, if there is any correlation between the payoff (whether it be equity, ED
future,..) and the financing rate, you will most certainly get a convexity adjustment, either
positive or negative
¨ OK, so back to swap and ED.
¨ You pay fixed on a swap, you are short convexity versus the rate (we know that this is not
exactly correct, you do not to know volatility to price a regular swap, the rate is the wrong x-
axis to look at the PV of a swap, but everybody does it, and it is true that the PV of a swap
plotted against the level of rate will exhibit some non-linearity, hence some convexity)
58

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Another summary – XV - t
¨ NOW you hedge that short duration by buying a strip of Eurodollar futures.
¨ NOW you are actually really short convexity on your long position in ED futures
¨ Note that the payoff of the future as a function of the rate is linear and a straight line, so you
might not think that there is any convexity, but there is for two reasons:
¨ 1) it is true that there is no convexity with respect to rate, but we kind of know by now that
we should look at zeros, not rates. That is why Libor in arrears payoff is linear with rates, but
exhibit some convexity, this is why a FRA or regular swaplet payoff is non-linear or convex as
a function of rates, but will exhibit no convexity since it is linear as a function of the Zeros, as
we saw in part II, III and IV). This also indicates that there will be a connection between the
Libor in arrears/Libor in advance convexity, and the ED future convexity
¨ 2) Even if you were to look at the payoff in rates (rates being the x-axis), because it is a
future contract with daily ACTUAL cash flows happening, any assumptions that rates do
move and that the financing rate is somewhat correlated with the future rate WILL produce
a non-zero accumulated PL, which will be the convexity adustment
59

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Another summary – XV - u
¨ So if you are long a ED future, you will be also short convexity (this time really)
¨ So again on a number of swap desks, you will hear “I am short convexity because I am
paying in swaps”, but that could mean a lot of things:
¨ “I am paying in swap, and I am plotting the PV of the swap as a function of the level of rate,
and that curve is negatively convex, and so I am short convexity even though I do not care
about the volatility to price a swap”
¨ “I am paying in swap, I know a swap is a linear function of the Zeros, no convexity there, but
I am hedging with being long ED futures, and there is a real short convexity there, so I am
short convexity, if the volatility increases, I will lose actual and real money on the position”
¨ And the last even more subtle one:
¨ “I am paying in swap, I know a swap is a linear function of the Zeros, no convexity there, and
I am NOT hedging with being long ED futures, BUT I am building my curve with ED futures,
and so I need to adjust the forward rate down from the future rate by an amount equal to
the convexity adjustment, and if the volatility increases, that adjustment will increase, the
forward rate will be even lower, I will lose actual and real money on the position, and so I am
short convexity”
60

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Another summary – XV – u2
¨ Or even more subtle and profound:
¨ “I am paying on a swap, I work at Salomon Brothers and have been using a 3 factors with
skew to build the swap curve since the 1970s or so, so of course any position I have on this
curve is going to have sensitivities to the volatility of each factors, as well as the correlation,
as well as the skew parameters”
¨ z
𝑑𝑥 = −𝑘C. 𝑥. 𝑑𝑡 + 𝜎C. ([). 𝑑𝑊C
𝑑𝑦 = −𝑘D. 𝑦. 𝑑𝑡 + 𝜎D. ([). 𝑑𝑊D
𝑑𝑧 = −𝑘.. (𝑥 + 𝑦 − 𝑧). 𝑑𝑡 + 𝜎.. ([). 𝑑𝑊.
¨ With: < 𝑑𝑊C. 𝑑𝑊C >= 𝜌. 𝑑𝑡
¨ And : 𝑟 𝑡 = 𝑅(𝑡, 𝑡, 𝑡) = 𝐼𝑅𝑀𝐴[𝑧 𝑡 + 𝜇 𝑡 ]
61

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Another summary – XV - v
¨ I do not want to be harsh, but my guess is sometimes like 10 years ago, and maybe still
today:
¨ 90% of swap traders would not even know what convexity is
¨ Of those who knew, 90% would think that paying fixed is short convexity because the
present value is negatively convex as a function of rates, and then would blank out when as
a follow up question you would ask how come they price swaps without a volatility
¨ I was fortunate enough that my first two bosses were some of the most expert swap traders
in the markets, and convexity has no secret for them
¨ Richard Robb at DKBFP
¨ Steven Mullaney at FujiCap
¨ Turns out that they both were fighting so much to be my boss that we ended up deciding to
merge DKBFP and FujiCap into Mizuho…yeah, that was the real reason behind the merger or
two of the largest banks in the world at the time, at least I think it was
62

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Another summary – XV – v - DKBFP
63

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Another summary – XV – v - FUJICAP
64

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Another summary – XV – v - MIZUHO
65

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Another summary – XV - w
¨ Oh also almost done since I am running out of letters on this digression.
¨ But another variant is that you will see a lot of swap traders or senior managers who would
be confused between “carry” and “convexity”, so they might tell you something like this:
¨ “There is positive carry in the book because I make money from the passage of time, so I
must be short convexity, just like in an option the time decay offset the expected gamma
gains from the market moving”
¨ Again carry is not the same as convexity.
¨ If you are short duration and valuing your swap where you pay fixed on a curve that is built
with ED futures and incorporate the convexity adjustment, it is true that the passage of time
will reduce the convexity adjustment, hence increase the predicted forward rates, and
produce a positive PL. That is true convexity, and a true time decay effect.
¨ If you are just receiving a fixed cash flow in the future, the passage of time will increase the
present value (assuming that rates are positive, again another of those assumptions that are
obvious until they are not anymore), and so you will have positive PL. That is carry, nothing
to do with any kind of time decay offsetting Gamma (Gamma is the PL that comes from the
convexity in a Taylor expansion of the PL)
66

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Another summary – XV - x
¨ All right before we go back to instantaneous forward and short rate, it is time for trivia just
to keep you guys awake.
¨ You recall the colors of the ED future packs ? White Red Green Blues Gold Purple Orange
Pink Silver Copper ?
¨ You know how they chose the colors ?
67

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Another summary – XV - y
¨ Answer: has to do with the colors of the chips when you go gambling at the casino
¨ See? I told you that you would learn something useful in this class
68

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Enough of the digression, back to our nice
“little” summary
69

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Another summary - XVI
¨ BACK TO SPOT RATES AND FORWARD RATES
¨ So we have defined our variables 𝑅 𝑡, 𝑡!, 𝑡" , 𝐿 𝑡, 𝑡!, 𝑡" , 𝑌 𝑡, 𝑡!, 𝑡" and 𝑌8 𝑡, 𝑡!, 𝑡" just to
name a few
¨ Remember, only the first variable 𝑡 is the “real” time over which stochastic variable will
evolve, and we will need to be super careful about stochastic calculus
¨ The other two time variables 𝑡! and 𝑡" are really parametrization of the zero or discount
curve, and we can do all the regular calculus we want on those
¨ IN PARTICULAR we can take the limit 𝑡" → 𝑡! (we are totally justified to do this).
70

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Another summary - XVII
¨ From the variable 𝑍𝐶 𝑡, 𝑡!, 𝑡" , we are absolutely free to define a bunch of other variables,
and we certainly did not deprive ourselves of doing so:
¨ Continuously compounded FORWARD : 𝑍𝐶 𝑡, 𝑡!, 𝑡" = exp −𝑅 𝑡, 𝑡!, 𝑡" . 𝜏 𝑡, 𝑡!, 𝑡"
¨ Simply compounded FORWARD: 𝑍𝐶 𝑡, 𝑡!, 𝑡" =
#
#$) &,&",&! .4 &,&",&!
¨ Annually compounded FORWARD : 𝑍𝐶 𝑡, 𝑡!, 𝑡" =
#
(#$5 &,&",&! )
# $,$",$!
¨ 𝑞-times per year compounded FOWARD: 𝑍𝐶 𝑡, 𝑡!, 𝑡" =
#
(#$
&
'
.5' &,&",&! )
'.# $,$",$!
¨ The function 𝜏 𝑡, 𝑡!, 𝑡" is the daycount fraction, will usually depends on what convention
(ACT/ACT, ACT/360, 30/360,…) you will choose, and potentially adjustment for holidays and
what holiday center
71

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Another summary - XVIII
¨ In the small 𝜏 𝑡, 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!, 𝑡" = 1 − 𝑅 𝑡, 𝑡!, 𝑡" . 𝜏 + 𝒪(𝜏E. 𝑅E)
¨ Simply compounded spot: 𝑍𝐶 𝑡, 𝑡!, 𝑡" = 1 − 𝐿 𝑡, 𝑡!, 𝑡" . 𝜏 + 𝒪(𝜏E. 𝑙E)
¨ Annually compounded spot: 𝑍𝐶 𝑡, 𝑡!, 𝑡" = 1 − 𝑌 𝑡, 𝑡!, 𝑡" . 𝜏 + 𝒪(𝜏E. 𝑦E)
¨ 𝑞-times per year compounded spot 𝑍𝐶 𝑡, 𝑡!, 𝑡" = 1 − 𝑌8 𝑡, 𝑡!, 𝑡" . 𝜏 + 𝒪(𝜏E. 𝑦8
E)
¨ So in the limit of small 𝜏 𝑡, 𝑡!, 𝑡" , (and also small rates), in particular when: 𝑡" → 𝑡!, all rates
converge to the same limit we call
¨ 𝐿𝑖𝑚 𝑡" → 𝑡! = lim
&!→&"
(
#G*+ &,&",&!
) &,&",&!
) that we will note Instantaneous Forward Rate
72

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Another summary - XIX
&!→&"
(
#G*+ &,&",&!
) &,&",&!
) that we will note Instantaneous Forward Rate
¨ 𝐼𝐹𝑤𝑅 𝑡, 𝑡!, 𝑡" = 𝐼𝐹𝑤𝑅 𝑡, 𝑡!, 𝑡!$ = 𝐼𝐹𝑤𝑅 𝑡, 𝑡! = lim
&!→&"
(
#G*+ &,&",&!
) &,&",&!
)
¨ In the small 𝜏 𝑡, 𝑡!, 𝑡" limit, (and also small rates) since really what matters is how small the
product of the defined rate by the daycount fraction, 𝑍𝐶 𝑡, 𝑡!, 𝑡" is close to 1.
&!→&"
(
#G*+ &,&",&!
) &,&",&!
) = lim
&!→&"
(
GHI(*+ &,&",&! )
) &,&",&!
)
¨ Usually most textbooks will assume without explicitly telling you that in that limit we will also
have:
¨ lim
&!→&"
(𝜏 𝑡, 𝑡!, 𝑡" ) = (𝑡" − 𝑡!), so that 𝐿𝑖𝑚 𝑡" → 𝑡! = lim
&!→&"
G HI *+ &,&",&!
) &,&",&!
73

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Another summary - XX
¨ 𝐼𝐹𝑤𝑅 𝑡, 𝑡!, 𝑡" = 𝐼𝐹𝑤𝑅 𝑡, 𝑡!, 𝑡!$ = 𝐼𝐹𝑤𝑅 𝑡, 𝑡! = lim
&!→&"
(
#G*+ &,&",&!
) &,&",&!
)
¨ 𝐼𝐹𝑤𝑅 𝑡, 𝑡! = lim
&!→&"
(
#G*+ &,&",&!
) &,&",&!
) = lim
&!→&"
(𝑅 𝑡, 𝑡!, 𝑡" ) = 𝑅 𝑡, 𝑡!, 𝑡!
&!→&"
(
#G*+ &,&",&!
) &,&",&!
) = lim
&!→&"
(𝐿 𝑡, 𝑡!, 𝑡" ) = 𝐿 𝑡, 𝑡!, 𝑡!
&!→&"
(
#G*+ &,&",&!
) &,&",&!
) = lim
&!→&"
(𝑌 𝑡, 𝑡!, 𝑡" ) = 𝑌 𝑡, 𝑡!, 𝑡!
&!→&"
(
#G*+ &,&",&!
) &,&",&!
) = lim
&!→&"
(𝑌8 𝑡, 𝑡!, 𝑡" ) = 𝑌8 𝑡, 𝑡!, 𝑡!
74

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Another summary - XXI
¨ 𝑅 𝑡, 𝑡!, 𝑡! = 𝐿 𝑡, 𝑡!, 𝑡! = 𝑌 𝑡, 𝑡!, 𝑡! = 𝑌8 𝑡, 𝑡!, 𝑡! = 𝑓(𝑡, 𝑡!) as per the notation in most
textbooks
¨ lim
&!→&"
(
#G*+ &,&",&!
) &,&",&!
) = lim
&!→&"
(
GHI(*+ &,&",&! )
) &,&",&!
)
¨ From bootstrap:
¨ ln(𝑍𝐶 𝑡, 𝑡!, 𝑡" = ln(𝑍𝐶 𝑡, 𝑡, 𝑡" − ln(𝑍𝐶 𝑡, 𝑡, 𝑡!
¨ lim
&!→&"
(
GHI(*+ &,&",&! )
) &,&",&!
) = − lim
&!→&"
HI(*+ &,&,&! GHI(*+ &,&,&"
) &,&",&!
= − lim
&!→&"
(
HI(*+ &,&,&! GHI(*+ &,&,&"
&! G &"
)
¨ lim
&!→&"
(
GHI(*+ &,&",&! )
) &,&",&!
) = −
JHI(*+ &,&,&"
J&"
75

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Another summary - XXII
¨ 𝑅 𝑡, 𝑡!, 𝑡! = 𝐿 𝑡, 𝑡!, 𝑡! = 𝑌 𝑡, 𝑡!, 𝑡! = 𝑌8 𝑡, 𝑡!, 𝑡! = 𝑓 𝑡, 𝑡! = −
JHI(*+ &,&,&"
J&"
¨ A lot of models loooove to use the Instantaneous Forward Rate (HJM)
¨ We can also take another limit, the Instantaneous Short Rate defined as:
¨ 𝐼𝑆ℎ𝑅 𝑡, 𝑡!, 𝑡" = 𝐼𝑆ℎ𝑅 𝑡, 𝑡+, 𝑡 + = 𝐼𝑆ℎ𝑅 𝑡 = lim
&!→&",&!→&
(
#G*+ &,&",&!
) &,&",&!
)
¨ 𝐼𝑆ℎ𝑅 𝑡 = lim
&"→&
𝐼𝐹𝑤𝑅 𝑡, 𝑡! = 𝑅 𝑡, 𝑡, 𝑡 = 𝐿 𝑡, 𝑡, 𝑡 = 𝑌 𝑡, 𝑡, 𝑡 = 𝑌8 𝑡, 𝑡, 𝑡 = 𝑓 𝑡, 𝑡 = 𝑟(𝑡)
¨ Most of the early models were built on the short rate, and then a lot of models were “affine
models” meaning that there were assumptions of linearity for a lot of the functions.
76

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Another summary - XXII
¨ Some cool relations on those new quantities:
¨ 𝑍𝐶 𝑡, 𝑡, 𝑡! = exp − ∫BK&
BK&"
𝑖𝑓𝑤𝑟 𝑡, 𝑢 . 𝑑𝑢 = exp − ∫BK&
BK&"
𝐿 𝑡, 𝑢, 𝑢 . 𝑑𝑢
¨ 𝑍𝐶 𝑡, 𝑡, 𝑡! = 𝑃 𝑡, 𝑡! = exp(− ∫BK&
BK&"
𝑓 𝑡, 𝑢 . 𝑑𝑢)
¨ Note that this one is just an integration of the following:
¨ 𝐿 𝑡, 𝑡!, 𝑡! = 𝑓 𝑡, 𝑡! = 𝑖𝑓𝑤𝑟(𝑡, 𝑡!) = −
JHI(*+ &,&,&"
J&"
¨ 𝔼&"
*+
𝑉 𝑡!, $𝐿 𝑡!, 𝑡!, 𝑡! , 𝑡!, 𝑡! |𝔉(𝑡) = 𝐿 𝑡, 𝑡!, 𝑡!
¨ 𝔼&"
*+
𝑉 𝑡!, $𝑟(𝑡!), 𝑡!, 𝑡! |𝔉(𝑡) = 𝑓 𝑡, 𝑡!
¨ That one is more complicated to show, but essentially in textbooks you will see it as (for
example Mercurio p. 34) “the expected value of any future instantaneous spot interest rate,
under the corresponding measure, is equal to the related instantaneous forward rate”
77

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Another summary - XXIII
¨ Combining the two equations we then have:
¨ 𝑍𝐶 𝑡, 𝑡, 𝑡! = exp − ∫BK&
BK&"
𝔼B
*+ 𝑉 𝑢, $𝐿 𝑢, 𝑢, 𝑢 , 𝑢, 𝑢 |𝔉(𝑡) . 𝑑𝑢
¨ 𝑃 𝑡, 𝑡! = exp − ∫BK&
BK&"
𝔼B
*+ 𝑉 𝑢, $𝑟(𝑢), 𝑢, 𝑢 |𝔉(𝑡) . 𝑑𝑢
78

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Another summary - XXIV
¨ Some more intuition on Instantaneous Forward Rates
¨ 𝑍𝐶 𝑡, 𝑡, 𝑡! = exp − ∫BK&
BK&"
𝐼𝐹𝑤𝑅 𝑡, 𝑢 . 𝑑𝑢 = exp − ∫BK&
BK&"
𝐿 𝑡, 𝑢, 𝑢 . 𝑑𝑢
¨ We also have by definition in the case of the continuously compounded rate
¨ 𝑍𝐶 𝑡, 𝑡!, 𝑡" = exp −𝑅 𝑡, 𝑡!, 𝑡" . 𝜏 𝑡, 𝑡!, 𝑡"
¨ In the case where (𝛼 = 1), which is equivalent of choosing to express the time in variable in
units of years (1 year = 1) and assuming what we could call an ACT/ACT daycount fraction,
𝜏 𝑡, 𝑡!, 𝑡" = 𝑡" − 𝑡!
¨ In particular:
¨ 𝑍𝐶 𝑡, 𝑡, 𝑡! = exp −𝑅 𝑡, 𝑡, 𝑡! . 𝜏 𝑡, 𝑡, 𝑡!
¨ 𝑅 𝑡, 𝑡, 𝑡! =
#
&"G&
∫BK&
BK&"
𝐼𝐹𝑤𝑅 𝑡, 𝑢 . 𝑑𝑢 =
#
&"G&
∫BK&
BK&"
𝐿 𝑡, 𝑢, 𝑢 . 𝑑𝑢 =
#
&"G&
∫BK&
BK&"
𝑓(𝑡, 𝑢). 𝑑𝑢
¨ The continuously compounded spot rate is the time weighted average of the instantaneous
forward rate (does not matter which one since they all tend to the same limit)
79

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Another summary - XXV
¨ 𝑅 𝑡, 𝑡, 𝑡! =
#
&"G&
∫BK&
BK&"
𝐼𝐹𝑤𝑅 𝑡, 𝑢 . 𝑑𝑢 =
#
&"G&
∫BK&
BK&"
𝐿 𝑡, 𝑢, 𝑢 . 𝑑𝑢 =
#
&"G&
∫BK&
BK&"
𝑓(𝑡, 𝑢). 𝑑𝑢
¨ ∫BK&
BK&"
𝐿 𝑡, 𝑢, 𝑢 . 𝑑𝑢 = 𝑡! − 𝑡 . 𝑅 𝑡, 𝑡, 𝑡!
¨
J
J&"
∫BK&
BK&"
𝐿 𝑡, 𝑢, 𝑢 . 𝑑𝑢 = 𝐿 𝑡, 𝑡!, 𝑡! = 𝑓 𝑡, 𝑡! =
J
J&"
. { 𝑡! − 𝑡 . 𝑅 𝑡, 𝑡, 𝑡! }
¨ 𝐿 𝑡, 𝑡!, 𝑡! =
J
J&"
. 𝑡! − 𝑡 . 𝑅 𝑡, 𝑡, 𝑡! = 𝑓 𝑡, 𝑡! = 𝑅 𝑡, 𝑡, 𝑡! + 𝑡! − 𝑡 .
J9 &,&,&"
J&"
¨ The instantaneous forward curve is equal to the continuously compounded spot rate curve
PLUS the first derivative of the continuously compounded spot rate curve with respect to
the maturity of said spot rate times the maturity
¨ Remember again that
¨ 𝑅 𝑡, 𝑡!, 𝑡! = 𝐿 𝑡, 𝑡!, 𝑡! = 𝑌 𝑡, 𝑡!, 𝑡! = 𝑌8 𝑡, 𝑡!, 𝑡! = 𝑓 𝑡, 𝑡! = −
JHI(*+ &,&,&"
J&"
80

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Another summary - XXVI
¨ ”It is always good to pay attention to the notation” Kurt Godel
¨ 𝑅 𝑡, 𝑡, 𝑡! is not 𝑅 𝑡, 𝑡!, 𝑡! which is also not 𝑅 𝑡, 𝑡!, 𝑡"
¨ 𝑅 𝑡, 𝑡, 𝑡! is the continuously compounded spot rate at time 𝑡 for the period [𝑡, 𝑡!]
¨ 𝑍𝐶 𝑡, 𝑡, 𝑡! = exp −𝑅 𝑡, 𝑡, 𝑡! . 𝜏 𝑡, 𝑡, 𝑡!
¨ 𝑅 𝑡, 𝑡, 𝑡! =
G#
) &,&,&"
. ln(𝑍𝐶 𝑡, 𝑡, 𝑡! )
¨ 𝑅 𝑡, 𝑡!, 𝑡" is the continuously compounded forward rate at time 𝑡 for the period [𝑡!, 𝑡"]
¨ 𝑍𝐶 𝑡, 𝑡!, 𝑡" = exp −𝑅 𝑡, 𝑡!, 𝑡" . 𝜏 𝑡, 𝑡!, 𝑡"
¨ 𝑅 𝑡, 𝑡!, 𝑡" =
G#
) &,&",&!
. ln 𝑍𝐶 𝑡, 𝑡!, 𝑡" =
G#
) &,&",&!
. ln{
*+ &,&,&!
*+ &,&,&"
}
¨ 𝑅 𝑡, 𝑡!, 𝑡" =
G#
) &,&",&!
. {ln 𝑍𝐶 𝑡, 𝑡, 𝑡" − ln(𝑍𝐶 𝑡, 𝑡, 𝑡! )}
¨ (*) Again, fairly certain that Kurt Godel never ever said that, but am trying to start a rumor
81

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Another summary - XXVII
¨ 𝑅 𝑡, 𝑡!, 𝑡! is the instantaneous forward rate at time 𝑡 for a maturity 𝑡!
¨ 𝑅 𝑡, 𝑡!, 𝑡! is sometimes noted 𝑓 𝑡, 𝑡! in a lot of textbooks
¨ 𝑅 𝑡, 𝑡!, 𝑡! = lim
&!→&"
(𝑅 𝑡, 𝑡!, 𝑡" )
¨ 𝑅 𝑡, 𝑡!, 𝑡! = −
JHI(*+ &,&,&"
J&"
¨ 𝑅 𝑡, 𝑡, 𝑡! =
#
&"G&
∫BK&
BK&"
𝑅 𝑡, 𝑢, 𝑢 . 𝑑𝑢
¨ 𝑅 𝑡, 𝑡!, 𝑡! =
J
J&"
. 𝑡! − 𝑡 . 𝑅 𝑡, 𝑡, 𝑡! = 𝑅 𝑡, 𝑡, 𝑡! + 𝑡! − 𝑡 .
J9 &,&,&"
J&"
82

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Another summary - XXVIII
¨ Again, am not trying to be overly pedantic here, but so many textbooks out there drop
variables left and right that I think it makes it more confusing than anything else.
¨ Have to say that Piterbarg is one of the few textbooks to be somewhat rigorous in the
notation, along with Hull. I have to say that some textbooks with Italian names in the title
seem to exhibit a certain “laissez-faire” when it comes to following some Germanic
discipline on the correct notation
¨ And yes I know that “laissez-faire” is a French expression
¨ Over the years, going back to the 3-time variables always helped me out potentially costly
mistakes.
83

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Another summary - XXIX
¨ Remember, when it comes to time (and especially time travel), it has been mathematically
proven that things always come in 3 with the famous triquerta
¨ So as a general rule, ALWAYS carry with you the 3 time variables 𝑡, 𝑡!, 𝑡" .
84

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Equivalences between SDE - I
From Zeros to Instantaneous Forwards
85

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From Zeros SDE to Instantaneous Forward SDE
¨ If we assume that we can write the dynamics for the Zeros as:
¨
L*+ &,&,&"
*+ &,&,&"
= 𝜇*+ 𝑡, 𝑡! . 𝑑𝑡 + 𝜎*+ 𝑡, 𝑡! . [ . 𝑑𝑊(𝑡)
¨ We can then apply Ito lemma to 𝑑𝑙𝑛(𝑍𝐶 𝑡, 𝑡, 𝑡! ) and 𝑑𝑙𝑛(𝑍𝐶 𝑡, 𝑡, 𝑡" ) since we are after
the dynamics of: 𝑅 𝑡, 𝑡!, 𝑡" . 𝜏 𝑡, 𝑡!, 𝑡" = ln 𝑍𝐶 𝑡, 𝑡, 𝑡! − ln(𝑍𝐶 𝑡, 𝑡, 𝑡" )
86

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From Zeros SDE to Instantaneous Forward SDE - II
¨ 𝑑 ln 𝑋 =
#
3
. ([). 𝑑𝑋 +
#
E
.
G#
33 . ([). (𝑑𝑋)E
¨ 𝑋 = 𝑍𝐶 𝑡, 𝑡, 𝑡"
¨ 𝑑 ln 𝑍𝐶 𝑡, 𝑡, 𝑡" =
#
*+ &,&,&!
. ([). 𝑑𝑍𝐶 𝑡, 𝑡, 𝑡" +
#
E
.
G#
*+ &,&,&!
3 . ([). (𝑑𝑍𝐶 𝑡, 𝑡, 𝑡" )E
¨ 𝑑𝑍𝐶 𝑡, 𝑡, 𝑡! = 𝜇*+ 𝑡, 𝑡! . 𝑍𝐶 𝑡, 𝑡, 𝑡! . 𝑑𝑡 + 𝜎*+ 𝑡, 𝑡! . 𝑍𝐶 𝑡, 𝑡, 𝑡! . ([). 𝑑𝑊 𝑡
¨ (𝑑𝑍𝐶 𝑡, 𝑡, 𝑡! )E= (𝜎*+ 𝑡, 𝑡! . 𝑍𝐶 𝑡, 𝑡, 𝑡! )E. 𝑑𝑡
¨ 𝑑 ln 𝑍𝐶 𝑡, 𝑡, 𝑡! = 𝜇*+ 𝑡, 𝑡! . 𝑑𝑡 −
#
E
𝜎*+ 𝑡, 𝑡!
E
. 𝑑𝑡 + 𝜎*+ 𝑡, 𝑡! . ([). 𝑑𝑊 𝑡
¨ Same for 𝑡"
¨ 𝑑 ln 𝑍𝐶 𝑡, 𝑡, 𝑡" = 𝜇*+ 𝑡, 𝑡" . 𝑑𝑡 −
#
E
𝜎*+ 𝑡, 𝑡"
E
. 𝑑𝑡 + 𝜎*+ 𝑡, 𝑡" . ([). 𝑑𝑊 𝑡
87

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From Zeros SDE to Instantaneous Forward SDE - III
¨ 𝑑 ln 𝑍𝐶 𝑡, 𝑡, 𝑡! = 𝜇*+ 𝑡, 𝑡! . 𝑑𝑡 −
#
E
𝜎*+ 𝑡, 𝑡!
E
. 𝑑𝑡 + 𝜎*+ 𝑡, 𝑡! . ([). 𝑑𝑊 𝑡
¨ 𝑑 ln 𝑍𝐶 𝑡, 𝑡, 𝑡" = 𝜇*+ 𝑡, 𝑡" . 𝑑𝑡 −
#
E
𝜎*+ 𝑡, 𝑡"
E
. 𝑑𝑡 + 𝜎*+ 𝑡, 𝑡" . ([). 𝑑𝑊 𝑡
¨ 𝑅 𝑡, 𝑡!, 𝑡" . 𝜏 𝑡, 𝑡!, 𝑡" = ln 𝑍𝐶 𝑡, 𝑡, 𝑡! − ln(𝑍𝐶 𝑡, 𝑡, 𝑡" )
¨ 𝑑𝑅 𝑡, 𝑡!, 𝑡" . 𝜏 𝑡, 𝑡!, 𝑡" = 𝑑 ln 𝑍𝐶 𝑡, 𝑡, 𝑡! − 𝑑(ln(𝑍𝐶 𝑡, 𝑡, 𝑡! ))
¨ 𝑑𝑅 𝑡, 𝑡!, 𝑡" . 𝜏 𝑡, 𝑡!, 𝑡" = 𝐴. 𝑑𝑡 + 𝐵. ([). 𝑑𝑊 𝑡
¨ 𝐴 = −
#
E
𝜎*+ 𝑡, 𝑡!
E
+
#
E
𝜎*+ 𝑡, 𝑡"
E
+ [𝜇*+ 𝑡, 𝑡! − 𝜇*+ 𝑡, 𝑡" ]
¨ 𝐵 = [𝜎*+ 𝑡, 𝑡! − 𝜎*+ 𝑡, 𝑡" ]
¨ Note that this should be familiar since we did it in the HJM section where we were working
in the risk-free measure and set:
¨ 𝜇*+ 𝑡, 𝑡! = 𝜇*+ 𝑡, 𝑡" = 𝑅(𝑡, 𝑡, 𝑡)
88

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From Zeros SDE to Instantaneous Forward SDE - IV
¨ But in this section, we do not assume any arbitrage, so the results hold, regardless of the
measure under consideration, and in particular we do not assume here that the markets are
free of arbitrage. This is really just stochastic calculus on some variables
¨ Once again we are going to assume that in the small time limit:
¨ lim
&!→&"
(𝜏 𝑡, 𝑡!, 𝑡" ) = (𝑡" − 𝑡!)
¨ And then we take the limit 𝐼𝐹𝑤𝑅 𝑡, 𝑡! = lim
&!→&"
(𝑅 𝑡, 𝑡!, 𝑡" ) = 𝑅(𝑡, 𝑡!, 𝑡!)
¨ lim
."→.!
5
6 .,.!,."
= lim
."→.!
7
(."9.!)
. −
7
;
𝜎/0 𝑡, 𝑡1
;
+
7
;
𝜎/0 𝑡, 𝑡2
;
+ [𝜇/0 𝑡, 𝑡1 − 𝜇/0 𝑡, 𝑡2 ]
¨ lim
."→.!
5
6 .,.!,."
= 𝜎/0 𝑡, 𝑡1 .
<
<.!
𝜎/0 𝑡, 𝑡1 −
<
<.!
𝜇/0 𝑡, 𝑡1
89

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From Zeros SDE to Instantaneous Forward SDE - V
¨ lim
&!→&"
M
) &,&",&!
= 𝜎*+ 𝑡, 𝑡! .
J
J&"
𝜎*+ 𝑡, 𝑡! −
J
J&"
𝜇*+ 𝑡, 𝑡!
¨ lim
&!→&"
N
) &,&",&!
= lim
&!→&"
#
(&!G&")
. 𝜎*+ 𝑡, 𝑡! − 𝜎*+ 𝑡, 𝑡" = −
J
J&"
𝜎*+ 𝑡, 𝑡!
¨ 𝑑𝑅 𝑡, 𝑡!, 𝑡" . 𝜏 𝑡, 𝑡!, 𝑡" = 𝐴. 𝑑𝑡 + 𝐵. ([). 𝑑𝑊 𝑡
¨ lim
&!→&"
𝑑𝑅 𝑡, 𝑡!, 𝑡" = 𝑑 lim
&!→&"
𝑅 𝑡, 𝑡!, 𝑡" = 𝑑𝑅 𝑡, 𝑡!, 𝑡! = 𝑑𝑓(𝑡, 𝑡!)
¨ lim
&!→&"
𝑑𝑅 𝑡, 𝑡!, 𝑡" = lim
&!→&"
M
) &,&",&!
. 𝑑𝑡 + lim
&!→&"
N
) &,&",&!
. ([). 𝑑𝑊 𝑡
90

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From Zeros SDE to Instantaneous Forward SDE - VI
¨ So if we assume the Zeros to follow a dynamics:
¨
L*+ &,&,&"
*+ &,&,&"
= 𝜇*+ 𝑡, 𝑡! . 𝑑𝑡 + 𝜎*+ 𝑡, 𝑡! . [ . 𝑑𝑊(𝑡)
¨ Using : 𝑅 𝑡, 𝑡!, 𝑡" . 𝜏 𝑡, 𝑡!, 𝑡" = ln 𝑍𝐶 𝑡, 𝑡, 𝑡! − ln(𝑍𝐶 𝑡, 𝑡, 𝑡" )
¨ And taking the limit 𝑡" → 𝑡!
¨ We can write the following dynamics for the Instantaneous Forward Rates
¨ 𝑑𝑅 𝑡, 𝑡!, 𝑡! = 𝑑𝑓 𝑡, 𝑡! = 𝜇OA9 𝑡, 𝑡! . 𝑑𝑡 + 𝜎OA9 𝑡, 𝑡! . [ . 𝑑𝑊(𝑡)
¨ We then have the following relations:
¨ 𝜇OA9 𝑡, 𝑡! = 𝜎*+ 𝑡, 𝑡! .
J
J&"
𝜎*+ 𝑡, 𝑡! −
J
J&"
𝜇*+ 𝑡, 𝑡!
¨ 𝜎OA9 𝑡, 𝑡! = −
J
J&"
𝜎*+ 𝑡, 𝑡!
91

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Equivalences between SDE - II
From Instantaneous Forwards
To
Instantaneous Short Rate
92

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From IFR (Instantaneous Forward Rate) SDE to Short Rate SDE
¨ Again, this should be somewhat familiar to us as we essentially derived it in the HJM section.
¨ We assume that we can write for the IFR (IfwR) the following dynamics:
¨ 𝑅 𝑡 + 𝛿𝑡, 𝑡1, 𝑡1 − 𝑅 𝑡, 𝑡1, 𝑡1 = ∫=>.
=>.?@.
𝜇ABC 𝑠, 𝑡1 . 𝑑𝑠 + ∫=>.
=>.?@.
𝜎ABC 𝑠, 𝑡1 . ([). 𝑑𝑊 𝑠
¨ So going forward in time (like how we would set up a Monte Carlo simulation), going from (𝑡 =
0) to 𝑡 = 𝑡! for a given 𝑡!:
¨ 𝑅 𝑡 + 𝛿𝑡, 𝑡!, 𝑡! − 𝑅 𝑡, 𝑡!, 𝑡! = ∫PK&
PK&$6&
𝜇OA9 𝑠, 𝑡! . 𝑑𝑠 + ∫PK&
PK&$6&
𝜎OA9 𝑠, 𝑡! . ([). 𝑑𝑊 𝑠
¨ 𝑅 𝑡!, 𝑡!, 𝑡! − 𝑅 0, 𝑡!, 𝑡! = ∫PKQ
PK&"
𝜇OA9 𝑠, 𝑡! . 𝑑𝑠 + ∫PKQ
PK&"
𝜎OA9 𝑠, 𝑡! . ([). 𝑑𝑊 𝑠
¨ We now have the explicit solution for the ISR, Instantaneous Short Rate 𝑅 𝑡!, 𝑡!, 𝑡! = 𝑟(𝑡!)
93

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From IFR SDE to Short Rate SDE - II
¨ Remember as we saw in Part IV that we can just plug (𝑡 = 𝑡!) in the equation:
¨ Because we would be missing out on the increment along the 𝑡! variable
¨ 𝑅 𝑡!, 𝑡!, 𝑡! − 𝑅 0, 𝑡!, 𝑡! = ∫PKQ
PK&"
PK&"
𝜎OA9 𝑠, 𝑡! . ([). 𝑑𝑊 𝑠
¨ 𝑑𝑅 𝑡!, 𝑡!, 𝑡! =
J
J&"
𝑅 0, 𝑡!, 𝑡! + ∫PKQ
PK&"
PK&"
𝜎OA9 𝑠, 𝑡! . ([). 𝑑𝑊 𝑠 . 𝑑𝑡!
¨ 𝑑𝑅 𝑡!, 𝑡!, 𝑡! =
J9 Q,&",&"
J&"
. 𝑑𝑡! + ˆ
‰
∫PKQ
PK&" J
J&"
𝜇OA9 𝑠, 𝑡! . 𝑑𝑠 +
∫PKQ
PK&" J
J&"
𝜎OA9 𝑠, 𝑡! . ([). 𝑑𝑊 𝑠 . 𝑑𝑡! + 𝜇OA9 𝑠 = 𝑡!, 𝑡! . 𝑑𝑡! + 𝜎OA9 𝑠 = 𝑡!, 𝑡! . ([). 𝑑𝑊 𝑡!
¨ Not sure what Thomas Bjork is doing on p.354, but am fairly convinced that he is wrong.
¨ Then again I could be the one who is wrong.
94

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From IFR SDE to Short Rate SDE - III
¨ So if can assume for the dynamics of the IFR something like:
¨ Then the IDR Instantaneous Short Rate follows the dynamics:
¨ 𝑑𝑅 𝑡!, 𝑡!, 𝑡! = 𝑑𝑓 𝑡!, 𝑡! = 𝑑𝑟(𝑡!) = 𝜇OR9 𝑡! . 𝑑𝑡! + 𝜎OR9 𝑡! . [ . 𝑑𝑊(𝑡!)
¨ With the following relations:
¨ 𝜎OR9 𝑡! = 𝜎OA9 𝑡!, 𝑡!
¨ 𝜇ADC 𝑡1 =
<C E,.!,.!
<.!
+ 𝜇ABC 𝑡1, 𝑡1 + ∫=>E
=>.! <
<.!
𝜇ABC 𝑠, 𝑡1 . 𝑑𝑠 + ∫=>E
=>.! <
<.!
𝜎ABC 𝑠, 𝑡1 . ([). 𝑑𝑊 𝑠
95
Bjork p.354 Missing from Bjork p.354

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Equivalences between SDE - III
From Instantaneous Short Rates SDE
To the Zeros SDE
In short rate model framework
96

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Short rate model: from short rate SDE to Zero SDE
¨ Let’s assume that our short rate model is given by the following SDE:
¨ 𝑑𝑅 𝑡, 𝑡, 𝑡 = 𝜇9 𝑡, 𝑅 𝑡, 𝑡, 𝑡 . 𝑑𝑡 + 𝜎9 𝑡, 𝑅 𝑡, 𝑡, 𝑡 . [ . 𝑑𝑊(𝑡)
¨ If we assume that 𝑍𝐶 𝑡, 𝑡, 𝑡! = 𝑍𝐶(𝑡, 𝑅 𝑡, 𝑡, 𝑡 , 𝑡!), meaning that the Zeros are a function of the
short rate, we then have:
¨ 𝑑𝑍𝐶 𝑡, 𝑡, 𝑡! =
J
J9
. 𝑍𝐶 𝑡, 𝑡, 𝑡! . ([). 𝑑𝑅 +
#
E
.
J3*+ &,&,&"
J93 . ([). (𝑑𝑅)E+
J
J&
. 𝑍𝐶 𝑡, 𝑡, 𝑡! . 𝑑𝑡
¨ Note that we do not have any term like
J
J&"
, because we are concerned with the evolution in the
“real” time 𝑡 of a stochastic variable. In that case as we have seen before the time variable 𝑡! is
really a parameter along the yield curve and can be viewed as fixed for our current purpose.
¨ 𝑑𝑍𝐶 𝑡, 𝑡, 𝑡1 =
</0
<C
. 𝜇 𝑡, 𝑅 𝑡, 𝑡, 𝑡 . 𝑑𝑡 + 𝜎 𝑡, 𝑅 𝑡, 𝑡, 𝑡 . [ . 𝑑𝑊 𝑡 +
7
;
.
<#/0
<C#
. 𝜎 𝑡, 𝑅 𝑡, 𝑡, 𝑡
;
. 𝑑𝑡 +
</0
<.
. 𝑑𝑡
97

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Short rate model: from short rate SDE to Zero SDE - II
¨ 𝑑𝑍𝐶 𝑡, 𝑡, 𝑡1 =
</0
<C
. 𝜇C 𝑡, 𝑅 𝑡, 𝑡, 𝑡 . 𝑑𝑡 + 𝜎C 𝑡, 𝑅 𝑡, 𝑡, 𝑡 . [ . 𝑑𝑊 𝑡 +
7
;
.
<#/0
<C#
. 𝜎C 𝑡, 𝑅 𝑡, 𝑡, 𝑡
;
. 𝑑𝑡 +
</0
<.
. 𝑑𝑡
¨ If we assume that we can write the dynamics for the Zeros as:
¨
L*+ &,&,&"
*+ &,&,&"
= 𝜇*+ 𝑡, 𝑅 𝑡, 𝑡, 𝑡 , 𝑡! . 𝑑𝑡 + 𝜎*+ 𝑡, 𝑅 𝑡, 𝑡, 𝑡 , 𝑡! . [ . 𝑑𝑊(𝑡)
¨ Then we have the following relations:
¨ 𝜎*+ 𝑡, 𝑅 𝑡, 𝑡, 𝑡 , 𝑡! =
SF &,9 &,&,&
*+ &,&,&"
.
J*+
J9
¨ 𝜇*+ 𝑡, 𝑅 𝑡, 𝑡, 𝑡 , 𝑡! =
#
*+ &,&,&"
. {
J*+
J&
+
J*+
J9
. 𝜇9 𝑡, 𝑅 𝑡, 𝑡, 𝑡 +
#
E
.
J3*+
J93 . 𝜎9 𝑡, 𝑅 𝑡, 𝑡, 𝑡
E
}
98

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Short rate model: from short rate SDE to Zero SDE - III
¨ Really to be more rigorous we should not say:
¨ If we assume that 𝑍𝐶 𝑡, 𝑡, 𝑡! = 𝑍𝐶(𝑡, 𝑅 𝑡, 𝑡, 𝑡 , 𝑡!)
¨ But: if we assume that there is a function 𝑓 𝑡, 𝑟, 𝑡! so that:
¨ 𝑍𝐶 𝑡, 𝑡, 𝑡! = 𝑓(𝑡, 𝑟 = 𝑅(𝑡, 𝑡, 𝑡), 𝑡!)
¨ So instead of :
¨ 𝑑𝑍𝐶 𝑡, 𝑡, 𝑡! =
J
J9
. 𝑍𝐶 𝑡, 𝑡, 𝑡! . ([). 𝑑𝑅 +
#
E
.
J3*+ &,&,&"
J93 . ([). (𝑑𝑅)E+
J
J&
. 𝑍𝐶 𝑡, 𝑡, 𝑡! . 𝑑𝑡
¨ We should really write:
¨ 𝑑𝑍𝐶 𝑡, 𝑡, 𝑡1 =
<G .,H,.!
<H
|H>C(.,.,.). ([). 𝑑𝑅 +
7
;
.
<#G .,H,.!
<H#
|H>C(.,.,.). ([). (𝑑𝑅);
+
<G .,H,.!
<.
|H>C(.,.,.). 𝑑𝑡
99

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Short rate model: from short rate SDE to Zero SDE - IV
¨ Similarly then, instead of writing :
¨ 𝜎*+ 𝑡, 𝑅 𝑡, 𝑡, 𝑡 , 𝑡! =
SF &,9 &,&,&
*+ &,&,&"
.
J*+
J9
¨ We should really be writing:
¨ 𝜎*+ 𝑡, 𝑅 𝑡, 𝑡, 𝑡 , 𝑡! =
SF &,9 &,&,&
*+ &,&,&"
.
JT &,@,&"
J@
|@K9(&,&,&)
¨ 𝜎*+ 𝑡, 𝑅 𝑡, 𝑡, 𝑡 , 𝑡! =
SF &,9 &,&,&
T &,@,&"
.
JT &,@,&"
J@
|@K9(&,&,&)
¨ 𝜎*+ 𝑡, 𝑅 𝑡, 𝑡, 𝑡 , 𝑡! = 𝜎9 𝑡, 𝑅 𝑡, 𝑡, 𝑡 .
GJHI(T &,@,&" )
J@
|@K9(&,&,&)
¨ If we define a function 𝑔 𝑡, 𝑟, 𝑡! = ln 𝑓 𝑡, 𝑟, 𝑡! = ln(𝑍𝐶 𝑡, 𝑡, 𝑡! )
¨ 𝜎*+ 𝑡, 𝑅 𝑡, 𝑡, 𝑡 , 𝑡! = 𝜎9 𝑡, 𝑅 𝑡, 𝑡, 𝑡 .
GJU &,@,&"
J@
|@K9(&,&,&)
100

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Short rate model: from short rate SDE to Zero SDE - V
¨ 𝜎*+ 𝑡, 𝑅 𝑡, 𝑡, 𝑡 , 𝑡! = 𝜎9 𝑡, 𝑅 𝑡, 𝑡, 𝑡 .
GJU &,@,&"
J@
|@K9(&,&,&)
¨ We also have :
¨ 𝜎OA9 𝑡, 𝑡! = −
J
J&"
𝜎*+ 𝑡, 𝑡!
¨ 𝜎OA9 𝑡, 𝑡! = −
J
J&"
(𝜎9 𝑡, 𝑅 𝑡, 𝑡, 𝑡 .
GJU &,@,&"
J@
|@K9(&,&,&))
¨ 𝜎OA9 𝑡, 𝑡! = 𝜎9 𝑡, 𝑅 𝑡, 𝑡, 𝑡 (
J3U &,@,&"
J@J&"
|@K9(&,&,&))
¨ Remember that: 𝑔 𝑡, 𝑟, 𝑡! = ln 𝑓 𝑡, 𝑟, 𝑡! = ln(𝑍𝐶 𝑡, 𝑡, 𝑡! )
¨ In the case of an affine model:
¨ 𝑍𝐶 𝑡, 𝑡, 𝑡! = exp(𝐴 𝑡, 𝑡! − 𝐵 𝑡, 𝑡! . 𝑅 𝑡, 𝑡, 𝑡 )
¨ 𝑔 𝑡, 𝑟, 𝑡! = ln 𝑓 𝑡, 𝑟, 𝑡! = ln 𝑍𝐶 𝑡, 𝑡, 𝑡! = 𝐴 𝑡, 𝑡! − 𝐵 𝑡, 𝑡! . 𝑅 𝑡, 𝑡, 𝑡
101

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Short rate model: from short rate SDE to Zero SDE - VI
¨ 𝑔 𝑡, 𝑟, 𝑡! = ln 𝑓 𝑡, 𝑟, 𝑡! = ln 𝑍𝐶 𝑡, 𝑡, 𝑡! = 𝐴 𝑡, 𝑡! − 𝐵 𝑡, 𝑡! . 𝑅 𝑡, 𝑡, 𝑡
¨ Or more rigorously:
¨ 𝑔 𝑡, 𝑟, 𝑡! = ln 𝑓 𝑡, 𝑟, 𝑡! = ln 𝑍𝐶 𝑡, 𝑡, 𝑡! = 𝐴 𝑡, 𝑡! − 𝐵 𝑡, 𝑡! . 𝑟|@K9(&,&,&)
¨ And so:
¨
J3U &,@,&"
J@J&"
|@K9(&,&,&) =
J
J&"
.
J
J@
𝑔 𝑡, 𝑟, 𝑡! |@K9(&,&,&) =
J
J&"
.
J
J@
(𝐴 𝑡, 𝑡! − 𝐵 𝑡, 𝑡! . 𝑟)|@K9(&,&,&)
¨
J3U &,@,&"
J@J&"
|@K9(&,&,&) =
J
J&"
. −𝐵 𝑡, 𝑡! |@K9 &,&,& = −
JN &,&"
J&"
102

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Short rate model: from short rate SDE to Zero SDE - VII
¨ And so:
¨ 𝜎OA9 𝑡, 𝑡! = 𝜎9 𝑡, 𝑅 𝑡, 𝑡, 𝑡 (
J3U &,@,&"
J@J&"
|@K9(&,&,&))
¨ Becomes:
¨ 𝜎OA9 𝑡, 𝑡! = −𝜎9 𝑡, 𝑅 𝑡, 𝑡, 𝑡 . (
JN &,&"
J&"
)
¨ And also:
¨ 𝜎*+ 𝑡, 𝑅 𝑡, 𝑡, 𝑡 , 𝑡! = 𝜎9 𝑡, 𝑅 𝑡, 𝑡, 𝑡 .
GJU &,@,&"
J@
|@K9(&,&,&)
¨ Becomes:
¨ 𝜎*+ 𝑡, 𝑅 𝑡, 𝑡, 𝑡 , 𝑡! = 𝜎9 𝑡, 𝑅 𝑡, 𝑡, 𝑡 . 𝐵 𝑡, 𝑡!
103

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Ho-Lee and Riccati
104

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Ho – Lee model
¨ This model is usually used to test and develop the intuition, as it offers many nice
properties:
¨ Affine
¨ Fits into the HJM model
¨ Can be made arbitrage free
¨ And is usually simple enough that the math does not come to obstruct the intuition
105

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Ho – Lee is an affine short rate model
¨ 𝑑𝑅 𝑡, 𝑡, 𝑡 = 𝜇 𝑡, 𝑅 𝑡, 𝑡, 𝑡 . 𝑑𝑡 + 𝜎 𝑡, 𝑅 𝑡, 𝑡, 𝑡 . [ . 𝑑𝑊(𝑡)
¨ 𝜇 𝑡, 𝑅 𝑡, 𝑡, 𝑡 = 𝜇Q 𝑡 + 𝜇# 𝑡 . 𝑅 𝑡, 𝑡, 𝑡
¨ 𝜎 𝑡, 𝑅 𝑡, 𝑡, 𝑡
E
= 𝛼Q 𝑡 + 𝛼# 𝑡 . 𝑅 𝑡, 𝑡, 𝑡
¨ You get the Ho-Lee model (1985) if you write
¨ 𝑟 𝑡 + 𝛿𝑡 − 𝑟 𝑡 = ∫&
&$6&
𝜃 𝑠 . 𝑑𝑠 + 𝜎 ∫&
&$6&
1. ([). 𝑑𝑊 𝑠
¨ 𝜇Q 𝑡 = 𝜃(𝑡)
¨ 𝜇# 𝑡 = 0
¨ 𝛼Q 𝑡 = 𝜎E
¨ 𝛼# 𝑡 = 0
106

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Ho-Lee is a simple Markov HJM model
¨ Most models do fit into the HJM framework.
¨ A very common one is the Ho-Lee model (1986)
¨ 𝑑𝑅 𝑡1, 𝑡1, 𝑡1 =
<C E,.!,.!
<.!
. 𝑑𝑡1 + D
E
∫=>E
=>.! <
<.!
𝑉 𝑠, 𝑡1, 𝑡1 .
<
<.!
𝑉 𝑠, 𝑡1, 𝑡1 . 𝑑𝑠 +
∫=>E
=>.! <
<.!
{
<
<.!
𝑉 𝑠, 𝑡1, 𝑡1 }. ([). 𝑑𝑊 𝑠 . 𝑑𝑡1 + 𝑉 𝑠 = 𝑡1, 𝑡1, 𝑡1 .
<
<.!
𝑉 𝑠 = 𝑡1, 𝑡1, 𝑡1 . 𝑑𝑡1 +
<
<.!
𝑉 𝑠 = 𝑡1, 𝑡1, 𝑡1 . ([). 𝑑𝑊 𝑡1
¨ We assume that:
¨ 𝜎 𝑡, 𝑡!, 𝑡! =
J
J&"
𝑉 𝑡, 𝑡!, 𝑡! = 𝜎
¨
J
J&"
J
J&"
𝑉 𝑠, 𝑡!, 𝑡! =
J
J&"
𝜎 = 0
¨ 𝑉 𝑠 = 𝑡!, 𝑡!, 𝑡! .
J
J&"
𝑉 𝑠 = 𝑡!, 𝑡!, 𝑡! = 0 because 𝑉 𝑠 = 𝑡!, 𝑡!, 𝑡! = 0
107

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Ho-Lee is a simple Markov HJM model - II
¨
J
J&"
𝑉 𝑠, 𝑡!, 𝑡! .
J
J&"
𝑉 𝑠, 𝑡!, 𝑡! =
J
J&"
𝑉 𝑠, 𝑡!, 𝑡! .
J
J&"
𝑉 𝑠, 𝑡!, 𝑡! + 𝑉 𝑠, 𝑡!, 𝑡! .
J3
J&"
3 𝑉 𝑠, 𝑡!, 𝑡!
¨
J
J&"
𝑉 𝑠, 𝑡!, 𝑡! .
J
J&"
𝑉 𝑠, 𝑡!, 𝑡! = 𝜎E
¨ ∫PKQ
PK&" J
J&"
𝑉 𝑠, 𝑡!, 𝑡! .
J
J&"
𝑉 𝑠, 𝑡!, 𝑡! . 𝑑𝑠 = ∫PKQ
PK&"
𝜎E. 𝑑𝑠 = 𝜎E. 𝑡!
¨ 𝑑𝑅 𝑡1, 𝑡1, 𝑡1 =
<C E,.!,.!
<.!
. 𝑑𝑡1 + D
E
∫=>E
=>.! <
<.!
𝑉 𝑠, 𝑡1, 𝑡1 .
<
<.!
𝑉 𝑠, 𝑡1, 𝑡1 . 𝑑𝑠 +
∫=>E
=>.! <
<.!
{
<
<.!
𝑉 𝑠, 𝑡1, 𝑡1 }. ([). 𝑑𝑊 𝑠 . 𝑑𝑡1 + 𝑉 𝑠 = 𝑡1, 𝑡1, 𝑡1 .
<
<.!
𝑉 𝑠 = 𝑡1, 𝑡1, 𝑡1 . 𝑑𝑡1 +
<
<.!
𝑉 𝑠 = 𝑡1, 𝑡1, 𝑡1 . ([). 𝑑𝑊 𝑡1
¨ 𝑑𝑅 𝑡1, 𝑡1, 𝑡1 =
<C E,.!,.!
<.!
. 𝑑𝑡1 + 𝜎;
. 𝑡1 . 𝑑𝑡1 + 𝜎. ([). 𝑑𝑊 𝑡1
¨ 𝑅 𝑡, 𝑡, 𝑡 = ∫.!>E
.!>. <
<.!
𝑅 𝑡1, 𝑡1, 𝑡1 . 𝑑𝑡1 = ∫.!>E
.!>.
𝑑𝑅 𝑡1, 𝑡1, 𝑡1 = ∫.!>E
.!>.
{
<C E,.!,.!
<.!
. 𝑑𝑡1 + 𝜎;
. 𝑡1 . 𝑑𝑡1 + 𝜎. ([). 𝑑𝑊 𝑡1 }
¨ 𝑅 𝑡, 𝑡, 𝑡 = 𝑅 0, 𝑡, 𝑡 − 𝑅 0,0,0 + 𝜎;
.
.#
;
+ 𝜎. ([). {𝑊 𝑡 − 𝑊 0 }
108

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Ho-Lee is a simple Markov HJM model - III
¨ We can also start from the explicit solution:
¨ 𝑅 𝑡!, 𝑡!, 𝑡! = 𝑅 0, 𝑡!, 𝑡! + ∫PKQ
PK&"
𝑉 𝑠, 𝑡!, 𝑡! .
J
J&"
𝑉 𝑠, 𝑡!, 𝑡! . 𝑑𝑠 + ∫PKQ
PK&" J
J&"
𝑉 𝑠, 𝑡!, 𝑡! . ([). 𝑑𝑊 𝑠
¨ 𝜎 𝑡, 𝑡!, 𝑡! =
J
J&"
𝑉 𝑡, 𝑡!, 𝑡! = 𝜎
¨ 𝑉 𝑡, 𝑡!, 𝑡! = 𝜎. (𝑡! − 𝑡) since 𝑉 𝑡, 𝑡, 𝑡 = 0
¨ ∫PKQ
PK&" J
J&"
𝑉 𝑠, 𝑡!, 𝑡! .
J
J&"
𝑉 𝑠, 𝑡!, 𝑡! . 𝑑𝑠 = ∫PKQ
PK&"
𝜎E. 𝑑𝑠 = 𝜎E. 𝑡!
¨ ∫PKQ
PK&"
𝑉 𝑠, 𝑡!, 𝑡! .
J
J&"
𝑉 𝑠, 𝑡!, 𝑡! . 𝑑𝑠 = ∫PKQ
PK&"
𝜎. (𝑡! − 𝑠). 𝜎. 𝑑𝑠 = 𝜎E. 𝑡!. 𝑡! −
#
E
. 𝑡!
E =
#
E
. 𝜎E. 𝑡!
E
¨ 𝑅 𝑡!, 𝑡!, 𝑡! = 𝑅 0, 𝑡!, 𝑡! + ∫PKQ
PK&"
𝑉 𝑠, 𝑡!, 𝑡! .
J
J&"
𝑉 𝑠, 𝑡!, 𝑡! . 𝑑𝑠 + ∫PKQ
PK&" J
J&"
𝑉 𝑠, 𝑡!, 𝑡! . ([). 𝑑𝑊 𝑠
109

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Ho-Lee is a simple Markov HJM model - IV
¨ 𝑅 𝑡!, 𝑡!, 𝑡! = 𝑅 0, 𝑡!, 𝑡! + ∫PKQ
PK&"
𝑉 𝑠, 𝑡!, 𝑡! .
J
J&"
𝑉 𝑠, 𝑡!, 𝑡! . 𝑑𝑠 + ∫PKQ
PK&" J
J&"
𝑉 𝑠, 𝑡!, 𝑡! . ([). 𝑑𝑊 𝑠
¨ 𝑅 𝑡!, 𝑡!, 𝑡! = 𝑅 0, 𝑡!, 𝑡! +
#
E
. 𝜎E. 𝑡!
E
+ 𝜎. ([). {𝑊 𝑡! − 𝑊 0 }
¨ We usually always assume that 𝑊 0 = 0
¨ 𝑅 𝑡!, 𝑡!, 𝑡! = 𝑅 0, 𝑡!, 𝑡! +
#
E
. 𝜎E. 𝑡!
E
+ 𝜎. ([). 𝑊 𝑡!
110

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Ho-Lee is a simple Markov HJM model - V
¨ We can also start from the SDE/SIE for the instantaneous forward rate:
¨ 𝑑𝑅 𝑡, 𝑡!, 𝑡! = 𝑉 𝑡, 𝑡!, 𝑡! .
J
J&"
𝑉 𝑡, 𝑡!, 𝑡! . 𝑑𝑡 +
J
J&"
𝑉 𝑡, 𝑡!, 𝑡! . ([). 𝑑𝑊 𝑡
¨ 𝑉 𝑡, 𝑡!, 𝑡! .
J
J&"
𝑉 𝑡, 𝑡!, 𝑡! = 𝜎. (𝑡! − 𝑡). 𝜎
¨ 𝑑𝑅 𝑡, 𝑡!, 𝑡! = 𝜎E. (𝑡! − 𝑡). 𝑑𝑡 + 𝜎. ([). 𝑑𝑊 𝑡
¨ 𝑑𝑅 𝑡!, 𝑡!, 𝑡! =
J9 Q,&",&"
J&"
. 𝑑𝑡! + 𝜎E. 𝑡! . 𝑑𝑡! + 𝜎. ([). 𝑑𝑊(𝑡!)
¨ The original Ho-Lee model had supposed:
¨ 𝑑𝑅 𝑡!, 𝑡!, 𝑡! = 𝜃(𝑡!). 𝑑𝑡! + 𝜎. ([). 𝑑𝑊(𝑡!)
¨ Within the HJM framework we have shown that in order to respect the arbitrage-free
relationship and fit the term structure of the initial yield curve, the function has no choice
but to be: 𝜃 𝑡! =
J9 Q,&",&"
J&"
+ 𝜎E. 𝑡!
111

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Ho-Lee is a simple Markov HJM model - VI
¨ This is a great illustration of the “art of the drift” to quote Bruce Tuckmann
¨ You can write: 𝑑𝑅 𝑡!, 𝑡!, 𝑡! = 𝜃(𝑡!). 𝑑𝑡! + 𝜎. ([). 𝑑𝑊(𝑡!)
¨ Enforcing the arbitrage free relationship that fits the initial term structure will NOT change
the instantaneous standard deviation term 𝜎. ([). 𝑑𝑊(𝑡!) but will change the advection
(drift) term, and will enforce:
¨ 𝜃 𝑡! =
J9 Q,&",&"
J&"
+ 𝜎E. 𝑡!
112

Luc_Faucheux_2020
Ho – Lee and Ricatti - I
¨ We know that it is an affine model so that we can start from the Ricatti set of equations:
¨ We know from the affine section in deck –IV that a model is said to possess and ATS (Affine
Term Structure) if we can express the Zeros as:
¨ Where we have: 𝑍𝐶 𝑡, 𝑡, 𝑡! = 𝑍𝐶 𝑡, 𝑡, 𝑡!, 𝑅 𝑡, 𝑡, 𝑡
¨ We then obtained the Ricatti set of equations:
¨ −
JN
J&
− 𝜇# 𝑡 . 𝐵 +
#
E
. 𝐵E. 𝛼# 𝑡 − 1 = 0
¨
JM
J&
− 𝜇Q 𝑡 . 𝐵 +
#
E
. 𝐵E. 𝛼Q 𝑡 = 0
¨ And the boundary conditions: 𝑍𝐶 𝑡!, 𝑡!, 𝑡! = 1 so we can choose the following conditions:
¨ 𝐴 𝑡!, 𝑡! = 0
¨ 𝐵 𝑡!, 𝑡! = 0
113

Luc_Faucheux_2020
Ho – Lee and Ricatti - II
¨ You get the Ho-Lee model (1985) if you write
¨ 𝑟 𝑡 + 𝛿𝑡 − 𝑟 𝑡 = ∫&
&$6&
𝜃 𝑠 . 𝑑𝑠 + 𝜎 ∫&
&$6&
1. ([). 𝑑𝑊 𝑠
¨ 𝜇Q 𝑡 = 𝜃(𝑡)
¨ 𝜇# 𝑡 = 0
¨ 𝛼Q 𝑡 = 𝜎E
¨ 𝛼# 𝑡 = 0
¨ So the Ricatti equations now become:
¨ −
JN
J&
− 0. 𝐵 +
#
E
. 𝐵E. 0 − 1 = 0 with boundary 𝐵 𝑡!, 𝑡! = 0
¨
JM
J&
− 𝜃(𝑡). 𝐵 +
#
E
. 𝐵E. 𝜎E = 0 with boundary 𝐴 𝑡!, 𝑡! = 0
114

Luc_Faucheux_2020
Ho – Lee and Ricatti - III
¨ So the Ricatti equations now become:
¨ −
JN
J&
= 1 with boundary 𝐵 𝑡!, 𝑡! = 0
¨
JM
J&
− 𝜃(𝑡). 𝐵 +
#
E
. 𝐵E. 𝜎E = 0 with boundary 𝐴 𝑡!, 𝑡! = 0
¨ Remember the actual functional dependency of the functions 𝐴 and 𝐵
¨ −
JN(&,&")
J&
¨
JM(&,&")
J&
− 𝜃(𝑡). 𝐵(𝑡, 𝑡!) +
#
E
. 𝐵(𝑡, 𝑡!)E. 𝜎E = 0 with boundary 𝐴 𝑡!, 𝑡! = 0
115

Luc_Faucheux_2020
Ho – Lee and Ricatti - IV
¨ Let’s see if we can solve:
¨ −
JN(&,&")
J&
¨ Looks like we should get:
¨ 𝐵 𝑡, 𝑡! = (𝑡! − 𝑡)
¨ Now onto:
¨
JM(&,&")
J&
− 𝜃(𝑡). 𝐵(𝑡, 𝑡!) +
#
E
. 𝐵(𝑡, 𝑡!)E. 𝜎E = 0 with boundary 𝐴 𝑡!, 𝑡! = 0
¨
JM(&,&")
J&
− 𝜃(𝑡). (𝑡! − 𝑡) +
#
E
. (𝑡! − 𝑡)E. 𝜎E = 0 with boundary 𝐴 𝑡!, 𝑡! = 0
¨ For now let’s assume that we do not know the function 𝜃(𝑡) (meaning we have not yet
enforced the arbitrage conditions that would yield:
¨ 𝜃 𝑡 =
J9 Q,&,&
J&
+ 𝜎E. 𝑡
116

Luc_Faucheux_2020
Ho – Lee and Ricatti - V
¨
JM(&,&")
J&
− 𝜃(𝑡). (𝑡! − 𝑡) +
#
E
. (𝑡! − 𝑡)E. 𝜎E = 0 with boundary 𝐴 𝑡!, 𝑡! = 0
¨
JM(&,&")
J&
= 𝜃 𝑡 . 𝑡! − 𝑡 −
#
E
. (𝑡! − 𝑡)E. 𝜎E with boundary 𝐴 𝑡!, 𝑡! = 0
¨
JM(P,&")
JP
= 𝜃 𝑠 . 𝑡! − 𝑠 −
#
E
. (𝑡! − 𝑠)E. 𝜎E
¨ ∫PK&
PK&" JM(P,&")
JP
. 𝑑𝑠 = [𝐴(𝑠, 𝑡!)]PK&
PK&"
= 𝐴 𝑡!, 𝑡! − 𝐴 𝑡, 𝑡! = −𝐴 𝑡, 𝑡!
¨ 𝐴 𝑡, 𝑡! = − ∫PK&
PK&" JM(P,&")
JP
. 𝑑𝑠
¨ 𝐵 𝑡, 𝑡! = (𝑡! − 𝑡)
¨ So we have the following equations for the Zeros in the general (no arbitrage enforced yet)
Ho-Lee affine model
117

Luc_Faucheux_2020
Ho – Lee and Ricatti - VI
¨ So there we have two options at our disposal:
¨ Option 1: we have done the work in the HJM framework, so just plug the functional for the
advection / drift term : 𝜃 𝑡 =
J9 Q,&,&
J&
+ 𝜎E. 𝑡
¨ Option 2: keep on carrying the advection 𝜃 𝑡 as is, and THEN enforce arbitrage on the
solutions for the Zeros we found, and check that we will recover indeed the drift condition
¨ We will start with option 1.
118

Luc_Faucheux_2020
Ho – Lee and Ricatti - VII
¨ 𝐴 𝑡, 𝑡! = − ∫PK&
PK&" JM(P,&")
JP
. 𝑑𝑠
¨ 𝐵 𝑡, 𝑡! = (𝑡! − 𝑡)
¨ 𝜃 𝑡 =
J9 Q,&,&
J&
+ 𝜎E. 𝑡
¨ 𝜃 𝑠 =
J9 Q,P,P
JP
+ 𝜎E. 𝑠
¨
JM(P,&")
JP
= 𝜃 𝑠 . 𝑡! − 𝑠 −
#
E
. (𝑡! − 𝑠)E. 𝜎E
¨ 𝐴 𝑡, 𝑡! = − ∫PK&
PK&"
{𝜃 𝑠 . 𝑡! − 𝑠 −
#
E
. (𝑡! − 𝑠)E. 𝜎E}. 𝑑𝑠
¨ 𝐴 𝑡, 𝑡! = − ∫PK&
PK&"
{[
J9 Q,P,P
JP
+ 𝜎E. 𝑠]. 𝑡! − 𝑠 −
#
E
. (𝑡! − 𝑠)E. 𝜎E}. 𝑑𝑠
119

Luc_Faucheux_2020
Ho – Lee and Ricatti - VIII
¨ 𝐴 𝑡, 𝑡! = − ∫PK&
PK&"
{[
J9 Q,P,P
JP
+ 𝜎E. 𝑠]. 𝑡! − 𝑠 −
#
E
. (𝑡! − 𝑠)E. 𝜎E}. 𝑑𝑠
¨ 𝑋 = − ∫PK&
PK&"
{𝜎E. 𝑠. 𝑡! − 𝑠 −
#
E
. (𝑡! − 𝑠)E. 𝜎E}. 𝑑𝑠
¨ 𝑋 = − ∫PK&
PK&"
{𝜎E. 𝑠. 𝑡! − 𝜎E. 𝑠E −
#
E
. (𝑡! − 𝑠)E. 𝜎E}. 𝑑𝑠
¨ 𝑋 = −[𝜎E.
P3
E
. 𝑡! −
#
V
. 𝜎E. 𝑠V +
#
E
.
#
V
. (𝑡! − 𝑠)V. 𝜎E]PK&
PK&"
¨ 𝑋 = −𝜎E[
P3
E
. 𝑡! −
#
V
. 𝑠V +
#
E
.
#
V
. (𝑡! − 𝑠)V]PK&
PK&"
¨ 𝑋 = −𝜎E. [
&"
3
E
. 𝑡! −
#
V
. 𝑡!
V +
#
E
.
#
V
. 𝑡! − 𝑡!
V − (
&3
E
. 𝑡! −
#
V
. 𝑡V +
#
E
.
#
V
. (𝑡! − 𝑡)V)]
¨ 𝑋 = −𝜎E. [
&"
3
E
. 𝑡! −
#
V
. 𝑡!
V −
&3
E
. 𝑡! +
#
V
. 𝑡V −
#
E
.
#
V
. 𝑡! − 𝑡 V]
120

Luc_Faucheux_2020
Ho – Lee and Ricatti - IX
¨ 𝑋 = −𝜎E. [
&"
3
E
. 𝑡! −
#
V
. 𝑡!
V −
&3
E
. 𝑡! +
#
V
. 𝑡V −
#
E
.
#
V
. 𝑡! − 𝑡 V]
¨ 𝑋 =
GS3
W
. [𝑡!
V − 3. 𝑡E . 𝑡! + 2. 𝑡V − 𝑡! − 𝑡 V]
¨ 𝑋 =
GS3
W
. [𝑡!
V − 3. 𝑡E . 𝑡! + 2. 𝑡V − (𝑡!
V − 𝑡V − 3. 𝑡!
E. 𝑡 + 3. 𝑡!. 𝑡E)]
¨ 𝑋 =
GS3
W
. [𝑡!
V − 3. 𝑡E . 𝑡! + 2. 𝑡V − 𝑡!
V + 𝑡V + 3. 𝑡!
E. 𝑡 − 3. 𝑡!. 𝑡E]
¨ 𝑋 =
GS3
W
. [−6. 𝑡E . 𝑡! + 3. 𝑡V + 3. 𝑡!
E. 𝑡]
¨ 𝑋 =
GS3
W
. 3. 𝑡. −2. 𝑡. 𝑡! + 𝑡E + 𝑡!
E =
GS3
E
. 𝑡. (𝑡 − 𝑡!)E=
GS3
E
𝑡(𝑡 − 𝑡!)E
¨ Let’s now deal with the other term : 𝑌 = − ∫PK&
PK&" J9 Q,P,P
JP
. 𝑡! − 𝑠 . 𝑑𝑠
121

Luc_Faucheux_2020
Ho – Lee and Ricatti - X
¨ 𝑌 = − ∫PK&
PK&" J9 Q,P,P
JP
. 𝑡! − 𝑠 . 𝑑𝑠
¨ We see our good old friend the Instantaneous forward 𝐼𝐹𝑤𝑅 0, 𝑠 = 𝑅 0, 𝑠, 𝑠 come back
¨ Since it looks like we are going to perform integration over the time variable 𝑠, it pays to be
a little rigorous and figure out exactly which one of the 3 time variables we are talking
about.
¨ Because remember, things always come in 3
¨ 3 seasons of Dark
¨ 3 worlds
¨ 33 year cycle
¨ 3 branches of the triquerta
¨ 3 branches of the tunnel
¨ 3 time variables when dealing with rates modeling in Finance
122

Luc_Faucheux_2020
Ho – Lee and Ricatti - XI
¨ Remember that:
¨ 𝑅 𝑡, 𝑡!, 𝑡" . 𝜏 𝑡, 𝑡!, 𝑡" = −ln(𝑍𝐶 𝑡, 𝑡!, 𝑡" )
¨ 𝑅 𝑡, 𝑡!, 𝑡! = lim
&!→&"
(𝑅 𝑡, 𝑡!, 𝑡" )
¨ 𝑅 𝑡, 𝑡!, 𝑡! = −
JHI(*+ &,&,&"
J&"
¨ 𝑅 𝑡, 𝑡, 𝑡! =
#
&"G&
∫PK&
PK&"
𝑅 𝑡, 𝑠, 𝑠 . 𝑑𝑠
¨ 𝑅 𝑡, 𝑡!, 𝑡! =
J
J&"
. 𝑡! − 𝑡 . 𝑅 𝑡, 𝑡, 𝑡! = 𝑅 𝑡, 𝑡, 𝑡! + 𝑡! − 𝑡 .
J9 &,&,&"
J&"
123

Luc_Faucheux_2020
Ho – Lee and Ricatti – XI - a
¨ Let me do a digression here because after part IV some of you came and asked me about the
correct interpretation of:
¨ 𝑅 𝑡, 𝑡, 𝑡! =
#
&"G&
∫PK&
PK&"
𝑅 𝑡, 𝑠, 𝑠 . 𝑑𝑠
¨ In particular, ”over which of the 𝑠 do you integrate?”
¨ Also, on the actual definition of 𝑅 𝑡, 𝑡!, 𝑡! = lim
&!→&"
(𝑅 𝑡, 𝑡!, 𝑡" ), and why we could not just
replace 𝑡" by 𝑡!.
¨ This exercise on the Ho Lee model is a great excuse to go over this again and make sure that
we are good on the notations and the math.
124

Luc_Faucheux_2020
Ho – Lee and Ricatti – XI - b
¨ First let’s go over:
¨ 𝑅 𝑡, 𝑡!, 𝑡! = lim
&!→&"
(𝑅 𝑡, 𝑡!, 𝑡" )
¨ 𝑅 𝑡, 𝑡!, 𝑡" . 𝜏 𝑡, 𝑡!, 𝑡" = −ln(𝑍𝐶 𝑡, 𝑡!, 𝑡" )
¨ 𝑅 𝑡, 𝑡!, 𝑡" =
GHI(*+ &,&",&! )
) &,&",&!
¨ When 𝑡" → 𝑡! we end up with the following:
¨ 𝑍𝐶 𝑡, 𝑡!, 𝑡! = 1 so ln 𝑍𝐶 𝑡, 𝑡!, 𝑡! = 0
¨ 𝜏 𝑡, 𝑡!, 𝑡! = 0
¨ So there is no issue in writing something like this:
¨ lim
&!→&"
(𝑍𝐶 𝑡, 𝑡!, 𝑡" ) = 𝑍𝐶 𝑡, 𝑡!, 𝑡!
125

Luc_Faucheux_2020
Ho – Lee and Ricatti – XI - c
¨ So there is no issue in writing something like this:
¨ lim
&!→&"
(𝑍𝐶 𝑡, 𝑡!, 𝑡" ) = 𝑍𝐶 𝑡, 𝑡!, 𝑡! = 1
¨ lim
&!→&"
(ln(𝑍𝐶 𝑡, 𝑡!, 𝑡" )) = ln 𝑍𝐶 𝑡, 𝑡!, 𝑡! = ln 1 = 0
¨ lim
&!→&"
(𝜏 𝑡, 𝑡!, 𝑡" ) = 𝜏 𝑡, 𝑡!, 𝑡! = 0
¨ The issue comes when we take the ratio of the two quantities above as we will end up with
something that is going to be
Q
Q
, always something that we want to be careful about.
126

Luc_Faucheux_2020
Ho – Lee and Ricatti – XI - d
¨ Good thing that we have some French mathematicians to help us
¨ This one is Guillaume de L’Hopital.
¨ Among other thing he is famous for proposing a way to deal with things that tend to the
limit of
Q
Q
or
X
X
127

Luc_Faucheux_2020
Ho – Lee and Ricatti – XI - e
¨ 𝑅 𝑡, 𝑡!, 𝑡" =
GHI(*+ &,&",&! )
) &,&",&!
¨ So if we were just to plug 𝑡" = 𝑡! in the expression above, we would get 𝑅 𝑡, 𝑡!, 𝑡" =
Q
Q
¨ This is why we need to be a little more careful
¨ We can use the L’Hopital rule, which is essentially (in the 0 case at hand here)
¨ If lim
&!→&"
𝑓 𝑡" = 0
¨ If lim
&!→&"
𝑔 𝑡" = 0
¨ If both functions are differentiable around 𝑡! and if lim
&!→&"
TY &!
UY &!
exists, then
¨ lim
&!→&"
TY &!
UY &!
= lim
&!→&"
T &!
U &!
128

Luc_Faucheux_2020
Ho – Lee and Ricatti – XI - f
¨ lim
&!→&"
TY &!
UY &!
= lim
&!→&"
T &!
U &!
¨ 𝑅 𝑡, 𝑡!, 𝑡" =
GHI(*+ &,&",&! )
) &,&",&!
¨ 𝑓 𝑡" = − ln 𝑍𝐶 𝑡, 𝑡!, 𝑡" = − ln
*+ &,&,&!
*+ &,&,&"
= −ln(𝑍𝐶 𝑡, 𝑡, 𝑡" ) + ln(𝑍𝐶 𝑡, 𝑡, 𝑡! )
¨ 𝑓Y 𝑡" = −
JHI(*+ &,&",&!
J&!
|&!K&"
= −
JHI(*+ &,&,&!
J&!
|&!K&"
¨ 𝑔 𝑡" = 𝜏 𝑡, 𝑡!, 𝑡" = (𝑡" − 𝑡!)
¨ 𝑔Y &! =
J) &,&",&!
J&!
|&!K&"
= 1
129

Luc_Faucheux_2020
Ho – Lee and Ricatti – XI - g
¨ So we now have using l’Hopital rule:
¨ 𝑅 𝑡, 𝑡!, 𝑡! = lim
&!→&"
(𝑅 𝑡, 𝑡!, 𝑡" ) = lim
&!→&"
(
GHI(*+ &,&",&! )
) &,&",&!
) = lim
&!→&"
(−
JHI(*+ &,&,&!
J&!
|&!K&"
)
¨ 𝑅 𝑡, 𝑡!, 𝑡! = lim
&!→&"
(−
JHI(*+ &,&,&!
J&!
|&!K&"
)
¨ That we can note as:
¨ 𝑅 𝑡, 𝑡!, 𝑡! = −
JHI(*+ &,&,&"
J&"
¨ Remember that the third time variable is “Newtonian” as Baxter says, all the variables are
only a stochastic process in the first variable that is “Brownian”
¨ Just take a moment to convince yourself that you can essentially write:
¨ lim
&!→&"
TY &!
UY &!
=
TY &"
UY &"
130

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