This document discusses the difference between Ito calculus and Taylor expansions in the context of calculating portfolio profit and loss (PL). While Taylor expansions deal with regular calculus and require differentiability, Ito's lemma looks similar to a Taylor expansion but is an exact equation, not an approximation. Ito's lemma describes how the chain rule works in stochastic calculus. The document uses the example of calculating option PL to illustrate the difference. A Taylor expansion can be used to calculate the first-order PL terms from changes in the stock price, volatility, and time. However, the full PL calculation involves additional non-market factors like funding costs, credit valuation adjustments, and cross-currency effects that

1. Luc_Faucheux_2021
Stochastic Calculus – ITO – III -a
ITO and Taylor/Leibniz
1

2. Luc_Faucheux_2021
In this section
¨ Based on some feedback I got from the previous deck
¨ We do here a little digression on the Taylor expansion and the ITO lemma
¨ A lot of textbooks sometimes say or imply: “ Hey ITO lemma is just a Taylor expansion, just
go up one more order in the expansion to capture the quadratic term, and you good”
¨ This is confusing because it is not true
¨ ITO lemma is NOT a Taylor expansion
¨ Note that it is somehow related to a Taylor expansion inside the definition of the integral, so
it is somewhat natural to get confused.
¨ ITO lemma formally looks like a Taylor expansion, but it is NOT a Taylor expansion
¨ Took me like 30 years to understand that, hopefully this deck will allow you to avoid
repeating my mistakes.
2

3. Luc_Faucheux_2021
ITO versus TAYLOR
The example of PL
3

4. Luc_Faucheux_2021
Ito vs Taylor, the example of PL - I
¨ Here we will illustrate the difference between ITO and Taylor through the example of the PL
explain/PL slice,..
¨ This ties out with our section on Risk management
¨ We will illustrate some of ways out there to compute, report and debug PL
4

5. Luc_Faucheux_2021
Ito vs Taylor, the example of PL - II
¨ I will try to be a little rigorous and stick to notations that will make more apparent the
difference between Ito and Taylor
¨ Roughly speaking:
¨ Taylor expansion deals with regular/Leibniz/Newton calculus, and in fact requires some
general statements about the function being differentiable to some extent
¨ The first thing they tell you about stochastic calculus is that it is NOT differentiable, so right
there, that should ring some alarm bells
¨ Ito lemma is really “how does the chain rule looks like in stochastic calculus”
¨ Turns out that it formally looks like a Taylor expansion going to the second order (delta +
gamma), and we are used to do Taylor expansion of PL of a portfolio, so we think that it
really is the same.
¨ It is not
¨ A Taylor expansion is an approximation (there is usually a term +𝒪() at the end)
¨ ITO lemma is an exact equation, it is not an approximation
5

6. Luc_Faucheux_2021
Ito vs Taylor, the example of PL - III
¨ 𝑋 𝑡 is a stochastic process, 𝑡 is the regular time, think of 𝑋 𝑡 as the stock price
¨ 𝑓 is a regular function of say variable 𝑥, we assume that 𝑓 is nicely continuous and
differentiable and “well behaved” from a math point of view
¨ If that helps, think of:
¨ 𝑋 𝑡 is the stock price process
¨ 𝑥 is the value NOW of the stock price
¨ 𝑓 is the call option price, 𝑓(𝑥, 𝜎, 𝑡!)
¨ Right now, 𝑋 𝑡! = 𝑥, and we are holding an option worth 𝑓(𝑥, 𝜎, 𝑡!), where we have either
input the volatility 𝜎 or calibrated it to the market (implied volatility)
¨ 𝑓(𝑥(𝑡!), 𝜎(𝑡!), 𝑡!)
¨ Even if it is a little cumbersome, we will keep the notation 𝑡!
6

7. Luc_Faucheux_2021
Ito vs Taylor, the example of PL - IV
¨ Right now at time 𝑡!, we are observing the stock price to be 𝑥(𝑡!)
¨ Our portfolio (say we are long only that option) is worth 𝑓(𝑥(𝑡!), 𝜎(𝑡!), 𝑡!)
¨ 𝑓(𝑥, 𝜎, 𝑡) is a regular function of the variables (𝑥, 𝜎, 𝑡) should we change to bump them and
has well behaved derivatives that we refer to as:
¨
"#
"$
DELTA
¨
"#
"%
VEGA
¨
"#
"&
THETA
¨ For the first order one, and
7

8. Luc_Faucheux_2021
Ito vs Taylor, the example of PL - V
¨
"!#
"$! GAMMA
¨
"!#
"%! VOLGA
¨
"!#
"&! THETA BLEED
¨
"!#
"$"%
VANNA
¨
"!#
"$"&
DELTA BLEED
¨
"!#
"&"%
VEGA BLEED
¨ Of course the order of derivation is irrelevant:
¨
"!#
"&"%
=
"!#
"%"&
=
"
"%
"#
"&
=
"
"&
(
"#
"%
)
8

9. Luc_Faucheux_2021
Ito vs Taylor, the example of PL - VI
¨ Historically for constant volatility models (the early ones), the process for the stock was
assumed to be something like:
¨ 𝑑𝑋 𝑡 = 𝑎 . . . 𝑑𝑡 + 𝜎. 𝑑𝑊(𝑡)
¨ And because of the quadratic property of this process:
¨ < 𝑑𝑋 𝑡 > = 𝑎 … . 𝑑𝑡
¨ < 𝑑𝑋 𝑡 ' > = 𝜎'. 𝑑𝑡
¨ The most important Greeks to consider are : DELTA, GAMMA, THETA
¨
"#
"$
DELTA
¨
"!#
"$! GAMMA
¨
"#
"&
THETA
9

10. Luc_Faucheux_2021
Ito vs Taylor, the example of PL - VII
¨ Of course, once people started calibrating their model models daily, they realized that they
had to change the value of the implied volatility and hence were observing a PL coming from
the change in the volatility
¨ The next Greek historically that become important was VEGA
¨
"#
"%
VEGA
¨ Note that of course since there is a change in 𝜎, obviously all the higher order start
contributing to the PL, but:
¨ A- was not necessarily hedgeable in an easy manner
¨ B- trading desks will still deal with it “after the fact” just the same way they would re-adjust
the DELTA coming from the passage of time
10

11. Luc_Faucheux_2021
Ito vs Taylor, the example of PL - VIII
¨ Note also that once a specific Greek becomes liquid enough to hedge in the market and the
market offers a product that is a good hedge for that Greek, that Greek will start to create its
own market dynamics, with the associated structural flows and short-term trading
idiosyncracies, and that dynamics will need to be incorporated back into the model.
¨ The sentence above packs a lot, and I would need an entire book to explain it fully, but trust
me on that one, if there is one important thing I say in those thousands of slides, that is it.
11

12. Luc_Faucheux_2021
Ito vs Taylor, the example of PL - IX
¨ Once people starting thinking of the volatility as a stochastic variable in its own right (and
not something that changes once a day after recalibrating your model, the same way time
would change once a day for example), the next Greek that became paramount were
VANNA and VOLGA
¨
"!#
"%! VOLGA
¨
"!#
"$"%
VANNA
¨ As the models become more and more refined and introduce more stochastic processes,
more and more higher order Greeks become important, to monitor, to have risk limits on, to
model in scenarios, to incorporate in the model through the implied dynamics of their
corresponding processes
¨ Because those quantities start to get monitored, with risk limits, this in turn create hedging
needs in the market, making those more relevant for pricing and modeling.
12

13. Luc_Faucheux_2021
Ito vs Taylor, the example of PL - X
¨ 𝑓(𝑥, 𝜎, 𝑡) is a regular function of the variables (𝑥, 𝜎, 𝑡)
¨ At time (𝑡 + 𝛿𝑡), the observed value of the stock is 𝑥(𝑡 + 𝛿𝑡) and the new implied volatility
is 𝜎(𝑡 + 𝛿𝑡)
¨ 𝑥 𝑡 + 𝛿𝑡 − 𝑥 𝑡 = 𝛿𝑥
¨ 𝜎 𝑡 + 𝛿𝑡 − 𝜎 𝑡 = 𝛿𝜎
¨ 𝑓 𝑥 𝑡 + 𝛿𝑡 , 𝑥 𝑡 + 𝛿𝑡 , 𝑡 + 𝛿𝑡 − 𝑓 𝑥, 𝜎, 𝑡 = 𝛿𝑓
¨ We are totally justified to write the usual Taylor expansion because we are in the world of
regular/Leibniz/Newton calculus, computing all the derivatives at the point (𝑥, 𝜎, 𝑡)
13

14. Luc_Faucheux_2021
Ito vs Taylor, the example of PL - XI
¨ 𝛿𝑓 = 𝐷𝐸𝐿𝑇𝐴. 𝛿𝑥 + 𝑉𝐸𝐺𝐴. 𝛿𝜎 + 𝑇𝐻𝐸𝑇𝐴. 𝛿𝑡
¨ And we think that did capture all the first order terms
¨ 𝛿𝑓 =
"#
"$
. 𝛿𝑥 +
"#
"%
. 𝛿𝜎 +
"#
"&
. 𝛿𝑡
¨ In the simple case of constant volatility (say from the 70s to the 80s) that reduces to:
¨ 𝛿𝑓 =
"#
"$
. 𝛿𝑥 +
"#
"&
. 𝛿𝑡
¨ That would be your option PL (to the first order)
¨ Note that I usually refer to that as the “Market PL”, that is the PL coming from changes in the
market observables.
14

15. Luc_Faucheux_2021
Ito vs Taylor, the example of PL - XII
¨ There are obviously a lot more components to the real PL if you were running an option
book. We will go into more details in the second deck on Risk Management but essentially
just to name a few:
¨ Funding/Financing PL: how do you fund yourself, how do you finance the fact that you are
holding an asset on your books. Did you borrow cash to buy that option? What interest do
you pay on the cash that you borrowed? Are you a Japanese bank who is getting funding in
Yen from retail depositors in Japan and buying US denominated assets? Do you put up any
collateral on the loan that you got in order to buy the option? If so what kind of collateral?
Do you have an obligation to repo that collateral every day in order to be in “good standing”
with the lender? If so what kind of collateral can you repo? Do you get the accruals back on
the collateral that you put up overnight (say a US bond)? Is there a weird language in your
CSA that floors the interest that you get back at 0? Can you put up collateral denominated
in a different currency than the loan, or the asset that you are holding? Can you put up your
own bonds as collateral? If you have more than one assets, can you commingle them for the
purpose of collateral posting? If the asset is traded on an exchange, what are the rules of
the exchange (watch Trading Places!), what is your margin agreement? What rate are you
being charged/credited on your margin? Can you put something else than cash?
15

16. Luc_Faucheux_2021
Ito vs Taylor, the example of PL - XIII
¨ Funding/Financing PL:…..if traded on a foreign exchange, do you have to put up as margin
cash in the local currency, or can you post up USD?
¨ So this is actually a super complicated question, leading to mysterious things like cross-
currency basis, term basis,..
¨ Usually all that is lumped under the FVA (funding valuation adjustment), essentially all those
“little behind the scenes” effects impact your actual PL
¨ Before 2008, no trading desk gave a hoot about it
¨ Ironically before the middle if the ‘90s when CSA became quite common, every trading desk
has a “cash manager” and did care a lot about this. That disappeared when CSA became
prevalent and essentially everyone was pricing funding at LIBOR flat and balance sheet was
infinite and free
16

17. Luc_Faucheux_2021
Ito vs Taylor, the example of PL - XIV
¨ Up until 2008, and even after, I can bet you a large amount of money that you could walk up
to any trading desk manager and ask them “What is your cash position?” and they would
not know what to say, and usually mumble some BS argument about “oh treasury does it for
me, it’s more efficient”
¨ And then if you want to keep on poking at them, then ask “if the market crashes and
liquidity becomes scarce, how does your cash position move?”
¨ That would be usually when they storm off waving arms and muttering something about the
fact that they are busy making money and they cannot waste their precious time with stupid
people like you
¨ FVA are super hard to compute, and there is also a legitimate argument to be made that
they should be computed centrally and then “redistributed” to the individual desks, which
leads to endless hours of arguments, because now you are actually showing to the desk an
actual amount of PL that they will realize they do not understand fully, leading to the usual
“you are killing my business”, “you are not franchise friendly”, “you are stealing money from
me” on one side, and “you are wasting the Firm’s money, you are destroying values, you are
not efficient” on the other…fun times…
17

18. Luc_Faucheux_2021
Ito vs Taylor, the example of PL - XV
¨ Others components of “non-market PL”
¨ Because you are holding an asset against someone else (exchange, clearing house,
counterparty,..) you should be sensitive to the credit of that counterparty, that is called CVA
(Credit Valuation Adjustment).
¨ CVA really only started to become important on trading desk in the years leading to the 2008
crisis
¨ CVA was usually before 2008 done on an ad hoc basis on large transactions and usually
managed separately on the credit desk
18

19. Luc_Faucheux_2021
Ito vs Taylor, the example of PL - XVI
¨ What comes after C? You guess it, D !
¨ If you are holding a liability, the other party is holding an asset against you, an has a CVA
adjustment.
¨ So surely you should also have an adjustment on your side to reflect your own credit ?
¨ That was called DVA (because D is after C)
¨ That became hugely important in the 2008 crisis, when firms like GS and Barclays escaped
bankruptcy by claiming a massive “DVA adjustment” on their quarterly earnings, essentially
reflecting the fact that their own credit deteriorated.
¨ To be fair, there is some legitimate reason to claim a positive DVA adjustment if your credit
deteriorates, but this is still subject to endless debates.
19

20. Luc_Faucheux_2021
Ito vs Taylor, the example of PL - XVII
¨ Also, as some of you astute readers would have anticipated by now, there is somewhat of a
loop there.
¨ If say the CVA adjustment is large, then I should adjust my PL for it, meaning that the value
of the portfolio changes, hence my FVA should also be adjusted for that.
¨ Then the CVA has to be adjusted again…and so on and so forth… as one of my most brilliant
boss was known to say.
¨ So the whole thing is super complicated because:
¨ It is centralized by nature because of exposure netting
¨ Modeling the adjustment is super hard because it will involve credit projection in the future,
market projection in the future (because that changes the amount of asset/liability at risk),
as well as all the products covered under the netting (could be equities, rates, FX,
commodities, but also credit itself, cash, derivatives,,..)
20

21. Luc_Faucheux_2021
Ito vs Taylor, the example of PL - XVIII
¨ Because of all that, all those adjustments are usually given to a team of super smart people
called the XVA team
¨ There is an awful lot of work begin done on XVA, how to compute it, how to compute it fast,
how to hedge it, and of course spending 90% of your time internally fighting the internal
battles of how to allocate the PL and who is going to pay for it.
¨ You can google the recent articles on Risk magazine on XVA to convince yourself that it is a
super hard problem
¨ Some people have used AAD (Automated Adjoint Differentiation) to make it super fast (like
Savine at Dansk, you have to read his book on AAD!)
¨ Some people also starting to use AI and Machine Learning to tackle the XVA problem (yours
truly hopefully)
21

22. Luc_Faucheux_2021
Ito vs Taylor, the example of PL - XIX
¨ Savine’s book on AAD, a great worked-put example using Black-Sholes !
22

23. Luc_Faucheux_2021
Ito vs Taylor, the example of PL - XX
¨ There are also plenty of other adjustments to the PL
¨ Like for example is the trading desk holding a reserve based on some Risk factors
¨ How is that reserve calculated? Is that computed daily? Does it have some market
sensitivity? In that case should you hedge that market sensitivity?
¨ All right back to our “simple” “Market PL”
23

24. Luc_Faucheux_2021
Ito vs Taylor, the example of PL - XXI
¨ 𝛿𝑓 =
"#
"%
. 𝛿𝑥 +
"#
"%
. 𝛿𝑡 + 𝑠𝑜𝑚𝑒𝑡ℎ𝑖𝑛𝑔
¨ In the case of a constant volatility (think about you are not hedging the volatility, or you do
not calibrate your model to the market and take a long view, something that is not without
merit for some long dated option books, that was an approach taken for a while at Salomon
as it reduced the “whipsawing” effect of your hedges)
¨ That is the first order Taylor expansion in 𝛿𝑥 and 𝛿𝑡
¨ The something is usually smaller and can be neglected (again there are assumptions there,
say if you are very close to the strike and near expiration, your higher orders are going to
blow up to infinity, so a Taylor expansion is no longer valid).
¨ Also this is usually true in the limit of “small” 𝛿𝑥 and 𝛿𝑡 and you would have to test that,
especially if there is a large market move. Usually the test of the the validity of the Taylor
expansion is done by looking at the amount of actual PL that you cannot explain from the
Taylor terms, or the unexplained PL
24

25. Luc_Faucheux_2021
Ito vs Taylor, the example of PL - XXII
¨ If we were to go to the second order we would get:
¨ 𝛿𝑓 =
"#
"$
. 𝛿𝑥 +
"#
"&
. 𝛿𝑡 +
(
'
"!#
"$! . 𝛿𝑥' +
(
'
"!#
"&! . 𝛿𝑡' +
"!#
"$"&
. 𝛿𝑥. 𝛿𝑡 + 𝑠𝑜𝑚𝑒𝑡ℎ𝑖𝑛𝑔
¨ 𝛿𝑓 = 𝐷𝐸𝐿𝑇𝐴. 𝛿𝑥 + 𝑇𝐻𝐸𝑇𝐴. 𝛿𝑡 + 𝐺𝐴𝑀𝑀𝐴. 𝛿𝑥! + 𝑇𝐻𝐸𝑇𝐴_𝐵𝐿𝐸𝐸𝐷. 𝛿𝑡! + 𝐷𝐸𝐿𝑇𝐴_𝐵𝐿𝐸𝐸𝐷. 𝛿𝑥. 𝛿𝑡 + 𝑠𝑜𝑚𝑒𝑡ℎ𝑖𝑛𝑔
¨ Usually (and again you should always check) the BLEED terms are usually small and so they
lumped into the “something”
¨ 𝛿𝑓 =
"#
"$
. 𝛿𝑥 +
"#
"&
. 𝛿𝑡 +
(
'
"!#
"$! . 𝛿𝑥' + 𝑠𝑜𝑚𝑒𝑡ℎ𝑖𝑛𝑔
¨ 𝛿𝑓 = 𝐷𝐸𝐿𝑇𝐴. 𝛿𝑥 + 𝑇𝐻𝐸𝑇𝐴. 𝛿𝑡 +
(
'
𝐺𝐴𝑀𝑀𝐴. 𝛿𝑥' + 𝑠𝑜𝑚𝑒𝑡ℎ𝑖𝑛𝑔
¨ People usually like the above because if the stock is assumed to follow a stochastic process,
¨ < 𝛿𝑥'> ~𝛿𝑡, and so the above has “only first order terms in 𝛿𝑡 and 𝛿𝑥”, and so traders loosely
think that they are covered with writing something like the above
¨ Read Nassim Taleb dynamic Hedging (that is when he was actually writing decent books and
before he became unsufferable) about the importance of higher orders when hedging and
especially dynamically hedging
25

26. Luc_Faucheux_2021
Ito vs Taylor, the example of PL - XXIII
¨ OK, to the “first order in stock and time”, the Taylor expansion of the PL looks like:
¨ 𝛿𝑓 =
"#
"$
. 𝛿𝑥 +
"#
"&
. 𝛿𝑡 +
(
'
"!#
"$! . 𝛿𝑥' + 𝑠𝑜𝑚𝑒𝑡ℎ𝑖𝑛𝑔
¨ NOW, a lot of people are getting this confused with the ITO lemma:
¨ 𝛿𝑓 =
"#
"$
. 𝛿𝑋 +
"#
"&
. 𝛿𝑡 +
(
'
.
"!#
"$! . 𝑏'. 𝛿𝑡 when we have a process following the Ito SDE:
¨ 𝑑𝑋 𝑡 = 𝑎 𝑡, 𝑋 𝑡 . 𝑑𝑡 + 𝑏 𝑡, 𝑋 𝑡 . ([). 𝑑𝑊
¨ Because it does in fact look like:
¨ Taylor expansion of the PL: 𝛿𝑓 =
"#
"$
. 𝛿𝑥 +
"#
"&
. 𝛿𝑡 +
(
'
"!#
"$! . 𝛿𝑥' + 𝑠𝑜𝑚𝑒𝑡ℎ𝑖𝑛𝑔
¨ Ito lemma: 𝛿𝑓 =
"#
"$
. 𝛿𝑋 +
"#
"&
. 𝛿𝑡 +
(
'
.
"!#
"$! . 𝛿𝑋'
26

27. Luc_Faucheux_2021
Ito vs Taylor, the example of PL - XXIV
¨ Taylor expansion of the PL: 𝛿𝑓 =
"#
"$
. 𝛿𝑥 +
"#
"&
. 𝛿𝑡 +
(
'
"!#
"$! . 𝛿𝑥' + 𝑠𝑜𝑚𝑒𝑡ℎ𝑖𝑛𝑔
¨ Ito lemma: 𝛿𝑓 =
"#
"$
. 𝛿𝑋 +
"#
"&
. 𝛿𝑡 +
(
'
.
"!#
"$! . 𝛿𝑋'
¨ First of all we should know that it is different because of the “something” term
¨ Also in Ito I have used a capital letter so that should also be an indication that something is afoot
¨ As we saw in the previous deck, the real rigorous way to write ITO lemma is :
¨ 𝑓 𝑋 𝑡" , 𝑡" − 𝑓 𝑋 𝑡# , 𝑡# = ∫
$%$#
$%$" &' (,$
&(
|(%* $ . [ . 𝑑𝑋 𝑡 + ∫
$%$#
$%$" &'((,$)
&$
|(%*($). 𝑑𝑡 +
-
!
. ∫
$%$#
$%$" &!' (,$
&(!
|(%* $ . 𝑏 𝑡, 𝑋 𝑡
!
. 𝑑𝑡
¨ 𝑑𝑋 𝑡 = 𝑎 𝑡, 𝑋 𝑡 . 𝑑𝑡 + 𝑏 𝑡, 𝑋 𝑡 . ([). 𝑑𝑊
27

28. Luc_Faucheux_2021
Ito vs Taylor, the example of PL - XXV
¨ 𝑓 𝑋 𝑡" , 𝑡" − 𝑓 𝑋 𝑡# , 𝑡# = ∫
$%$#
$%$" &' (,$
&(
|(%* $ . [ . 𝑑𝑋 𝑡 + ∫
$%$#
$%$" &'((,$)
&$
|(%*($). 𝑑𝑡 +
-
!
. ∫
$%$#
$%$" &!' (,$
&(! |(%* $ . 𝑏 𝑡, 𝑋 𝑡
!
. 𝑑𝑡
¨ 𝑑𝑋 𝑡 = 𝑎 𝑡, 𝑋 𝑡 . 𝑑𝑡 + 𝑏 𝑡, 𝑋 𝑡 . ([). 𝑑𝑊
¨ Let’s point out that the above is an EXACT equation, there is no “+something” like there is in the
Taylor expansion
¨ Sometimes the above SIE is written in SDE form:
¨ 𝛿𝑓 =
"# $,&
"$
|$NO & . 𝛿𝑋 +
"#($,&)
"&
|$NO(&). 𝛿𝑡 +
(
'
.
"!# $,&
"$! |$NO & . 𝑏'. 𝛿𝑡
¨ Or sometimes to be even more liberal with the notation:
¨ 𝛿𝑓 =
"#
"$
. 𝛿𝑋 +
"#
"&
. 𝛿𝑡 +
(
'
.
"!#
"$! . 𝑏'. 𝛿𝑡 and since 𝛿𝑋'~𝑏'. 𝛿𝑡
¨ 𝛿𝑓 =
"#
"$
. 𝛿𝑋 +
"#
"&
. 𝛿𝑡 +
(
'
.
"!#
"$! . 𝛿𝑋'
28

29. Luc_Faucheux_2021
Ito vs Taylor, the example of PL - XXVI
¨ So indeed this is somewhat rather similar:
¨ Taylor expansion of a function 𝑓(𝑥, 𝑡) by keeping first order terms and also for some reason
(which we are totally authorized to do) keeping also the second term in 𝑥:
¨ 𝛿𝑓 =
"#
"$
. 𝛿𝑥 +
"#
"&
. 𝛿𝑡 +
(
'
"!#
"$! . 𝛿𝑥' + 𝑠𝑜𝑚𝑒𝑡ℎ𝑖𝑛𝑔
¨ As opposed to :
¨ 𝑋(𝑡) is a stochastic process of the form: 𝑑𝑋 𝑡 = 𝑎 𝑡, 𝑋 𝑡 . 𝑑𝑡 + 𝑏 𝑡, 𝑋 𝑡 . ([). 𝑑𝑊
¨ 𝑓(𝑥 = 𝑋 𝑡 , 𝑡) is also a stochastic process. Which SIE/SDE does this process follow? Well, Ito
lemma (in Ito calculus) tells us the exact way to transform the process for 𝑋(𝑡) into the process
for 𝑓(𝑥 = 𝑋 𝑡 , 𝑡). Formally being loose with notation it reads like:
¨ 𝛿𝑓 =
"#
"$
. 𝛿𝑋 +
"#
"&
. 𝛿𝑡 +
(
'
.
"!#
"$! . 𝛿𝑋'
¨ Being rigorous in the notation it reads:
¨ 𝑓 𝑋 𝑡" , 𝑡" − 𝑓 𝑋 𝑡# , 𝑡# = ∫
$%$#
$%$" &' (,$
&(
|(%* $ . [ . 𝑑𝑋 𝑡 + ∫
$%$#
$%$" &'((,$)
&$
|(%*($). 𝑑𝑡 +
-
!
. ∫
$%$#
$%$" &!' (,$
&(! |(%* $ . 𝑏 𝑡, 𝑋 𝑡
!
. 𝑑𝑡
29

30. Luc_Faucheux_2021
Ito vs Taylor, the example of PL - XXVII
¨ As a side note:
¨ In the Taylor expansion in regular calculus, I know what “.” means, it is the regular
multiplication when writing:
¨ 𝛿𝑓 =
"#
"$
. 𝛿𝑥 +
"#
"&
. 𝛿𝑡 +
(
'
"!#
"$! . 𝛿𝑥' + 𝑠𝑜𝑚𝑒𝑡ℎ𝑖𝑛𝑔
¨ In stochastic calculus I do not know what “.” means when I write:
¨
"#
"$
. 𝛿𝑋
¨ It would be better if I wrote:
"#
"$
. ([). 𝛿𝑋
¨ It would be even better if I only wrote integrals and I defined the ITO integral as:
¨ ∫
&N&P
&N&Q
𝑓 𝑋 𝑡 . ([). 𝑑𝑋(𝑡) = lim
R→T
{∑UN(
UNR
𝑓(𝑋(𝑡U)). [𝑋(𝑡UV() − 𝑋(𝑡U)]}
30

31. Luc_Faucheux_2021
Ito vs Taylor, the example of PL - XXVIII
¨ In stochastic calculus, as soon as you see a “.” in front of a 𝑑𝑋 or 𝑑𝑊 term, you are in
trouble.
¨ As always the right notation is 95% of the work, as Godel used to say
31

32. Luc_Faucheux_2021
Ito vs Taylor, the example of PL - XXIX
¨ I think in practice the confusion can also be cleared if you think about:
¨ 𝑓(𝑥, 𝑡) is the value of an option on the stock that has value 𝑥 at my observation time 𝑡
¨ At time 𝑡 + 𝛿𝑡, the stock is now at 𝑥 𝑡 + 𝛿𝑡 = 𝑥(𝑡) + 𝛿𝑥
¨ The value of my option is now:
¨ 𝑓(𝑥 + 𝛿𝑥, 𝑡 + 𝛿𝑡)
¨ I can perform any kind of Taylor expansion on that, it is a regular function, and I can write
things like:
¨ 𝛿𝑓 =
"#
"$
. 𝛿𝑥 +
"#
"&
. 𝛿𝑡 +
(
'
"!#
"$! . 𝛿𝑥' + 𝑠𝑜𝑚𝑒𝑡ℎ𝑖𝑛𝑔
32

33. Luc_Faucheux_2021
Ito vs Taylor, the example of PL - XXX
¨ Which is different from: I sit now at time 𝑡 and I look at a stock worth 𝑥(𝑡)
¨ I want to look at the FUTURE process of this stock 𝑋(𝑡W) with 𝑡W > 𝑡 in the future
¨ I can look at the process for a claim on that stock 𝑓(𝑥 = 𝑋 𝑡W , 𝑡W)
¨ If I choose to write SDE, I will write equations involving terms that will look formally like 𝑑𝑋
and 𝑑𝑓, and there will be something called Ito lemma that will look like:
¨ 𝛿𝑓 =
"#
"$
. 𝛿𝑋 +
"#
"&
. 𝛿𝑡 +
(
'
.
"!#
"$! . 𝛿𝑋'
¨ But is is worth reminding ourselves that that time 𝑡 is in the future, where 𝑋(𝑡) is not set yet
to the observed value of the stock.
¨ So 𝑓 in Ito lemma is really a stochastic process
¨ But 𝑓 in the Taylor expansion of the PL is a regular function
33

34. Luc_Faucheux_2021
Ito vs Taylor, the example of PL - XXXI
¨ Note that adding somewhat to the confusion, they are related since writing the stochastic
process for 𝑓(𝑥 = 𝑋 𝑡W , 𝑡W), and using hedging arguments and risk-neutral arguments,
allowed us to solve for 𝑓(𝑥, 𝑡)
¨ 𝑓(𝑥, 𝑡) is a regular function that is a solution of a PDE
¨ 𝑓 𝑥 = 𝑋 𝑡W , 𝑡W = 𝐹(𝑋 𝑡W , 𝑡W) is a stochastic process, because it is a function of a
stochastic process
¨ Sometimes for call option, I will write:
¨ 𝑐(𝑠, 𝜎, 𝑡) for the value of a call option for observed stock price 𝑠 at time 𝑡
¨ And 𝐶 𝑆, 𝜎, 𝑡 = 𝐶(𝑆 𝑡 , 𝜎, 𝑡) for the stochastic process that is modeled in the future
¨ Sometimes just to make it even more obvious
¨ 𝐶 𝑆 𝑡 , 𝜎, 𝑡 = 𝐶 𝑆 𝑡 𝑆 𝑡! = 𝑠X, 𝜎, 𝑡 > 𝑡! = 𝐶 𝑆 𝑡 , 𝜎, 𝑡|𝔉(𝑡!)
34

35. Luc_Faucheux_2021
What is the “something”
35

36. Luc_Faucheux_2021
What is the something - I
¨ When we do the Taylor expansion of a regular function say 𝑓(𝑥, 𝑡) we write things like:
¨ 𝛿𝑓 =
"#
"$
. 𝛿𝑥 +
"#
"&
. 𝛿𝑡 +
(
'
"!#
"$! . 𝛿𝑥' + 𝑠𝑜𝑚𝑒𝑡ℎ𝑖𝑛𝑔
¨ More explicitly say if we confine ourselves to the 𝑥 variable:
¨ 𝛿𝑓 =
"#
"$
. 𝛿𝑥 +
(
'
"!#
"$! . 𝛿𝑥' + 𝑠𝑜𝑚𝑒𝑡ℎ𝑖𝑛𝑔
¨ 𝛿𝑓 𝑥 = 𝑓 𝑥 + 𝛿𝑥 − 𝑓(𝑥) =
"#
"$
|$. 𝛿𝑥 +
(
'
"!#
"$! |$. 𝛿𝑥' + 𝑠𝑜𝑚𝑒𝑡ℎ𝑖𝑛𝑔
¨ Hopefully you want the 𝑠𝑜𝑚𝑒𝑡ℎ𝑖𝑛𝑔 to be small, or controlled
¨ Usually sometimes that something will be noted as 𝒪() to denoted that this quantity is
something that should scale with the order inside the parenthesis
36

37. Luc_Faucheux_2021
What is the something - II
¨ 𝛿𝑓 𝑥 = 𝑓 𝑥 + 𝛿𝑥 − 𝑓 𝑥 =
"#
"$
|$. 𝛿𝑥 + 𝑠𝑜𝑚𝑒𝑡ℎ𝑖𝑛𝑔
¨ 𝛿𝑓 𝑥 = 𝑓 𝑥 + 𝛿𝑥 − 𝑓(𝑥) =
"#
"$
|$. 𝛿𝑥 + 𝒪(𝛿𝑥')
¨ 𝛿𝑓 𝑥 = 𝑓 𝑥 + 𝛿𝑥 − 𝑓(𝑥) =
"#
"$
|$. 𝛿𝑥 +
(
'
"!#
"$! |$. 𝛿𝑥' + 𝑠𝑜𝑚𝑒𝑡ℎ𝑖𝑛𝑔
¨ 𝛿𝑓 𝑥 = 𝑓 𝑥 + 𝛿𝑥 − 𝑓(𝑥) =
"#
"$
|$. 𝛿𝑥 +
(
'
"!#
"$! |$. 𝛿𝑥' + 𝒪(𝛿𝑥Y)
¨ There is actually an exact formulation for this quantity, useful to know
37

38. Luc_Faucheux_2021
What is the something - III
¨ 𝑓 𝑥 = 𝑓 𝑥! + ∫
UN$"
UN$
𝑓W 𝑘 . 𝑑𝑘
¨ Integrating by parts:
¨ 𝑓 𝑥 = 𝑓 𝑥! − [𝑓W 𝑘 . (𝑘 − 𝑥)]UN$"
UN$
+ ∫
UN$"
UN$
𝑓WW 𝑘 . (𝑘 − 𝑥). 𝑑𝑘
¨ 𝑓 𝑥 = 𝑓 𝑥! + 𝑥 − 𝑥! . 𝑓′(𝑥!) + ∫
UN$"
UN$
𝑓WW 𝑘 . (𝑘 − 𝑥). 𝑑𝑘
¨ And iterating we are starting to see something that looks like the regular Taylor expansion:
¨ ∫
UN$"
UN$
𝑓WW 𝑘 . (𝑘 − 𝑥). 𝑑𝑘 = −[𝑓WW 𝑘 .
(UZ$)!
'
]UN$"
UN$
+ ∫
UN$"
UN$
𝑓WWW 𝑘 .
(UZ$)!
'
. 𝑑𝑘
¨ More generally:
¨ ∫
UN$"
UN$
𝑓([V() 𝑘 .
(UZ$)#
[!
. 𝑑𝑘 = −[𝑓([V() 𝑘 .
(UZ$)#$%
[V( !
]UN$"
UN$
+ ∫
UN$"
UN$
𝑓([V() 𝑘 .
(UZ$)#$%
[V( !
. 𝑑𝑘
38

39. Luc_Faucheux_2021
What is the something - IV
¨ ∫
UN$"
UN$
𝑓([V() 𝑘 .
(UZ$)#
[!
. 𝑑𝑘 = −[𝑓([V() 𝑘 .
(UZ$)#$%
[V( !
]UN$"
UN$
+ ∫
UN$"
UN$
𝑓([V() 𝑘 .
(UZ$)#$%
[V( !
. 𝑑𝑘
¨ ∫
UN$"
UN$
𝑓([V() 𝑘 .
(UZ$)#
[!
. 𝑑𝑘 = 𝑓([V() 𝑥! .
($Z$")#$%
[V( !
+ ∫
UN$"
UN$
𝑓([V() 𝑘 .
(UZ$)#$%
[V( !
. 𝑑𝑘
¨ And so not only do we recover the regular expression, but we also have a formula for the
higher order residual:
¨ 𝑓 𝑥 = ∑]N!
]N^
𝑓(]) 𝑥! .
($Z$")&
]!
+ ∫
UN$"
UN$
𝑓(^V() 𝑘 .
(UZ$)'$%
^V( !
. 𝑑𝑘
¨ 𝑓 𝑥 = ∑]N!
]N^
𝑓(]) 𝑥! .
($Z$")&
]!
+ 𝒪 𝛿𝑥^V(
¨ 𝒪 𝛿𝑥^V( = ∫
UN$"
UN$
𝑓(^V() 𝑘 .
(UZ$)'$%
^V( !
. 𝑑𝑘
¨ Pretty nifty
¨ Can be easily generalized to more than one variable
39

40. Luc_Faucheux_2021
Do not lose the information about the mesh
40

41. Luc_Faucheux_2021
Mesh information - I
¨ As we saw in the previous decks, integrals are defined usually as the limits of a sum
¨ Let’s place ourselves in regular calculus and assume a regular variable 𝑡, and a function 𝑓(𝑡)
¨ We can define a partition of the interval [𝑡P, 𝑡Q], we note it {𝑡U}
¨ Let’s assume that 𝑡RV( = 𝑡Q and 𝑡( = 𝑡P
¨ It is true that no matter what:
¨ 𝑓 𝑡Q − 𝑓 𝑡P = ∑UN(
UNR
𝑓 𝑡UV( − 𝑓(𝑡U)
¨ We can now obviously Taylor expand the quantity {𝑓 𝑡UV( − 𝑓(𝑡U) } around a point inside
the bucket [𝑡U, 𝑡UV(], which we can call mesh, and define by 𝑀([𝑡U, 𝑡UV(])
41

42. Luc_Faucheux_2021
Mesh information - II
¨ Some simple mesh would be:
¨ 𝑀 𝑡U, 𝑡UV( = 𝑡U
¨ 𝑀 𝑡U, 𝑡UV( = 𝑡UV(
¨ 𝑀 𝑡U, 𝑡UV( =
(
'
(𝑡UV(+𝑡U)
¨ Or any other function that you can imagine
¨ So we have in the usual calculus again:
¨ 𝑓 𝑡Q − 𝑓 𝑡P = ∑UN(
UNR
𝑓 𝑡UV( − 𝑓(𝑡U)
¨ 𝑓 𝑡UV( = 𝑓 𝑀 𝑡U, 𝑡UV( + 𝑡UV( − 𝑀 𝑡U, 𝑡UV( .
"#
"&
|_ &(,&($%
+ 𝒪((𝑡UV(−𝑡U)')
¨ 𝑓 𝑡U = 𝑓 𝑀 𝑡U, 𝑡UV( + 𝑡U − 𝑀 𝑡U, 𝑡UV( .
"#
"&
|_ &(,&($%
+ 𝒪((𝑡UV(−𝑡U)')
42

43. Luc_Faucheux_2021
Mesh information - III
¨ 𝑓 𝑡Q − 𝑓 𝑡P = ∑UN(
UNR
𝑓 𝑡UV( − 𝑓(𝑡U)
¨ 𝑓 𝑡UV( = 𝑓 𝑀 𝑡U, 𝑡UV( + 𝑡UV( − 𝑀 𝑡U, 𝑡UV( .
"#
"&
|_ &(,&($%
+ 𝒪((𝑡UV(−𝑡U)')
¨ 𝑓 𝑡U = 𝑓 𝑀 𝑡U, 𝑡UV( + 𝑡U − 𝑀 𝑡U, 𝑡UV( .
"#
"&
|_ &(,&($%
+ 𝒪((𝑡UV(−𝑡U)')
¨ 𝑓 𝑡Q − 𝑓 𝑡P = ∑UN(
UNR
𝑓 𝑡UV( − 𝑓(𝑡U)
¨ 𝑓 𝑡) − 𝑓 𝑡* = ∑+,-
+,. /0
/1
|2 1!,1!"#
. {𝑡+4- − 𝑀 𝑡+, 𝑡+4- − 𝑡+ + 𝑀 𝑡+, 𝑡+4- } + 𝒪((𝑡+4-−𝑡+)5
)
¨ 𝑓 𝑡Q − 𝑓 𝑡P = ∑UN(
UNR "#
"&
|_ &(,&($%
. {𝑡UV( − 𝑡U} + 𝒪((𝑡UV(−𝑡U)')
43

44. Luc_Faucheux_2021
Mesh information - IV
¨ 𝑓 𝑡Q − 𝑓 𝑡P = ∑UN(
UNR "#
"&
|_ &(,&($%
. {𝑡UV( − 𝑡U} + ∑UN(
UNR
𝒪((𝑡UV(−𝑡U)')
¨ We know take the limit : 𝑁 → ∞
¨ lim
.→7
𝑓 𝑡) − 𝑓 𝑡* = 𝑓 𝑡) − 𝑓 𝑡* = lim
.→7
∑+,-
+,. /0
/1
|2 1!,1!"#
. 𝑡+4- − 𝑡+ + lim
.→7
∑+,-
+,.
𝒪((𝑡+4-−𝑡+)5
)
¨ 𝑓 𝑡Q − 𝑓 𝑡P = lim
R→T
∑UN(
UNR "#
"&
|_ &(,&($%
. 𝑡UV( − 𝑡U + lim
R→T
∑UN(
UNR
𝒪((𝑡UV(−𝑡U)')
¨ In that case:
¨ lim
R→T
∑UN(
UNR
𝒪((𝑡UV(−𝑡U)') ~ lim
R→T
(𝑁.
(
R!)~ lim
R→T
(𝑁Z() = 0
¨ And so :
¨ 𝑓 𝑡Q − 𝑓 𝑡P = lim
R→T
∑UN(
UNR "#
"&
|_ &(,&($%
. 𝑡UV( − 𝑡U
44

45. Luc_Faucheux_2021
Mesh information - V
¨ 𝑓 𝑡Q − 𝑓 𝑡P = lim
R→T
∑UN(
UNR "#
"&
|_ &(,&($%
. 𝑡UV( − 𝑡U
¨ We can now formally define the usual Riemann integral by keeping the formalism on the
mesh as :
¨ ∫
&N&8
&N&9 "#
"&
𝑡 . (𝑀). 𝑑𝑡 = lim
R→T
∑UN(
UNR "#
"&
|_ &(,&($%
. 𝑡UV( − 𝑡U
¨ Now because in all cases, this is equal to 𝑓 𝑡Q − 𝑓 𝑡P , keeping the exact information
about the mesh is useless, because no matter what the mesh definition is, we will always get
the same value
¨ We then can get rid of the mesh information and formally write:
¨ 𝑓 𝑡Q − 𝑓 𝑡P = ∫
&N&8
&N&9 "#
"&
𝑡 . 𝑑𝑡
¨ This is the celebrated chain rule (Leibniz rule) of usual calculus
45

46. Luc_Faucheux_2021
Mesh information - VI
¨ 𝑓 𝑡Q − 𝑓 𝑡P = ∫
&N&8
&N&9 "#
"&
𝑡 . 𝑑𝑡
¨ Unfortunately things are not the same in stochastic calculus
¨ Because again stochastic calculus is NOT the same as regular calculus
¨ We got so used in regular calculus about not keeping track of the mesh information that we
get confused once we get to the world of stochastic calculus.
46

47. Luc_Faucheux_2021
Mesh information - VII
¨ Instead of regular calculus and assuming a regular variable 𝑡, and a function 𝑓(𝑡)
¨ We now have a stochastic process 𝑋(𝑡) and we want to say something about 𝑓(𝑋(𝑡))
¨ We can still define a partition of the interval [𝑡P, 𝑡Q], we note it {𝑡U}
¨ Let’s assume that 𝑡RV( = 𝑡Q and 𝑡( = 𝑡P
¨ It is true that no matter what:
¨ 𝑓 𝑋(𝑡Q) − 𝑓 𝑋(𝑡P) = ∑UN(
UNR
𝑓 𝑋(𝑡UV() − 𝑓(𝑋(𝑡U))
¨ So things so far are somewhat similar to regular calculus
¨ We can still do a Taylor expansion of 𝑓 𝑋(𝑡UV() and 𝑓 𝑋(𝑡U) around some point inside the
bucket (nothing prevents us to do that, even if that point is not on the path 𝑋(𝑡), but a
function somewhat of the trajectory 𝑋(𝑡))
47

48. Luc_Faucheux_2021
Mesh information - VIII
¨ Some simple mesh would be:
¨ 𝑀 𝑋(𝑡U), 𝑋(𝑡UV() = 𝑋(𝑡U) ITO MESH
¨ 𝑀 𝑋(𝑡U), 𝑋(𝑡UV() = 𝑋(𝑡UV() KLIMONTOVITCH MESH
¨ 𝑀 𝑋(𝑡U), 𝑋(𝑡UV() =
(
'
(𝑋(𝑡UV() + 𝑋(𝑡U)) STRATANOVITCH MESH
¨ But you can obviously come up with something more complicated if you wanted
48

49. Luc_Faucheux_2021
Mesh information - IX
¨ In regular calculus we had:
¨ 𝑓 𝑡Q − 𝑓 𝑡P = ∑UN(
UNR
𝑓 𝑡UV( − 𝑓(𝑡U)
¨ 𝑓 𝑡UV( = 𝑓 𝑀 𝑡U, 𝑡UV( + 𝑡UV( − 𝑀 𝑡U, 𝑡UV( .
"#
"&
|_ &(,&($%
+ 𝒪((𝑡UV(−𝑡U)')
¨ 𝑓 𝑡U = 𝑓 𝑀 𝑡U, 𝑡UV( + 𝑡U − 𝑀 𝑡U, 𝑡UV( .
"#
"&
|_ &(,&($%
+ 𝒪((𝑡UV(−𝑡U)')
¨ We still have:
¨ 𝑓 𝑋(𝑡Q) − 𝑓 𝑋(𝑡P) = ∑UN(
UNR
𝑓 𝑋(𝑡UV() − 𝑓(𝑋(𝑡U))
49

50. Luc_Faucheux_2021
Mesh information - X
¨ HOWEVER, the issue is that we will encounter the problem that:
¨ In regular calculus:
¨ lim
R→T
∑UN(
UNR
𝒪((𝑡UV(−𝑡U)') ~ lim
R→T
(𝑁.
(
R!)~ lim
R→T
(𝑁Z() = 0
¨ But in stochastic calculus:
¨ lim
R→T
∑UN(
UNR
𝒪((𝑋(𝑡UV() − 𝑋(𝑡U))') ~ lim
R→T
𝑁.
(
R% ~ lim
R→T
𝑁Z! ~ lim
R→T
1 <> 0
¨ So we need to go up one more level in the Taylor expansion
¨ Note that I am not saying that Ito lemma is just a Taylor expansion where you go up one
more level. I am saying that in order to define an integral I need to do a Taylor expansion to
at least one more level in order to get some convergence of the sum to something defined
50

51. Luc_Faucheux_2021
Mesh information - XI
¨ We did that in the section on [𝛼] calculus, and in the deck I on stochastic integrals
¨ This is also why some people get confused between ITO lemma and a Taylor expansion,
because in order to define a stochastic integral you do a Taylor expansion indeed, and you
do indeed go up one more level
¨ HOWEVER, for example, STRATANOVITCH lemma looks formally like the usual chain rule, and
you do not hear people saying “STRATANOVITCH lemma is like a Taylor expansion, but you
just stay at the first order”
¨ In stochastic calculus you need to keep some information about the exact mesh that you
used, because the exact mesh being used does change the end result
51

52. Luc_Faucheux_2021
Mesh information - XII
¨ To make things easier we choose a simple definition of the mesh:
¨ 𝑀` 𝑋(𝑡U), 𝑋(𝑡UV() = 𝑋(𝑡U) + 𝛼. 𝑋(𝑡UV() − 𝑋(𝑡U)
¨ That way we just carry the term (𝛼) as encoding the information about the mesh
¨ Without redoing the derivation of the previous decks, we then define the mesh specific
integral as:
¨ ∫
&N&P
&N&Q "#
"$
. ([𝛼]). 𝑑𝑋(𝑡) = lim
R→T
∑UN(
UNR "#
"$
|_ O(&(),O(&($%) . 𝑋(𝑡UV() − 𝑋(𝑡U)
¨ This is just if you want to think about it, a way to simplify writing things like:
lim
R→T
∑UN(
UNR "#
"$
|_ O(&(),O(&($%) . 𝑋(𝑡UV() − 𝑋(𝑡U)
¨ And simplifying it by defining a new notation using ∫(𝑠𝑜𝑚𝑒𝑡ℎ𝑖𝑛𝑔), but that does not mean
that the interpretation of that symbol is the usual one that are used to in regular calculus
52

53. Luc_Faucheux_2021
Mesh information - XIII
¨ ∫
&N&P
&N&Q "#
"$
. ([𝛼]). 𝑑𝑋(𝑡) = lim
R→T
∑UN(
UNR "#
"$
|_ O(&(),O(&($%) . 𝑋(𝑡UV() − 𝑋(𝑡U)
¨ With some simplifying mesh:
¨ 𝑀` 𝑋(𝑡U), 𝑋(𝑡UV() = 𝑋(𝑡U) + 𝛼. 𝑋(𝑡UV() − 𝑋(𝑡U)
¨ We showed that those sums have actually different limits
¨ 𝑓 𝑋 𝑡Q − 𝑓 𝑋 𝑡P = ∫
&N&P
&N&Q "#
"$
. ([𝛼]). 𝑑𝑋(𝑡) +
(
'
− 𝛼 . ∫
&N&P
&N&Q "!#
"$! 𝑋 𝑡 . 𝑏 𝑡, 𝑋 𝑡
'
. 𝑑𝑡
¨ For a process:
¨ 𝑑𝑋 𝑡 = 𝑎 𝑡, 𝑋 𝑡 . 𝑑𝑡 + 𝑏 𝑡, 𝑋 𝑡 . (𝛼). 𝑑𝑊(𝑡)
¨ Note that only the drift will change going from different (𝛼), so there is no ambiguity as to
what 𝑏 𝑡, 𝑋 𝑡 we should pick
53

54. Luc_Faucheux_2021
Mesh information – XIII - a
¨ Just to illustrate it another way:
¨ What is the limit of ∑UN(
UNR "#
"$
|_ O(&(),O(&($%) . 𝑋(𝑡UV() − 𝑋(𝑡U) when 𝑁 → ∞ ?
¨ lim
.→0
∑1%-
1%. &'
&(
|2 *($"),*($"#$) . 𝑋(𝑡13-) − 𝑋(𝑡1) = 𝑓 𝑋 𝑡" − 𝑓 𝑋 𝑡# −
-
!
− 𝛼 . ∫
$%$#
$%$" &!'
&(! 𝑋 𝑡 . 𝑏 𝑡, 𝑋 𝑡
!
. 𝑑𝑡
¨ If we were to find a more convenient way to write this expression, because the end result
depends on the value of 𝛼 , we need to keep somewhere that notation.
¨ Remember for the simple mesh that we chose to be of the form:
¨ 𝑀` 𝑋(𝑡U), 𝑋(𝑡UV() = 𝑋(𝑡U) + 𝛼. 𝑋(𝑡UV() − 𝑋(𝑡U)
¨ We would not get a similar result for a different function that we would choose for the mesh
¨ So we cannot drop the 𝛼 and write :
¨ lim
R→T
∑UN(
UNR "#
"$
|_ O(&(),O(&($%) . 𝑋(𝑡UV() − 𝑋(𝑡U) = ∫
&N&P
&N&Q "#
"$
. ([𝛼]). 𝑑𝑋(𝑡)
54

55. Luc_Faucheux_2021
Mesh information – XIII - b
¨ So we cannot drop the 𝛼 and write :
¨ lim
R→T
∑UN(
UNR "#
"$
|_ O(&(),O(&($%) . 𝑋(𝑡UV() − 𝑋(𝑡U) = ∫
&N&P
&N&Q "#
"$
. 𝑑𝑋(𝑡)
¨ We need to keep that information somewhere and write:
¨ lim
R→T
∑UN(
UNR "#
"$
|_ O(&(),O(&($%) . 𝑋(𝑡UV() − 𝑋(𝑡U) = ∫
&N&P
&N&Q "#
"$
. ([𝛼]). 𝑑𝑋(𝑡)
¨ ∫
&N&P
&N&Q "#
"$
. ([𝛼]). 𝑑𝑋(𝑡) = 𝑓 𝑋 𝑡Q − 𝑓 𝑋 𝑡P −
(
'
− 𝛼 . ∫
&N&P
&N&Q "!#
"$! 𝑋 𝑡 . 𝑏 𝑡, 𝑋 𝑡
'
. 𝑑𝑡
¨ What a mesh !
55

56. Luc_Faucheux_2021
Mesh information - XIV
¨ 𝑓 𝑋 𝑡Q − 𝑓 𝑋 𝑡P = ∫
&N&P
&N&Q "#
"$
. ([𝛼]). 𝑑𝑋(𝑡) +
(
'
− 𝛼 . ∫
&N&P
&N&Q "!#
"$! 𝑋 𝑡 . 𝑏 𝑡, 𝑋 𝑡
'
. 𝑑𝑡
¨ ∫
&N&P
&N&Q "#
"$
. ([𝛼]). 𝑑𝑋(𝑡) = 𝑓 𝑋 𝑡Q − 𝑓 𝑋 𝑡P −
(
'
− 𝛼 . ∫
&N&P
&N&Q "!#
"$! 𝑋 𝑡 . 𝑏 𝑡, 𝑋 𝑡
'
. 𝑑𝑡
¨ ∫
&N&P
&N&Q "#
"$
. ([𝛼]). 𝑑𝑋(𝑡) = lim
R→T
∑UN(
UNR "#
"$
|_ O(&(),O(&($%) . 𝑋(𝑡UV() − 𝑋(𝑡U)
¨ Because 𝑓 𝑋 𝑡Q − 𝑓 𝑋 𝑡P is a constant:
¨ ∫
&N&P
&N&Q "#
"$
. ([0]). 𝑑𝑋(𝑡) = 𝑓 𝑋 𝑡Q − 𝑓 𝑋 𝑡P −
(
'
. ∫
&N&P
&N&Q "!#
"$! 𝑋 𝑡 . 𝑏 𝑡, 𝑋 𝑡
'
. 𝑑𝑡
¨ ∫
&N&P
&N&Q "#
"$
. ([
(
'
]). 𝑑𝑋(𝑡) = 𝑓 𝑋 𝑡Q − 𝑓 𝑋 𝑡P
¨ We recover obviously:
¨ ∫
&N&P
&N&Q "#
"$
. ([
(
'
]). 𝑑𝑋(𝑡) = ∫
&N&P
&N&Q "#
"$
. ([0]). 𝑑𝑋(𝑡) −
(
'
. ∫
&N&P
&N&Q "!#
"$! 𝑋 𝑡 . 𝑏 𝑡, 𝑋 𝑡
'
. 𝑑𝑡
56

57. Luc_Faucheux_2021
Mesh information - XV
¨ 𝑓 𝑋 𝑡Q − 𝑓 𝑋 𝑡P = ∫
&N&P
&N&Q "#
"$
. ([𝛼]). 𝑑𝑋(𝑡) +
(
'
− 𝛼 . ∫
&N&P
&N&Q "!#
"$! 𝑋 𝑡 . 𝑏 𝑡, 𝑋 𝑡
'
. 𝑑𝑡
¨ That is an EXACT formula, it is NOT a TAYLOR expansion up to some order
¨ There is no “+something” on either side
¨ If we use the ITO mesh we will get the ITO lemma:
¨ 𝑓 𝑋 𝑡Q − 𝑓 𝑋 𝑡P = ∫
&N&P
&N&Q "#
"$
. ([0]). 𝑑𝑋(𝑡) +
(
'
. ∫
&N&P
&N&Q "!#
"$! 𝑋 𝑡 . 𝑏 𝑡, 𝑋 𝑡
'
. 𝑑𝑡
¨ 𝑓 𝑋 𝑡Q − 𝑓 𝑋 𝑡P = ∫
&N&P
&N&Q "#
"$
. ([). 𝑑𝑋(𝑡) +
(
'
. ∫
&N&P
&N&Q "!#
"$! 𝑋 𝑡 . 𝑏 𝑡, 𝑋 𝑡
'
. 𝑑𝑡
¨ Which you find sometimes written in loosy goosy notation as :
¨ 𝛿𝑓 =
"#
"$
. 𝛿𝑋 +
(
'
.
"!#
"$! . 𝛿𝑋'
57

58. Luc_Faucheux_2021
Mesh information - XVI
¨ That is another weird property of the stochastic calculus: where you take the derivative
inside that bucket that gets infinitesimally small actually does matter, on the left, on the
right or in the middle, or as sometimes you see in textbooks, before the jump, after the
jump or in the middle of the jump
¨ In essence you cannot define an integral without keeping some information about what
mesh convention you took
¨ There are an infinite number of mesh you could choose
¨ The two most often used are:
¨ ITO: 𝛼 = 0
¨ 𝑀` 𝑋(𝑡U), 𝑋(𝑡UV() = 𝑋(𝑡U) + 𝛼. 𝑋(𝑡UV() − 𝑋(𝑡U) = 𝑋(𝑡U)
¨ STRATANOVITCH: 𝛼 = 1/2
¨ 𝑀` 𝑋(𝑡U), 𝑋(𝑡UV() = 𝑋(𝑡U) + 𝛼. 𝑋(𝑡UV() − 𝑋(𝑡U) =
(
'
𝑋(𝑡U + 𝑋(𝑡UV()}
58

59. Luc_Faucheux_2021
Mesh information - XVII
¨ So in a sense in regular calculus we dropped the mesh information because all sums were
converging to the same value
¨ 𝑓 𝑡Q − 𝑓 𝑡P = lim
R→T
∑UN(
UNR "#
"&
|_ &(,&($%
. 𝑡UV( − 𝑡U
¨ ∫
&N&8
&N&9 "#
"&
𝑡 . (𝑀). 𝑑𝑡 = lim
R→T
∑UN(
UNR "#
"&
|_ &(,&($%
. 𝑡UV( − 𝑡U
¨ And we can drop the mesh from the above, does not matter where you compute the
quantity
"#
"&
|_ &(,&($%
, on the left, on the right, in the middle, wherever you want inside
the bucket, as 𝑁 → ∞, they all converge to the same value so you can drop it
¨ ∫
&N&8
&N&9 "#
"&
𝑡 . 𝑑𝑡 = lim
R→T
∑UN(
UNR "#
"&
|_ &(,&($%
. 𝑡UV( − 𝑡U , whatever 𝑀 𝑡U, 𝑡UV(
¨ Not so in stochastic calculus, you have to drag with you the exact definition of the mesh
59

60. Luc_Faucheux_2021
PL slice, PL scallop, PL explain
All the Taylor expansion tricks around PL
60

61. Luc_Faucheux_2021
PL explain - I
¨ Back to PL and the usual ways to compute it that you might encounter
¨ Again, we are only looking at the “market PL”, no CVA, no DVA, no FVA, no XVA, no reserve,
no MVA….you know…the simple stuff
¨ You can look into the deck “SKEW” for a nice application of the Taylor expansion for the
Vanna/Volga and the corresponding skew (correlation vol/stock) and smile (vol of vol)
¨ But for our purpose for illustration we will keep at constant vol (so no Vega PL, but you can
easily add those).
¨ This is more to illustrate the points on how to compute Market PL and the fact that it is a
Taylor expansion in the good old usual calculus
61

62. Luc_Faucheux_2021
PL explain - II
¨ PL explain is essentially the “proof in the pudding”
¨ If you cannot explain your PL without an acceptable degree of accuracy, there is something
wrong, and you have to go dig into what is wrong
¨ A lot of things could be wrong
¨ A deal was misbooked
¨ An option exercise was missed
¨ A model calibration went nuts
¨ Some market data are corrupted
¨ Maybe your PL batch did not run or aggregate properly and actually nothing is wrong
¨ So a PL explain with the ability to drill down at a deal level is ESSENTIAL for any trading desk
62

63. Luc_Faucheux_2021
PL explain - III
¨ PL slice or PL scallop is NOT PL explain
¨ PL slice is just slicing the PL into different components.
¨ Let’s go back to the example of the one option 𝑐(𝑥, 𝜎, 𝑡)
¨ Portfolio value at time 𝑡: 𝑐(𝑥(𝑡), 𝜎(𝑡), 𝑡)
¨ Portfolio value at time 𝑡 + 𝛿𝑡: 𝑐(𝑥(𝑡 + 𝛿𝑡), 𝜎(𝑡 + 𝛿𝑡), 𝑡 + 𝛿𝑡)
¨ Market PL: 𝑃𝐿 𝑡 + 𝛿𝑡, 𝑡 = 𝑐(𝑥(𝑡 + 𝛿𝑡), 𝜎(𝑡 + 𝛿𝑡), 𝑡 + 𝛿𝑡) -𝑐(𝑥(𝑡), 𝜎(𝑡), 𝑡)
¨ You have all the rights in the world to write:
¨ 𝑃𝐿 𝑡 + 𝛿𝑡, 𝑡 = 𝑐(𝑥(𝑡 + 𝛿𝑡), 𝜎(𝑡 + 𝛿𝑡), 𝑡 + 𝛿𝑡) −𝑐(𝑥(𝑡), 𝜎(𝑡), 𝑡)
¨ 𝑃𝐿 𝑡 + 𝛿𝑡, 𝑡 = 𝑐(𝑥(𝑡 + 𝛿𝑡), 𝜎(𝑡 + 𝛿𝑡), 𝑡 + 𝛿𝑡) −𝑐(𝑥(𝑡), 𝜎(𝑡), 𝑡)
¨ And break it down (slice/scallop) into the following:
63

64. Luc_Faucheux_2021
PL explain - IV
¨ 𝑃𝐿 𝑡 + 𝛿𝑡, 𝑡 = 𝑐 𝑥 𝑡 + 𝛿𝑡 , 𝜎 𝑡 + 𝛿𝑡 , 𝑡 + 𝛿𝑡 − 𝑐(𝑥(𝑡), 𝜎(𝑡), 𝑡)
¨ 𝑃𝐿 𝑡 + 𝛿𝑡, 𝑡 = 𝑐(𝑥(𝑡 + 𝛿𝑡), 𝜎(𝑡 + 𝛿𝑡), 𝑡 + 𝛿𝑡) − 𝑐(𝑥(𝑡), 𝜎(𝑡 + 𝛿𝑡), 𝑡 + 𝛿𝑡)
¨ + 𝑐(𝑥(𝑡), 𝜎(𝑡 + 𝛿𝑡), 𝑡 + 𝛿𝑡) − 𝑐(𝑥(𝑡), 𝜎(𝑡), 𝑡 + 𝛿𝑡)
¨ +𝑐(𝑥(𝑡), 𝜎(𝑡), 𝑡 + 𝛿𝑡) − 𝑐(𝑥(𝑡), 𝜎(𝑡), 𝑡)
¨ You are also totally free to call each line whatever you want like for example:
¨ 𝑃𝐿 𝑡 + 𝛿𝑡, 𝑡 = 𝐷𝐸𝐿𝑇𝐴fghij + 𝑉𝐸𝐺𝐴fghij + 𝑇𝐻𝐸𝑇𝐴fghij
¨ 𝐷𝐸𝐿𝑇𝐴fghij = 𝑐(𝑥(𝑡 + 𝛿𝑡), 𝜎(𝑡 + 𝛿𝑡), 𝑡 + 𝛿𝑡) − 𝑐(𝑥(𝑡), 𝜎(𝑡 + 𝛿𝑡), 𝑡 + 𝛿𝑡)
¨ 𝑉𝐸𝐺𝐴fghij = 𝑐(𝑥(𝑡), 𝜎(𝑡 + 𝛿𝑡), 𝑡 + 𝛿𝑡) − 𝑐(𝑥(𝑡), 𝜎(𝑡), 𝑡 + 𝛿𝑡)
¨ 𝑇𝐻𝐸𝑇𝐴fghij = 𝑐(𝑥(𝑡), 𝜎(𝑡), 𝑡 + 𝛿𝑡) − 𝑐(𝑥(𝑡), 𝜎(𝑡), 𝑡)
64

65. Luc_Faucheux_2021
PL explain - V
¨ Note that the order matters.
¨ Usually the 𝑇𝐻𝐸𝑇𝐴fghij is the first one (happens overnight with the assumptions that
markets do not move overnight)
¨ 𝑇𝐻𝐸𝑇𝐴fghij = 𝑐(𝑥(𝑡), 𝜎(𝑡), 𝑡 + 𝛿𝑡) − 𝑐(𝑥(𝑡), 𝜎(𝑡), 𝑡)
¨ Here the 𝑉𝐸𝐺𝐴fghij happens “before” the 𝐷𝐸𝐿𝑇𝐴fghij but nothing prevents you to do it in
another order.
¨ This does not EXPLAIN the PL, it only slices it into components that hopefully should be
related to the actual names we gave
¨ We would expect for example that:
¨ 𝑇𝐻𝐸𝑇𝐴fghij = 𝑐 𝑥 𝑡 , 𝜎 𝑡 , 𝑡 + 𝛿𝑡 − 𝑐 𝑥 𝑡 , 𝜎 𝑡 , 𝑡
¨ 𝑇𝐻𝐸𝑇𝐴fghij = 𝑐 𝑥 𝑡 , 𝜎 𝑡 , 𝑡 +𝛿𝑡.
"i($ & ,% & ,&)
"&
+something − 𝑐 𝑥 𝑡 , 𝜎 𝑡 , 𝑡
¨ 𝑇𝐻𝐸𝑇𝐴fghij = 𝛿𝑡.
"i($ & ,% & ,&)
"&
+ something
65

66. Luc_Faucheux_2021
PL explain - VI
¨ 𝑇𝐻𝐸𝑇𝐴fghij = 𝛿𝑡.
"i($ & ,% & ,&)
"&
+ something
¨ So in the way we defined it, the 𝑇𝐻𝐸𝑇𝐴fghij, up to higher orders in the Taylor expansion, is
indeed related to the starting point first derivative with respect to time (THETA) times the
amount of time elapsed
¨ So this seems to make some sense
¨ Note however that this is not accurate anymore after the first one
¨ The way we defined it for example, the second one was the 𝑉𝐸𝐺𝐴fghij
¨ 𝑉𝐸𝐺𝐴fghij = 𝑐(𝑥(𝑡), 𝜎(𝑡 + 𝛿𝑡), 𝑡 + 𝛿𝑡) − 𝑐(𝑥(𝑡), 𝜎(𝑡), 𝑡 + 𝛿𝑡)
¨ If we Taylor expand, we get:
¨ 𝑉𝐸𝐺𝐴fghij = 𝑐(𝑥(𝑡), 𝜎(𝑡), 𝑡 + 𝛿𝑡) +𝛿𝜎.
"i($(&),%(&),&Vk&)
"%
+ something − 𝑐(𝑥(𝑡), 𝜎(𝑡), 𝑡 + 𝛿𝑡)
¨ 𝑉𝐸𝐺𝐴fghij = 𝛿𝜎.
"i($(&),%(&),&Vk&)
"%
+ something
66

67. Luc_Faucheux_2021
PL explain - VII
¨ 𝑉𝐸𝐺𝐴fghij = 𝛿𝜎.
"i($(&),%(&),&Vk&)
"%
+ something
¨ So up to higher orders in the Taylor expansion (and again you need to check that those
higher orders can kept under control, which is not always the case), the 𝑉𝐸𝐺𝐴fghij is
proportional to the change in volatility times something that looks like the Vega
¨ It is NOT the starting Vega, however, and is polluted by VEGA_BLEED
¨ You can again Taylor expand
"i($(&),%(&),&Vk&)
"%
as:
¨
"i($(&),%(&),&Vk&)
"%
=
"i($(&),%(&),&)
"%
+ 𝛿𝑡.
"
"&
"i($(&),%(&),&)
"%
+ 𝑠𝑜𝑚𝑒𝑡ℎ𝑖𝑛𝑔
¨
"i($(&),%(&),&Vk&)
"%
= 𝑉𝐸𝐺𝐴 + 𝛿𝑡. 𝑉𝐸𝐺𝐴_𝐵𝐿𝐸𝐸𝐷 + 𝑠𝑜𝑚𝑒𝑡ℎ𝑖𝑛𝑔
¨ 𝑉𝐸𝐺𝐴fghij = 𝛿𝜎.𝑉𝐸𝐺𝐴 + 𝛿𝑡. 𝛿𝜎. 𝑉𝐸𝐺𝐴_𝐵𝐿𝐸𝐸𝐷 + something
67

68. Luc_Faucheux_2021
PL explain - VIII
¨ It gets even more complicated as you go to the next slice as this one gets polluted by
DELTA_BLEED, VANNA, VEGA_BLEED,…
¨ Will do it if I have time
68

69. Luc_Faucheux_2021
PL explain - IX
¨ A PL slice or scallop is really just a sequential somewhat arbitrary dissection of the actual PL
¨ It is useful but explains little
69
𝑐(𝑥(𝑡), 𝜎(𝑡), 𝑡)
𝑐(𝑥(𝑡), 𝜎(𝑡), 𝑡 + 𝛿𝑡)
𝑐(𝑥(𝑡), 𝜎(𝑡 + 𝛿𝑡), 𝑡 + 𝛿𝑡)
𝑐 𝑥 𝑡 + 𝛿𝑡 , 𝜎 𝑡 + 𝛿𝑡 , 𝑡 + 𝛿𝑡
𝑃𝐿 𝑎𝑥𝑖𝑠

70. Luc_Faucheux_2021
PL explain - X
¨ A PL slice or scallop is really just a sequential somewhat arbitrary dissection of the actual PL
¨ It is useful but explains little
¨ By definition (unless you messed up), there will be ZERO residual to a PL slice
¨ I was at a shop where 2 systems (one from London, one from NY) were competing for global
adoption. Quite frankly the NY system was far superior, and had a real PL explain.
¨ The LN system only had a PL slice, but somehow those beefeaters went up and down the
management chain saying “we have 0 PL unexplained using the LN system, but the NY guys
have still unexplained PL”
¨ Because management was too stupid or ignorant, or were playing political games to favour
London, that was a key argument to choose the London system globally
¨ But hey the 2nd law of thermodynamics has to be respected, meaning that order and
efficiency will always devolve into sh.., so maybe that was the reason at the time why
management chose the London system
¨ Fairly quickly no one could explain their PL anymore, they could just partition the number
into arbitrary buckets, and that was kind of it….
70

71. Luc_Faucheux_2021
A true PL explain
71

72. Luc_Faucheux_2021
A true PL explain - I
¨ The true PL explain is not a slice, it is a Taylor expansion.
¨ The residual term is the unexplained
¨ Checking that unexplained is the daily job (or should be the daily job) or any respectable
trader, in order to ensure that the book is properly risk managed.
¨ If the residual is too large, then the PL explain has to go one more order in the Taylor
expansion in order to explain the PL move
¨ Again, this can be challenging for short dated options close to the strike, or in general books
with diverging higher order of the Greeks.
¨ For Gamma books (short-dated), usually the usefulness of the Taylor expansion and the PL
explain gets lost, and it is better in those cases to revert to a full reval.
¨ That is one of the few examples where the PL slice actually is the better pragmatic choice
compared to the PL explain and Taylor expansion
72

73. Luc_Faucheux_2021
A true PL explain - II
¨ 𝛿𝑓 =
"#
"$
. 𝛿𝑥 +
"#
"&
. 𝛿𝑡 + 𝑠𝑜𝑚𝑒𝑡ℎ𝑖𝑛𝑔
¨ 𝛿𝑓 = 𝐷𝐸𝐿𝑇𝐴. 𝛿𝑥 + 𝑇𝐻𝐸𝑇𝐴. 𝛿𝑡 + 𝑠𝑜𝑚𝑒𝑡ℎ𝑖𝑛𝑔
¨ 𝛿𝑓 = 𝑓 𝑥 + 𝛿𝑥, 𝑡 + 𝛿𝑡 − 𝑓(𝑥, 𝑡)
¨ A picture is worth a thousand words
73

74. Luc_Faucheux_2021
A true PL explain - III
74
𝑥 𝑥 + 𝛿𝑥
𝑡
𝑡 + 𝛿𝑡
𝑓(𝑥, 𝑡)
𝑓 𝑥 + 𝛿𝑥, 𝑡 + 𝛿𝑡
𝑓(𝑥, 𝑡) +
𝜕𝑓
𝜕𝑥
. 𝛿𝑥
𝑓(𝑥, 𝑡) +
𝜕𝑓
𝜕𝑡
. 𝛿𝑡 𝑓 𝑥, 𝑡 +
𝜕𝑓
𝜕𝑡
. 𝛿𝑡 +
𝜕𝑓
𝜕𝑥
. 𝛿𝑥
𝑟𝑒𝑠𝑖𝑑𝑢𝑎𝑙

75. Luc_Faucheux_2021
A true PL explain - IV
¨ If the residual is too big, then start expanding to go to higher orders
¨ So go from:
¨ 𝛿𝑓 =
"#
"$
. 𝛿𝑥 +
"#
"&
. 𝛿𝑡 + 𝑠𝑜𝑚𝑒𝑡ℎ𝑖𝑛𝑔
¨ To:
¨ 𝛿𝑓 =
"#
"$
. 𝛿𝑥 +
(
'
"!#
"$! . 𝛿𝑥' +
"#
"&
. 𝛿𝑡 + 𝑠𝑜𝑚𝑒𝑡ℎ𝑖𝑛𝑔
¨ Note that the “something” is of course different between the two equations
75

76. Luc_Faucheux_2021
A true PL explain – V
¨ EX-ANTE: means that you compute the derivatives at the starting point
¨ EX-POST: means that you compute the derivatives at the end point
¨ You can play all the tricks of Taylor expansion that you want, combining in some cases ex-
post and ex-ante, the goal is to get the unexplained PL under something “reasonable” so
that you can sleep easy at night
¨ Does not have to be on actual market moves, that could be run on extreme scenarios, again
to ensure that you are controlling the trading book as you should
¨ Also it is IMPERATIVE that you should be able to drill down this PL explain at the level of
individual trades. That often does not happen in banks as sometimes this is a large batch
that runs overnight and the database cannot handle all the computations at a trade level,
and save only the aggregate numbers. That is a shame, and renders the PL explain quite
useless
76

77. Luc_Faucheux_2021
Leibniz and the divine machine
77

78. Luc_Faucheux_2021
Leibniz and the divine machine - I
¨ Leibniz was an absolute genius
¨ He invented regular differential calculus, even though Newton tried to torpedo him.
¨ It pays to read and re-read Leibniz, especially now in the age of a possible emergence of AI.
78

79. Luc_Faucheux_2021
Leibniz and the divine machine - II
¨ Butchering what he said:
¨ Man-made machine (computers, automats,.._) are always discrete in nature, there is always
a finite scale
¨ God-made machine (or what he calls divine machine), have no limit to how small you go. As
you keep breaking them down and looking at them in further details, you never encounter a
finite block, it keeps going
¨ That was somewhat related to his discovery of differential calculus (also at the time called
”infinitesimal calculus”, people had issue with small things going to 0 adding up to
something not 0)
¨ At the time, there was some real controversy as to what was the limit of a slope (first
derivative), and if that was even defined.
¨ So you can say that in a way ITO (because it is better suited for discrete processes) is human,
and STRATANOVITCH (better suited for continuous processes) is divine
79

80. Luc_Faucheux_2021
Leibniz and the divine machine - III
¨ A great read on Leibniz and his philosophy of biological machines (divine machines,
continuous a la STARATANOVITCH) and man-made machines (discrete a la ITO)
80

81. Luc_Faucheux_2021
Leibniz and the divine machine - IV
¨ Blockbuster: “To play is human, to rewind is divine”
¨ Leibniz: “Man-made machine are discrete, divine
machine are continuous”
¨ Luc: “ITO is human, STRATANOVITCH is divine”
81

82. Luc_Faucheux_2021
Summary
¨ So hopefully this section addressed some of the feedback I received.
¨ Again, running the risk of repeating myself:
¨ ITO lemma is NOT a Taylor expansion
¨ Stochastic calculus is NOT like regular calculus
¨ In particular, you need to keep track of the mesh that you used to compute the sum that you
are using to define the integral as a limit of this sum
¨ For a given mesh you will get a given stochastic calculus
¨ Take the point on the left you get ITO
¨ Take the point in the middle you get STRATANOVITCH
¨ Take the point on the right you get KLIMONTOVITCH
82

83. Luc_Faucheux_2021
So at least for now…..
83