Risk Managing a Fixed Income business

1. Luc_Faucheux_2021
RISK MANAGEMENT 101
Risk Managing a Fixed-Income business
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2. Luc_Faucheux_2021
Why that deck?
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3. Luc_Faucheux_2021
Why that deck - I
¨ Those decks started when I taught a two-weeks class in Fixed Income at DRW in Chicago in
2018
¨ I noticed that I had plenty of notes that I had accumulated over my career, starting in 1995,
so roughly 25 years or so.
¨ Those notes were mostly handwritten
¨ I also noticed that I was at times redoing the same derivation or calculations because it was
faster than looking for the notepad where I remembered having done the calculations
¨ So from a selfish perspective I started putting all those notes in easily accessible (for me)
Powerpoint format, backed up on the cloud, and slideshare
¨ There are now like 20 or so of those decks available for your perusing
¨ https://www.slideshare.net/lucfaucheux
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4. Luc_Faucheux_2021
Why that deck - II
¨ There is also not any great textbooks out there on Risk Managing a Fixed-Income franchise.
¨ There are usually bits and pieces in many different textbooks.
¨ The 2 best ones in my opinion are Piterbarg volume III and Darbyshire
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Why that deck - III
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Why that deck - IV
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7. Luc_Faucheux_2021
Why that deck - V
¨ Over the course of my career, I estimated that I generated, implemented, designed,
reviewed or merely looked at more than 10,000 risk reports.
¨ So I have plenty of stories or “tricks” that are not easily put in a textbook format
¨ I will try to tell some of those stories, or tricks in this deck
¨ Hopefully you can learn from my mistakes
¨ Also when I was teaching the class at DRW, telling those little stories was a great way to
capture the student’s attention and generate questions and ideas and discussion, as
opposed to a more formal exposition
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8. Luc_Faucheux_2021
Why that deck - VI
¨ Gladwell said that you need 10,000 hours to become an expert in your field.
¨ I have studied at least more than 10,000 risk report
¨ I have certainly spent that much time on it (including writing those decks)
¨ So hopefully I know what I am talking about
¨ I did the same math and unfortunately I do not think that I spent more than 10,000 hours on
a tennis court
¨ That is maybe why I have not won Wimbledon yet
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9. Luc_Faucheux_2021
Why that deck - V
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10. Luc_Faucheux_2021
Why that deck - VI
¨ Also, since having left in March from my previous job, I have started interviewing at hedge
funds and other financial institutions
¨ A natural fit for me would be on the Risk Management side, at a shop that is mostly equity
based and is expanding into the Macro and Fixed Income side
¨ During those interviews, I noticed that I was essentially explaining this deck (how to risk
manage a fixed income business), and so this is my pet peeve: when I do the same thing a
couple of times, I try to automate it and make it more formal and automated
¨ So this deck is essentially also an interview pitch if you want to apply for a job in Risk
Management
¨ If you know such a job, email me
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11. Luc_Faucheux_2021
Why that deck - VII
¨ Caveat: Risk management is NOT Risk control
¨ To me, trading is essentially risk managing. A lot of departments in banks are called Risk
Management, but they really do not manage anything, they just produce risk reports, so
they should really be called “Risk Control”
¨ But hey, everyone wants to have “manager” in their title
¨ If that makes them happy, why not.
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12. Luc_Faucheux_2021
Why that deck - VIII
¨ So I hope that you will enjoy it.
¨ I will not be able to put anything and everything (hey I cannot tell you everything otherwise
people will not hire me)
¨ I will try to be logical and complete
¨ I will also try to get stories around the campfire to illustrate and entertain
¨ If needed I will break this deck in pieces, or just refer to another section
¨ I will also refer a lot of the materials that we covered in the previous decks
¨ Most likely after this one I will have another deck on Risk Management of a Macro business,
since Fixed-Income and funding is central to all the other assets. You can always add
equities PM to a Fixed-Income fund, but the other way around is quite difficult, much harder
to add a Fixed-Income PM to an equity fund. We will explain why, and how to do it properly.
¨ I will also try to be a little provocative in order to motivate reactions and thoughts
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13. Luc_Faucheux_2021
Why that deck - IX
¨ Due to the large size and number of slides, I will have to make this deck into multiple parts,
but wanted the whole logical arc in the first deck, so there will be gaps that I will fill in the
next decks.
¨ Hope that you enjoy, any comments, agreement, disagreement please do not hesitate to
reach out !
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14. Luc_Faucheux_2021
The structure of this deck
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15. Luc_Faucheux_2021
The structure of the deck - I
¨ We start with the yield curve
¨ How to build a yield curve
¨ How to bump a yield curve
¨ Short story if you do not want to go through a bunch of slide:
¨ NEVER WORK IN SPOT SPACE, ALWAYS WORK IN FORWARD SPACE
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16. Luc_Faucheux_2021
The structure of the deck - II
¨ We then go over models
¨ How to calibrate a model
¨ How to bump a model
¨ Again, the lesson there is:
¨ ALWAYS BETTER TO WORK WITH A SIMPLER MODEL.
¨ You can always modify a simpler model to accurately capture assumptions not included in
your model
¨ We will show and explain why
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17. Luc_Faucheux_2021
The structure of the deck - III
¨Ladders and scenarios
¨ We then go over how to define scenarios and ladders that make sense, bumping both curves
and market inputs, as well as model parameters
¨ We use market inputs and model parameters to value trades
¨ A lot of the useful scenarios include bumping the trades (i.e. modifying the trade details in
order to obtain a sensitivity to it, could be counterparty rating, call date, strike,…)
¨ So in essence you have 3 things
¨ Market inputs
¨ Model and model parameters
¨ Trade details
¨ MEASURING RISK IS JUST BUMPING MARKET INPUTS, MODELS AND TRADES, AND
COMBINATIONS OF THOSE
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18. Luc_Faucheux_2021
The structure of the deck - IV
¨ You want to bump all those things in order to measure your risk
¨ Some of the questions will be
¨ Do you bump market inputs and fully recalibrate your model?
¨ Or do you just bump the model parameters?
¨ Wat size for the bump is appropriate? How do you avoid pollution from numerical noise and
higher order in the case of a convex portfolio, or a model highly sensitive to inputs
¨ How do you bump things together (say either market inputs being bumped together, or
market inputs and model parameters)
¨ Do you bump sequentially one by one, or do you bump in a cumulative fashion (Piterbrag
calls it a waterfall approach)
¨ The story there is usually:
¨ BUMPING IS OK, JUST KNOW WHAT YOU ARE DOING
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19. Luc_Faucheux_2021
The structure of the deck - V
¨ The next question is then logically:
¨ How do you aggregate risk ?
¨ We go over some of the pitfalls of aggregation (for example can you add the delta coming
from the mortgage desk who runs on a lognormal model, and the delta from the option
desk who runs on a completely different skew assumption)
¨ We explain how we propose to deal with this issue, that we implemented with some varying
degrees of success at some of my previous shops.
¨ Some firms are open to listen, some are not, but hey that is their choice.
¨ The story usually there is:
¨ HEDGE WITH A MODEL ACROSS SCENARIOS WITH A COMMON SET OF HEDGING
INSTRUMENTS, AGGREGATE ACROSS DESK BY ADDING THE HEDGING INSTRUMENTS, NOT
THE RISK REPORTS PRODUCED ON EACH DESK
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20. Luc_Faucheux_2021
The structure of the deck - VI
¨ After the aggregation of risk, we go over Valuation adjustment
¨ This is usually done a posteriori because it is usually something that is impossible to do from
the start at the trade valuation level
¨ The issue is usually the fact that this adjustment itself has some risk
¨ So what do you do with this Risk ? Do you hedge it ? Can you hedge it ?
¨ We will discuss some of the issues around the Bermuda / European basis
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21. Luc_Faucheux_2021
The structure of the deck - VII
¨ We then explore some issues around Valuation Adjustment with market risk
that you cannot hedge
¨ In particular in that case, we will go over some stories on cross gammas for swap desk
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22. Luc_Faucheux_2021
The structure of the deck - VIII
¨ We then go back so trade lifecycle and some of the issues around it
¨ In particular a funny story about put-call parity, and how sometimes something that looks
great on paper is a disaster in real life (to the tune of tens of millions of $)
¨ The story there usually is :
¨ BOOK THE TRADE AS IT IS, STUPID.
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23. Luc_Faucheux_2021
The structure of the deck - IX
¨ We then tie all this together in the very important tool that is PL explain
¨ Note that a PL slice (or PL scallop as one irritating colleague of mine used to call it) is NOT a
PL explain
¨ We will go over the difference
¨ The PL explain is really the most powerful tool to manage and control your risk
¨ PL explain can be run over realized market moves (every day) or over ladders and scenarios
¨ Again the story there is:
¨ FOLLOW THE MONEY
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The Yield Curve (s)
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25. Luc_Faucheux_2021
The Yield Curve(s) - I
¨ In short, build the yield curve HOWEVER you want it
¨ I don’t care and neither should you.
¨ The point is that the curve construction should be independent from the curve manipulation
¨ Being able to change real-time how the curve is being constructed (number of ED futures,
3year treasury or not, convexity adjustment method, adding discrete events like turn-of-the-
year or Central Banks meetings,…) is crucial, and should be always tested continuously as it
allows you to :
¨ 1) fit the market
¨ 2) check that you are not too reliant on a specific input or assumption by observing the final
curve and pricing your portfolio against it
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26. Luc_Faucheux_2021
Always work in Forward space
Never work in Spot space (almost never)
TOP TEN REASONS WHY
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27. Luc_Faucheux_2021
Always work in Forward space
REASON #1
Separate the curve construction from the risk
representation
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28. Luc_Faucheux_2021
Always work in Forward Space – Reason #1 - I
¨ In the deck I on rates we went over the yield curve construction.
¨ Here is what you should do:
¨ Construct the yield curves HOWEVER you want to do it, use cash rates, use futures prices,
plus convexity adjustment, use forward space rates, use spot swap rates, or treasury prices
plus spread, use ANY kind of interpolation / extrapolation you want (Hey if you are at
Salomon Brothers you can even have your interpolation/extrapolation done with a 3-factor
short rates model, meaning that your yield curve will have sensitivity to volatility, now that is
cool)
¨ The point is that the way you build the curve should be COMPLETELY independent of all the
downstream processes, in particular, RISK, Var, FRTB, …
¨ Because if it is not, everytime you change anything on the curve format you will impact
everything downstream, and so you will end up never changing anything, and you will end
up having a useless curve, that will lead to mismarking
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29. Luc_Faucheux_2021
Always work in Forward Space – Reason #1 - II
¨ At Citi, we could change the number of futures used in building the curve in real-time “on
the fly”, same at Fuji, same at UBS
¨ The curve was then stored usually as a vector of dates and discount factors, with different
frequency, daily, monthly, quarterly or yearly, depending on how much size in memory you
can use
¨ Usually very granular in the front end of the curve, then sparse as you go further out the
curve
¨ Since you have the daily discounts, you can recreate the yield curve, and choose to
manipulate this curve however you see fit (shift the curve, bump some buckets,…)
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30. Luc_Faucheux_2021
Always work in Forward Space – Reason #1 - III
¨ At Lehman, we had a way to minimize the storage of the curve by storing only the inputs
(spot rates), and then rebuild the curve when needed.
¨ But then the US treasury discontinued the 3 year, and we had to change the format of the
curve, and we realized that using the same code did not work on the new curve, so we had
to also change the code needed to rebuild the curve, and now you had to have 2 pieces of
different code, one for curves who had a 3y input, and one for the curves who did not have
a 3y input.
¨ So sometimes simple is smarter, because it is more robust.
¨ Sometimes less smart is smarter…
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31. Luc_Faucheux_2021
Always work in Forward Space – Reason #1 - V
¨ The next slide is the usual combination of instruments being used to build the yield curve
¨ You will notice that Bloomberg allows in the drop-down option to change on-the-fly the
number of ED futures used to build the curve
¨ You will also notice that Bloomberg allows you different choices on the convexity adjustment
method
¨ This is another reason why you want the curve construction to be independent from the
curve manipulation.
¨ Because if it is independent, you can, and you should, tweak with the construction all the
time to ensure that:
¨ 1) you fit the market
¨ 2) you observe the impact of your assumptions on the portfolio and the final curve, which
will alert you if there is overreliance or sensitivity to some parameters or interpolation
method that you should really look into, instead of blindly relying on the same thing for 20
years…but hey the earth is really flat after all it seems.
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32. Luc_Faucheux_2021
[ICVS][GO]
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33. Luc_Faucheux_2021
Always work in Forward space
REASON #2
Bumping in spot space creates MASSIVE
non-intuitive results
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34. Luc_Faucheux_2021
Always work in Forward Space – Reason #2 - I
¨ The reason why we build curves is so that we can price trades, and measure some risk by
“bumping” something.
¨ The main 3 things that can be bumped are the curves (market inputs, includes vol surfaces),
the models (options models, but also say convexity adjustment in curve building,
interpolation method, anything that has a parameter that is not a direct market observable),
and the portfolio (the representation of the trades booked in your portfolio)
¨ We briefly went over that in deck I, so copying a couple of the slides from that deck.
¨ But really you should have a look a deck I
¨ https://www.slideshare.net/lucfaucheux/lf-2020-ratesi-237330935
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35. Luc_Faucheux_2021
Why build a curve in the first place?
To bump parameters and Risk Manage!
Quick overview of Risk Management
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36. Luc_Faucheux_2021
Risk Management 101
¨ Why do we go through the trouble of building a curve?
¨ SO that we can price rates derivatives
¨ SO that we can trade rates derivatives (hopefully at the right price)
¨ SO that we can book those trades
¨ SO that we can risk manage a book of such trades
¨ The curve construction (like the volatility surface or the models used) have huge implications
for the pricing, but also for the ongoing risk management and risk representation
¨ We will use the example of the yield curve we constructed to illustrate some of the issues
around risk management
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Principles of Risk Management
¨ Market, Model, Portfolio
¨ The market observables {𝑀!} are used to derive a yield curve. Note that the same applies
when dealing with options. Option prices in the market are used to calibrate the volatility
surface, or the skew surface, or to calibrate the option pricing model to recover the market
prices
¨ The curve can be formally viewed as input to the pricing model
¨ For example when parametrized in forward space, or bootstrapped, the curve is then
defined by another set of parameters than the market observables. Similarly, a volatility
cube will be fed into an option pricing model.
¨ The model parameters are denoted {𝑃"}
¨ Note that usually 𝑖 ≠ 𝑗, and the number of market observables is different from the number
of model parameters
¨ The model itself is the “code”, or the function call that takes the model parameters, the
trade details and returns a value
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Market, Model, Trade
¨ Market observables
¨ Swap Rates
¨ Bond prices
¨ Eurodollar futures
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Discount Curve
}
Reprice Market
¨ This was the schematic to build the discount curve, either through explicit
bootstrapping or implicit minimization solver

39. Luc_Faucheux_2021
Market, Model, Trades II
¨ Stating the obvious, if the trade details should be identical to one of the market observables,
we should expect to recover the price of that market observable that we use to calibrate the
model (obtain the values of the market inputs)
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¨ Market observables
¨ Swap Rates
¨ Bond prices
¨ Eurodollar futures
Model inputs
}
Price
Trade details
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Market, Model, Trades III
¨ Market observables {𝑀!}
¨ Model parameters {𝑃"}
¨ Portfolio composed of a number of positions or trades Π
¨ If the ”market moves”, the model is usually recalibrated and the portfolio repriced
¨ This could happen real-time, high-frequency or less frequent, daily or weekly
¨ Challenge of getting “accurate” market data, at the same time and that can be “trusted”
¨ Most financial institutions run a “official close” or “end-of-day” PL, meaning that an official
close is created to mark the books and records of the firm
¨ Sometimes this ”closing time” is conveniently aligned to an exchange close but often is not
¨ Usually this official closing curve (or closing model) is verified and validated by non-trading
functions like Market risk or Financial Controls
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41. Luc_Faucheux_2021
Official PL
¨ There is usually a record of the Market observables chosen for the close and of the model
parameters
¨ Usually stored in a central location
¨ Used for marking the books and record
¨ Used subsequently for VAR calculations, backtesting, FRTB, and analysis
¨ Challenge of keeping and retaining record easily accessible
¨ Challenge of backwards compatibility when for example the curve structure changes, the
actual model changes and now the historical stored values cannot be extracted in a
meaningful manner anymore
¨ Trust me, that sounds stupid, but that happens everyday
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Risk Management 101
¨ Most people would want to compute the risk of their portfolio
¨ Either to decide to hedge it, or some of it
¨ Even if not hedged, the portfolio risk is usually computed and reported either to
management or investors, or regulators
¨ For example, even if you only trade on an exchange, the exchange will compute some risk
measure on your portfolio in order to adjust the margin that you need to post to the
exchange
¨ The bottom line about risk management is to measure the risk, and the way to do this is to
“bump” something, whether you bump the market, you bump the model or you bump the
portfolio
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43. Luc_Faucheux_2021
Bumping the Market Observables
¨ You can chose an actual (historical) observed market move
¨ You can chose a pre-defined set of scenarios
¨ Sometimes this scenario is a “ladder” or a grid of shifts in the market observables
¨ Those shifts could be relative or absolute
¨ Two methods: ”Full reval” or “parametric”
¨ “Full reval” : the model is fully recalibrated for each bump in the market observables and the
portfolio is fully repriced. Brute force and overkill, does not sound too smart and elegant,
but simple, robust and in the end the smart thing to do
¨ “parametric”: risk sensitivities are computed on the portfolio (usually an overnight batch),
and those risk measures are used to estimate the PL impact from the observed market
move, or used for risk calculation (”parametric VAR” for example)
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44. Luc_Faucheux_2021
Bumping the model parameters
¨ The number of model parameters is usually much smaller than the number of market
observables
¨ The model parameter are also usually more reliable than the market observables (issues of
volume, reliability, trading time,..) and also have been “filtered” and checked against
previous historical values
¨ People get used to their models, it is usually a way to gauge the market and make an
informed decision
¨ Similar to market observables, two approach, “full reval” or parametric
¨ Unlike bumping market observables, bumping the model parameters offers the advantage of
not re-calibrating the model, and avoid potential issues coming from the minimization
problem or bootstrapping, with unstable solution
¨ So this is usually the preferred approach, to bump the model parameters and reprice the
portfolio
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45. Luc_Faucheux_2021
Bumping the portfolio
¨ The portfolio itself is “bumped”.
¨ This sounds weird at first, but is crucial.
¨ Sometimes the counterparty details would get changed and the portfolio repriced with the
new value. Those changes could be:
– Credit of counterparty changes
– Counterparty defaults
– Collateral posted by the counterparty changes
– Counterparty decides to early exercise options
– Resets are applied to the trades
– Triggers are exercised
A number of reports (reset report, strike report, notional report, expiry report,..) will also
usually be produced to “slice and dice” the portfolio across all the relevant risk
dimensions
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46. Luc_Faucheux_2021
Always work in Forward Space – Reason #2 - II
¨ OK, so now you have a curve and you want to “bump” it, meaning you will manipulate the
curve in a pre-determined manner, and measure the impact of this manipulation on the final
curve, and on the portfolio of trades valued using that curve
¨ The reason why you should never use spot bumps, is that they create MASSIVE and NON-
INTUITIVE deformations in forward space.
¨ Let’s go through a simple example that will show how terrible it is to bump the curve in spot
space (note that we will show that you can STILL express the risk in terms of equivalent spot
swaps, there is a way to do it which does NOT involve bumping the spot input of the curve)
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47. Luc_Faucheux_2021
Always work in Forward Space – Reason #2 - III
¨ So say that you have a curve defined in terms of spot rates, like {1y,2y,3y,4y,…}
¨ So usually the inputs are bumped by say 1 basis point up
¨ There is a lot of numerical games you can play about bumping up, then down, then
averaging in order to remove the convexity (but keep the odd higher orders), or bump with
different sizes and plot the results as a function of the bump size, or the order in which you
bump
¨ There is also different orders of bumping
¨ Individual bump: one individual input is bumped, all the others kept at their original values
¨ Cumulative, or sequential, or waterfall bump: inputs are bumped one a time, but once
bumped they stay at that value
¨ Numerically the order and size and direction will all create different results (see pdf notes)
¨ But let’s focus on one of the simplest illustration
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Always work in Forward Space – Reason #2 - IIIa
¨ Some tricks used to isolate
different orders in the
waterfall.
¨ Check out the date on
those pdf…
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Always work in Forward Space – Reason #2 - IIIb
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50. Luc_Faucheux_2021
Always work in Forward Space – Reason #2 - IIIc
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51. Luc_Faucheux_2021
Always work in Forward Space – Reason #2 - IV
¨ You are long a digital floor starting in 7y from now for one year, struck say 5 basis point
under the current forward
¨ You have a curve built using spot swap rates spaced every year
¨ Your risk infrastructure is tied to your curve construction, and the only thing that you can do
when manipulating the curve is to bump the curve inputs that are the yearly spot swap rates
¨ You decide that it would make sense to bump those individually up by 1 basis point.
¨ Roughly speaking (assuming that rates are pretty close to 0%, and forgetting any other
instabilities that could come from your choice of interpolation/extrapolation, like for
example using a curve spline or any kind of solver on a non-trivial function), you will get the
following sequence of bumps:
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52. Luc_Faucheux_2021
Always work in Forward Space – Reason #2 - V
¨ Initial curve {1y,2y,3y,4y,5y,6y,7y,8y,9y,10y}
¨ Individual bumps: 1y {1y+1bp,2y,3y,4y,5y,6y,7y,8y,9y,10y}
¨ Individual bumps: 2y {1y,2y +1bp,3y,4y,5y,6y,7y,8y,9y,10y}
¨ Individual bumps: 3y {1y,2y,3y +1bp,4y,5y,6y,7y,8y,9y,10y}
¨ Individual bumps: 4y {1y, 2y,3y,4y +1bp,5y,6y,7y,8y,9y,10y}
¨ Individual bumps: 5y {1y, 2y,3y,4y,5y +1bp,6y,7y,8y,9y,10y}
¨ Individual bumps: 6y {1y, 2y,3y,4y,5y,6y +1bp,7y,8y,9y,10y}
¨ Individual bumps: 7y {1y, 2y,3y,4y,5y,6y,7y +1bp,8y,9y,10y}
¨ Individual bumps: 8y {1y, 2y,3y,4y,5y,6y,7y,8y +1bp,9y,10y}
¨ Individual bumps: 9y {1y, 2y,3y,4y,5y,6y,7y,8y,9y +1bp,10y}
¨ Individual bumps: 10y {1y, 2y,3y,4y,5y,6y,7y,8y,9y,10y +1bp}
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53. Luc_Faucheux_2021
Always work in Forward Space – Reason #2 - VI
¨ You should REALLY look at how those curves look like in your risk engine.
¨ Some Risk systems, again at some place I worked at that shall not be named, were so obtuse
that it was impossible to extract the manipulated curves from inside the risk batch and look
at them
¨ And then the risk management was like “why do you need to look at those?”
¨ Fun times.
¨ Anyways, if you look at those curves, especially in forward space you will see that (in the
approximation of 0% rate so that every PV01 of a yearly forward swap is 100$/bp for 1mm
notional). If non –zero rate, then what I am saying will be scaled by the actual PV01 of the
yearly forward swaps
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54. Luc_Faucheux_2021
Always work in Forward Space – Reason #2 - VII
¨ You will roughly get something like that:
¨ Initial curve {1y,2y,3y,4y,5y,6y,7y,8y,9y,10y}
¨ Individual bumps_1: 1y {1y+1bp,2y,3y,4y,5y,6y,7y,8y,9y,10y}
¨ Individual bumps_2: 2y {1y,2y +1bp,3y,4y,5y,6y,7y,8y,9y,10y}
¨ Individual bumps_3: 3y {1y,2y,3y +1bp,4y,5y,6y,7y,8y,9y,10y}
¨ Individual bumps_4: 4y {1y, 2y,3y,4y +1bp,5y,6y,7y,8y,9y,10y}
¨ Individual bumps_5: 5y {1y, 2y,3y,4y,5y +1bp,6y,7y,8y,9y,10y}
¨ Individual bumps_6: 6y {1y, 2y,3y,4y,5y,6y +1bp,7y,8y,9y,10y}
¨ Individual bumps_7: 7y {1y, 2y,3y,4y,5y,6y,7y +1bp,8y,9y,10y}
¨ Individual bumps_8: 8y {1y, 2y,3y,4y,5y,6y,7y,8y +1bp,9y,10y}
¨ Individual bumps_9: 9y {1y, 2y,3y,4y,5y,6y,7y,8y,9y +1bp,10y}
¨ Individual bumps+10: 10y {1y, 2y,3y,4y,5y,6y,7y,8y,9y,10y +1bp}
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55. Luc_Faucheux_2021
Always work in Forward Space – Reason #2 - VIII
¨ Initial curve {1y,2y,3y,4y,5y,6y,7y,8y,9y,10y}
¨ Individual bumps_1: 1y spot is bumped up by 1bp, all other spot rates unchanged
¨ Individual bumps_2: 2y spot is bumped up by 1bp, all other spot rates unchanged
¨ Individual bumps_3: 3y spot is bumped up by 1bp, all other spot rates unchanged
¨ Individual bumps_4: 4y spot is bumped up by 1bp, all other spot rates unchanged
¨ Individual bumps_5: 5y spot is bumped up by 1bp, all other spot rates unchanged
¨ Individual bumps_6: 6y spot is bumped up by 1bp, all other spot rates unchanged
¨ Individual bumps_7: 7y spot is bumped up by 1bp, all other spot rates unchanged
¨ Individual bumps_8: 8y spot is bumped up by 1bp, all other spot rates unchanged
¨ Individual bumps_9: 9y spot is bumped up by 1bp, all other spot rates unchanged
¨ Individual bumps_10: 10y spot is bumped up by 1bp, all other spot rates unchanged
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56. Luc_Faucheux_2021
Always work in Forward Space – Reason #2 - IX
¨ Initial curve {1y,2y,3y,4y,5y,6y,7y,8y,9y,10y}
¨ Individual bumps_1: 0yx1y forward is bumped up by 1bp, 1yx2y bumped down by
1bp, all other forward rates unchanged
¨ Individual bumps_2: 1yx2y spot is bumped up by 2bp, 2yx3y bumped down by
2bp, all other forward rates unchanged
¨ Individual bumps_3: 2yx3y spot is bumped up by 3bp, 3yx4y bumped down by
3bp, all other forward rates unchanged
¨ Individual bumps_4: 3yx4y spot is bumped up by 4bp, 3yx4y bumped down by
4bp, all other forward rates unchanged
¨ And so on and so forth, as a mighty manager of mine was known to use and abuse as an
expression…
56

57. Luc_Faucheux_2021
Always work in Forward Space – Reason #2 - X
¨ The reason if you look at the T spot bumped, is that
¨ The (T-1) has to be unchanged
¨ The (T) has to be bumped up by 1 basis point
¨ So the forward swap starting at (T-1) and ending at T has to be bumped up by T basis points.
¨ And since the (T+1) spot has to be kept unchanged, the forward swap starting at T and
ending at (T+1) has to be bumped DOWN by also T basis point in a symmetrical manner.
¨ A picture is worth a thousand words
57

58. Luc_Faucheux_2021
Always work in Forward Space – Reason #2 - XI
¨ Suppose that we have the following starting yield curve.
¨ We will have the following relationship between the forward rates and the spot rates
58

59. Luc_Faucheux_2021
Always work in Forward Space – Reason #2 - XII
¨ The spot rates will be the PV01 weighted average of the forward rates
¨ 𝑆𝑝𝑜𝑡𝑅𝑎𝑡𝑒(𝑇). [∑ 𝐷 𝑖 ] = ∑ 𝐷 𝑖 𝑓 𝑖
¨ Where:
¨ 𝑆𝑝𝑜𝑡𝑅𝑎𝑡𝑒(𝑇) spot rate
¨ 𝐷 𝑖 : discount factor at time I
¨ 𝑓 𝑖 : forward rate for the period [i,i+1]
¨ Here for the sake of simplicity we assumed same frequency on both side of the swap and
same unit daycount fraction.
¨ This does not invalidate the conclusion, just make the math and notation a little more
complicated if you keep track of the exact frequency, roll, holidays and daycount fraction
¨ 𝑆𝑝𝑜𝑡𝑅𝑎𝑡𝑒(𝑇) =
∑ $ ! % !
[∑ $ ! ]
59

60. Luc_Faucheux_2021
Always work in Forward Space – Reason #2 - XIIa
¨ 𝑆𝑝𝑜𝑡𝑅𝑎𝑡𝑒(𝑇) =
∑ $ ! % !
[∑ $ ! ]
¨ The approximation that we made earlier is essentially saying
¨ ∑!()
!(*
𝐷(𝑖) ~𝑇
¨ 𝐷 𝑖 ~1
¨ Please discuss amongst yourselves and convince yourselves that it is really not that bad of an
approximation, especially these days of low if not negative rates
60

61. Luc_Faucheux_2021
Always work in Forward Space – Reason #2 - XIII
61

62. Luc_Faucheux_2021
Always work in Forward Space – Reason #2 - XIV
¨ So the manipulated yield curve is quite deformed, and certainly by more than 1 basis point.
¨ In the example where you are long a 7x8 digital floor struck at 5bps under the forward, in
this specific bump you will see an increase in value
¨ So your risk management system will return the result that you are long rates (short the
market) on the 7year spot rate plot with this portfolio, whereas in reality you are long the
floor, hence long the market (short rates).
¨ This will give you the WRONG RISK
¨ That reason, and that reason alone, should tell you that you should NEVER use spot risk
when dealing with portfolio or trades that have significant convexity (in particular options)
62

63. Luc_Faucheux_2021
Always work in Forward Space – Reason #2 - XV
¨ Since we are at it, something that we will use later on, for example the risk representation in
forward and spot space (both in units of notional and $/bp) for say a 7x8 forward swap and
a spot swap, with the assumption to simplify that :
¨ ∑!()
!(*
𝐷(𝑖) ~𝑇
¨ 𝐷 𝑖 ~1
¨ In order to simplify the discussion (otherwise use the actual PV01)
63

64. Luc_Faucheux_2021
Always work in Forward space
REASON #3
The Gamma matrix in spot space is absolutely
and completely FUBAR
64

65. Luc_Faucheux_2021
Always work in Forward Space – Reason #3 - I
¨ All right, if you thought that the Delta Risk was an issue when you use bumps in spot space,
buckle up buttercup because it gets ever worse in Gamma space.
65

66. Luc_Faucheux_2021
Always work in Forward space
REASON #4
Historical backtesting becomes impossible in
spot space when the curve format changes
66

67. Luc_Faucheux_2021
Always work in Forward space
REASON #5
The moments of the risk are much better
ordered in forward space
67

68. Luc_Faucheux_2021
Always work in Forward Space – Reason #5 - I
¨ One of the interesting thing to look at when you have a vector of sensitivities
¨ For each instrument {i} we can calculate the sensitivity of the portfolio Π to the market
instrument (spot or forward) noted 𝑅! and compute the value
+
,-!
and note this 𝛿!
¨ FIRST moment : DELTA ∑ 𝛿!
¨ SECOND moment: FLATTENER / STEEPENER ∑ 𝑖. 𝛿!
¨ THIRD moment: BUTERFLY ∑ 𝑖.. 𝛿!
¨ An outright “Delta” position (say long or short the market) will have the DELTA as the first
non-zero moment on the bucketed risk
¨ A flattener or a steepener will have zero DELTA but non-zero second moment
¨ A butterfly will have zero DELTA, zero second moment but non-zero third moment
68

69. Luc_Faucheux_2021
Always work in Forward Space – Reason #5 - II
¨ In some ways, you can think for example of the second moment as the analytical (not full
reval) PL for a curve deformation that is linear
¨ Technically we should really call the first moment the zero moment and offset by one
(second moment is first moment), and so on and so forth because:
¨ FIRST moment : DELTA ∑ 𝛿! = ∑ 𝑖). 𝛿!
¨ SECOND moment: FLATTENER / STEEPENER ∑ 𝑖. 𝛿! = ∑ 𝑖/. 𝛿!
¨ THIRD moment: BUTERFLY ∑ 𝑖.. 𝛿! = ∑ 𝑖.. 𝛿!
69

70. Luc_Faucheux_2021
Always work in Forward Space – Reason #5 - III
¨ In any case now you have a measure of some curve deformation
¨ You can define limits for your PM
¨ You can also instead of choosing 𝑖) and higher powers of, you could use the PV01 of the
instrument 𝑅!
¨ You could also decide to use the PCA eigenvectors for the curve deformation
¨ In that case you would have the analytical PL on the PCA
¨ You can also define a limit not as a hard limit, but as penalty by defining a distance from 0 of
those moments, which can be used as a reserve in order to motivate your PM to reduce the
risk, or if taking a position, put his/her money wher his/her mouth is, and be willing to put
aside as a reserve the penalty
¨ You can then look at the ratio of the trader target to the size of the penalty see if that makes
sense
70

71. Luc_Faucheux_2021
Always work in Forward Space – Reason #5 - IV
¨ So again, never work in spot space, because forwards are so much more clean when looking
at the moments of the sensitivity against curve deformation
¨ For a position that corresponds to the moment, both notional and dv01 in forward space are
non-zero for this moment and zero for all moments below
¨ That is NOT true for risk representation in spot space
71

72. Luc_Faucheux_2021
Always work in Forward Space – Reason #5 - V
¨ Say it another way, long 100mm of 7y spot and short 100mm of 6y spot is NOT a curve
position, it is an outright delta long in 6x7 forward
¨ Another way to say it, is that being long 100mm of a 7x8 forward, and short 100mm of a 8x9
forward, is INDEED a curve position as we expect with a net delta of 0 (in the approximation
we made, if you use actual PV01, adjust the notionals to make it delta neutral)
72

73. Luc_Faucheux_2021
Always work in Forward space
REASON #6
PCAs and correlations are much better
defined in forward space, and more
meaningful
73

74. Luc_Faucheux_2021
Always work in Forward space
REASON #7
Bump sizes are better controlled in forward
space
74

75. Luc_Faucheux_2021
Always work in Forward Space – Reason #7 - I
¨ In forward space, say we bump every yearly forward by 1 basis point. Those will be fairly
consistent because they have roughly similar durations, so the impact on the PL will be
commensurate
¨ HOWEVER in spot space, bumping every spot rates by the same amount creates large
inconsistencies (first in forward space as we saw) but also because they have different
durations
¨ You can say, well I do not care because I am expressing my risk in term of dv01, so that will
be ok
¨ Fair enough on that point, I really needed to get to 10 reasons
75

76. Luc_Faucheux_2021
Always work in Forward space
REASON #8
Forward representation is much closer to the
models (LMM, HJM, BGM…)
76

77. Luc_Faucheux_2021
Always work in Forward Space – Reason #8 - I
¨ Most models out there (BGM, LMM,..) are models of the FORWARD rates
¨ There is really no models of the spot rates
¨ So looking at the risk in forward space is much more closely aligned to the model (and most
of those are numerical, not closed form), so will create a lot less numerical noise and
distortion
¨ This is no coincidence
¨ A lot of the products (caps, callable, callable accreeters, swaptions) are mostly ALL very
forward in nature, and so it makes sense that the models would reflect this.
¨ Forward space is much more consistent with the products, and the models, than spot space
77

78. Luc_Faucheux_2021
Always work in Forward space
REASON #9
Rolling the curve makes sense in forward
space. In spot space….not so much…
78

79. Luc_Faucheux_2021
Always work in Forward Space – Reason #9 - I
¨ Already at 130MB file, and I need a number of slides to explain in details the issues around
rolling the curve, so that will be for another deck
¨ Am starting to realize that those decks are not sequential, but have gaps
¨ That is cool, it is kind of like the language maps in “Arrival”…or the IRMA SQUID function !
79

80. Luc_Faucheux_2021
Always work in Forward space
REASON #10
You can easily recover the risk in terms of spot
instruments if you wanted to, and more
accurately…
80

81. Luc_Faucheux_2021
Always work in Forward Space – Reason #10 - I
¨ We are going to illustrate this point in the next section (always use a simpler model), but
essentially:
¨ Create scenarios (PCAs, simple ladders, anything you want)
¨ Those are better controlled in forward space, so do it in forward space
¨ Create a map of PV, and compute the Greeks from there
¨ Define a vector of spot hedges if you want
¨ Solve for the weights of that vector that will minimize a distance that you will have defined
on those scenarios of your portfolio + hedges (could be worst loss, could be local delta,
could be Var, could be any kind of distance that you think makes sense to look at)
¨ Boom, voila..you have your spot hedges
¨ For a linear portfolio, if you define the buckets to be 0 as your distance you will recover the
spot risk, and you can go between spot and forward risk by inverting the Jacobian
81

82. Luc_Faucheux_2021
Always work in Forward Space – Reason #10 - II
¨ By the way, anyone who is just telling you “spot and forwards are the same, just invert the
Jacobian”
¨ Has either
¨ Never inverted a Jacobian themselves
¨ Never encountered the numerical noise that you sometimes get out of doing that
¨ Is usually more interest in sounding smart than being pragmatic
82

83. Luc_Faucheux_2021
Always work in Forward Space – Reason #10 - III
¨ This is another example (like bootstrapping the curve) of something that sounds exact and
rigorous, but is not flexible once you start changing anything, and will break on you
¨ Biology and flexible pragmatism always win over autocratic rigidity, or as we know from
Jurassic Park, “Nature always find a way”
83

84. Luc_Faucheux_2021
Always use a simpler model
Example: recovering skew and smile from a
simple lognormal Black-Sholes model
84

85. Luc_Faucheux_2021
Always use a simpler model - I
¨ We want through that example in the “Skew” deck, but here it is again in context
¨ This example is for a simple European option on rate
¨ Suppose that the only tools we have at our disposal are:
¨ Lognormal Black-Sholes option pricer, so no skew and no smile, meaning that all strike are
priced at the same lognormal price volatility
¨ We know that in the market they do trade at different implied volatility from the at-the-
money: this is what we call skew and smile
¨ We only have Excel at our disposal and a random number generator
¨ We know some market dynamics: we know that rates do move (are stochastic), we observe
that the at-the-money volatility does change with time (maybe stochastic, maybe a function
of the at-the-money rate, but it does look to us to be more like a stochastic process with its
own volatility, the VolofVol, and a correlation with the stochastic rate process)
85

86. Luc_Faucheux_2021
Always use a simpler model - II
¨ We also observe in the markets something that is crucial for the dynamics: people do tend
to HEDGE their risk
¨ In practice, market participants do observe risk limits, either self-imposed, or externally
imposed (margin calls that are essentially synthetic risk limits).
¨ Market participants also tend to measure risk in a similar fashion (like running a Callable
Vega book, for example second-tier banks would usually not have a limit more than 10mm
of $ per 1% move in the lognormal volatility input to their models, this is known by the first
tier banks, either by interviewing people and getting knowledge, or from observing how
second tier banks behave in the market)
¨ Based on that, we CAN accurately predict and recover the skew and smile observed in the
market by running the simple model we have a simulation over time that reproduces the
market dynamics
86

87. Luc_Faucheux_2021
Always use a simpler model - III
¨ This is in many cases very close to reality (a little like the chicken and the egg problem)
because the observed market dynamics sometimes originates from participants relying on a
simple model (wither for historical reason, computational reasons, pragmatic reasons).
¨ So a more complicated model would be an overkill, would not be intuitive, is usually prone
to numerical instabilities that are very hard to control, or deep hidden assumptions that no
one really understand
¨ On to the slides from the “skew” deck to refresh our memory
87

88. Luc_Faucheux_2021
88
Modeling the Skew and Smile using Monte-Carlo
¨ It works
¨ It matches the market observed skew and smile
¨ Very flexible (jumps, hedging strategies,…)

89. Luc_Faucheux_2021
89
Modeling the Smile and Skew Premium
¨ A recipe to reproduce market observed smile/skew:
¨ Generate stochastic rate
¨ Generate stochastic volatility
¨ Incorporate the correlation between rate and volatility
¨ Hedge the rates and volatility exposure
¨ The residual PnL coming from Vega hedging is the additional premium we are willing to
pay/receive: this is the smile/skew

90. Luc_Faucheux_2021
90
How to Hedge Rates and Volatility
¨ Since we do not know the option price, we do not know how to compute the Delta and Vega
¨ As a first iterative guess, we choose to compute the Greeks in a simpler model with no
correlation and no stochastic volatility (Black-Scholes)

91. Luc_Faucheux_2021
91
Stochastic Rate
0
1
2
3
4
5
6
0 50 100 150 200 250 300 350 400
days
Rate (%)

92. Luc_Faucheux_2021
92
Stochastic Volatility
0
20
40
60
80
100
120
0 50 100 150 200 250 300 350 400
days
Volatility (%)

93. Luc_Faucheux_2021
93
Smile and Skew Premium
¨ Smile premium: additional PnL coming from hedging the changes in Vega resulting from
changes in volatility
(Volga PnL)
¨ Skew premium: additional PnL coming from hedging the changes in Vega resulting from
changes in rates
(Vanna PnL)

94. Luc_Faucheux_2021
94
Smile and Skew Premium
¨ We obtain the deviation from the ATM volatility (in %) by dividing these PnL by the Vega of
the option
Skew [%] = Vanna Pnl [$] / Vega [$/%]
Smile [%] = Volga Pnl [$] / Vega [$/%]

95. Luc_Faucheux_2021
95
Smile and Skew Premium
-3.5
-3
-2.5
-2
-1.5
-1
-0.5
0
0.5
1
0 50 100 150 200 250 300 350 400
days
Skew Premium (%)
Smile Premium (%)

96. Luc_Faucheux_2021
Getting some intuition from running different paths
¨ Running at-the-money paths.
96

97. Luc_Faucheux_2021
Running some high strike paths
97

98. Luc_Faucheux_2021
Running some more high strike paths
98

99. Luc_Faucheux_2021
Running some more high strike paths
99
-12
-10
-8
-6
-4
-2
0
2
4
6
8
10
0 50 100 150 200 250 300 350 400
rate (%)
volga pnl (%)
vanna pn (%)l
Strike

100. Luc_Faucheux_2021
A couple of observations to gain intuition:
¨ Paths that do not stay “on target to the strike” quickly become deep away from the strike
options, with no change in Vega, either from rates move or volatility move, and the
accumulated PL from Vanna and Volga “flat line”
¨ So when averaging over many paths in order to get the average expected PL from skew and
smile (and scaling that into the implied volatility by dividing by the initial Vega), it seems
sensible that we should pay attention to the paths that “stay on target”
¨ One of those paths is the simplistic “straight path” that we will revisit later in order to gain
some intuition on the skew versus backbone issue
¨ Volga accumulated PL always tend to be positive, which makes sense as it is quadratic.
¨ However not always for options close to the money (when doing the actual calculation the
Volga for at-the-money options is actually slightly negative)
¨ On average though, Volga impact is always positive (hence a quadratic formula)
100

101. Luc_Faucheux_2021
101
Repeating Over Many Paths
¨ For the particular path shown in the previous slide, the smile hedging (Volga hedging)
resulted in a positive 0.6% change in vol, the skew hedging (Vanna hedging) resulted in a
negative –3% change in vol
¨ For that path, in order to reflect the cost of both Vanna and Volga hedging, we have to
adjust the implied volatility down by 2.4%
¨ We average over many paths for a given strike
¨ We repeat over different strikes

102. Luc_Faucheux_2021
102
Example of Generated Smile
¨ T=1year, s=50%, F=2.35%, Correlation=0, ss=50%
Skew/Smile (%)
0.00
5.00
10.00
15.00
-200 -100 0 100 200
moneyness (bp)

103. Luc_Faucheux_2021
103
Example of Generated Skew
¨ T=1year, s=50%, F=2.35%, Correlation=-0.5, ss=50%
Skew/Smile (%)
-5.00
0.00
5.00
10.00
15.00
-200 -100 0 100 200
moneyness (bp)

104. Luc_Faucheux_2021
104
EDZ4 Revisited
¨ T=1.61y, s=48.2%, F=2.58%, Correl=-0.52, Ss=26%
Skew (% )
-6.00
-4.00
-2.00
0.00
2.00
4.00
6.00
8.00
10.00
-200 -100 0 100 200
moneyness (bp)

105. Luc_Faucheux_2021
105
EDU3 Revisited
¨ T=0.36y, s=45%, F=1.21%, Correl=-0.25, Ss=111%
Sk ew (% )
-20.0 0
0.0 0
20.0 0
40.0 0
60.0 0
80.0 0
100.0 0
-100 0 100 20 0 300 4 00 500
mo ne yne ss (bp)

106. Luc_Faucheux_2021
106
Implied versus Realized
CME contract Implied r Implied ss Historical r Historical ss
EDU3 -53% 178% -34% 170%
EDH5 -55% 31% -56% 40%
(June 29 2003)

107. Luc_Faucheux_2021
107
Skew and Smile Come from Vega Hedging
¨ Smile and skew can be explained simply with:
- Vega hedging
- Stochastic volatility
- Correlation between rate and volatility

108. Luc_Faucheux_2021
108
Smile and Skew
¨ Smile effect when: - Stochastic volatility
- Options with high Volga
¨ Skew effect when: - Correlation between underlying and volatility
- Options with high Vanna

109. Luc_Faucheux_2021
109
How to Generate the Smile/Skew
¨ Pick a simple model (like Black-Scholes)
¨ Simulate Vega hedging within this model using “real world” stochastic volatility and
correlation
¨ Calculate the residual PnL resulting from this hedging
THIS IS THE SMILE / SKEW !

110. Luc_Faucheux_2021
110
What Did we Get ?
¨ This explanation fits the market
– Qualitatively (intuition)
– Quantitatively (option pricing, market making,…)
¨ Calibration to market or historical data
– Richness/cheapness of the smile/skew
– Compare skew and smile across different option classes (Eurodollar versus
caps versus swaptions)

111. Luc_Faucheux_2021
111
What Else Can we Get?
¨ Effects of jumps (short-dated options)
¨ Effects of mean reversion (long dated options)
¨ Extension to non-European options (Bermuda, American,…)
¨ Extension to term structure models
¨ Different choices of hedging strategies/hedging costs

112. Luc_Faucheux_2021
Always use a simpler model
Extending the approach to large portfolios
112

113. Luc_Faucheux_2021
Always use a simpler model - IV
¨ In the previous example we had a single option with a closed form model (Black-Sholes)
¨ This is not always the case especially if you have large books (say callable accreters).
¨ The approach stays the same
¨ INSTEAD of calling the closed form within the simulation (which you might not even have),
you create SCENARIOS of your large portfolio by bumping rates and vol (in our specific
example, but could be any asset)
¨ Those batches usually take a really long time so usually they run over th weekend or
overnight
¨ You want to run them regularly so that you can catch any mistake in the process, and also
have enough backups if needed (NEVER rely on the fact that your computer will turn on as
expected tomorrow morning, and that all the batch processes overnight did run smoothly)
113

114. Luc_Faucheux_2021
Always use a simpler model - V
¨ You can then run simulations on that surface (instead of calling the function, you pick the
value from the surface, and with differences compute all the Greeks that you need for Vega
hedging and Delta hedging)
¨ This works especially well (trust me) on really large books whose profile does not change too
much over time (think very large callable accreters books)
114

115. Luc_Faucheux_2021
Always use a simpler model - VI
¨ The next few slides are examples of scenarios, greeks and the simulations that one can run
defining the desired hedging method.
¨ Note that the hedging process (or absence of) changes the DISTRIBUTION of PL, but does
not change the expected (mean, average), PL.
¨ We know from Black-Sholes that this would be the risk-free rate
¨ Hedging does not change the expected, just the actual shape of the distribution
¨ Canonical example: you are long an out-of-the-money call option, and do not delta hedge it,
the most that you can lose is the premium, so the distribution of PL is floored at (-option
premium)
¨ HOWEVER if you decide to delta hedge, you could end up in a case where say the market
rallies every day gently up to the strike, and you delta hedge all the way up, you will lose a
lot more than the option premium
¨ On AVERAGE though your expected PL does not change
115

116. Luc_Faucheux_2021
Always use a simpler model - VII
¨ Example of the MAP of a typical book of Callable options
¨ Instead of a 2 dimensional map of bumping rates and volatility, the graphs below are just
slices of that two-dimensional surface
¨ At the time (check out the date), we did feel that bumping the vol up and down by 1% was
enough, the big one was the rate
¨ Bear in mind that I did not tell you which vol we are bumping, but you can assume for the
prupose of this illustration that it is lognormal (hint: it is not, it is a smart IRMA vol, which
was super stable, which is why we did not feel the need to bump it up and down by a lot,
saving us a lot of CPU time)
116

117. Luc_Faucheux_2021
Always use a simpler model - VIII
¨ We can then using those calculate all the Greeks that we desire
117

118. Luc_Faucheux_2021
Always use a simpler model - IX
118

119. Luc_Faucheux_2021
Always use a simpler model - X
119

120. Luc_Faucheux_2021
Always use a simpler model - XI
¨ For some added insight, here is that Callable portfolio with its hedge of European options,
you can see that the net risk has been sharply reduced, but the higher order risk becomes
more unstable, this is usually the typical trade off when hedging, making sure that you
control your higher order risk.
¨ As much as I do not really understand all the noise around Taleb, his book “Dynamic
Hedging” is a must read and explores a lot of those issues
120

121. Luc_Faucheux_2021
Always use a simpler model - XII
121

122. Luc_Faucheux_2021
Always use a simpler model - XIII
122

123. Luc_Faucheux_2021
Always use a simpler model - XIV
¨ We can now run our simulation on this surface by defined the market parameter dynamics
that we want (volofvol, correlation), and the desired (or expected from market participants)
hedging strategy, hedge every certain number of days, or hedge when you run up against a
limit, or every time the market move more than certain amount,..see below an example)
123

124. Luc_Faucheux_2021
Always use a simpler model - XV
¨ We can then bucket the results of this simulation onto something that is useful for the
traders and risk managers (run below on 2,000 paths). For example the paths were rates
rally 100bps and vol goes up 1% would result in a loss of 14m, with 0.66% probability. Then
the real educated discussion between traders and Risk can start
124

125. Luc_Faucheux_2021
Always use a simpler model - XVI
¨ Illustration that the average PL does not change, only the distribution
¨ (need to run on lot more path to get good convergence of the mean, goes as 1 over the
square root, if I have time will run it on a lot more paths)
¨ Hedging in this case reduces the standard deviation of PL outcomes
125

126. Luc_Faucheux_2021
Always use a simpler model - XVII
¨ If you do not have a MonteCarlo simulator, you can also do the “straight path” hedging,
which is actually not that stupid of an idea, it gives you some intuition f the market goes in a
straight line to the outcome.
126

127. Luc_Faucheux_2021
Always use a simpler model - XVIII
¨ FINALLY, usually you have a set of “pre-defined” hedges
¨ You have run the scenarios on those hedges
¨ You can now add those hedges to your portfolio and define any kind of risk measure that
you want to reduce, and use a solver to give you the set of optimal hedges
¨ This is why a simple model is better
¨ This is also how you aggregate risk between different desks who might have different
models (say the option desk is on normal model, and the mortgage desk is using a lognormal
model, surely even though the Delta produced by those two desks have the same units, you
are not going to just add those up right ???)
¨ Same as you can always use for those hedges in rates spot swaps for example, and minimize
the risk. Another reason why you should never use spot, always forwards
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128. Luc_Faucheux_2021
Good old friends….
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129. Luc_Faucheux_2021
So at least for now…..
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