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Mathematical Finance: What Are Financial Derivatives & Valuation? - Lecture 2 – A. Sokol - CompatibL — Transcript

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0:00all right so so let's get started so the
0:02second lecture tool uh of the course
0:04introduction to mathematical finance by
0:06compatible my name is alexander sokol
0:08i'm the head of quant research
0:10comparable and the company founder
0:12welcome so today's lecture is on
0:14derivatives and validation
0:16uh first of all we'll introduce
0:18derivatives and over the country
0:19instruments and these are not opposite
0:21to each other these categories overlap
0:24and then we'll learn more about equity
0:26derivatives and about interest rate
0:28derivatives
0:30first of all what is the derivative
0:32a derivative is a financial instrument
0:34whose payments are based on or derived
0:36from other financial instruments
0:38some derivatives say securities
0:42namely you know they are trading uh on
0:44exchange or for example futures or
0:46deliveries like that but most are
0:48over-the-counter instruments and we'll
0:50talk about what this means
0:52a financial instrumental asset uh on
0:55which payments of a derivative depend is
0:57cool is underlying right so the
0:59underlying maybe security or bilateral
1:01instrument
1:02and bilateral instrument is the contract
1:04between two parties right so it's not an
1:06asset to buy a cell
1:08it's a contract
1:09sometimes the underlying is not even a
1:11financial instrument it may be a
1:13commodity for example oil grain you know
1:15price of rice
1:17and
1:18the diagram below shows a first
1:19generation derivative based on cash and
1:22an asset such as stock bond or commodity
1:27the second generation derivative
1:29can be based on this first derivative
1:31first generation derivative and cash
1:32right so in this diagram we have
1:35security and cash
1:37and based on them we have first
1:38derivative which is based on them
1:41and then if you combine first derivative
1:43with cache you can then build on top of
1:45it second uh you know so second
1:48generation derivative uh based on the
1:50first generation derivative which is
1:52based on something that's not a
1:53generator so
1:54you know this cycle can be very long
1:58and uh some banks and financial market
2:00participants got into trouble trading
2:03very very long chains of derivatives
2:04because the more derivatives you can
2:06have in the chain the mode risk but you
2:08know first and second generation are
2:10totally safe very mainstream uh and uh
2:13you know that's what we'll talk about
2:15today first or second generation
2:16derivatives
2:18so generator securities securities are
2:20not opposites some derivatives are
2:22securities uh and uh you know some and
2:26first or even second generation
2:27derivatives are traded on exchanges just
2:30like stock for example even though you
2:32know sometimes one exchange will trade
2:34stocks and derivatives sometimes there
2:36is a specialized derivatives exchange
2:38for example many derivatives trades on a
2:40trade on the chicago mercantile exchange
2:42which is uh today you know most of the
2:44time just called base abbreviation cme
2:48and they are considered securities and
2:50they're considered securities because
2:51they meet the definition of securities
2:54that we learned about
2:56yesterday you know two days ago in the
2:57pre in the first lecture namely they're
2:59standardized
3:00uh they're fundable right so they're
3:02tradable or negotiable and they have a
3:04public price record
3:06so for example cmea trades u.s treasury
3:09futures
3:10which is the first generation derivative
3:12for which the underlying is a u.s
3:14treasury bond
3:15and options on u.s treasury futures
3:17which is a second generation derivative
3:20for which the underlying is u.s treasury
3:22futures
3:23the diagram below shows how these
3:25derivatives are related to each other
3:26right so the future and bond is based on
3:28bond and cash
3:30and then option on future is based on
3:32future and bond
3:34and cash
3:35uh anastasia can you confirm we're
3:37recording actually you can see here that
3:38where okay good yes
3:40so the meeting is being recorded all
3:42right
3:43so now uh
3:46securities derive advantages from their
3:48standardization but they're also
3:50disadvantages to standardization
3:52financial market participants require
3:54financial instruments suited to their
3:56specific objectives and economic
3:58situation
3:59and the closer financial instrument
4:01matches the specific objectives of a
4:03single market participant the more
4:05difficult it is for the exchange to
4:07match buyers and sellers
4:09right so some financial instruments are
4:10bespoke they're customized so strongly
4:12to the needs of one market participant
4:14that they're not of interest to other
4:16market participants right in others of
4:19interest to
4:21multiple market participants
4:23but not standard enough to trade loan
4:25exchanges
4:26for example uh if sometimes uh it's a
4:29kind of a classic generator example uh
4:32let's say your corporation
4:34and you're planning to to build a
4:36factory in the foreign country
4:38and to build this factory you need to
4:40borrow money let's say you plan to start
4:42a year from now and you will be building
4:44for the next two years and which stage
4:47of construction you will need to borrow
4:49additional amounts of money but you
4:52would like to have the option of not
4:54doing the borrowing because you have not
4:56decided yet if you definitely build the
4:58factory or not so that creates a
5:00derivative that's very highly customized
5:02right it has to have a specific option
5:04and maybe
5:05uh this option of build or not to build
5:07is linked to let's say price of the
5:08stock uh or some other foreign exchange
5:10rate uh and you need specific dates on
5:13machine the money so that's a highly
5:15customized bespoke financial instrument
5:18in bespoke uh you know basically when
5:20you say bespoke suit it means a suit
5:22that the tailor creates for you
5:25from clothes
5:27right and uh you know just like
5:28financial instruments bespoke financial
5:30instruments are created for a single
5:31market participant
5:33uh by the dealer or a bank
5:36now over the counter financial
5:37instruments uh you know when when
5:40financial instruments attract investor
5:42interest but do not meet the
5:44requirements to become securities or
5:46mean some of the requirements but not
5:48others there are three right so there is
5:49uh
5:50you have to be fundable it has to be
5:52negotiable and it has to be a price
5:54record it has to be liquid
5:56so when an instrument meets
5:58none or some but not all of these
6:00requirements they may be traded over the
6:03counter
6:04right so over the counter meaning you
6:06come to a shop you know you talk to a
6:09salesperson and over the shop counter
6:11you get a financial instrument right as
6:13opposed to an exchange
6:15so an over the counter financial
6:17instrument is a bilateral contract
6:19between two parties to exchange
6:21contractually agreed payments with each
6:22other
6:23right so the parties to an otc
6:25instrument are called counterparties and
6:27the payments are called cash flows even
6:30if some of them are involved exchange of
6:32something other than cash like for
6:34example
6:35shares of stock or commodity
6:38sometimes they're called
6:40asset laws uh and sometimes they're
6:42called um
6:43you know just cash flows is more common
6:47an otc instrument has a term sheet which
6:50is a summary description of contractual
6:52obligations of the parties
6:54and term sheet can be used to determine
6:56the price of an otc instrument and the
6:59process of doing so is called valiation
7:02so derivatives valuation or generally
7:05being a valuations analyst is one of the
7:07drops in financial markets and that's
7:09one of the career choices for people who
7:11study mathematical finance
7:13and uh you know there are lots of jobs
7:15like that compatible that's one of the
7:17things that the company does
7:18but also uh there are jobs like that for
7:21banks they're also banks
7:23jobs like that the corporations that
7:24trade with banks
7:25uh and uh generally it's one of the um
7:29key uh professions uh that that uh you
7:31know for guidance of mathematical
7:33finance
7:34all right uh let me just uh take uh one
7:36brief moment uh i wanna stop uh share
7:38for a second uh i want to see if i can
7:40stop the
7:41uh
7:43the sound when people join
7:45uh the meeting because people are still
7:47joining one second shift can easily
7:49if anybody knows how to uh how to
7:52disable the notification let me know
7:56play sound when someone drains or leaves
7:58i think that's it all right
8:01okay so we're back and can share the
8:03screen
8:05all right so let's continue
8:07all right uh
8:08by the way just like in the last lecture
8:11if you have any questions uh please type
8:13it into the chat
8:14my colleagues will see the chat and
8:16alert me
8:18either answer it directly or will alert
8:20me and then at the appropriate time
8:22during the presentation i will answer
8:24these questions
8:26all right so uh now for over the counter
8:29uh sorry for uh
8:32you know securities and otc instruments
8:33right securities and over the country
8:35instruments are opposite right so
8:36derivatives are not opposite of
8:38securities some derivatives and
8:40securities and some are not
8:41but securities of no other country
8:43instruments are opposites right and over
8:46the country instruments
8:47are not or not completely standardized
8:50right so some are uniquely totally
8:52unique bespoke others are partially
8:54standardized for example forward rate
8:56agreements and vanilla swaps and we'll
8:58talk about vanilla swaps later today
9:00have standard contractual terms but
9:02different start dates and fixed rate
9:04right and the meaning of vanilla is
9:06plain in this case right so vanilla of
9:08course is a ice cream flavor
9:10but in this case uh you know then you're
9:12in english you say plain vanilla which
9:14means plain or simple so meaning of
9:16vanilla here means plain quite simple
9:19other country instruments may uh you
9:21know are not tradable right and meaning
9:24non-negotiable
9:25namely for counterparty to otc is to
9:28over the country what you see
9:29instruments to sell the role in a
9:31contract to someone else the other party
9:34has to agree and may ask for payment in
9:36exchange
9:37and the process of this sale is called
9:38the innovation right it's not like a
9:40stock i buy the stock of amazon i don't
9:43need permission of amazon to buy it
9:46then i may sell the stock to someone
9:47else i don't need uh permission of um
9:51uh
9:52for amazon to sell it
9:54right so in our
9:56you know stock is functional right so
9:58otc instruments are contracts between
10:00two parties and these are two specific
10:02parties
10:03unless specifically provided in the
10:05contract
10:06you cannot just say well i'm going to
10:08give my obligations to someone right so
10:09sometimes actually the contract says
10:11that you can
10:12but in most cases it doesn't
10:14uh in the process of doing so is called
10:16innovation but you need to agree uh with
10:18the other party to do that
10:20if the contract doesn't allow it up
10:21front
10:23and finally the meaning of being liquid
10:25is different for securities and for over
10:27the country instruments because of the
10:28different nation so for securities being
10:30liquid means being easy to buy and sell
10:33right so a lot of buyers a lot of
10:34sellers
10:35if you want to sell a security liquid
10:38security means that uh on any given day
10:42a lot of buyers and sellers and you can
10:44sell your shares for other over the
10:47country instruments it's a contract
10:48between two parties which exchange cash
10:50flows
10:51so there's no asset here to buy or sell
10:54being liquid in this case means that
10:56it's easy to find a counterparty that
10:58would agree to enter into this contract
10:59with you
11:01or agree to terminate the contract that
11:02you request for a payment representing
11:04the current value of your contractual
11:06obligation right so you know it's uh you
11:08know a swap is also quite liquid
11:11but
11:12the liquidity of swap means that it's
11:14easier to find someone who would agree
11:16to entry and to swap with you
11:18and it's also easy to uh either
11:20terminate the swap or to find someone
11:22who would enter with you into an
11:24offsetting swap right so now so if
11:26you're receiving floating payments and
11:28again so we'll talk about this mean in
11:30the future but uh their payments based
11:32on uh
11:34interest rate that changes um uh from
11:36time to time
11:37so fixed payment that's interest rate
11:40that doesn't so one way that you can
11:43sell quote unquote right or exit
11:45financial instrument which is over the
11:47country is to find someone else who will
11:49enter with you into an offsetting
11:52instrument that reverses your cash so
11:54any money you receive from one party you
11:56pay to the other party
11:57so you no longer have
11:59uh
12:00exposure to changes of for example
12:02interest rates right so liquidity has
12:04different meaning for over the country
12:06instruments
12:07compared to um
12:09securities they can still be liquid or
12:11illiquid but the definition is different
12:15now we're going to look at
12:17two specific types of uh derivatives
12:20today or specific what's called asset
12:22classes all right so the first asset
12:24class is equity derivatives
12:26in the second asset class is interest
12:28rate derivatives
12:29we will look at equity derivatives first
12:34and again so this concludes the first
12:37section so i wanted to remind if there
12:38is anything in the chat uh please let me
12:40know i can see there are two messages in
12:42chat but i don't see anything
12:44that's not questions right they are not
12:46questions right okay perfect
12:48all right so uh
12:50now in this you know now now we're going
12:52to
12:53learn about equity generators right and
12:55then after that uh we learn about uh
12:57interest rate derivatives
12:59okay so the simplest equity derivative
13:02is single name equity forward single
13:04limited forward is an over the country
13:06derivative which stock shares is the
13:09underlying
13:10there's a very similar
13:12security which is a different slightly
13:14different type of derivative which is
13:16called equity futures
13:17uh so it's uh beyond the scope of this
13:20lecture to uh to discuss um uh you know
13:24the difference between them they're very
13:26similar
13:26uh one is uh traded over the country
13:29that's the one we will study
13:31the other one is traded on an exchange
13:33and there are very tiny difference
13:35between them but particularly linked to
13:38related to standardization
13:40and the way that you uh prove to the
13:43other party that you are creditworthy
13:45right so so in our you know one one of
13:48the ways is to post some money
13:51called the margin to
13:53prove to them that you have enough money
13:54enough money to
13:57meet your financial obligations
13:59and uh you know the way that this works
14:01is different but
14:03more or less they're very similar we
14:04will study here equity forward
14:06so single name equity forward is an
14:08emergency derivative which talkshares
14:10are then blank
14:11all right so uh let me uh answer the
14:14question right so uh is your main idea
14:17that many derivatives mostly illiquid
14:19well you know i don't know how to define
14:21many uh i think
14:23probably i would say the opposite is
14:25true
14:26namely most of derivatives are liquid
14:29simply because uh liquid derivatives are
14:32those that are more popular trade it
14:34more frequently
14:35and therefore there are more of them
14:37right so in some so i think in some way
14:39uh i would think the opposite is true
14:41right so most of the generators are
14:42liquid but there are a lot of
14:44derivatives that are illiquid
14:47bespoke derivatives are liquid in a
14:49sense that they have very particular
14:51structure and interest only one market
14:53participant so you cannot by definition
14:56find others uh who are interested in
14:58them because they're tailored to the
15:00needs of one right so the bespoke are
15:02illiquid
15:03but most of them are liquid right swaps
15:05are liquid so some of the largest uh you
15:07know volume uh
15:09in interest rate derivatives are swaps
15:11they're liquid
15:12so they are very liquid uh some of the
15:15largest uh or the largest volume in
15:18equity derivatives are forwards and
15:22you know simple uh
15:24options like uh calls and plots they're
15:26also very liquid so actually no it's
15:29opposite right so there are more liquid
15:30derivatives than in liquid but both
15:32exist
15:34all right so single name equity forward
15:36is an odc derivative uh over the country
15:38derivative which stock shares is the
15:40underlying
15:41and single name means that underlying is
15:43shares of a single stock not a group of
15:46stocks such as basket or index so there
15:48are other types of forwards for example
15:52forward una aggregate or average price
15:54of the group of stocks
15:57which form a stock index uh are index
16:00forwards
16:01they're also basket forwards when it's
16:03not an index that a company like dow
16:06jones or someone else
16:07or standard and poor's right you know
16:09put together is the best case that you
16:11define as market participant
16:13but we will talk about single name
16:15equity forward but we'll emit a single
16:17name you know and uh
16:19that's the forward and language to stop
16:21okay another question
16:23how are derivatives prices evaluated on
16:25stock exchange you know stock exchange
16:27derivative prices are evaluated a very
16:29good question
16:31exactly the same way as prices of stocks
16:33themselves right so on a stock exchange
16:36derivatives prices are evaluated by
16:39market makers
16:41who
16:42establish market clearing price for a
16:44derivative if it's traded on a stock
16:46exchange like for example call option
16:49or forward so they establish market
16:51clearing price
16:53such that at any given point on time
16:55there is the same number of buyers and
16:57sellers for that price right if the
16:59price is very high
17:00nobody would would buy everybody would
17:02sell if the price is very low nobody
17:04will sell
17:06everybody would buy
17:07so there is somewhere in between the
17:08price at least will be approximately the
17:11same number of buyers and sellers and if
17:13it gets a little bit out of balance the
17:15market makers move the price so it gets
17:17back in balance in the market maker only
17:20temporarily accumulates a little bit of
17:22what's called position or on ownership
17:24of this security
17:26uh their objective is to set the price
17:28such that they unloaded
17:29and at the end of the day there is
17:31something called the closing auction to
17:32make sure that uh all of the positions
17:34are closed out
17:35and the market maker is left with no uh
17:38you know position whatsoever
17:40until the next trading session
17:42however
17:43even if uh the derivative was traded on
17:46exchange you still need derivatives
17:47valuation because stock
17:50is really a fundamental price right so
17:53stock is something that
17:55um
17:56essentially you know fundamental
17:57analysis which is the only way to in the
18:00prices talk independently of financial
18:02markets
18:03gives you an answer that can be
18:06you know significantly different
18:08and for some of the most exciting stocks
18:11like things like tesla or amazon and so
18:13forth they're like multiple times
18:15different from what fundamental analysis
18:16tells you because investors build
18:18castles in the air
18:21for derivatives
18:22there's a little bit less of that
18:24because
18:25derivative price
18:27has some very definite connection
18:29to the stock price
18:31there is also an unknown and derivative
18:33price and we'll talk about it later
18:35which is separate from the stock prices
18:37the volatility right so derivative price
18:38is determined by
18:40stock price volatility or variation of
18:43stock price from day to day
18:45and also what investors think about risk
18:48so there is a big element of uh
18:52price discovery there as well in other
18:54words uh what fundamental analysis is to
18:57stock
18:58derivative relation is two options right
19:00so in other words you can
19:03look at historical volatility you can
19:05look at
19:06stock price you can look at
19:08other things and say that's what the
19:10price option price should be
19:13but
19:14in reality it will be different from
19:16that right so and uh
19:19the way to
19:21you know the price of derivatives is
19:23set for on exchange is price discovery
19:27the greatest value of derivative
19:28evaluation
19:30is to price bespoke derivatives because
19:32bespoke derivatives or illiquid
19:33derivatives
19:35they
19:36are not traded on exchange so you cannot
19:38do price discovery in exchange
19:40and when you cannot do price discovery
19:42in exchange what derivative relation
19:43does is that it observes prices of
19:46options which are liquid
19:48which are related or similar in some way
19:50for example there may be options on the
19:51same stock
19:53and then build a model so model is
19:56basically a mathematical
19:58you know algorithm for coming up with
19:59the price
20:00which takes is input prices which i
20:03discovered by trading on exchange and
20:06produces price of a bespoke or a liquid
20:08instrument
20:09so on the exchange
20:11just price discovery by matching buyers
20:13and sellers
20:14but for something that's not traded on
20:16exchange we just you know basically
20:18either liquid or does not meet the other
20:20requirements that's where derivative
20:22relation becomes the primary method
20:24of foliation
20:26and when it uses
20:28derivatives prices
20:30discovered by matching buyers and
20:32sellers on exchange as input it becomes
20:34a lot more accurate compared to models
20:36that don't right so uh so there's a you
20:39know a lot of value in other words
20:40people who trade simple derivatives in
20:42the market
20:44they do it because
20:46you know they need to trade this
20:47instruments for their own purposes but
20:49they also provide a public service
20:52by helping to price other instruments
20:55because we can use the price discovery
20:56that they occur during trading to price
20:59other instruments so it's like a public
21:01service basically done by market
21:02participants to help
21:04people who don't stroke trade on
21:06exchange but still want to participate
21:07in markets
21:09any other questions before i continue
21:13not yet
21:14okay good
21:15all right so now uh so uh back to equity
21:18forward right so single name means uh
21:20underlying is a single stock and equity
21:22forward uh is an obligation to buy a
21:24specific number of stock shares on this
21:26maturity date of the forward right so
21:28stock does not have maturity date it you
21:30know
21:31we hope if you buy a stock that will
21:33last forever it is long enough but
21:34forward does right so forward has a
21:36maturity date
21:38so equity forward is an obligation to
21:40buy a specified number of stock shares
21:42in a maturity
21:43a predetermined price
21:45for one counterparty and the obligation
21:47to sell it
21:48on the same terms for the other
21:49counterparty right so now it's two
21:51parties two market participants one
21:53agrees to buy
21:55no matter what right so it's not an
21:56option it's basically it's an obligation
21:58right so one
22:00has an obligation to buy the other has
22:01an obligation to sell
22:03at the predetermined price that this
22:05party is agreed to in advance and that's
22:06called the forward price
22:09right and the can the party that has the
22:11obligation to buy is said to have the
22:12long position
22:14in the underlying and the contemporary
22:15that has the obligation to sell is the
22:17short position right so long position is
22:20um uh something like um
22:23a long position is essentially something
22:25like owning an asset right but in this
22:27case it's an obligation to buy it later
22:29as opposed to
22:30outright ownership
22:31in short position is obligation to sell
22:33but you can think of it as obligation to
22:35buy a negative amount right so it's a
22:37kind of mathematical
22:38way to think about selling is buying a
22:40negative amount right
22:42okay so uh now for the equity forward
22:46there may be physical settlement or cash
22:48settlement so physical settlement means
22:50that it's really you know one party
22:52agrees to acquire the asset and the
22:54other one to agrees to
22:57sell this asset and that can be stroke
22:59or commodity and so forth at the forward
23:01price
23:02and physical settlement
23:04for stocks is a little bit of a nuisance
23:07because basically as the time approaches
23:09or in advance you need to buy the stock
23:12if you have an obligation to
23:13sell
23:14you know you have to buy it in advance
23:16and then um you know you sell it to
23:18someone
23:19or if you have an obligation to buy then
23:21you will receive it from the other party
23:22and then if that's not the stock that
23:24you want to invest in you have to sell
23:26it right so a lot of uh futures and
23:29forward contracts
23:30and we're talking about forwards here
23:32a lot of these contracts are um
23:37a lot of these contracts are
23:39purely for financial
23:41means right so now it's a lot of this
23:43contracts is
23:44you know the objective is to get the
23:46financial or you know cash equivalent
23:49of doing this transaction you don't
23:50necessarily want to buy the stock if you
23:52get the stock as part of the transaction
23:54you will sell it right
23:55in this case it's easier to do cash
23:57settlement meaning the parties do not
23:59exchange any actual shares of stock and
24:00one party will simply pay the other
24:02what the cost of physical settlement
24:04would have been
24:06cash settlement is even more valuable
24:08for underlying which are not stocks for
24:12example and the like maybe grain
24:14right
24:15so
24:16you would have you can have a future
24:19on the price of price
24:21and in case of physical settlement you
24:23actually have to get
24:25a real car full of rice
24:28to someone to the other party
24:31you may not have a real card of rice
24:33right desire party may not need it right
24:35all they want is to protect themselves
24:37against price fluctuations in rice
24:39that's like remember the example from
24:41the first lecture about the samurai
24:43right
24:44so uh in fact uh you know with this uh
24:46futures contracts um you know the point
24:49of this contracts back in 16 or so in
24:5117th century
24:53uh was to protect against price
24:55fluctuations you don't actually need the
24:57you you don't have the rice from
24:59the next two harvests in the future
25:01right so you have a future
25:03with maturity of two years
25:05you will need to
25:06in this there's a physical settlement
25:08you will need to either store the rice
25:10you need to find the rice you don't want
25:12to do it it's even worse when it's
25:14cattle right there is a futures one pork
25:17bellies right pork bellies are pigs
25:19right this cattle
25:21so when you're buying and selling cattle
25:23you have to feed the cattle you have to
25:25support the cattle
25:27uh not an easy drop right so cash
25:29element simply that the parties exchange
25:31the cash that's equivalent to
25:33what they would have
25:34you know money would they would have
25:36received by selling the underlying it's
25:37a lot more convenient
25:39right
25:40well you know the purpose of uh
25:42having both is because uh
25:44you know some sometimes uh people
25:46actually do want the underlying
25:48right but uh you know a lot more it's a
25:50lot more convenient usually to have cash
25:52flow
25:53all right so uh now the uh in another
25:57example just in passing right so
25:59sometimes for stocks cash settlement is
26:01useful because um in some countries
26:05for example like china
26:06uh there are restrictions on foreign
26:08ownership of nationally significant
26:11companies for example defense
26:13so if you want to participate in the
26:15financial success of the company
26:17but government is not just china you
26:19know there are many
26:20germanium
26:22countries where there are similar
26:23restrictions
26:24so if you want to participate in the
26:26financial success of a company
26:29but
26:30government regulations prevent you from
26:32actually owning the shares
26:34then uh cash settlement is one way that
26:37you can participate in a financial
26:38success
26:39uh and bet essentially on what the price
26:42uh will go but without
26:44having to own the shares and then you
26:46you can do it even if you're not
26:47permitted to actually buy the shares
26:49so we'll assume cash flow right here all
26:52right now uh
26:54we came to a very critical point and
26:57i'll spend a lot of time on the next two
26:58slides because uh the slides uh describe
27:02a very important notion that's the
27:04underpinning of derivatives valuation
27:07theory
27:08and that notion is efficient markets
27:11theory of efficient market hypothesis
27:13and no arbitrage conditions on the
27:16financial market
27:17so efficient markets theory sometimes
27:19it's all hypothesis but it's really not
27:21a hypothesis anymore you know it's
27:22really a theory right
27:24it's proven right that it stays in
27:26financial markets there are no
27:28opportunities for guaranteed risk create
27:30profit
27:32okay so all of us have seen
27:34advertisements saying here you know
27:37strategy trade forex right we will teach
27:39you how to trade forex without risk and
27:41make a lot of money usually with a
27:42picture of someone on the yacht
27:44okay people who follow this
27:46advertisement usually end up not on a
27:48yacht but maybe on a yacht as a waiter
27:51right so as a seller
27:53because there is no such thing as
27:54risk-free profit in financial markets
27:56right so this profit is called an
27:58arbitrage opportunity
28:00in its substance there's no arbitrage
28:03and the reason it doesn't exist is
28:05because if
28:06riskless arbitrage opportunities existed
28:09then people would find them and people
28:11who find them called the arbitrage
28:13and bankrupt those who offer this
28:15opportunity
28:17for example
28:18suppose that
28:19you decided to sell stock at one dollar
28:22less than the price on exchange right
28:25arbitrage would see that you're selling
28:28stock cheaper than the price on exchange
28:30and will simultaneously buy a share from
28:32you and sell one on exchange they will
28:34earn a dollar you will lose a dollar
28:36right so then they do it again and again
28:39and again
28:40until
28:41the seller runs out of stock runs out of
28:44money to buy stock right because every
28:46time this happens
28:55all the markets
28:56creating the opportunity for his close
28:58profit
28:59would simply you know a lot a lot of
29:00people
29:01will jump on the opportunity
29:03and bankrupt
29:05the person who is selling
29:07right uh uh you can still hear me
29:10i think
29:18okay so all right
29:20now uh this drawing uh below shows that
29:24there is a couple who walked past a pile
29:26of money on the ground
29:28right
29:29because they believe in the efficient
29:31market hypothesis they knew that if
29:32there is a pile of money in the ground
29:34this riskless profit opportunity
29:37and they were absolutely correct not to
29:40pick it up because they knew it's an
29:41illusion
29:42of course on the real street it's not an
29:43illusion
29:45but in financial markets when you see a
29:47pile of money and it seems that there is
29:49no risk and you can just take it
29:51it's an illusion you're missing
29:52something right so there is a risk you
29:54don't realize
29:55uh maybe there's a transaction cost you
29:57don't realize
29:59if you see in financial markets the
30:00opportunity to make money without risk
30:04is an illusion right on a street is not
30:06but
30:07these people in financial markets they
30:09were correct right and this other person
30:11who is picking it up is up for a
30:13surprise right so maybe there is some
30:15risk
30:17maybe you know there is a
30:19you know some transaction cost uh
30:22something is really
30:24you know wrong here and uh this
30:26opportunity cannot exist in financial
30:28markets
30:29okay but there's a chicken neck problem
30:31here right because if there is no
30:33arbitrage there can be no arbitrators
30:36right and if there are no arbitragers
30:38who would then ensure that there is no
30:40arbitrage
30:41well the reason uh you know this uh
30:44you know
30:45this really seemingly you know a
30:47contradiction is because in fact tiny
30:49very tiny arbitrage opportunities exist
30:52with profits just barely sufficient for
30:54a betrayal to make money
30:56and i betray yourself
31:06the
31:07trading strategy and the people tell you
31:09that uh it's risk-free and you make a
31:11lot
31:14not true right there is a risk there
31:16maybe you make money for some time right
31:17but there is a risk there that
31:19is not recognized
31:21if you're a bank or corporate you still
31:23don't have the technology
31:26to banks and corporates still don't have
31:27the technology
31:28to take advantage of arbitrage i bet
31:31reverse are very highly specialized
31:32terms some of them are called high
31:34frequency traders and they make trades
31:37with execution time in microseconds not
31:40not even milliseconds but microseconds
31:42these are the people
31:43you know i was once talking to the
31:45arbiter treasurer
31:47uh and he was complaining about how slow
31:49the speed of light is
31:50because he was arbitraging trades
31:52between chicago and new york and it
31:54takes a few microseconds right those who
31:56are physicists you know that takes a few
31:57microseconds for light to get through
31:59the fiber optic cable between new york
32:02uh and chicago
32:04and that's long enough that people who
32:06have better cable that has the faster
32:08speed of light it depends on the density
32:10right so if you go from microwave tower
32:12through the air
32:13your light gets there like your signal
32:15gets there like a microscope consumer
32:17and you can make the trade first so
32:18these are the people with technology who
32:20can make arbitrage in stock trading
32:23other types of trading uh arbitrage
32:25don't need technology like that but they
32:26need other technology there is something
32:28called statistical arbitration so
32:30normal market participants
32:33they can assume that there is no
32:34arbitrage opportunities because there
32:35are so tiny and they get eliminated by
32:37specialized firms i betrayed so quickly
32:41that when we are pricing generators we
32:43can assume that there is no arbitrage
32:45right
32:45and arbitrary source actually they
32:47perform amazing public service
32:49and they're very brave
32:51because uh for example um the
32:54uh you know one very well known uh in
32:56financial markets very well known uh you
32:58know disaster
33:00happened to
33:01an asset manager hedge fund called ltc
33:03long-term capital management which
33:05employed some extremely smart and
33:07qualified people
33:09several of them nobel prize winners and
33:11one of these nobel prize winners
33:13in trying to explain to investors why
33:15this firm makes money and takes no risk
33:18would take a nickel five cent coin
33:20out of the pocket and say we can pick up
33:23nickels five cent coins from the floor
33:26where nobody can see them and that's how
33:27we make money
33:28well the thing is that
33:30when you're picking up nickels from the
33:32floor you should make sure that there is
33:33no steamroller just behind you because
33:35if you're focusing on picking them up
33:37you can get run over by a stimulator and
33:39that's your risk right so this guy here
33:41you know doesn't see the steam level so
33:43that's exactly what happens to ltcm so
33:45they were picking up nickels
33:47and one day the steam roller just got
33:49behind them
33:51and what they thought was a completely
33:52riskless trait
33:54of spread difference in interest between
33:56two different types of bones
33:58instead of disappearing as they
33:59anticipated
34:01with the
34:02you know adoption of the euro it
34:03actually temporarily widened
34:05in the fund when bankrupt and lost money
34:07for the investors so arbitrage
34:10sometimes it's not actually truly
34:12riskless what they do right there's
34:14still some risk maybe a low risk but a
34:16very high loss and they perform this
34:18amazing public service such that we can
34:21have the rest of the market participants
34:23have a reliable way to price derivatives
34:26and that's a you know basically a
34:27benefit to everybody so so they are
34:29heroes who you know help us
34:32help financial markets to function
34:33efficiently okay there is a question and
34:36by the way these two slides were
34:37absolutely critical point on which a lot
34:39of other
34:40um uh you know
34:41we should wish basically all of the
34:43pricing models derivatives variation
34:44models are based so if there are any
34:46questions you have about these two
34:47slides
34:48please ask them and i'll answer
34:50all right in the meantime there's
34:51already a question there are three forms
34:52of different market hypothesis weak
34:54semi-strong and strong according to
34:56experience which one is correct okay
34:58that's actually a brilliant question
35:00it's not the introductory math finance
35:02question uh i will be very happy to
35:04answer so please um send a message to um
35:07uh you know
35:09with your contacts to the chat or
35:10private to anastasia uh i will answer
35:13after the lecture unfortunately uh
35:15that's the
35:16question which i cannot answer quickly
35:19it's also a very high level question
35:21uh and uh this will distract us from the
35:23slides that we need to go through and i
35:25don't think we have time so so brilliant
35:27question but uh but it's not really an
35:29introductory math finance question and
35:31more like advanced
35:33uh any questions about specifically
35:35arbitrage when the pastor slice
35:39okay so let's continue right all right
35:42so uh now how do we use uh no arbitrage
35:46or
35:48efficient market hypothesis and
35:49derivative variation well one way to use
35:51it is called static replication so
35:53static application is evaluation
35:55technique which involves contracts
35:57constructing a static replicating
35:58portfolio from a long
36:00positive right or short negative
36:02position so you're long being uh you
36:04know buying positive amount right short
36:06meaning buying negative amount
36:08positions in the instrument underlying
36:10in cash
36:11the static replicating portfolio is
36:13designed to make exactly the same
36:15payments as the derivative instrument
36:18it replicates in all future states of
36:20the world
36:21and static replication is not always
36:23possible but when it is possible right
36:25it's not possible when uh
36:27it's what's called an option right so
36:29when the
36:30one of the parties has a choice to do
36:32something or not to do something but
36:34it's often possible when there is no
36:36choice when the two parties like for the
36:37forward so one party agrees to buy the
36:40other party agrees to sell there's no
36:41choice right so that's one of the cases
36:43when static replication is possible
36:45so it's possible
36:47norbit rush condition right to efficient
36:49market hypothesis means that the price
36:51of a derivative instrument
36:53is the same is the price of investing in
36:55the replicating portfolio right because
36:58if the replicating portfolio
37:01would have different price
37:02of entering into it than the price of
37:05the derivative
37:06in this case there will be arbitrage
37:08between
37:09buying a derivative or entering into a
37:11derivative and then
37:13uh
37:14entering into the reverse position in
37:16the replicating portfolio in that
37:18arbitrage is exactly the same as with
37:20buying and selling the same stock at a
37:22different price
37:23so no arbitrage condition tells us that
37:25derivative price is equal
37:27to the price of the replicating
37:29portfolio and all of the derivative
37:31pricing models are based on replication
37:34right at the static application we're
37:36talking about here
37:37or dynamic replication which we'll talk
37:39about a few slides later
37:41so when calculating the price of
37:43investing uh
37:45you know in something right so long
37:46position is treated as adding the price
37:48and short position is subtracting to the
37:50price right or adding a negative amount
37:52in because the price of the replicating
37:54portfolio is the linear function of the
37:56underlying price an instrument for which
37:58static replication is possible is called
38:00the linear
38:02instrument all right so variation of
38:05equity forward by static replication
38:06right so let's say let's
38:08put us put this to real use right unless
38:11calculate the price of the forward
38:12derivative or derivative of
38:15the forward um uh forward
38:17uh instrument
38:18let's assume that the link is one share
38:21of stock that currently trades at the
38:22price um
38:24uh of um
38:26at the currently trade trade trades at
38:27price uh s right so uh i hope one second
38:31so i hope this window is not blocking it
38:33right so um uh it currently trades at
38:36price s and the forward price is 100.
38:39let's also assume that the stock does
38:40not pay dividends to its owner
38:43and let's assume that there is discount
38:45factor is equal to one or in other words
38:47interest rate is zero right just for
38:49simplicity in fact of course they're not
38:51equal to one so discount factor is not
38:53one because there are interest rates
38:55but
38:56for this simple model we will first
38:58assume is equal to one
39:00the static replicating portfolio for the
39:03forward consists of a long
39:05positive position and one share of stock
39:07and short negative position in a hundred
39:09dollar cash
39:11so the price of entering into the
39:13replicating portfolio into essentially
39:15acquiring the financial position
39:18is buying one share of stock for s
39:21dollars
39:22and then
39:26basically um uh making an obligation to
39:29give someone 100 cash or rather
39:31more precisely you're buying the stock
39:33right today and then you're making an
39:35obligation to then give her to someone
39:37so it's s minus 100.
39:40in the equity forward uh price
39:43or the price of this replicating
39:44portfolio which no arbitrage tells us
39:46should be the same as equity forward
39:47price
39:48does not depend on the volatility of
39:50volatility as we started in the previous
39:53lecture
39:54is the amount average amount or mean
39:57square of the
39:59change in the stock price
40:01over overnight from one day to the next
40:04all right so daily volatility of all is
40:06financial markets or squads or you know
40:08people who build uh derivatives models
40:11uh call it daily volatility what they
40:13abbreviated to vol
40:15daily wool is the mean square mean
40:17square
40:19average of the
40:21change in stock price from one day to
40:23the next
40:25but
40:25for forwards
40:27it does not enter the forward price
40:30because we only need to know the price
40:31of the replicating portfolio today not
40:33in the future right because in every
40:35state of the world the static
40:36replication so this portfolio that we
40:38construct today
40:40we don't need to change it in every
40:42possible state of the world
40:44independently of what the stock price
40:46will be it will be equivalent to afford
40:49right so
40:50what it means is that the price of the
40:51forward is the same as the price of the
40:53replication portfolio in s minus 100 so
40:56when the stock price today is 100
40:59it costs nothing to enter into the
41:00forward so the price is zero
41:03if the price today is 105 or 95 then you
41:06know if it's 105 it's five dollars
41:09otherwise it's minus five dollars right
41:11so meaning that uh five dollars means
41:13that if you find someone to take over
41:15your position they will give you five
41:16dollars
41:18negative means that in order for someone
41:20to take over your position because
41:21remember derivatives are bilateral
41:24contracts would rather
41:25sorry over the country derivatives have
41:27let all contracts you cannot just sell
41:29it
41:30you need to find another party to assume
41:32your position
41:33so if the price of the forward is five
41:36you
41:37need to you get five dollars from
41:39someone who will assume your position
41:40maybe a little bit less because uh
41:42that's called a bit ask spread right
41:44because the party who takes over your
41:46position has to be paid for the service
41:48of uh doing this for you
41:50uh so this is a tiny difference but it's
41:52five dollars minus a tiny difference
41:55if the price of the forward is minus
41:57five it means that
41:58uh you will find someone to take over uh
42:02the position and you have to give them
42:04five dollars
42:05for the service plus again a tiny bit
42:07more
42:08uh you know because they're doing it for
42:10you
42:12all right so and the linear instrument
42:13is because the relationship between
42:15stock price and the price of the forward
42:17is a straight line right so it's a
42:19linear relationship that's what's that's
42:20one of the reasons it's called the
42:22linear instrument
42:24all right so now we're going to options
42:26again if there are any questions please
42:28let me know
42:30and
42:31the
42:32options uh we talk about single name
42:35equity option
42:37is an over the country derivative for
42:38which just like for the forward
42:41stock price is the underlying
42:43the ocean buyer
42:45or first contemporary has a right but
42:47not an obligation to buy a specified
42:49number of structures on each maturity
42:51date you can option but not the stock
42:53has the maturity date for predetermined
42:55price called the strike price the act of
42:58exercising the right to buy is called
43:00option exercise
43:02the option seller or the second counter
43:04party has the obligation to sell on the
43:06same terms as the other counterparty but
43:09only if the option is the exact exactly
43:11exercised by the buyer
43:13so the first party the option buyer or
43:15the option owner
43:17the party which has the option has the
43:19right but not an obligation
43:21the other party has an obligation
43:24but this obligation will only come to
43:26pass right you know should only be
43:27required
43:29if the first party decides to do it
43:32so option buyer has the right
43:34option seller has the obligation if
43:37option buyer chooses to exercise
43:39the right
43:41the option is called
43:42call option
43:44when the buyer has the right to buy
43:45shares so it's like call you know come
43:47to me right
43:49when option buyer has the right to sell
43:50shares it's called the put option so i
43:52take something that i have i put it away
43:54i don't want anymore right so it's ocean
43:56to sell
43:57and the option cannot be priced by
43:59static replication because in some
44:00states of the world the buy option buyer
44:02will exercise the option in others they
44:04will not
44:06so depending on whether or not the
44:07option is exercised a replicating
44:09portfolio is different
44:11while static replication requires it to
44:13be the same
44:16and
44:17static replication
44:18only works for some derivatives
44:20but dynamic replication
44:22always works that's a kind of
44:23interesting thing right so static
44:25replication only works for linear
44:26derivatives
44:28but you can prove and you know the proof
44:30is
44:31quite complex but you can prove that
44:32dynamic replication also always works
44:35in the way dynamic replication works is
44:37it involves selecting a time horizon
44:40constructing a different replicating
44:42portfolio in each state of the world for
44:43this horizon
44:45and averaging portfolio prices or the
44:47probability distribution for the horizon
44:51and because you take the average
44:53over the probability distribution and
44:55what is the probability distribution of
44:57right stock price in our case so the
44:59option depends on stock price
45:01stock price which is 100 dollars today
45:05it may be 105 in the future it may be 95
45:08and maybe 80 or 120
45:10so there are different stock prices in
45:13the future
45:14and there is probability of these stock
45:16prices in the future
45:18when you do dynamic replication you
45:20construct a replicating portfolio for
45:23each of the future possible stock price
45:26and then average this prices of
45:28replicating portfolio over this
45:30probability distribution
45:32and because of this averaging the
45:34derivative price
45:36when there is an option
45:38when it's not a linear instrument when
45:39there's an option somebody has an option
45:41to do something uh by you know to
45:44basically exercise or not the rate of
45:46price uh option price depends on the on
45:48the volatility because uh if the
45:50volatility is low
45:51the future price is close to 100 initial
45:54price
45:55if the volatility is high it could be
45:57very different
45:58this will affect the averaging this will
46:00affect the possible probability of each
46:02stock price in the future and will
46:04affect the
46:06price of the derivative which is the
46:07average
46:09so generally to perform variation for an
46:10equity option by dynamic replication
46:12would have to start at maturity which is
46:14the last date that option still exists
46:17and work backwards in small time
46:19increments until we get to time equals
46:21zero right so if there is a generic kind
46:23of derivative instrument
46:25we would need to start from the maturity
46:27of the instrument
46:28and at that point we can construct
46:30replicating portfolio from the
46:32underlying and cache
46:35and then
46:36we will go back one day
46:38and in this case we construct the
46:39portfolio from underlying cash and the
46:43price of the instrument one day later
46:45and go backward again and backward again
46:47so in the most general case
46:50when this exercise decisions depend not
46:52only on the underlying price but also on
46:54the future price there are options for
46:56example called american or bermuda
47:00uh where you choosing between exercising
47:03right away or exercising in the future
47:05and to make this choice you need to know
47:07what the derivative price in the future
47:08will be so the replicating portfolio may
47:11sometimes depend on the future
47:14derivative price and by backward
47:16induction you're working from maturity
47:18of the instrument let's say one year
47:20from now
47:21back to today and at every step you're
47:23using what you just computed for this
47:25following day
47:26but our option only has exercise and
47:29cash flows at maturity
47:31and we only need to replicate the
47:32maturity date so what i'm
47:34going to describe is a simplification
47:37and this simplification only applies uh
47:39to options with european exercise uh in
47:42european exercise and you know it's
47:43simply because in europe uh this type of
47:46options they're called european and
47:47american because um european exchange
47:49has traded one type of options primarily
47:52with only one day american uh exchange
47:54is traded another type of ocean and
47:56bermuda is a island in the middle
47:58right so that's how historical names
48:00appeared of course today both types of
48:01options are traded everywhere
48:04we have a question when we replicate a
48:05portfolio are we doing this from the
48:07investment bank side or uh the
48:09investment funds use this yeah so they
48:11both both use this right so when you
48:12replicate the portfolio uh you do it
48:15from your side right so if your
48:16financial market participant if you're
48:18at a
48:19with the investment fund you replicate
48:21from your site if your
48:24investment fund trading with the bank
48:25you replicate it from the your side
48:27which is the opposite of the other side
48:29and for a linear instrument one price is
48:32equal to minus the other price but when
48:34there is also an option
48:36right you know in this case you have to
48:38remember that there are cash flows you
48:39change there is also option you own
48:42but still any derivative instrument
48:44is a contract between two parties
48:47and it's a zero sum contract
48:49so if the derivative price for one party
48:51is x
48:52the derivative price for the other party
48:54is minus x so if your investment fund
48:56you can also do it from the bank side
48:58just don't forget to put minus in front
49:01all right
49:02so uh because our option calendar has
49:04exercise cash flows and maturity we only
49:06need to replicate maturity date right so
49:08uh and let's now do the uh pricing in
49:11the simplified two-state model right
49:14so this two-step model uh
49:16is uh we're going to assume that
49:18the today the price is 100
49:21and the price at maturity has 50
49:23probability to be 105
49:25and 50 probability to be 95 right so
49:28it's a very simplified model
49:30where future stock price one year from
49:32now can only have two values
49:3495 and 105
49:37and each value has 50 probability
49:41so the price of the equity option is the
49:43average of replicating portfolio cost
49:45for these two states of the world right
49:47so if the stock price at maturity is one
49:50of five
49:51is more than one hundred so the option
49:53buyer can make five dollars by
49:55exercising the option at the price so
49:58strike price is 100
49:59we have the right but not an obligation
50:01to exercise
50:03if we exercise
50:04we make five dollars so we will why not
50:07make five dollars right
50:08so that means that you know that's
50:10called uh expiration in the money so in
50:12the money means uh the option holder or
50:15the option buyer decides to exercise and
50:18makes money from that right because
50:19otherwise why would they exercise
50:21so ocean payment and this replicating
50:23portfolio in this case is the same as
50:25for the forward because the exercise
50:26happens so in the state of the world
50:28when the price is 105 the replicating
50:30portfolio is stock
50:32minus 100
50:34and
50:35the profit is five dollars because
50:38you're selling you're basically
50:39receiving the stock that's worth 105
50:42but you're paying 100.
50:44so you made five dollars by exercising
50:47and in this state of the world you earn
50:49five
50:50if the stock price is 95 the option
50:54buyer will lose
50:55five dollars by exercising the options
50:57so they will decline to exercise right
50:59why would you exercise and lose five
51:00dollars you have the right but not an
51:03obligation so you say thank you but no
51:04thanks
51:05right and then that's called expiration
51:08out of the money right
51:09so when the option expires out of the
51:12money you wasted the money spent to no
51:14option right
51:15you had
51:16option you know i said you had the
51:18option if you had the right the right
51:19was valuable to you but in the end you
51:21didn't use the right
51:23however uh you know if you uh
51:26exercise you lose even more money right
51:28in this case you lose the price and you
51:30also lose extra money so in this case
51:32you say well you know i paid for this
51:34right but i didn't use it and that's
51:35fine i i'm glad you had it i had it
51:37that's right
51:39so
51:40you will not exercise in this case the
51:42price of replicating portfolio is zero
51:43and the replicating portfolio is just
51:45not having any portfolio
51:48so without dynamic replication we have
51:5150 probability of five dollar profit 57
51:55probability of zero
51:57profit
51:58and again um
52:00there are a couple of simplifications
52:01here so one of these probabilities are
52:03actually not um
52:05actual probabilities but something
52:06called risk neutral probabilities which
52:08also adjust for
52:10the risk that uh you know investors take
52:12but again this is not an introductory
52:14lecture topic
52:15right
52:16we will however cover this topic a
52:18little bit later um uh
52:21uh today when when we talk about um the
52:23straight the next lecture apologize so
52:26so we will cover uh the market price
52:28risk uh later in this course but for
52:30today uh we'll just assume that these
52:32are probabilities uh without specific
52:35you know what kind of probabilities and
52:36these probabilities are each 50 percent
52:38so we'll make on average two and a half
52:40dollars
52:41and by no arbitrage it means that the
52:43option is worth
52:45in this model
52:46in this world where stock can go to
52:49105 or 95 an issue with 50 probability
52:52the option price will be two and a half
52:53dollars
52:55okay so now let's uh calculate how the
52:58option price depends on the stock price
53:00right so previous calculation was uh for
53:02100.
53:05uh the previous uh
53:07previous uh calculation was assuming
53:09that this price today is 100 let's
53:11assume that the price today could be
53:12different
53:13if the price is 95 i'm going to go
53:15quickly but you can verify it right so
53:18if the price is 95
53:21in this case there is a 50 probability
53:23of a price 100 and 50 probability of 90
53:26in both cases you don't want to exercise
53:29right because uh
53:30in one case you just
53:32get zero in other case you lose ten
53:33dollars so the option price is zero so
53:35if the stock price today is 95 the ocean
53:37price is zero
53:39if the stock price today is 105
53:41same calculation
53:43the option price is 5.
53:45is the stock price is 110
53:47then
53:49price is 10
53:51and
53:52when you do it for a couple of other
53:53prices you will notice that for any
53:56price below 95
53:59the option always expires at the money
54:01basically if the price today is below 95
54:04then
54:05both two tools the price is right so
54:07we're assuming that the model says that
54:09the price has 50
54:1050 probability of going up five dollars
54:1350 percent probability of going down
54:15five dollars so the price is below 95 it
54:18never reaches the strike price
54:20so the option is always worthless
54:23below 95
54:24below the stock price
54:26level of 95 dollars today
54:28the option is worthless
54:30so the price is zero
54:33if the price is
54:35above 105 today
54:37then you exercise in both cases and in
54:39this case the price is the same as the
54:41forwardness linear
54:43and in the middle is something uh
54:45basically it's also a straight line but
54:47with a lower slope so it looks like this
54:49right so uh ocean price below 95 is zero
54:52above 105 is same as the forward and
54:56here it's also linear but with a lower
54:58slope
55:00okay so let's now
55:02improve our model and assume that there
55:03are four state models because the
55:04previous model is not realistic it was
55:07not realistic in one way so the
55:08direction of the move in the stock price
55:11was random
55:13but the
55:14um
55:16magnitude of the move was always five
55:18dollars in reality of course the
55:20magnitude of the move over one year is
55:22not exactly five dollars it's also
55:24random
55:25so we will improve the model by assuming
55:27the stock has 25 probability to change
55:29by
55:30seven and a half
55:31or two and a half dollars
55:33in either direction
55:34so in this case we have random direction
55:37and also random magnitude
55:38more realistic model
55:40so in this case if we repeat the same
55:42calculations we will see that
55:44it's also
55:450 below 95. it's also linear above 105.
55:49and here
55:51if you mean instead of two segments so
55:52instead of one segment there are
55:55several smaller segments
55:56uh which all uh basically have slightly
55:59different slope
56:01and they kind of provide kind of sort of
56:02a smooth transition
56:04between this limits right and
56:08if we keep doing it and assuming even
56:10more realistic models in which the stock
56:12moves in very small increments daily
56:15and instead of discrete
56:17binomial or quite like you know our
56:19first model had binomial distribution
56:21rate two possible states right so our
56:23second model has a quadro right so
56:25there's like four possible states so now
56:27we assume that there is a continuous
56:28distribution if you uh assume that the
56:31stock has
56:33moves uh
56:35continuously
56:36and the moves uh
56:39are not correlated it's like a random
56:40work basically there's no correlation
56:42between the moves then you will assume
56:44that then the distribution there will be
56:46a random walk
56:47or winner process
56:49and the distribution of the end point
56:50will be gaussian
56:52right and that's exactly what happens
56:54when the stock moves daily driven by
56:56tiny news right so you know something
56:59happens in the world uh industry you
57:01know good news about the industry bad
57:03news about the industry
57:05good news about the inflation bad news
57:06about the inflation so every day there
57:08are some news that move the stock
57:10and if we assume that this is a random
57:12walk or winner process then the end
57:14point will have a gaussian distribution
57:16and to be precise is the change of log
57:18of the price of this gaussian right so
57:20again so
57:21unfortunately it's beyond the scope of
57:23this lecture
57:24uh just you know to for to provide some
57:26context we're trying to
57:28uh compress uh you know a multi-year
57:31mathematical finance uh education into
57:33four lectures
57:35so
57:35regretfully
57:36i cannot do it uh you know the same they
57:39say i cannot cover all of the same
57:40materials that mathematical finance
57:42graduates
57:44uh you know would learn right so uh so
57:46we cannot go into like normal models and
57:49ways to look and not the price itself is
57:51gaussian
57:52but uh it's not a huge difference right
57:55so you know the reality is gaussian is
57:58the look
57:59but in our simple model we assume that
58:01gaussian you know prices gaussians
58:03itself
58:04when the change is not very large
58:05they're almost the same because uh
58:07logan's you know locally logos close to
58:10linear
58:11right so we'll ignore this distinction
58:13in this case it will look something like
58:14this right so our
58:16four transitions our two transitions
58:17become four transactions now it becomes
58:19kind of a continuous smooth curve which
58:21approaches well in this case you know
58:23with this tiny increments i actually
58:25will be above 105
58:27the stock can be load 95 so the option
58:30price in this case
58:33is a function of stock price it will
58:34never exactly be equal to the forward
58:36they'll approach it when the stock price
58:37is high
58:39it will never be exactly zero but will
58:41approach zero when the stock price is
58:42very low
58:43and we'll have a kind of a continuous
58:44transition which looks like this and
58:47that's the price of equity option
58:51and now we're ready to go
58:53to interest rate derivatives on which we
58:55spend the remaining half an hour of this
58:58lecture
58:59do you have any questions about equities
59:00it will be a good time to ask
59:07okay so no question so far but you know
59:09feel free to type them in a shot
59:11all right now interest rate derivatives
59:13right so we start from the simplest
59:15interest rate derivative which is uh
59:17rather instru trade instrument that's
59:19not even a derivative yet
59:20called the zero coupon bond
59:22this instrument
59:24is traded on the financial markets
59:25especially for short maturities
59:28but uh it's not the most popular
59:30instrument
59:31but it's absolutely essential and very
59:33important instrument for constructing uh
59:35interest rate derivative evaluation
59:37models
59:38so we'll study this instrument not
59:40because of its prevalence even though
59:42they are traded for short maturities but
59:44because of the
59:46importance
59:47of this instrument
59:48for model construction
59:51all right so zero command bond is a bond
59:54that has only the principal
59:56but no interest rate payments so for a
59:58fixed rate bond the issuer says the
1:00:00fixed rate so that the bond price is 100
1:00:03and this cannot be done for zero coupon
1:00:05bond because it does not pay interest
1:00:07right so remember we started fixed rate
1:00:09bond in the previous lecture and we
1:00:11concluded that this like the clock which
1:00:13uh
1:00:14sorry
1:00:15you know basically because the issue of
1:00:17the bond is setting interest
1:00:20such that the price at uh
1:00:22when it's issued first is equal to the
1:00:24principal
1:00:26uh
1:00:27the the price of the fixed rate bond uh
1:00:30is equal to the principal when it's
1:00:31insured and also at maturity
1:00:34with zero coupon bond it cannot be done
1:00:35because there is no rate to set right so
1:00:37it does not pay interest it only pays
1:00:39the principal
1:00:40so instead
1:00:42three months let's say there is a zero
1:00:44coupon bond with three month maturity so
1:00:46zero coupon bond investor makes money
1:00:48only difference between the cost of
1:00:49buying the bond from the issuer address
1:00:51price
1:00:53and then receiving the principal of 100
1:00:56later right so today you buy the bond
1:00:59at whatever price there is there is
1:01:02and three months later you receive the
1:01:03principal
1:01:05so you don't receive any interest you
1:01:06make money you lose money on the
1:01:08difference between these two amounts
1:01:12so this charge shows the possible price
1:01:14trajectories over zero coupon bonds
1:01:16without the associated random noise
1:01:18right so normally you know the bond
1:01:20price would go up and down so this thing
1:01:22let's assume that it just uh
1:01:24interest rates remain static
1:01:26so unlike for fixed rate bonds zero
1:01:28coupon bond is a clock which shows
1:01:30correct time only once a day so remember
1:01:32fixed-rate bond is like a broken clock
1:01:34which shows correct time twice a day
1:01:37for zero coupon
1:01:38bond it only is like a clock which was
1:01:40created like 24 hour clock it shows
1:01:42correct time only once a day
1:01:44namely it's only equal to the principal
1:01:46at maturity right so uh
1:01:48at origin uh the price is different
1:01:51because there is no interest
1:01:53and now we have the poll
1:01:56right so
1:01:57can do you think the price of zero
1:01:59coupon bond can exceed its principle
1:02:01right so the price of zero of discount
1:02:03bond with one hundred principle at
1:02:05maturity is 100.
1:02:07can the price of a zero coupon bond
1:02:09exceed 100 100
1:02:12before maturity and the possible answers
1:02:15to the poll are yes it can be above or
1:02:17below 100
1:02:18or it can be only below 100 by the way
1:02:21don't pay attention to the chart it just
1:02:23shows below 100 but you know there are
1:02:25two possible answers so let me launch
1:02:27the poll all right so uh
1:02:30i shared the screen
1:02:32so who thinks it's one right raise your
1:02:34hand it can be above or below 100.
1:02:41well
1:02:4210
1:02:4310 11
1:02:45people in this variant
1:02:48okay and who thinks uh
1:02:50okay so let's lower the hands right and
1:02:52who thinks it's always below 100.
1:03:02eight
1:03:04eight plus one nine
1:03:06nine pretty close okay yeah very close
1:03:08all right okay so
1:03:10sounds good right so um okay so
1:03:12the answer yes it can
1:03:15be higher than 100 if the interest rates
1:03:17become negative
1:03:19so the first answer was correct until
1:03:21i would say not recently but back in the
1:03:2570s 80s uh even 90s the first answer was
1:03:28absolutely correct in fact even moral
1:03:31for the interest rate derivatives
1:03:32variation predicted negative rates it
1:03:34was considered to be actually a bad
1:03:35thing
1:03:36right
1:03:37so
1:03:38it was assumed that interest rates can
1:03:40never get become negative
1:03:42and uh zero common price in this case uh
1:03:45would always be below principal but
1:03:48since then
1:03:49uh uh central bankers have decided that
1:03:52negative interest rates are is a thing
1:03:54and uh that they get the appropriate in
1:03:56certain situations i'll talk about a bit
1:03:58later
1:03:59and uh what is the negative rate means
1:04:01right it means that bond investor is not
1:04:04rewarded for lending money to the issuer
1:04:05but has to pay for it right
1:04:08and the reason this exists they exist is
1:04:10because negative interest rates are
1:04:11created deliberately by central banks to
1:04:13get the economy out of a crisis so the
1:04:15idea is relatively recent
1:04:17and there are even uh kind of
1:04:20negative rates on steroids which you
1:04:22call quantitative easing when the banks
1:04:24not only set the rate below zero but
1:04:25also they start actually buying um
1:04:28securities uh from people
1:04:30so uh so essentially the idea is that uh
1:04:34when you lend money to someone you also
1:04:37give them
1:04:38additional money negative interest
1:04:41for safe keeping your money
1:04:43right and the reason
1:04:45why investors accept negative rates when
1:04:47they can exist right
1:04:49is because uh
1:04:50the expense of accepting your deposit if
1:04:53you're a bank they accept the expense of
1:04:55accepting your deposit guarding your
1:04:56deposit against the hackers paying for
1:04:59branch locations clerk salaries you come
1:05:01to the bank you say give me you know i
1:05:03would like withdraw my money the clerk
1:05:04says yes you know no problem
1:05:07it exceeds the amount of profit
1:05:09sometimes it exceeds the amount of
1:05:10profit the bank can make lending you
1:05:12deposit to third parties
1:05:14right but
1:05:16other than for small amounts
1:05:18you cannot or rather should not maybe
1:05:20you can but you should not store the
1:05:22cash under your mattress
1:05:24even if you did you would probably hire
1:05:26a security guard you would probably
1:05:27install an alarm system and it cost
1:05:29money
1:05:30so when the central bank offered loans
1:05:33at nearly zero cost
1:05:35to restart the economy after a crisis
1:05:37the banks also have to lower the rates
1:05:39and profit from lending may fall below
1:05:41the cost of accepting deposits right so
1:05:43in other words uh the interest rates uh
1:05:45that the bank earns
1:05:48by taking money from you and giving it
1:05:50to some people for example who want to
1:05:52mortgage right
1:05:54uh
1:05:55if you subtract all of the expenses they
1:05:57don't earn enough right so they have to
1:05:59in this case the invest depositors
1:06:01must partially subsidize the storage of
1:06:03the deposit right so think of it uh
1:06:05basically you give money to someone
1:06:08and they say well i need to pay for a
1:06:09security guard and for the alarm system
1:06:10right so you say well here's some extra
1:06:12money to pay for the security guard so
1:06:14normally they buy money to you because
1:06:16they can make even more by taking your
1:06:18money and lending them out to someone
1:06:19who needs to buy a house for example or
1:06:21you know
1:06:22finance a company
1:06:24but when the economy is coming out of
1:06:26the crisis and the central bank says
1:06:28here come you know we have infinite
1:06:29resources where the government collected
1:06:31taxes maybe we printed some money here
1:06:33we can give you money to you for free
1:06:34right and then you know the banks in
1:06:36order to for someone to come to them
1:06:38they have to lower the rates almost to
1:06:39zero
1:06:41uh and
1:06:42uh
1:06:43you know that may be shocking if you're
1:06:45in europe especially in eastern europe
1:06:47uh but uh in the us uh
1:06:49in a lot of countries uh in western
1:06:51europe and the european union uh
1:06:54until recently no longer right but until
1:06:56a couple year ago
1:06:58uh you could practically uh borrow money
1:07:01for buying a house at almost zero uh
1:07:03percentage rate but no longer because
1:07:05inflation is here
1:07:07so uh the depositors when the interest
1:07:09rates are low must partially subsidize
1:07:10the storage with the deposit and rates
1:07:13can go negative right negative rates are
1:07:15usually small a fraction of one percent
1:07:17anything higher and the investors will
1:07:19look for other ways to store the money
1:07:20long-term deposits
1:07:22risky but interest paying corporate
1:07:23bonds foreign currency purchases stable
1:07:26coins you know also as we recently
1:07:29discovered with luna right it's very
1:07:30risky
1:07:31so uh however uh negative rates they
1:07:34cannot be vain they cannot be minus five
1:07:36percent for as long as there is cash
1:07:38but they can be like minus one percent
1:07:40or minus half percent
1:07:42so
1:07:43raising zero unborn by static
1:07:44application right so zero common bond
1:07:46has one cash flow that we need to value
1:07:48the payment of uh uh you know 100
1:07:51principle uh at maturity of three months
1:07:54and uh the
1:07:57other cash flow is a payment of this
1:07:58price
1:08:00so
1:08:01we previously defined the previous
1:08:02lecture the discount factor is the value
1:08:04today of a fixed dollar payment
1:08:06at time t capital
1:08:09and that's exactly the zero coupon bond
1:08:11except times 100 right so here we have a
1:08:13fixed 100 payment so the price of this
1:08:16euro coupon bond is the principal 100
1:08:19times the discount factor
1:08:21and note that maturity
1:08:23time t capital
1:08:25and time t small when you observe the
1:08:27price they're not the same but they're
1:08:28different
1:08:30okay now
1:08:32now that we've done this euro coupon
1:08:33bond which turns out to be the same as
1:08:35the discount factor and the price is
1:08:37equal to this discount factor times
1:08:38principle
1:08:40let's look at more complex instruments
1:08:42the cash deposit and then their interest
1:08:44rates form
1:08:45so for the cash deposit again so to
1:08:48understand a derivative you we must
1:08:50begin from its underlying
1:08:52for interest rate derivatives right for
1:08:54equity derivatives we started earlier
1:08:56and lying is the stock
1:08:58for interest rate derivatives the
1:08:59underlying is the floating interest rate
1:09:02paid by banks on short-term deposits
1:09:04so floating as opposed to fixed means
1:09:06the rate can go up and down depending on
1:09:08market conditions
1:09:10and in the past the rates used as
1:09:12underlying to interest rate derivatives
1:09:14were mostly for borrowing over the three
1:09:16months or six months term
1:09:18in this race had names like libor uribar
1:09:20tiber right
1:09:22and the word for example liber is an
1:09:24acronym where ibor means interbank offer
1:09:27rate right ib interbank over or offer
1:09:31our rate
1:09:32namely the rates that the bank offer on
1:09:34three or six month deposits or by the
1:09:36maturities three and six months are most
1:09:39common
1:09:40in the first letter
1:09:42in this case l usually refers to the
1:09:44city with the rate of set for l stands
1:09:46for london t stands for tokyo
1:09:48uriber stands for the euro rate right
1:09:51it's not
1:09:52named after particular city but it's
1:09:54named after currency
1:09:56now
1:09:57recently and it was like a long
1:09:59transition process but at the end of
1:10:012021 so past december
1:10:04libraries were mostly discontinued
1:10:06discontinued for new instruments
1:10:08in the market switch to using more
1:10:09reliable overnight rates is underlying
1:10:12in this race of names like software euro
1:10:14eurostr or sonia right and the overnight
1:10:17rates
1:10:18are fixed or measured for borrowing
1:10:20period of one day right so libraries
1:10:23uh
1:10:24thai bar uribe and so forth we measure
1:10:26it for three months or six month deposit
1:10:28or rates were measured for are measured
1:10:31for overnight deposit
1:10:34and
1:10:35the reason this happened is because uh
1:10:38well there are a couple of reasons one
1:10:39of uh that the process for setting this
1:10:42three months or six months rate was not
1:10:43reliable
1:10:45but also
1:10:46when you borrow when you set the rate
1:10:49based on three month deposit
1:10:52some of this race
1:10:54is also related to how credit worth is
1:10:56the other part is right so it's not just
1:10:58the time value of money it's not just
1:11:00the value of dollar today versus the
1:11:02value of dollar three months from now
1:11:04it also depends on
1:11:06and we started in the first lecture
1:11:09on the credit spread right on the only
1:11:11survival probability in other words it's
1:11:13not just the um um
1:11:16it's not just the
1:11:17uh
1:11:18uh the
1:11:20uh time value of money but also
1:11:23you pay less for a dollar three months
1:11:25from now if
1:11:27you need because you need this dollar
1:11:28you know it will be better to have the
1:11:30dollar for the three months right but
1:11:31also decide a party may not give it back
1:11:33to you because they may go bankrupt so
1:11:35the longer the rate the bigger is the
1:11:37component that has to do with not
1:11:38interest but with credits so by making
1:11:40it overnight
1:11:42it becomes the rate that's exclusively
1:11:44almost exclusively with very high
1:11:46precision is related to time value of
1:11:48manual interest
1:11:50because if the bank is okay today it's
1:11:52very unlikely it will default tomorrow
1:11:54it's possible but it's very unlikely
1:11:56whereas for three or six months a lot
1:11:57can happen right some you know basically
1:11:59world prices may happen
1:12:01question why interbank where no
1:12:02different rate right well with first
1:12:04part why interbank well because uh when
1:12:06banks lend to each other that's a kind
1:12:09of a standardized rate and it was
1:12:11assumed that the probability of bank
1:12:13default is small so it was like you know
1:12:16this rate has a
1:12:18lower at lower component
1:12:20of credit or
1:12:23there is that somebody would not return
1:12:25the money compared to buying to lending
1:12:27to corporates so when the bank
1:12:29lends to corporates
1:12:30to corporation
1:12:32to you know a company right
1:12:35they charge higher rate because
1:12:36corporation
1:12:38is less regulated more risky is more
1:12:40likely it will go bankrupt
1:12:42previously it was assumed that banks can
1:12:44never go bankrupt
1:12:46lehman brother did
1:12:48back uh you know in uh during the
1:12:50financial crisis ten years ago
1:12:53and that's when started this whole
1:12:55process that eventually led to the
1:12:56elimination of library because people
1:12:58realized
1:12:59market participants realized
1:13:02that
1:13:02the um
1:13:03banks can also go bankrupt
1:13:05right and why not a different rate well
1:13:08you know it is a different trade so so
1:13:09with the library also there was one
1:13:11problem with um calculating it because
1:13:13it was not based on actual transactions
1:13:16library rate is based on what the banks
1:13:17report
1:13:19is what they would do as opposed to what
1:13:21they actually did
1:13:23and
1:13:25there was an accusation that the banks
1:13:27manipulated this rate so the banks
1:13:29agreed with each other that they will
1:13:30all report slightly wrong rate
1:13:32to make the business look better and to
1:13:34make uh you know to basically change
1:13:36[Music]
1:13:38the perception of how much risk they
1:13:39take
1:13:40so especially during the financial
1:13:41crisis and even before uh there was
1:13:43what's called a liberal scandal right or
1:13:46you know basically uh when
1:13:48bankers were caught uh changing the rate
1:13:51not what they would
1:13:53lend but
1:13:54basically reporting to this committee
1:13:56that was publishing the rate
1:13:58a different amount so again
1:14:00you know we're going to already cutting
1:14:01pretty close we'll probably finish um
1:14:03five minutes late and that's okay
1:14:05because we also started five minutes
1:14:06late but if i keep talking about it
1:14:09we'll be even more late
1:14:10so uh you know this is all in wikipedia
1:14:13and so forth uh you know it's interest
1:14:15uh has some historical interest but the
1:14:17bottom line is that
1:14:19uh rates such as libraries based on
1:14:22deposits of three to six months
1:14:25were eliminated as underlying to
1:14:27interest rate financial instruments and
1:14:30replaced by rates which are based on
1:14:32overnight ledging but but
1:14:34the payments still occur every three to
1:14:37six months because there will be a lot
1:14:38of administration overhead to pay every
1:14:41day so the rate on which the payment are
1:14:43based is observed every day
1:14:46in an either average to compounded means
1:14:48reinvested
1:14:49for the payments to occur at the old
1:14:52frequencies of three to six months right
1:14:53so only the rate observation is for
1:14:55overnight but the payments are still
1:14:57have three to six months frequency
1:15:00now the cash deposit the cash deposit
1:15:03right is uh an otc derivative whose
1:15:05purpose and cash flows are similar to
1:15:06the zero coupon bond right
1:15:09and the way it works is that in the
1:15:11beginning you calculate so there's a
1:15:13calculator
1:15:14icon here to indicate that that's when
1:15:16calculation is happening
1:15:17you calculate uh what the uh interest
1:15:21rates are uh what the prices of zero
1:15:23command once are and then you agree that
1:15:25one party pays the other the principal
1:15:28and then after three months this party
1:15:30pays the principal in fixed interest and
1:15:31this interest is calculated to fix in
1:15:34the beginning right so you agree today
1:15:37what you would pay three months from now
1:15:39which consists of basically you paid the
1:15:41principal of one hundred dollars
1:15:43and you agree that at the end of three
1:15:45month period you receive back the
1:15:47principle of 100
1:15:48plus a fixed interest amount which would
1:15:51determine today
1:15:52so it's just like the euro coupon bond
1:15:54right so here basically you receive you
1:15:56pay one amount today we receive
1:15:58hopefully a large amount
1:16:01three months from now if the rate is
1:16:02negative it's actually a small amount
1:16:04right but somebody you know we should
1:16:06pay for safekeeping your money
1:16:08and uh otherwise it's equivalent right
1:16:11okay so now uh the
1:16:14instruments or cash deposit is a
1:16:16different is financially equivalent to
1:16:18zero coupon bond except the bond is a
1:16:20security and uh
1:16:22basically is a functional but the cash
1:16:23deposit is a derivative
1:16:25interest rate swap
1:16:27is different and it consists of cash
1:16:29deposits a series of cash deposits
1:16:32some of which start today and some of
1:16:35which start in the future
1:16:37now intro vanilla interest rate swap or
1:16:39interest rate swap and vanilla meaning
1:16:41you know again plain or simple interest
1:16:43rate swap
1:16:44is one of the most
1:16:45widely used
1:16:47the counter derivative so it's one of
1:16:48the most
1:16:49frequently used liquid uh derivatives
1:16:52with the highest volume volume meaning
1:16:54the amount of money invested in this
1:16:56derivative right or the
1:16:58highest principal amount
1:17:01and its purpose is to change the stream
1:17:03of fixed payments for stream of payments
1:17:05linked to a floating interest rate
1:17:07and the reason is so popular is because
1:17:09they serve as a very specific and
1:17:10extremely important business purpose
1:17:13if you're a bank right
1:17:15and this diagram below shows uh you know
1:17:17how the bank works
1:17:19if you're a bank
1:17:21you have two parts of your business
1:17:23one is that you take
1:17:25money from uh account holders right
1:17:28and
1:17:29that's the you know principle that the
1:17:31money the account holder or depositor
1:17:33right puts the money in the bank
1:17:36and the depositor receives floating
1:17:39interest based on the current percentage
1:17:40rate so in most banks the interest they
1:17:43pay you is based on uh whatever the
1:17:46interest rates are today so maybe one
1:17:48year ago they were low now they're
1:17:50higher
1:17:51so interest paid by the bank to the
1:17:52depositors changes
1:17:55okay so the bank also has another line
1:17:57of business which is to provide
1:17:59mortgages right financing for buying a
1:18:02house a loan to buy a house
1:18:04most mortgages are based on a fixed rate
1:18:07right because people have a fixed salary
1:18:10so if the amount of money that uh was um
1:18:14paid on a mortgage
1:18:17was based on the floating interest rates
1:18:19a lot of people would either go bankrupt
1:18:21when the interest rates rise
1:18:24or if they they're conservative they
1:18:26would have to buy a lot smaller house or
1:18:28not buy a house at all
1:18:30because uh they would say well uh today
1:18:33the payment would be let's say um
1:18:36you know five hundred dollars per month
1:18:38but if the interest rates rise and if
1:18:40the payment was floating it would go to
1:18:42a thousand to 1500
1:18:44and the salary may not cover it right so
1:18:46in this case you say well i'm going to
1:18:47buy a much smaller house so even if the
1:18:50rates go up i still can afford it that's
1:18:52not good right because you know you want
1:18:53to have some predictability
1:18:56so
1:18:57banks receive
1:18:59money and pay interest which is floating
1:19:02however when they lend money for buying
1:19:04houses or financing factories and so
1:19:06forth most not always but most of the
1:19:09time the receive interest which is fixed
1:19:11so the bank has a problem right because
1:19:13now the bank has to worry about
1:19:14bankruptcy right so the bank
1:19:16made it right for the
1:19:18homeowner
1:19:19by making the interest predictable so
1:19:21someone you can say well my salary is uh
1:19:23session session you know i can afford
1:19:24500 dollars per month in a mortgage
1:19:26right
1:19:27but now the bank is the risk right so
1:19:29the bank removed the risk normally you
1:19:31know financial markets it's about buying
1:19:33and selling risk we'll talk about it in
1:19:35the next lecture
1:19:36so the bank took the risk away from the
1:19:38homeowner
1:19:40and now the homeowner has no risk but
1:19:42the bank does
1:19:44and the bank now has huge exposure
1:19:47because if the interest rates go up
1:19:49you know they will be paying a lot more
1:19:51interest than they receive and the bank
1:19:53may go bankrupt
1:19:55so the bank would also like to transfer
1:19:57this risk right again financial markets
1:19:59is about
1:20:00reducing risk and giving risk to people
1:20:03who wish to take risk and they paid for
1:20:05it
1:20:06and
1:20:07helping other people who don't want the
1:20:08risk who want predictability to pay a
1:20:11little bit and have predictability so
1:20:14the bank will go to a dealer which is
1:20:16also a bank but much larger and better
1:20:18finance bank right think about like
1:20:20citibank right as opposed to your local
1:20:22bank
1:20:23and the
1:20:25bank uh you know the dealer right
1:20:28will enter into a swap with the bank
1:20:30that takes deposit and uh
1:20:32you know and gives loans for homes
1:20:35mortgages
1:20:36in the swap with basically uh the bank
1:20:39would pay fixed interest that is
1:20:40receiving from the mortgages to the
1:20:41dealer
1:20:42and the dealer would pay floating
1:20:44interest which the bank here on top
1:20:46right so so you know it's you know fixed
1:20:48interest goes to the dealer floating
1:20:50interest is coming
1:20:52to the bank and then it's paid to the
1:20:53account holder and this instrument which
1:20:55exchanges fixed interest
1:20:58for floating and stress it's called
1:21:00fixed lag is the leg or part of the swap
1:21:03which is so swap stands on two legs
1:21:04right so one leg is a fixed leg where
1:21:07there is a receiver so the dealer is um
1:21:09so the bank is paying fixed interest
1:21:12and the bank is receiving floating
1:21:13interest
1:21:14so this instrument is so popular because
1:21:17most of the commercial banks that take
1:21:19deposits from people and give mortgages
1:21:21basically issue loans to buy houses are
1:21:24in the same situation and dealers help
1:21:25them to reduce the risk and that's what
1:21:27makes it supportable
1:21:30okay now in order to price it we need to
1:21:32count cash flow so this here's a little
1:21:33poll i'm not even going to try it launch
1:21:35it
1:21:36so put your hand up if you think that
1:21:38when software engineer or you know
1:21:41derivatives valuation analyst says
1:21:42something has priority one
1:21:44do they mean is the most important
1:21:46priority or the second most important
1:21:48priority
1:21:49first answer raise your hand if you
1:21:51think is the most important priority
1:21:52priority one
1:21:59three people okay who thinks it's the
1:22:02second most important priority if it's
1:22:03priority one
1:22:06aha okay clearly winning okay right so i
1:22:08can see the last here exactly they know
1:22:11this right there
1:22:12all right yeah well you're absolutely
1:22:13right so software engineers and uh as i
1:22:16mentioned before many of the generators
1:22:18analysts also have to be software
1:22:20engineers because you write the code on
1:22:21python
1:22:23a plus plus if you're in high frequency
1:22:25or c even if you're on high frequency
1:22:27but normally
1:22:28these days in python
1:22:29to do models
1:22:31and in python as well as most other
1:22:33programming languages
1:22:35the first index is zero right so when
1:22:37we're counting remember that the first
1:22:38time is has index zero
1:22:41in the second time has index one right
1:22:44so you so it's in zero based right you
1:22:46start from zero when you're counting
1:22:48so
1:22:48uh watch the indexes because
1:22:51you know it's very important to account
1:22:53that correctly so vanilla swap
1:22:55is
1:22:56an agreement between two parties in this
1:22:58case the bank and this dealer to
1:23:00periodically for example every three
1:23:01months exchange fixed interest rate
1:23:03payments for floating interest payments
1:23:05and the floating payment is based on
1:23:07overnight rate which is observed daily
1:23:10and then either compounded or averaged
1:23:12for example if there are 20 business
1:23:14days a month is either reinvested 20
1:23:16times or averaged from 20 observations
1:23:20in the first payment covers the initial
1:23:22period from t 0 to g 1
1:23:24and the next payment covers the next
1:23:25period from t1 to t2
1:23:28and
1:23:29swap normally is struck just like the
1:23:32bond issuer would set the interest such
1:23:35that the bond is equal to the principal
1:23:38at issuer
1:23:39for the swap the dealer would set the
1:23:41fixed trade such that
1:23:43no payments change hands in the
1:23:45beginning right so there was two parties
1:23:48enter into a contract
1:23:50without any upfront cost to exchange
1:23:52floating interest for fixed interest
1:23:55and typically typically
1:23:58the
1:23:59payments are different for example in
1:24:01the beginning fixed is more than
1:24:03floating in the end floating more is
1:24:05more than fixed right so typically
1:24:06depending on
1:24:07how the discount factor depends on time
1:24:10usually uh in the beginning there is
1:24:13primarily one direction of payments in
1:24:14the end there is primarily another
1:24:16direction of payments
1:24:18but on average they are the same based
1:24:20on our replication model that we will
1:24:23build in this in a moment
1:24:26and there is no cost to entry into the
1:24:28swap the fixed rate that you pay is set
1:24:30such that the
1:24:32swap is fair right so that makes the
1:24:34floating leg uh equal to the fixed lag
1:24:36and price right and the total price of
1:24:38swap zero
1:24:40so uh that's the last slide uh and
1:24:44here we're going to build a replication
1:24:45model for the swap
1:24:47uh and this is probably the most complex
1:24:49formula in the whole lecture today
1:24:51so you should go for it right so first
1:24:53of all the swap is equal to principle
1:24:56times the difference between floating
1:24:58and fixed payment and there are i is the
1:25:00index of the payment right so for
1:25:02example if it's quarterly for 30 years
1:25:05it would be 120 payments so i would go
1:25:08from 0 to 119 or you know basically
1:25:11because we're counting from zero
1:25:14okay floating rate is equivalent right
1:25:17it's based on observations
1:25:20but
1:25:20is equivalent
1:25:22and obviously objective of it is to
1:25:25provide the financial equivalent of
1:25:27borrowing money for the period
1:25:30that we're talking about for example it
1:25:31may be a three month period starting 10
1:25:32years from now from 10 years so ti
1:25:36will be
1:25:3710 years
1:25:38in ti plus one will be ten years plus
1:25:40three months
1:25:42so floating rate
1:25:44is based on the summer night
1:25:45observations
1:25:46but is
1:25:48specifically designed
1:25:50in such a way that it's financially
1:25:52equivalent to borrowing money for this
1:25:53three-month period starting in the
1:25:54future
1:25:56and
1:25:57by static replication it's equivalent to
1:26:00zero bond price expiring at 10 years
1:26:04minus euro bond price expiring at 10
1:26:06years minus three months and we already
1:26:08knew know that the price of zero bond is
1:26:10discount factor times the principle
1:26:13so the floating leg is discount factor
1:26:15of ti minus this common factor of ti
1:26:17plus one
1:26:18times the principle
1:26:20okay now the fixed leg
1:26:23is a fixed amount right so you obtain
1:26:26the
1:26:27interest equal to fixed rate times year
1:26:29fraction what's the year fraction the
1:26:31year fraction is because uh you know the
1:26:33interest rate fixed rate is usually
1:26:35measured annually
1:26:36so if you are paying interest over three
1:26:38months you're receiving one quarter of
1:26:40the rate right
1:26:42and
1:26:43then you're receiving it at the end of
1:26:45the period together with the second uh
1:26:47you know cash flow
1:26:49so the price by static application of
1:26:52this fixed payment is
1:26:54amount of interest which is principal
1:26:56times fraction times the fixed rate
1:26:59and you receive it at t i plus one so
1:27:02you value the value today to you
1:27:04of the dollar at time t a plus one is
1:27:07equal to discount factor of ti plus one
1:27:09so that's the fixed rate so that's the
1:27:11swap formula
1:27:13and
1:27:13it appears complex but we just assembled
1:27:16it from zero bond prices
1:27:19in zero bond prices we proved that by
1:27:21static replication that they are equal
1:27:23to discount factor times the principle
1:27:25so
1:27:26we took several steps and in the end we
1:27:28arrived to fairly complex formula and
1:27:30that's the formula that professionals
1:27:32actually use to price the swaps
1:27:34with a few you know
1:27:36adjustment right so there are it's not
1:27:38the exact formula right it's very close
1:27:39to the formula
1:27:41uh what's the differences right from the
1:27:43formula that banks actually use to this
1:27:47simplified version written here
1:27:48first of all uh the times and fractions
1:27:51are calculated using complex rules that
1:27:53involve holidays and day counting
1:27:55conventions
1:27:56back in the day where computers were not
1:27:58prevalent and people were literally
1:28:00using uh
1:28:01abacuses and you know basically
1:28:03mechanical calculators
1:28:05uh to
1:28:07do interest calculations
1:28:09uh it was easier to assume that every
1:28:10month has exactly 20 days
1:28:13so there is a convention for the fixed
1:28:14life typically which is called 30 over
1:28:17360. it means that the year has 360 days
1:28:20in the month has 30 days that
1:28:23essentially is just a simple way of
1:28:25convoluted rather you know roundabout
1:28:27way of saying that each interest payment
1:28:29is equal even if some months are a
1:28:31little longer and some months a little
1:28:32shorter some have 31 days some have 30
1:28:35and some have 28 to 29 right
1:28:37so
1:28:38that's just because you know with
1:28:39computers of course you don't care it's
1:28:41very easy right but before computers
1:28:43have made it easier and this convention
1:28:45still exists
1:28:46also
1:28:47sometimes when there is a holiday
1:28:49national holiday right you moved the
1:28:53time t
1:28:54one day ahead or one day behind
1:28:56so approximately
1:28:58t is equal to i times tau with hours
1:29:00three months or six months
1:29:02frequency and fraction is approximately
1:29:04equal to tau right expressed in years so
1:29:06for example for three months tau is 0.25
1:29:09so t in years is i times 0.25
1:29:13fraction is approximately tau
1:29:15up to a few percent accuracy and if
1:29:17you're doing it professionally you need
1:29:19to take into account and you know
1:29:21there's a document that describes how
1:29:24and the formula for floating
1:29:26is only exact this formula that i have
1:29:29here when the warning raise is
1:29:30compounded and certain other conditions
1:29:32are met for example sometimes the rate
1:29:34that something could block out where the
1:29:36overnight rate is not observed for the
1:29:37last two days because if your
1:29:40back office or you know people who are
1:29:42responsible for sending the payment
1:29:43already starting to calculate and verify
1:29:45the payments and they're still waiting
1:29:47to observe this rate they will be
1:29:49delayed so sometimes the overnight rate
1:29:52is not observed in the last two days uh
1:29:54of the three-month period so if we
1:29:56ignore all of that then this expression
1:29:59and only if it's compounded when it's
1:30:01not compounded but average now which the
1:30:04interest is collected but don't reinvest
1:30:06it it will be slightly but not very
1:30:08different again typically this
1:30:09difference just a few percent
1:30:11so with that uh it's uh one mole 37 so
1:30:14what two minus one
1:30:17uh plant uh 90 minute time so i need to
1:30:20wrap up this is the last slide for today
1:30:23and the next lecture is the same time on
1:30:26tuesday of next week
1:30:28we will we will learn about risk
1:30:31and the final lecture will be on
1:30:33thursday of next week at the same time
1:30:35and in that final lecture we'll learn
1:30:37about other asset classes so thank you
1:30:40any final questions i'll stay in line a
1:30:41little longer than you than two days ago
1:30:44if there are any questions please ask
1:30:58going once
1:30:59going twice
1:31:01any questions
1:31:02no it seems that no question
1:31:05all right sounds good so uh well you
1:31:07know if you think of any question uh you
1:31:09know you're welcome to ask at the next
1:31:10lecture
1:31:12or you're also very welcome to email
1:31:14them to training at compatible.com so
1:31:18we'll also review them and answer your
1:31:21next lecture
1:31:22that's right yeah and also i wanted to
1:31:24remind that after the completion of the
1:31:26four lectures uh we'll have an online
1:31:29quiz and based on that we'll issue a
1:31:31certificate so um you know if you're
1:31:33interested in receiving a certificate
1:31:34for this course so you know please stay
1:31:36on uh through the last lecture and we'll
1:31:38have information about how to receive it
1:31:40uh at the end so everybody has a
1:31:42wonderful weekend and see you next
1:31:44tuesday thank you

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