basics of derivative pricing and valuation now we'll spend about 10 minutes on learning basics on derivatives derivative is a financial instrument which derives its value based on the value of underlying asset so derivative is a financial instrument which derives value based on the value of underlying asset and we know that there are roughly four categories of derivative that we have to look at for the level 1 exams the first one is forward contract forward contract is a TC contract or it is a OTC instrument what is the meaning of OTC over-the-counter that means two parties will sit together and then they will enter into an agreement to either buy or sell goods at a predetermined price on a predetermined date and the moment a contract is OTC one of the big concern that parties will have is the default risk the next category of derivative is future contract future contracts are essentially exchange-traded instrument so these are exchange-traded contracts and future contract will have no default risk because of whom yes they will have no default risk because of clearing houses which are of course the part of exchanges because Clearing House takes the responsibility of ensuring that there is no default and this is typically achieved by asking investors or speculators to deposit margin in their accounts the next category of derivative were option contracts and the most important word which summarizes the entire option contracts is is a symmetry of the contracts extremely important word when you would learn when you go to say cfa level 2 and level 3 you realize that most of the mathematical models which are developed for option valuation are because of our symmetry of the contract which means at one stage it will only have a right or only have a obligation whereas in case of forward and future we had right an obligation together so options are further into two categories a call option and put option and next category of derivatives now yep option contracts are the OTC or e.t.c option contract are both they are OTC as well as etc' and swap contracts these are essentially agreement between two parties to exchange pre specified set of cash flows based on performance of certain asset classes and in most of the cases these are OTC contracts I will find so how you deal with this syllabus we will go as per the yellowest fashion however I will be reshuffling few of the yellowest out of order so that it becomes more logical so the first point of discussion which is required is understanding the concept of arbitrage then understanding the concept of replication and understanding the concept of risk neutral aiding arbitrage replication and risk neutrality now you would have substantially better understanding of this particular Louis by the time we finish the reading as of now I will just give you an overview of how it works now the idea of arbitrage is let us say we have an asset a and we have a derivative on asset a so let me call this as derivatives one and we have another category of derivative on asset a let me call this as derivative two now these derivatives are structured in such a way that if you make a portfolio of d1 and d2 then the behavior of that portfolio will always be equal to a that means if a increases d1 plus d2 portfolio will increase if a decreases development d2 will decrease then the idea is that at any part of time the value of a should be equal to d1 plus d2 ab and if it is not then anyone can exploit an arbitrage which means they will buy whatever is cheaper and they will sell whatever is expensive so then this arbitrage gives us an opportunity to value derivative for example let's say we know the value of a we know the value of D – we can always say D 1 would be equal to a – Stephen so when we do options and certain category of forwards you will have a better understanding of what were those C 1 and D 2 the second one is replication which is again the same concept that portfolio a can be replicated by using a combination of portfolio D 1 and D 2 we find with this and the last one risk neutrality here we would make an assumption so to understand risk neutrality we need to go a little bit into portfolio management there are three category of investors risk averse risk loving and risk neutral risk averse investor means essentially the ones who dislikes risk but that does not mean that risk averse investor will never take risk okay so as far as exams is concerned you want to think of these investors as the rational investor whereas both risk loving and risk neutral are not necessary in the rational category and what rationality mean here is that if the risk increases then a risk averse investor will demand excess returns okay so if an investor is more risk-averse then he will demand even more excess written for given level of risk this is the meaning of risk aversion then risk loving investor he would not worry about what is he return on he would select the portfolio select a portfolio tell me with highest risk okay so a risk level or risk loving investor derives maximum utility by additional level of risk and a risk neutral investor he does not worry about the level of risk okay so irrespective of what is the level of risk it does not matter to him because he is neutral on risk and how this concept would be used in derivative pricing is we are going to assume that no matter what asset class we are talking about a risk neutral investor will always want to earn risk-free rate of return why because even if it's a risky asset it doesn't matter to him he's a neutral investor and therefore any category of asset the assumption would be the investor wants one risk-free rate of return and this concept would particularly be helpful when we learned when we will learn option pricing model using the binomial tree this is where we would learn to calculate risk neutral probabilities should be good the next three Aloise are related to forward and future contract so first we will start with the forward contracts we know that forward contract is essentially an agreement where one party which is long will have right and obligation to buy and the other party which is short will have right and obligation to sell a predetermined quantity on a predetermined date at a predetermined price and if you are long on this contract then essentially you're you have a bullish position because you would want these prices to go up and if you are short on the contract then you have a bearish position because you want underlying asset prices to go down should be good so let us say now we have signed a contract the price of an underlying asset today was 100 and we decided to take a long position on the same underlying asset that after one year we are committing to buy that underlying asset at a price of 1 1 0 okay now this spot price will keep on fluctuating for the whole year maybe it will go down or it will go up so let us say by the end of the earth spot price was 150 if you are long on this contract is it a profit or loss is it a profit or loss of it because we have a right and obligation to buy at a price of hundred and ten and if is in the market is 150 we are buying cheaper we have a profit of 40 this profit of 40 is called the value of forward contract and how I want you to think of value is value is simply the amount of profit that we are earning at any part of time now here in this example we have calculated value on the last day we can also calculate value in between but what value essentially means the amount of profit or loss earned on the contract as against that this 110 is called the price of forward contract so finding out what amount is to be quoted inside the contract would be called as price and finding out the amount of profit or loss would be called as value I will okay you now we will spend some time on understanding how to price forward contract so heading in your notes would be pricing or forward contract now let us say spot prices hundred risk-free rate of return 10 percent maturity for forward contract is one year and the forward contract price this is what we want to learn to calculate but for the time being I'm going to assume that this forward contract price is 130 now ideally this price should have been some other value but if it is 130 we can use this scenario to our benefit and try to earn some arbitrage profit out of it and this type of arbitrage would be called as the type of arbitrage that we can exploit in this scenario would be called as cash-and-carry arbitrage it's now observe carefully what we're doing here these are the justment these are the step of actions you will perform today and these are the actions you will perform when you from now step number one what you do now is go to an bank and take a loan of 100 at the rate of 10% then step number 2 by underlying at spot price which is how much 100 that means you go to a bank take hundred rupees and use a hundred rupee n-body underlying asset in step number three take a short position in the forward contract short position means write an obligation to sell which means after one year irrespective of what happens to the price of the asset we have to sell that asset at a price of 130 and from where will we get the asset we already have it we already purchase an asset which means no matter what what happens in the market one year from now we are going to sell the asset at a price of 130 but to the bank we are required to deposit how much 110 and that means in this process you are going to earn an arbitrage of 20 rupees so what we are understanding here is in order to avoid this arbitrage the price which which would have done that had this contract price win 1 1 0 since the contract price was more than that we said forward is overvalued and therefore we took a short position in the forward contract is it fine should we go it now the opposite scenario spot is 100 again our FRS 10% maturity is one year and forward price now is 80 now in this kind of scenario the arbitrage that we will exploit would be called as reverse cash-and-carry so what do you do in Reverse cash-and-carry step number one take a short position on underlying asset short position on underlying asset 100 now this is a naked short position it's a theoretical assumption what it meant is that you went to a friend of yours who was a broker you said can I borrow a stock from you and then you sold that stock in the market with an assumption that we have received this 100 today with us is that fine without having the stock we have sold the stock step number two we will deposit that hundred in a bank now again I'm saying Bank it should not be a bank ideally because it's an RFI but I'm seeing Bank just to simplify the understanding so deposit hundred in a bank at the rate of 10% and step number three take a long position in forward contract so long position in forward contract means write an obligation to buy and if you have an obligation to buy how much amount do we need 80 but do we have 80 will we have 80 rupees after one year yes in fact we will have more than 80 we will have 110 isn't it so after one year after one year Bank will give us 110 out of which 80 we will use for the contract purpose which will give us a stock how much is left for ourselves 30 and what will we do with that stock we will give it back to the broker so again to avoid the arbitrage in this scenario the forward price should have been how much 110 had the price been 1 1 0 they wouldn't have any name any arbitrage but since the price was less than 80 so it since the price was less than 110 which was 80 we the forward was undervalued and therefore we decided to take a long position is that making sense so for your exams you are not expected to remember how to exploit this arbitrage but what you would be expected to remember is which arbitrage is to be used when so just make this summary with me if the forward contract is overvalued then we short forward since it is expensive we should sell it and when we shot forward it is called as cash-and-carry arbitrage when the forward contract is undervalued we take a long position on the forward and this scenario is called as reverse cash-and-carry so here we've made a theoretical resolution so the question is us asking is that when you borrow stock won't there be a cost to borrow that stock plus it is okay for us to invest at RFR but can we we borrow at RF our only government can borrow at RF are so therefore what calculations were doing as of now these are called as theoretical Norbit Raj price and this is a base case scenario to understand what should be the minimum price of forward contract so based on this in the previous example what we said is when the spot was 100 n was 1 our effort was 10% no arbitrage forward price should have been 110 but the arbitrage told us that suit that gives us a formula which says forward price is equal to spot price into 1 plus interest rate raised to n in our example n was 1 so let's practice a couple of questions on this let us say spot price of an asset is 2500 risk-free rate of return is 6% continuously compounded maturity of the contract is 180 days find out what should be D value or price of forward are we the formula that we have learnt was that the formula for price or was that the formula for value the formula that we've learned is that for Christ's result for value that was for price which means we cannot calculate value as of now the only thing that we will calculate this price what should be the amount that should be quoted in the contract solve 6% continuously compounded to 575 or let's all together we learned how to use continuous compounding when we were in school forward price is equal to spot into 1 plus interest rate raised to n but since we have continuous compounding here I am going to change the formula slightly so now I'm going to say spot into e raised to R T and how will we use a raise to R T 2500 into e raised to 6% into 180 divided by 365 and how to do this on your financial calculator please do this with me 6% 6% into 180 divided by 365 second arrays to X multiplied with 2500 how much is that 2 5 7 5 let me just live it up 7 6 you don't have to do that this is see we don't this is actually the conversion into discrete rate is the same thing right so there is no shortcut to this you have to learn continuous compounding half of your level 2 syllabus works on how continuous compounding is to be used so if you don't know know this sit for two hours and practice 100 questions where we've required thousand but make sure you know how to calculate this let's do one more we are still on pricing forward contract spot is 1,750 risk free rate of return 8% or random if nothing is mentioned it's always annual compounding let us say expiry of the contract or maturity of the contract is three months find out Norbit rush forward price 1784 so forward is equal to spot into one plus interest rate rest to N 1 7 5 0 into 1 plus 8% raise to 3 divided by 12 so that would give us 1 7 8 4 point some value so 1784 is your Norbit ROG forward price surprising is straightforward and simple you just need to know the formula now in this case you could have also done this using your TV mrow so let's do it quickly second-tier TV M 1750 present value 8's iy3 by 12 equal to is n compute future value have you got to word let us now learn valuation of future contract let us say spot price today is hundred risk free rate of return 10% n was one we decided to enter into long side of forward contract so this is time zero this is year one on year one we have right and obligation to buy underlying asset what should be the price of the forward contract 110 right now how to calculate value of the contract at maturity okay so at maturity we've done this example earlier if the spot price is 150 then it straight forward for us we have a right and obligation to buy it 1 1 0 price in the market 150 that would give us value of 40 so I want you to write down this example quickly we are buying it 1 1 0 market price 150 value is 40 so when we are when we are at maturity value of the contract is simply difference between the spot price and the forward price future forward boot profit value means profit value of any contract at inception value of any contract at inception has to be 0 imagine if the value is not 0 if the value was +10 then maybe you would want to sit and sign infinite number of contracts because you are getting 10 without doing anything and if the value is minus 10 then of course no one would sign the contract isn't it which means that inception value to both the parties has to be zero now we have to learn valuation when we are in between the two sets so let us say that after three months three months means how many years 0.25 years now the spot price has become 140 if you are long on the contract will you be happy or unhappy about it happy why because prices have increased so you are going to make a profit now the question is how to determine how much profit we are going to make and there are three different formulas through which you can get the value but the good news is you don't have to remember any of it once you understand what is the logic behind it so you would say spot is 140 if if I wanted to find out that if I have to sign a new contract for same underlying asset but for 0.75 years can we find out what should be the price of that new contract we want to find out the price of a new contract that will mature from today in 0.75 years so how will we do it same formula forward price is equal to 140 into 1 plus 10% but this time it will not be raised to 1 it would be raised to 0.75 one 150 point three seven okay so today's spot price is 140 you give a call to dealer you said what if I want to enter into a new forward contract which expires 0.75 years from now a dealer will quickly perform these calculations here we tell you you can do that at a price of one fifty point three seven now let us say you decided that you want to extract profit of this out of this contract today itself how will you do it after one year you have after one year you have right and obligation to buy right and obligation to buy at one one zero you can enter into short sight on this new contract you would go to the same dealer you would say on the new contract I want to enter on to the opposite side and opposite side means we will have right and obligation to sell at a price of 150 point three seven which means point seven five years from now on that particular day I will go to the original party the first party with whom I have signed this contract I will give him hundred and ten take the asset then I will go to the second party I will give him the asset and from him I will take how much 150 point three seven and this is going to happen no matter what is the price of underlying asset because both of these are contractual obligations which means on that day your profit is going to be confirmed for two point three seven is that correct but this profit you will have 0.75 years from now so what is the amount of profit today so this 0.75 this value we have to discount backwards for 0.75 so we would say forty point three seven divided by plus 10% raise to 0.75 and that will give us how much thirty seven point five eight and this is the value of the contract today difficult you're not you're not understanding have you understood this so I'm going to repeat the process again on time zero spot price 100 on time zero spot price 100 you went to a dealer you said I want to enter into long said of a forward contract one year from now so your dealer use a simple formula spot into 1 plus RF R raise to n and he told you that forward price is going to be 1 1 0 you took a long position on the forward contract so this long position was signed here back now once this long position was signed after 0.25 years the price of the underlying asset was 140 so you called up the dealer again you said what if I want to sign a new contract but that would be how many years from now point seven five years from now so he used the same formula he told you that price would be 150 point three seven you decided fine on this new contract I want to take a short position you already have a long position on 1 1 0 that means 0.75 years from today on this particular day you will buy the underlying asset at 1 1 0 you will sell the asset at 150 point 3 7 irrespective of what is the price in the market which means on that day you are going to 140 point 3 7 but this profit you will have 0.75 years from now so present value of that profit today is going to be 37 point 5 8 which is the value of contract and how is this value put to use instead of doing this entire story would just call the counterparty and you will see see I can do this so why don't you pay me 30 7.58 today and let's clear out the contract following this so this process would be called as cash settlement and to find out how to find out that cash settlement value we have to perform this calculation so this gives us a formula can I clear up the screen so the formula is value of a forward contract now at level 1 exams it is unlikely that you would be required to calculate this because the command word in the LOA says explain it does not say calculate which means you are not calculating this but you should have if you know how to calculate you will have a better understanding of this so value of the forward contract is equal to forward price for future value future value of spot – forward price divided by one plus interest rate raised to n now this formula is same as a calculation we perform but there is also shortcut formula available which is spot – present value of future price or forward price that would have also got us the same answer hope you write down both the formulas I'll show you how it would get as the same answer so in the same example this was hundred this was at 0.25 it was how much 140 and we had entered into a contract price at 1 1 0 other way of solving business you can just take present value of 1 1 0 and reduce that here so 1 1 0 divided by 1 plus 10% raise to point no dot yeah raise to 0.75 how much is this 1 0 2 point 4 1 and 140 minus 1 0 2 point 4 1 37.6 so even this process will get you the same answer and there is one more shortcut another ways you can take this 104 words for 0.25 years and take the difference so 100 into 1 plus 10% raise to 0.25 how much point for one and take difference of 1 0 to 0.40 and 140 that will again give you same answer 37.6 it's simple time value of money you're just moving numbers yeah that was for understanding purpose you can just use any of the three formulas this is a least preferred formula because not necessary you will have this prize always so maybe we will summarize the entire circus for you here so this is spot at time zero this is spot at 0.25 and this is where we have a forward price the first mechanism was take this here let me call this as let me call this as a then take the difference of F minus a let me call this as B and then take this be backwards this was the first method okay the first calculation that we did second method is this is s 0 this is s 0.25 this is forward simply take this forward backwards let me call this SC s minus C is your answer directly which is V you're bringing backwards that means taking present value and just calculating the difference the third method so let me do it on the first flow chart itself third method take this forwards take this forwards let me call this as H and this minus H will again give you the value all the three methods will fetch you the same answer and as far as exams are concerned you need not know anyone of them you just need to know that value is the amount of profit or loss at any point of time I will find you next concept next concept I feel is particularly important for your syllabus because this is a new addition to the we know that derivatives syllabus has changed and this particular Louis wasn't there in the syllabus in the previous derivative section generally anything that changes in the syllabus is a high probability concept so understand this now what we said is that forward price or a no arbitrage forward price was equal to spot into 1 plus interest rate raised to n why because we saw that otherwise there can be a cash-and-carry arbitrage that means buy this spot and carry it for 1 year so instead of writing it like this I am writing it in this fashion spot plus interest cost so when you are dealing with a financial security the only cost you will incur to carry the asset is the financial cost but think of a scenario when you are looking at a forward price on a commodity let's say something like gold so you purchase the goal so we will have spot plus that gold will have an opportunity cost our amount will have an opportunity cost so we will have interest cost but because there is a possibility that that gold might be solar or something might go wrong you might want to also have a bank Locker isn't it to store that gold for one year so another cost that we will incur is storage cost which is a part of carrying constants we will have storage cost and maybe there is still this state could still be attempt in a bank Locker who knows so interest cause storage cost plus we might also want to ensure that gold so we will say insurance cost but the good news is so let's say this was gold jewelry and one of your friend was one of your friends was getting married and he or she wanted to rent that jewelry gold is a leasable I said you can give it on rent so when you give it on rent it helps you recover some part of the cost so therefore we will say – whatever are the benefits we will have and one of the benefit is lease yield and you would have one more benefit of holding the asset and that benefit would be called as convenience yield okay so spot plus interest cost plus storage cost plus insurance costs plus any other costs required to carry data set – leeze yield convenience yield or any other benefit we receive by holding the asset leisurely straightforward giving the asset on rent and then earning tailed what would be the meaning of convenience yield so convenience yield how you want to think of this is think of two factories let's say factory a and factory B both of these factories are operating in two food and beverages industry and what they do is they make some snack out of soya beans so their input is so er so Evan Caesar input in the manufacturing facility now every month the typical quantity which is sold is 100 tons and 100 tons the policy of company is or factory is that at any part of time the quantity that they will store would be hundred tons for the next month for the next month's requirement they will have a forward contract signed that means amount of inventory kept in the factory at any part of time is only hundred tons based on the consumption pattern whereas policy of factory B is at any part of time they store 500 tons that means they store for five months in advance now imagine out of a sudden a large customer comes up and the customer requires an immediate customer requires immediate output let us say of 400 tons will factory a be able to produce that so factory a might lose that customer but factory B will still be able to execute that deal because it's taught excess inventory so by storing excess inventory we are incurring the larger opportunity cost but then we have that intangible benefit that if a large customer comes to us that we can execute that order immediately and this is precisely the definition of convenience yield the amount of benefit derived by holding the asset as against entering into a forward or future contract okay so please write down the definition convenience yield convenience yield amount of benefit amount of benefit in bracket write down in tangible in bracket in tangible bracket close amount of benefit derived amount of benefit derived by holding the asset by holding the asset as against as against entering into entering into long side entering into long side of forward or future contract and we find next sentence convenience yield cannot be shorted convenience yield cannot be shorted and therefore therefore it is difficult to exploit arbitrage therefore it is difficult to exploit arbitrage on forward contracts okay you cannot have a short position on convenience yield you can always have a long position if you have commodity you are having convenience yield but you cannot sell that convenience you don't have a short position so if your forward price of a commodity contract does not come out as per the formula then it is difficult to exploit arbitrage as against had the underlying asset been a financial asset okay if you have not understood this sentence just remember it that we cannot exploit arbitrage easily on commodity forward contracts because it is intangible credit so we'll stop for a small break five minutes then we'll start again

# 2015- CFA level I- Basic of Derivative Pricing and Valuation- Part I (of 4)

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