Mathematical Models of Financial Derivatives: Oxford Mathematics 3rd Year Student Lecture

[Music] okay good morning welcome to 2020 I guess this is the first lecture for at least some of you so we're here today for mathematical models of financial derivatives I'm Sam Cohen I'm one of the lecturers here in math finance and what we're going to do is in this course we are going to try and understand some of the contracts that are commonly traded in finance we're going to try and ask ourselves how can we use mathematics to understand them better how can we understand the risk how can we manage it how can we price it etc so before we begin we really should be asking ourselves why are we wanting to do this and it's a it's a pertinent question because mathematical models of financial derivatives have been more than a little bit controversial now you guys may not remember it as well as I do but twelve thirteen years ago there was a very large financial crisis the GFC and one of the things that have been blamed for this was an over reliance on mathematical models of financial derivatives you've got the the article the formula that killed Wall Street so is this a sensible thing for us to be studying at all it's an open question it's a serious one I would argue it is for a variety of reasons as we'll see financial derivatives have always existed pretty much since agriculture we have had financial derivatives and because of that we we can't close our eyes to the fact that these things exist that they are traded that they are real and building models of them trying to understand their risk is a sensible thing to do we also need to understand where those models are going to fail we need to understand the restrictions of what we're doing and so that's something I'm hoping that we will touch on quite a lot in this course where is the model going to go wrong where are the restrictions where have we made abstractions or simplifications that are a little too brave a little too far from reality and I'm hoping that we'll cover some of those things as we go along so what are we trying to actually do in this course we are trying as I say I've got three broad goals the first is to build models of contracts traded on financial markets so that's the the basic idea of what we're going to be doing we're going to try and build models of some of these contracts we're going to try and understand their payoffs and then from that we want to understand where they fail we want to understand how to use them where they fail and where we need to take particular care well we need to be careful and as a third point we want to develop the technical proficiency to build understand and use better models so this course is not the end point we're not going to get to the advanced things that one can do in the world of mathematical finance but we're going to try and get enough technical proficiency that we could at least move on to that step at the end of this course okay so when we're building a model in finance what are the key concerns now the usual thing people think is that you're trying to build a model so that you make money sounds good the truth of the matter is that usually making money is not our first concern usually our key concern in much financial modeling so key concern is don't be exploited we are more worried about if we're trying to give a price for something that we will give a price that allows someone else to come and rip us off that is actually our first concern and this is going to come up again and again to the concept of arbitrage that if we're giving prices that allow us to be exploited then we're not going to do very well and so this is our first concern our second concern is we need to understand and manage our risk it's not enough to have something which pays off well if it's so risky that we're going to lose before we win and so we need to understand and manage our risk and then a third goal is make a profit but in what we do in this course this is actually not going to be a key aim we're going to be trying to understand these steps because only once we've done these that this is even possible to even talk about making profit unless you can understand your risk and you're sure you're not going to be exploited there's no way you can turn a profit okay so what are the economic assumptions that underlie most around modeling let's try and start moving towards actually building a model of something so we'll begin with a fairly simple set of assumptions about how financial markets work and these are deliberately simplified but they allow us to start making some conclusions so the first assumption that we're going to use in a lot of the models is that there is a riskless investment now what do I mean by riskless I mean that if I put money into it now I know how much money I'm going to have in a year's time so this is a fairly common assumption you can think of this as being cash or more usually as being a bond or a bank account with a fixed interest rate I can invest in it today and I know how much money I'm going to have in the future so we'll often refer to this either as a bank account or a bond if you're not sure a bond is just a agreement to pay in the future if I'd thought of it I would have I have one in my office an old physical bond it's a contract where what you do is a company or government will issue bonds they sell them in the market and they give the holder of the bond the right to certain cash flows so if you buy a bond you then can go to the company and every month or however often the payment is they will give you money now to make things simple this grows at a constant continuously compounded interest rate which will denote by a lowercase R now the fact it's constant just simplifies things a lot we'll come back to whether this is a good assumption at various points the continuous compounding that's more of a just a technical simplification the fact that something is compounding continuously means that I can figure out the value of my riskless investment at any point in time without worrying about when exactly the interest is being determined now in many cases interest is calculated on a day to day basis so at the end of the day and what that means is that this is only an approximation it's the first point where we've already started making bad assumptions some assumptions that are not technically true but they're good enough if I'm pricing something which is a long way in the future say a few months the fact that interest is being computed daily rather than continuously makes very very little difference in most cases okay so what does this practically mean mathematically well it means that if a quantity MT is invested at time T then it grows to how much M big t which is e to the R big t minus little T MT at time T and so this is this continuous compounding you get an exponential coming in okay a guaranteed amount B T to be paid at time T is therefore worth be little T which is e to the minus R T so if this is true if I can invest at time little T so today I invest into the future and my amount that I've invested grows by multiplying by this coefficient e to the R big t minus little T the time duration well that means if I want to get a contract which will pay me in the future a fixed amount B big t and how much do I have to invest today when you rearrange this formula and you get it but with a negative exponent we assume borrowing and lending rates are the same well there's an assumption it's clearly not true but if you're a big financial institution it gets close to being true that the the amount that you're paying to borrow and to lend is very similar and so we're just going to assume that they're the same but we were going to remember that's an assumption which we know is sketchy okay so this is the first thing that we've got we've got this riskless investment in the world where we can invest now for the future and it's going to grow at a fixed rate that we know beforehand okay what are the assumptions about the market because so far we've said nothing about a market we've just said dairies and investment so we're going to make two assumptions which are sort of our basic modeling assumptions about financial markets so there are no transaction costs so what does that mean it means that if you want to buy something or if you want to sell it you're going to pay the same price to make either side of the trade so you pay the same price you pay the same price to buy or sell and for any quantity so the price that you would pay for buying one unit of an asset is the same as what you'd get for selling one unit of an asset which is one hundredth of what you'd get for buying or selling 100 units of the asset now again is this assumption reasonable well it's a reasonable first approximation but it's not technically true we all know that if I came to you and said I'd like to sell you a TV maybe you'd pay me one price if I said I'd like to buy a TV you'd give me a different price and if I said I'd like to sell you a thousand TVs you'd look at me strangely and ask where have I gotten a thousand TVs from and why am I trying to sell them so the the quantity effects are important but we're just going to ignore them as a first approximation okay a related assumption is that since our infinitely divisible okay you can own 0.2 shares this is more a technical assumption than anything else it means that we don't have to worry about discrete numbers of shares that you've got to buy one share or two shares you can't buy one and a half shares practically it doesn't matter because once you're working on a large scale if you're working on a deal for 10,000 shares the difference between buying eight thousand and eight thousand and one shares is pretty small it's a discretization error but it's small enough that we don't need to worry about that as a first approximation okay something a bit more controversial than those is that we're usually going to build models where short selling is allowed okay a little bit of terminology whenever I say short I mean selling something whenever I say long I mean owning something so what's a short sale that's where I own a negative quantity of an asset now is this possible well it depends on what you mean it's certainly true that in many markets what you can do saying there the equity market for big stocks if I want to hold a negative amount of some big stock what can I do I can go and I can borrow that stock from someone else I pay a fee we'll ignore the fee but I pay a fee to borrow the stock I then hold the stock I can sell the stock now what that means is I have an obligation in the future to buy the stock again and return it to the person I borrowed it from but in the meantime I hold a negative quantity of stock why is that if the stock go up in price I lose the stock goes down I win because I'm going to have to buy it in the future now that is what's called a covered short sale I have gone I have borrowed the stock from someone so the stock is physically there and I'm able to sell it on when I say physically I don't mean physically pretty much everything's done electronically these days but don't let that worry you there is also what's called a naked short sale a naked short sale is basically where I go into the market and say I'll tell you some stocks and you say alright I'll buy them and I go great and I know that the contract gives me two days to make good on the deal so if you've bought stock from me I know that I have two days in which I in two days time or by two days time I have to deliver the stock so I am now currently short the stock because the stock goes up in value I'm in trouble but there's the idea that I could go and if I do this if I make a naked short sale I say I'm going to sell it I could then go and buy it in the market or borrow it from someone after I've made the deal this is a naked short sale this has been very controversial in the past few years why is it because if I haven't already identified the stock that I'm going to be borrowing or if I haven't already borrowed it it's very easy for me to start selling things I don't own it's very easy if I'm allowed to do that for me to go and say I'm going to sell ten thousand ten million however many stocks I want and that looks like lots of people are trying to sell a lot of stock even though I don't own it and so that's been linked to the idea that it's going to push prices down and so it's being quite controversial we're going to assume that it's allowed we're not going to go into the details of how short selling is actually manipulated but we're just going to say you can hold a negative quantity of an asset okay so these are the basic assumptions of how we are going to model markets unless I say otherwise we're going to take these as given even though we know that most of them are sort of not true but this is true in most mathematical modeling we have to abstract we approximate and we hope that we've captured the key features of what we're trying to understand okay so how do we proceed well we haven't yet given any principles which will allow us to find a price of anything but we have said one thing earlier here that we're going to avoid being exploited and the most basic form of exploitation is arbitrage so what's an arbitrage an arbitrage is a deal that is too good to be true so an arbitrage let's get the definition right so is an investment which costs nothing to set up at time T I if I think of the cost or value of this investment it starts off being negative or at most zero but at a later time later time big teeth we have a zero probability of it being negative so the probability that X at big T is less than zero zero because if it was worth a negative amount that would be costly for me in the future and have to pay it so this is a deal which costs me nothing to set up in the future I can't lose the probability of me losing is zero and it has a strictly positive probability of a strictly positive payoff so I mean that the probability that XT is greater than zero should be positive so an arbitrage if we think of simple games is something like the game where we flip a coin if it's heads I'll give you a pound if it's a tails nothing happens now if that doesn't cost you anything to enter into it's a very good game to play because well maybe you win something maybe you don't but you're never going to lose so that is an arbitrage and we are going to basically assume that arbitrage is shouldn't exist so our the way we're going to build models is based on the principle of no arbitrage now that mean arbitrage is actually don't exist well no they do exist but on well-functioning markets with lots of players whenever an arbitrage exists someone will exploit it and someone will exploit it quickly and if someone exploits it they start changing prices and when they change prices they do so in a way which gets rid of the arbitrage sort of eat up all of the money that can be made out of the arbitrage by changing the price so we're going to assume there are no arbitrage as when we build models what does that mean it means we're not building a model to try and detect and exploit other trush we're going to be building models which say let's assume we're not fast enough we're not aiming to exploit this what can we then say about the market ok so we assume no arbitrage no arbitrage has exist ok so we've got some basic assumptions about markets and how they work but we haven't actually considered any contracts yet so where we'll begin is with probably the oldest form of financial contract this is what's called a forward and what it is I'll give you a simple example imagine for a moment that you are an exporter so you are based here in the UK you are selling things in the US for example so in one year's time you will make a sale you've already signed the deal today to make a sale in the US and you will receive some quantity of US dollars you know that you're going to receive them on that day you will have to pay all your costs in the UK all those costs have to be paid in pound sterling okay you have calculated that if the exchange rate is as it is today then this is a good deal you will make money but you're worried you've looked at the last couple of years in the price of pound sterling versus the US dollars has fluctuated quite a lot and you're worried that in one year's time when you receive a whole lot of US dollars but have a whole lot of expenses in pounds that you won't have the money to cover it and you might lose even though you think today this is a good deal so what could you do well what you do is you can go to a bank and you say to them look this is my situation I would like an agreement with you to sell US dollars into pound sterling in one year's time now there's nothing strange about this contract all it is is in one year I will be giving you say 1 million u.s. dollars and I would like you to give me however many it is pound sterling and what you try and do is you go to the bank and say could we please agree today on the price at which we'll do this deal okay so we're going to agree today on the exchange rate at which we will trade in one year's time and that's the basic idea of a forward it's about taking a price today at which we will trade in the future the question is what price should we trade at so how do we determine the price today okay so let's try and write this out okay so a forward is an agreement to trade on a fixed date in the future for a price determined today now one thing to notice this is an agreement to trade all parties must trade on that date there's no way I can get out of this once I've signed the contract with the bank I am legally bound to deliver US dollars in one year's time and they are legally bound to deliver pound sterling in one year's time no one is allowed to back out of the deal so both parties to the contract have the obligation to trade so how are we going to find a price for this okay let's think about this how could I what other options could I do well I know that I'm going to be I'm gonna have to sell US dollars in one year to get pound sterling so what I could do hey this is called the short position from the UK's perspective because I am selling the asset the asset is US dollars okay so for example I suppose we wish to sell 1 USD in 1 here for simplicity we assume no interest is paid for holding US dollars although we'll come back to that assumption so let's begin as soon as you hold any US dollars there's no interest being paid on that so what could I do well one thing I could say is I could let's suppose a price of one US dollar today is s little T so that's the price at which I can buy one US dollar in pounds right now and we've assumed no transaction cost so I could buy or sell at the same price so what could I do well I could consider borrowing s T today buying one u.s. dollar and holding it for a year the payoff for doing this is well what's the value of my US dollar that I've bought in one year's time its st okay that's the future value of one US dollar I don't know its value now I'll only know that at time Big T but I've also had to borrow money and we remember by borrowing SC today I've had to pay interest on that so I've got e to the r t minus T this is my interest rate on the money that I've borrowed in pounds times the amount that I borrowed which is s little T okay if I hold this and the forward which has pay off let's think about this what's the forward contract worth if I have agreed a price today okay then I've got well my US dollar in one year's time I can sell it for some amount okay which is the forward price okay okay agreed so agreed price ft so if I've agreed that price what's the value of this contract well in one year I have agreed to trade it F little T the asset the US dollar is worth s big T so I have a payoff which is f little T which is what I'm actually getting - s big T which is what the price would be if I didn't have the contract I didn't have the forward ok so if I hold this and the forward my total cash flow is f t minus e to the R a s T so this is my cash flow sorry total profit everyone see where I've got that problem so I've got my terminal pay off this is what I paid for the amount that I borrowed I've got my forward contract which I agree a price on today but I'm effectively losing out because I'm not getting the value at the stock in the future but if I add these two together I just get this payoff here the interesting thing is well the problem of figuring out the value of this was that I don't know the value of s T yet that I'll only know in one year's time at the terminal point of my contracts but this depends on things that I know today ok there's nothing risky in here I've got this is the amount that I've borrowed I know what I'm going to pay for that this is the price I'm agreeing on today so there's my total profit which I know today by no arbitrage this cannot have a positive value that has to be negative because if that was positive this would be free money I'd go and do this deal I'd enter into the forward contract I would borrow money I would buy US dollars and in one year's time I would have money it costs me nothing today forward contracts I'm not paying you anything today we're just agreeing a price it's no cash exchange today and so if this were a positive amount then I'd be able to make money for nothing so that's an arbitrage which we've excluded from our models so we certainly know that this quantity must be negative Oh what about the other side so this was if I was if I was thinking as myself but what I could do is I could instead think what if I was my bank so what's my bank's position they're doing everything opposite to me so they're selling me they're agreeing the price to do the trade the other way and they could borrow money they could effectively take money invested and sell US dollars okay so instead of buying a US dollar they short sell a u.s. dollar instead of borrowing money they invest money so they do everything with the opposite sign to me so considering the long forward with the direction of trades reversed forget that because if it was possible for my banker to just go and make money then that's an arbitrage for them and in fact if they're offering me a deal which okay there's money on the table I can make money for doing nothing well we do this that's an arbitrage so we're going to assume this can't happen either but now we put those two together and we find that the forward price is equal e to the R to take T times s T that is the current price of one US dollar so the price at which we agree to trade in the future depends on the current exchange rate well the current exchange rate and the interest rate payments okay so just before we continue we should just think a little bit about this this payoff here so the payoff of a forward is so if this is the stock price okay this is the payoff well if I go for the short forward which is what I've described here so that's an agreement to sell the asset then the payoff looks like this on the other hand if I look at the other side of the deal an agreement to buy the asset in one year's time then I get the opposite this is the long forward okay so what are some peculiar things that we notice when we're doing this observations the forward value forward price is based on current prices and interest payments it does not depend on whether st is a fair price or the asset I have made no assumption anywhere that the current exchange rate is a sensible exchange rate and there's something a little bit weird going on here that I haven't even assumed anything about the future evolution of the exchange rate we're making an agreement for what's going to happen in one year's time and I have told you nothing about what I think the exchange rate will be in one year's time that has not entered into our calculations at all so even though we've got a contract the payoff of which depends on the exchange rate in one year's time okay whether it's profitable so whether I make a profit or a loss depends on what the exchange rate in one year's time is we have not used any statements about what I think this is what I think the exchange rate will be now if you've got an efficient market then this should be a fair price and there is a secret assumption it's not so secret we made it up here there are no transaction costs and I can buy and sell any amount for the same price so because of that if there was money on the table if this was a bad price this would be a very strange assumption to be making because what it's saying is I can start just buying as much as I like without changing the price anyway but our calculation for the forward price actually doesn't depend on whether we have a fair price for the asset at all hey we did not need to model st didn't need a model for what the future exchange rate will be so if interest rates are positive which is usually a good assumption okay and there is no cost or benefit to holding the asset then ft you look at this and you say well this is a positive this is a number well if R is positive this quantity is going to be bigger than 1 if I made a silly mistake someone then ft is a set price okay but we've made an assumption in there so question what if interest is and on holding so let's quickly in the last minute so what happens if interest is earned on holding US dollars at a rate R hat well okay the thing is here we made our contract about 1 US dollar but if I'm going to earn interest on holding US dollars in this deal I buy 1 US dollar and I hold it to the end so how about I change this to holding e to the minus R hat do you want to see US dollars why is that amount well this is the amount of u.s. dollars I need to buy today to have one u.s. dollar in a year's time or in the future so if I hold if I do this I get the same payoff okay but what's it gonna cost me well I have to correct this because I no longer how much do I borrow well I get e to the minus R hat I've got to change the amount i buy these combined okay so you get now this is e to the R minus R hat to take tea nestea so this changes and you end up with e to the R minus R hat do you take T s T and this is the sort of formula you get whenever there's a cost or a benefit for holding the asset now four words are traded on many things including commodities that was the the classic example is buying agricultural products they've been traded since antiquity so there's evidence for them in the code of hammurabi which is 18th century BC there's examples of them in Aristotle in the politics on the other hand trading on financial products like interest rates exchange rates is much more recent that's really since the 1970s so they we're out of time we'll continue talking about this tomorrow morning where we will also discuss what are the criticisms of what we've done where is what we've done gone wrong you

Channel: Oxford Mathematics

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Keywords: Oxford Mathematics Student Lectures, Mathematical Models of Financial Derivatives, Maths and Finance, Sam Cohen

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Length: 49min 13sec (2953 seconds)

Published: Wed Feb 12 2020