---------------------------------------------------------------------- Welcome to Lecture 3 of Notes on Financial Mathematics, a lecture in three acts, by Scot Adams and Fernando Reitich. In our last two lectures, we studied a derivative whose underlying is the Euro. The derivative is financed by a hedging portfolio whose starting value is ---------------------------------------------------------------------- question mark dollars, and whose ending value, after one month, is ---------------------------------------------------------------------- 5 dollars on an uptick in the Euro, or ---------------------------------------------------------------------- nothing on a downtick. Our assumption, on the Euro itself, is that it starts out the month at ---------------------------------------------------------------------- 1 dollar, and ends up either at ---------------------------------------------------------------------- a dollar five on an uptick, or at ---------------------------------------------------------------------- 95 cents on a downtick. Meanwhile, we have a risk-free market, called the bank, in which an investment of ---------------------------------------------------------------------- 1 dollar will grow to ---------------------------------------------------------------------- a dollar one, whether the Euro goes up or down. The hedging portfolio we constructed consists of ---------------------------------------------------------------------- x Euros and ---------------------------------------------------------------------- y dollars ---------------------------------------------------------------------- borrowed from the bank and, again, this portfolio ---------------------------------------------------------------------- pays for the derivative. More professionally, we say that it "replicates" the derivative. From the graphic you see in front of you, we obtained ---------------------------------------------------------------------- these three equations in the three unknowns question mark, x and y. In the last lecture, we used risk-neutral pricing to show that ---------------------------------------------------------------------- question mark is 2.97, with*out* solving the system of equations. Let's now say that ---------------------------------------------------------------------- we're pricers, so we know this price. Let's also say ---------------------------------------------------------------------- we want to get hedge. That is, we want to learn how to find ---------------------------------------------------------------------- x and y with*out* solving the system of equations. Since we know that ---------------------------------------------------------------------- question mark is equal to 2.97, we see, from ---------------------------------------------------------------------- this, that x minus y is 2.97, so, if we know x *or* y, then we can figure out the other, by solving a single equation, not a system. In this lecture, we'll explain Delta-hedging, which is a trick for computing ---------------------------------------------------------------------- x. ---------------------------------------------------------------------- This graphic shows the possible evolution of the exchange rate over the month. The ---------------------------------------------------------------------- difference between 1.05 and point 95 is ---------------------------------------------------------------------- point 1. ---------------------------------------------------------------------- The option starts out worth question mark dollars, and pays off, after one month, at 5 dollars or zero. The ---------------------------------------------------------------------- difference between 5 and zero is ---------------------------------------------------------------------- 5. ---------------------------------------------------------------------- Here again is the system of equations. If we take ---------------------------------------------------------------------- the first equation ---------------------------------------------------------------------- minus ---------------------------------------------------------------------- the second ---------------------------------------------------------------------- then these cancel, and we get ---------------------------------------------------------------------- this. ---------------------------------------------------------------------- Solving, ---------------------------------------------------------------------- and computing the numerator and denominator, we see that x is just ---------------------------------------------------------------------- the option difference over the Euro difference. So that's one way to get x. Here's a second approach. Let's ---------------------------------------------------------------------- back up to the graphic that gave us the system of equations, and then ---------------------------------------------------------------------- put in the differences. Note that the bank's difference is ---------------------------------------------------------------------- zero. A difference can be thought of as a measure of volatility, and the bank has no volatility at all; it's completely ---------------------------------------------------------------------- predictable. Now we can see that ---------------------------------------------------------------------- x times ---------------------------------------------------------------------- point 1 ---------------------------------------------------------------------- minus y times ---------------------------------------------------------------------- zero ---------------------------------------------------------------------- is 5. Let's ---------------------------------------------------------------------- write that down. ---------------------------------------------------------------------- This term is zero, so we ---------------------------------------------------------------------- drop it. ---------------------------------------------------------------------- Solving for x, we again get ---------------------------------------------------------------------- the option difference over the Euro difference. In some sense, though, neither of these two approaches is satisfactory, since we're not really using ideas from finance to get at x. Here's a third approach. Let's examine carefully a general portfolio ---------------------------------------------------------------------- with a Euros and b dollars invested in the bank. Graphically, a Euros looks like ---------------------------------------------------------------------- this, and ---------------------------------------------------------------------- we add a b dollar bank investment to get ---------------------------------------------------------------------- this portfolio which starts out worth ---------------------------------------------------------------------- a plus b dollars, and ends up worth either ---------------------------------------------------------------------- this or ---------------------------------------------------------------------- this, depending on whether the Euro ticks up or down. ---------------------------------------------------------------------- Here's that portfolio again. We calculate its ---------------------------------------------------------------------- difference, or volatility. To do this, we need to take ---------------------------------------------------------------------- this minus ---------------------------------------------------------------------- this. ---------------------------------------------------------------------- These cancel, and ---------------------------------------------------------------------- 1 point 05 minus ---------------------------------------------------------------------- point 95 is point 1, so the difference is ---------------------------------------------------------------------- point 1 times a. Next, we expose a ---------------------------------------------------------------------- theorem which states that ---------------------------------------------------------------------- if two portfolios have equal differences ---------------------------------------------------------------------- then they have the same number of Euros. Intuitively, this is clear: If two portfolios exhibit the same level of volatility, then they should have the same amount of the only volatile asset, which, in this simple economy, is the Euro. More formally, here's a ---------------------------------------------------------------------- proof. Form a portfolio of ---------------------------------------------------------------------- a Euros and b in the bank, and another portfolio of ---------------------------------------------------------------------- c Euros and d in the bank. We saw that the first portfolio has a ---------------------------------------------------------------------- difference of point 1 times a, and a similar calculation shows that the second portfolio has a ---------------------------------------------------------------------- difference of point 1 times *c*. To say that they have ---------------------------------------------------------------------- equal differences is to say that ---------------------------------------------------------------------- point 1 times a is equal to ---------------------------------------------------------------------- point 1 times c. ---------------------------------------------------------------------- Let's write that down, and ---------------------------------------------------------------------- cancel point 1 from both sides, leaving ---------------------------------------------------------------------- a equals c. In other words, ---------------------------------------------------------------------- the number of Euros in the first portfolio is equal to ---------------------------------------------------------------------- the number of Euros in the second, so they do ---------------------------------------------------------------------- have the same number of Euros, ---------------------------------------------------------------------- completing the proof. Now ---------------------------------------------------------------------- here's a picture again of Alice's hedging portfolio. It ends up with a ---------------------------------------------------------------------- difference of 5. Our goal is to figure out how many Euros it has, and we denoted this unknown number by ---------------------------------------------------------------------- x. ---------------------------------------------------------------------- Here's what a portfolio of *one* Euro looks like. It ends up with a ---------------------------------------------------------------------- difference of point 1. These two differences ---------------------------------------------------------------------- are far apart, since ---------------------------------------------------------------------- 5 is 50 times bigger than ---------------------------------------------------------------------- point 1. So, thinking about applying our theorem, let's look at ---------------------------------------------------------------------- 50 Euros, which looks like ---------------------------------------------------------------------- this, and has a ---------------------------------------------------------------------- difference of 5. Now the ---------------------------------------------------------------------- two differences are the same. By our theorem, the ---------------------------------------------------------------------- two portfolios have the same number of Euros. The number of Euros in the hedging portfolio is ---------------------------------------------------------------------- x, while the number of Euros in the 50 Euro portfolio is, of course, ---------------------------------------------------------------------- 50, and we conclude that ---------------------------------------------------------------------- x is 50. We summarize this argument as a ---------------------------------------------------------------------- fact that says that if a porfolio has ---------------------------------------------------------------------- difference equal to the ---------------------------------------------------------------------- difference of a 50 Euro portfolio, then it must have, in it, ---------------------------------------------------------------------- 50 Euros. Intuitively, they both have the same level of volatility, and since volatility can only be attributed to Euros, they must have the same number of Euros. This ---------------------------------------------------------------------- Fact is part of a ---------------------------------------------------------------------- general observation that says the following. ---------------------------------------------------------------------- Suppose we're studying a portfolio that consists of ---------------------------------------------------------------------- a risk-free asset, like a bank investment, and ---------------------------------------------------------------------- a risky asset, like Euros. Suppose it ---------------------------------------------------------------------- has difference equal to ---------------------------------------------------------------------- the difference of a portfolio of N units of the risky asset. ---------------------------------------------------------------------- Then this portfolio we're studying also has N units of the risky asset. This --------------------------------------- observation depends on knowing that the risk-free asset has zero volatility, whereas the risky asset has nonzero volatility. In this kind of situation, the difference of a portfolio tells us how much of the risky asset is to be found in it. So, in three different ways, we've now seen that ---------------------------------------------------------------------- x is the option difference over the Euro difference. A quotient of two differences is called a ---------------------------------------------------------------------- difference quotient. Remember that, for Alice and Dan, the ---------------------------------------------------------------------- Euro market is the ---------------------------------------------------------------------- *underlying* market, and the ---------------------------------------------------------------------- option market is the ---------------------------------------------------------------------- *derivative* market, so x can also be described as ---------------------------------------------------------------------- the derivative difference over the underlying difference. This difference quotient is also called ---------------------------------------------------------------------- the "Delta" of the derivative. Its significance lies in the ---------------------------------------------------------------------- Principle of Delta-Hedging, which says that ---------------------------------------------------------------------- the Delta of a derivative ---------------------------------------------------------------------- is the amount of the underlying needed in the hedging portfolio. The hedging portfolio is sometimes called the ---------------------------------------------------------------------- "hedge" for short. In the above example, 50 Euros are needed in the hedge. Any less, and we don't get the needed amount of volatility to replicate the derivative's payoff. Any more, and we get too much. ---------------------------------------------------------------------- The term, or period of time, for Dan's option was one month, and we'll be interested in exploring what happens as portfolios are adjusted more and more frequently, which would make that ---------------------------------------------------------------------- period of time tend to zero. By general ideas in calculus, as the denominator of a ---------------------------------------------------------------------- difference quotient tends to zero, the difference quotient itself tends to a rate of change. Applying that to this situation, ---------------------------------------------------------------------- Delta limits on the instantaneous rate of change in the derivative price ---------------------------------------------------------------------- per unit change in the underlying price. General calculus also teaches us that rates of change are called derivatives, and, applying *that* to this situation, we see that Delta tends to ---------------------------------------------------------------------- the derivative of the derivative price ---------------------------------------------------------------------- with respect to the underlying price. Of course, this is confusing, because the word "derivative" is being used in two different ways. ---------------------------------------------------------------------- Here, "derivative" is being used in the sense of calculus, as a rate of change, whereas ---------------------------------------------------------------------- here, it's being used in the sense of finance, as a financial instrument which is derived out of some underlying instrument. I suppose there's a weak connection in that the derivative of a function is derived out of the underlying function, through a mathematical process called differentiation. However, differentiation of functions is clearly not the same as the financial process of developing new markets out of old ones. ---------------------------------------------------------------------- Here, again, is our system of equations, and we now focus on ---------------------------------------------------------------------- this one ---------------------------------------------------------------------- and forget the others. We now know, by the principle of Delta-hedging, that ---------------------------------------------------------------------- x is 50. As pricers, we know about risk-neutral pricing, so we know that ---------------------------------------------------------------------- ? is 2.97. Plugging these in, we get ---------------------------------------------------------------------- this single equation in y. Solving this equation ---------------------------------------------------------------------- we find that y is ---------------------------------------------------------------------- 47.03. So we pricers have finally gotten hedge! We can now redisplay ---------------------------------------------------------------------- our earlier graphic, representing three equations in three unknowns. We ---------------------------------------------------------------------- plug in the values of x, y and ?, and we leave it as an exercise for you to get out your calculator, and check that the resulting three equations are all true, . . ., at least, up to rounding error. To recap, Alice offers Dan a financial instrument, called an option, which ---------------------------------------------------------------------- looks like this, and she hedges it by a ---------------------------------------------------------------------- strategy of holding 50 Euros, and taking out a 47.03 bank loan. This is nice, but is ---------------------------------------------------------------------- $2.97 really the *right* price? Could someone else, by some other method, decide to price the option differently? What's so special about pricing via setting up a hedging portfolio anyway? Of course, as long as Alice is willing to sell the option at 2.97, a competitor can't get away with a higher price. Still, could someone *under*cut Alice's price? In fact, remember how Alice and Cathy had a falling out? ---------------------------------------------------------------------- Suppose Cathy, in a moment of rage, decides to compete with Alice, and offers the same option at $2.50. So, what Cathy's offering ---------------------------------------------------------------------- looks like this. Of course, Alice can get angry, but, even better, given that she believes in her own $2.97 price, she has a way to get even. ---------------------------------------------------------------------- Alice's response would come in three steps. ---------------------------------------------------------------------- At the start of the month, she ---------------------------------------------------------------------- first, buys Cathy's option. The general precept is that one should always buy low, and if one thinks that an asset is underpriced, the first step is always to buy it. In this case, Alice thinks Cathy has underpriced the option, so Alice jumps in, and buys. Next, Alice looks at ---------------------------------------------------------------------- the hedging strategy that she uses when *selling* the option. She's now *buying* from Cathy, and so she negates her seller's strategy. That is, instead of buying *50* Euros and taking out a 47.03 loan *from* the bank, ---------------------------------------------------------------------- she buys *minus 50* Euros and ---------------------------------------------------------------------- invests $47.03 *into* the bank, in a savings account. You may wonder how Alice can ---------------------------------------------------------------------- buy -50 Euros, and the answer is that she can borrow 50 Euros from somewhere, and sell them for dollars. At $1 per Euro, this brings in $50, but she incurs a 50 Euro debt, which she'll pay back a month later. The cost to her, at that time, will be 5% more or 5% less, depending on whether the Euro goes up or down over that month. In her --------------------------- seller's hedge, Alice *spent* 50 dollars at the beginning of a month, in order to acquire 50 Euros, and then cashed them out for dollars at the end of the month, for 5% more or 5% less. By contrast, she's here receiving 50 dollars at the beginning of a month, by selling 50 borrowed Euros, and then losing 5% more or 5% less, at the end, in order to repay the loan. This process of borrowing an asset and selling it, as opposed to selling something you already own, is called ---------------------------------------------------------------------- "short selling", and we say that the second step of Alice's response is to ---------------------------------------------------------------------- short 50 Euros. For simplicity, we'll assume that short selling is available on any asset we study in this lecture, *and* is available with*out* transaction costs. Again remember that the short sale brings in $50, but will result in a *negative* dollar cash flow a month later, when Alice buys the Euros and pays them back. This process of buying and paying back the Euros is called ---------------------------------------------------------------------- "closing out the short position". ---------------------------------------------------------------------- Remember this graphic which represents three equations valid up to rounding error. It says that there's a portfolio of ---------------------------------------------------------------------- Euros and ---------------------------------------------------------------------- bank that starts out worth ---------------------------------------------------------------------- $2.97, and that ends up at ---------------------------------------------------------------------- $5 or zero, depending on whether the Euro ticks up or down. If we decide to set up this portfolio, it results in ---------------------------------------------------------------------- an initial cash flow of ---------------------------------------------------------------------- *minus* 2.97, since we have to pay $2.97 to buy the portfolio. On the other end, if we sell the portfolio at the end of the month, it ---------------------------------------------------------------------- gives back $5 or zero depending on uptick or downtick. Important to remember is ---------------------------------------------------------------------- this minus sign. The ---------------------------------------------------------------------- initial cash flow is the negative of the ---------------------------------------------------------------------- initial portfolio value, whereas ---------------------------------------------------------------------- *final* cash flow is *equal* to the ---------------------------------------------------------------------- final portfolio value. Now let's ---------------------------------------------------------------------- multiply this portfolio by minus 1. On the left hand side, we'll negate ---------------------------------------------------------------------- these two numbers, while on the right hand side, we'll negate ---------------------------------------------------------------------- all three of them. ---------------------------------------------------------------------- Done. You should check that this negates each of the three equations represented in the original graphic, and so these new equations are also true, up to rounding. Now remember ---------------------------------------------------------------------- Alices's response. Note that ---------------------------------------------------------------------- steps 2 and 3 are to ---------------------------------------------------------------------- short 50 ---------------------------------------------------------------------- Euros and invest ---------------------------------------------------------------------- 47.03 ---------------------------------------------------------------------- in the bank, so ---------------------------------------------------------------------- this shows the results of steps 2 and 3. ---------------------------------------------------------------------- Note the negative numbers. The initial cost ---------------------------------------------------------------------- of minus $2.97 indicates that the initial cash flow from setting up Steps 2 and 3 will be a *positive* $2.97. So, if Alice does only Steps 2 and 3, she'll get $2.97 with which to buy, say, a fancy cup of coffee. Let's break that $2.97 down. The ---------------------------------------------------------------------- short sale of the 50 Euros, at ---------------------------------------------------------------------- one dollar per Euro, brings in $50, but Alice ---------------------------------------------------------------------- invests $47.03 of that $50 into the bank, leaving $2.97 for her coffee. Of course, she has to worry that, a month later, she might ---------------------------------------------------------------------- incur a negative $5 cash flow, so this isn't a free lunch, or, rather, it isn't free coffee. In any case, ---------------------------------------------------------------------- this graphic does describe ---------------------------------------------------------------------- Steps 2 and 3. ---------------------------------------------------------------------- Here's Cathy's competing offer, which is ---------------------------------------------------------------------- Step 1 in Alice's response above. Adding, we can get ---------------------------------------------------------------------- Alice's response in total, which consists of ---------------------------------------------------------------------- Step 1 combined with Steps 2 and 3. ---------------------------------------------------------------------- Adding this to this, ---------------------------------------------------------------------- we get negative 47 cents. ---------------------------------------------------------------------- For the case of an uptick, ---------------------------------------------------------------------- we add minus 5 dollars to 5 dollars, ---------------------------------------------------------------------- and get zero. ---------------------------------------------------------------------- For the case of a downtick, ---------------------------------------------------------------------- we add zero to zero ---------------------------------------------------------------------- and get zero. So ---------------------------------------------------------------------- this graphic ---------------------------------------------------------------------- totals all three steps. ---------------------------------------------------------------------- The negative 47 cents represents a positive cash flow of 47 cents that comes from the initial setup of all three steps. Alice can enjoy, perhaps, a small chocolate mint. ---------------------------------------------------------------------- These two zeroes represent the lack of any possible debit (or credit) at the end of the month. In other words, the setup of this portfolio brings in 47 cents, and it has no future risk. Evidently, there is such a thing as a free mint. One little thing. Alice has made an assumption of 5% up or 5% down on the Euro, an assumption that is sometimes called a volatility assumption. If that assumption should fail, then Cathy might well end up with the last laugh. So here, much depends on Alice's confidence in her own sense of the market, since volatility assumptions are predictions of the future, and are not objectively verifiable. Let's say, though, that Alice has a hazy crystal ball that does tell her that her volatility assumption will hold, though it doesn't specifically say which way the Euro will go. Then Alice's three-step response wins. Let's call a portfolio ---------------------------------------------------------------------- an arbitrage portfolio if it starts out at a ---------------------------------------------------------------------- negative value, and ends up ---------------------------------------------------------------------- at zero in all scenarios. In other words, it brings in a positive cash flow immediately, and has no future risk. A broader definition of arbitrage is sometimes used, but this one'll work for us for now. Before Cathy started competing, the two basic instruments in our economy were Euros and bank, and Alice did her work with portfolios made up of those two. Then Cathy injected something new. Recall that a general portfolio of ---------------------------------------------------------------------- a Euros and b dollars in the bank ---------------------------------------------------------------------- looks like this. ---------------------------------------------------------------------- Note that no such portfolio is an arbitrage portfolio. In fact, ---------------------------------------------------------------------- if both of these expressions are zero, then a and b are both zero, so ---------------------------------------------------------------------- this expression cannot be negative. The point is that ---------------------------------------------------------------------- Cathy, by competing with Alice, introduced arbitrage into an otherwise arbitrage-free economy. The net effect of this, in the end, was to give 47 cents to Alice. If Cathy does this a few times, then Alice can send her kids to expensive colleges, or maybe even rent herself a small apartment in New York City. So ---------------------------------------------------------------------- joy to those who find arbitrage, for they can take advantage of those who introduce it. Poor Cathy. Remember that, in the happier world of Lecture 2, ---------------------------------------------------------------------- Cathy sells Fred a two-month option on 100 Euros at a strike of $100. The single word "strike" is short for "strike price". Remember that the ---------------------------------------------------------------------- Euro exchange rate starts at ---------------------------------------------------------------------- 1 over point 95 dollars per Euro, and then ---------------------------------------------------------------------- increases or decreases 5%, and then ---------------------------------------------------------------------- increases or decreases another 5%. Remember that the value of Cathy's hedging portfolio ---------------------------------------------------------------------- evolves like this. It starts out costing ---------------------------------------------------------------------- $8.02, which is what she charges Fred. It ends up worth ---------------------------------------------------------------------- $16.05 or 5 dollars or zero, depending on what happens to the exchange rate over the two month term. For example, ---------------------------------------------------------------------- if there are two upticks, then the portfolio will give her the ---------------------------------------------------------------------- $16.05 needed to meet her obligation to Fred. That $16.05 is the dollar cost of 100 Euros less the 100 dollar strike, at ---------------------------------------------------------------------- this exchange rate. In other words, if we take 100 times ---------------------------------------------------------------------- this, and then subtract 100, then, to two decimals, we get ---------------------------------------------------------------------- 16.05. A more accurate calculation would give ---------------------------------------------------------------------- this. If there's ---------------------------------------------------------------------- uptick-downtick or ---------------------------------------------------------------------- downtick-uptick, then Cathy's portfolio will end up with ---------------------------------------------------------------------- $5, which is exactly what she needs to cover 100 Euros for 100 dollars at ---------------------------------------------------------------------- a dollar five per Euro. Finally, if there are ---------------------------------------------------------------------- two downticks, then Fred will *not* exercise, and Cathy needs ---------------------------------------------------------------------- nothing. In fact, if, for some reason, Fred should make the mistake of exercising in this ---------------------------------------------------------------------- downtick-downtick scenario, then Cathy'll have an unexpected windfall, since, at ---------------------------------------------------------------------- 95 cents apiece, the 100 Euros she owes him cost *less* than the 100 dollar strike that he'd give her on exercising. Suppose that, in the first month, there's an ---------------------------------------------------------------------- uptick in the exchange rate, and, immediately after that, Fred decides he wants to sell his option on the options market. In that case, he'll be offering up, for sale, an asset that will be worth ---------------------------------------------------------------------- $16.05 or 5 dollars depending on whether there's an ---------------------------------------------------------------------- uptick or a ---------------------------------------------------------------------- downtick in the *second* month. The question we ask now is: ---------------------------------------------------------------------- What price can Fred obtain for his option after this ---------------------------------------------------------------------- one uptick? Remember, from Lecture 1, that ---------------------------------------------------------------------- $11.52 is Beth's price for *exactly* the same deal. For $11.52, she offers up, for sale, a contract that will become worth ---------------------------------------------------------------------- $16.05 or 5 dollars, one month later, depending on which way the Euro goes. So Fred is competing with Beth in selling his option. Clearly, he can't sell the option for more than Beth's price of $11.52, because everyone'll choose to work with her, if he tries to offer a higher price. On the other hand, we leave it as an exercise for you to verify that, if Fred tries to undercut Beth, then he'll introduce arbitrage in the market, and she can take advantage of that by buying his option at his cutrate price, and negating her own seller's hedging strategy. So, ultimately, to avoid introducing arbitrage, Fred should sell at ---------------------------------------------------------------------- $11.52. So the $11.52, which we think of as the value of Cathy's hedging portfolio, is also the value of the option itself on the options market. We conclude that ---------------------------------------------------------------------- the value of the hedging portfolio ---------------------------------------------------------------------- is equal to the value of the option. So, we can replace the words ---------------------------------------------------------------------- "portfolio value" by ---------------------------------------------------------------------- "option value" in honor of the fact that the numbers on the ---------------------------------------------------------------------- right give the values of the option on the options market as time evolves. Remember also that, in Cathy and Fred's case, the ---------------------------------------------------------------------- Euro is the underlying, so ---------------------------------------------------------------------- instead of Euro value, we can say ---------------------------------------------------------------------- underlying value, if we like. Similarly, the ---------------------------------------------------------------------- option is the derivative, so ---------------------------------------------------------------------- instead of option value, we can say ---------------------------------------------------------------------- derivative value, if we like. Do remember that there are now multiple ways of talking about the same thing: The underlying is the Euro, the derivative is the option and the option value is the value of the hedging portfolio. Suppose now a ---------------------------------------------------------------------- downtick happens in the first month of the option. Then the option value is ---------------------------------------------------------------------- $2.97 and, after the second month, it might change ---------------------------------------------------------------------- either to 5 dollars or to zero, which gives a ---------------------------------------------------------------------- derivative difference of ---------------------------------------------------------------------- five. After a ---------------------------------------------------------------------- downtick, the value of the underlying Euro is ---------------------------------------------------------------------- $1 and, after the second month, might change ---------------------------------------------------------------------- either to $1.05 or to 95 cents, which gives an ---------------------------------------------------------------------- underlying difference of ---------------------------------------------------------------------- point 1. The quotient of ---------------------------------------------------------------------- these two differences is the ---------------------------------------------------------------------- Delta of the option, and is equal to five over point 1, which gives ---------------------------------------------------------------------- 50. That is, ---------------------------------------------------------------------- after one downtick, 50 ---------------------------------------------------------------------- Euros are needed to match the volatility of ---------------------------------------------------------------------- the option, and so the hedging portfolio must have 50 Euros in it, if it's to replicate the option. Let's track the Delta of the option ---------------------------------------------------------------------- over here. We've just calculated that ---------------------------------------------------------------------- after a downtick, Delta is ---------------------------------------------------------------------- 50. Let's analyze Delta if there's an ---------------------------------------------------------------------- uptick in the first month. In this case, the value of the derivative is ---------------------------------------------------------------------- 11.52, but might change to ---------------------------------------------------------------------- 16.05 or 5, ---------------------------------------------------------------------- which gives a derivative difference of ---------------------------------------------------------------------- 11.05. The value of the underlying is ---------------------------------------------------------------------- this, but might change to ---------------------------------------------------------------------- one of these ---------------------------------------------------------------------- giving an underlying difference of ---------------------------------------------------------------------- point 1105. The quotient of the differences is ---------------------------------------------------------------------- the Delta, and it computes to ---------------------------------------------------------------------- 100, and we fill that in ---------------------------------------------------------------------- over here. So, in the event of an uptick in the exchange rate, Cathy should respond by adjusting her hedge so that it has 100 Euros. Incidentally, she might have guessed this, because, if there's an ---------------------------------------------------------------------- uptick in the first month, then, ---------------------------------------------------------------------- whatever happens in the second month, we know that Fred will exercise, so we know that Cathy will have to give him the promised underlying of 100 Euros. It's a general fact that, when an option enters a range where exercise is unavoidable, then the hedge will contain the full promised underlying. We'll see something similar when we get to obligation derivatives, later in this lecture. Initially, the derivative value is ---------------------------------------------------------------------- $8.02 and will change to ---------------------------------------------------------------------- either $11.52 or $2.97 ---------------------------------------------------------------------- which gives a derivative difference of ---------------------------------------------------------------------- $8.55. The initial underlying value is ---------------------------------------------------------------------- this, and will change to ---------------------------------------------------------------------- one of these two, ---------------------------------------------------------------------- which gives an underlying difference of ---------------------------------------------------------------------- point 10526. The quotient of the differences is ---------------------------------------------------------------------- the Delta, and it computes to ---------------------------------------------------------------------- 81.23, and we fill that in ---------------------------------------------------------------------- over here. A couple of remarks are in order at this time. First, ---------------------------------------------------------------------- there is no Delta at the end of the option, because there are no more differences to be calculated. Moreover, if we think of Delta as telling Cathy how many Euros to hold in the month *after* it's computed, then there's no need for Delta at the end of two months, because that's the moment that she'll close out her portfolio for dollars, and settle with Fred. Our second remark concerns the accretive nature of rounding errors. Remember that this ---------------------------------------------------------------------- $16.05 is more accurately calculated as ---------------------------------------------------------------------- this number. The perspective we'll adopt in these lectures is that the the number ---------------------------------------------------------------------- $16.05 is really just an abbreviation for the ---------------------------------------------------------------------- longer number that it approximates, which, in turn, abbreviates a number with infinitely many decimal places. With some calculation, you can verify that ---------------------------------------------------------------------- 11.52 abbreviates ---------------------------------------------------------------------- this, and that ---------------------------------------------------------------------- 2.97 abbreviates ---------------------------------------------------------------------- this, and that ---------------------------------------------------------------------- 8.02 abbreviates ---------------------------------------------------------------------- this. Redoing ---------------------------------------------------------------------- this subtraction, we see that 8.55 abbreviates ---------------------------------------------------------------------- this. Continuing, ---------------------------------------------------------------------- point 10526 is an abbreviation for ---------------------------------------------------------------------- this. Redoing ---------------------------------------------------------------------- this division, we get ---------------------------------------------------------------------- this number, which, to two decimals, is *not* 81.23, but, rather, ---------------------------------------------------------------------- 81.19. That number belongs ---------------------------------------------------------------------- over here. ---------------------------------------------------------------------- We fix that. Incidentally, deciding on exactly the right hedge really depends on knowing how Cathy's broker and bank handle rounding errors. In these lectures, we'll assume that everyone does their calculations to infinitely many decimals, but we display rounded values in the slides. So when we say that Cathy starts out holding ---------------------------------------------------------------------- 81.19 Euros, she actually tells her broker to buy ---------------------------------------------------------------------- 81 point 18 81 19 62 Euros, and, in fact, she communicates even more accurately than that. Similarly, we'll be assuming that exchange rates are always worked out to infinitely many decimals, even if we only show a rounded abbreviation on our slides. The principle of Delta-hedging tells us that ---------------------------------------------------------------------- Delta is the number of Euros in the hedge. The other hedging parameter is ---------------------------------------------------------------------- the bank loan. Again, hedging amounts are forward-looking, and ---------------------------------------------------------------------- there's no need to have a hedge going forward after Cathy cashes out the portfolio. ---------------------------------------------------------------------- After an uptick, ---------------------------------------------------------------------- the hedge has 100 Euros, each worth ---------------------------------------------------------------------- this many dollars. This combines with ---------------------------------------------------------------------- some unknown bank loan to make a portfolio value of ---------------------------------------------------------------------- 11.52. Thinking of all these numbers as abbreviations for more accurate numbers, solving for the ---------------------------------------------------------------------- unknown, and then rounding to two decimals, we get ---------------------------------------------------------------------- $99.01 as the bank loan after an uptick. ---------------------------------------------------------------------- After a downtick, a similar calculation gives ---------------------------------------------------------------------- $47.03. A similar calculation gives ---------------------------------------------------------------------- $77.44 initially. This completes our analysis of the option that Cathy sells Fred, so with regrets, we bid them adieu. ---------------------------------------------------------------------- In the remainder of this lecture, ---------------------------------------------------------------------- we'll practice more pricing and hedging, we'll define and study something we call ---------------------------------------------------------------------- obligation derivatives, we'll learn about ---------------------------------------------------------------------- put-call parity, and about two other kinds of derivatives, called ---------------------------------------------------------------------- forwards and futures. Finally, we'll set up ---------------------------------------------------------------------- an option whose hedging strategy involves millions of adjustments. For now, though, ---------------------------------------------------------------------- let's take a break. ---------------------------------------------------------------------- Welcome to the second act of Lecture 3 of Notes on Financial Mathematics, by Scot Adams and Fernando Reitich. Let's remember, from Lecture 2, our friends ---------------------------------------------------------------------- Gail and Harry. Harry's the CEO of XYZ corporation, and he buys a stock option from Gail. Specifically, she sells him a three-month option to buy 1000 shares of XYZ at a strike of 970 with the spot price at $1 per share. Into ---------------------------------------------------------------------- a template, we filled the ---------------------------------------------------------------------- share prices, and the ---------------------------------------------------------------------- values of Gail's hedging portfolio. Now we understand that that ---------------------------------------------------------------------- portfolio value is also the price that the option would fetch if sold on the options market. We can therefore call the red numbers the ---------------------------------------------------------------------- option value. Let's now focus on the hedging parameters: ---------------------------------------------------------------------- the number of XYZ shares in the hedging portfolio, and ---------------------------------------------------------------------- the dollar bank loan. Note again that we often refer to the "hedging portfolio" simply as the ---------------------------------------------------------------------- hedge. ---------------------------------------------------------------------- There is no hedging at the end, because hedging parameters are forward-looking. Let's start by working out the hedging parameters ---------------------------------------------------------------------- here in the middle of the template. The ---------------------------------------------------------------------- number of shares in the hedge will be the Delta of the option, which is a difference quotient. It is the difference of ---------------------------------------------------------------------- these two numbers, ---------------------------------------------------------------------- which is this, divided by the difference of ---------------------------------------------------------------------- these two numbers, ---------------------------------------------------------------------- which is this. Using a calculator, we see that this difference ---------------------------------------------------------------------- quotient is equal to ---------------------------------------------------------------------- 884.69, to two decimals. Or is it? ---------------------------------------------------------------------- This denominator is exactly correct, but the numbers in the numerator are more accurately described ---------------------------------------------------------------------- by these numbers, which would then give ---------------------------------------------------------------------- this Delta, which ---------------------------------------------------------------------- rounds to 884.72. So, following the standards of these lectures, the right answer is ---------------------------------------------------------------------- 884.72. ---------------------------------------------------------------------- You may worry that ---------------------------------------------------------------------- this equation wouldn't pass the calculator test -- in that if you pull out your calculator, and work out the answer, you'll get 884.69, not 884.72. However, we ask you to remember, throughout these lectures, that, in any equation, each number may stand in as an approximation for a more accurate number. In particular, ---------------------------------------------------------------------- 58.42 and 15.07 are simply abbreviations for ---------------------------------------------------------------------- these longer numbers which, in turn abbreviate numbers with infinitely many decimals. Similarly, ---------------------------------------------------------------------- 884 point 72 is an abbreviation for ---------------------------------------------------------------------- this longer number. In any event, we fill in ---------------------------------------------------------------------- 884 point 7 2 here. Let's now work on ---------------------------------------------------------------------- this hedging parameter. This number is a ---------------------------------------------------------------------- bank *loan*, ---------------------------------------------------------------------- so it ---------------------------------------------------------------------- contributes ---------------------------------------------------------------------- negatively to the value of the hedge. The hedge also has ---------------------------------------------------------------------- 884 point 7 2 Euros which are each worth ---------------------------------------------------------------------- 98 cents and this contributes ---------------------------------------------------------------------- positively to the value of the hedge. The total value of the hedge is ---------------------------------------------------------------------- 48.77. Each node has an equation, and this is the equation of ---------------------------------------------------------------------- this node. We now solve for the unknown, represented by a ---------------------------------------------------------------------- blue rectange. Remember, though, that ---------------------------------------------------------------------- 884.72 and ---------------------------------------------------------------------- 48.77 are just abbreviations of longer numbers. Using these longer numbers, solving and rounding to two decimals, we get ---------------------------------------------------------------------- 818.25 which we fill in ---------------------------------------------------------------------- here. Note that the ---------------------------------------------------------------------- black numbers, the values of the underlying, were calculated ---------------------------------------------------------------------- forward in time, from left to right. The ---------------------------------------------------------------------- red numbers, the values of the derivative, were calculated ---------------------------------------------------------------------- backward in time, from right to left. ---------------------------------------------------------------------- The green and blue numbers, the hedging parameters, can be calculated in any order, and, to make that point, we started ---------------------------------------------------------------------- here, in the middle. We now leave it as an exercise for you to calculate ---------------------------------------------------------------------- the rest of these numbers, in whatever order you prefer. Some of the numbers on this slide are abbreviations of longer numbers ---------------------------------------------------------------------- which we display here, in case you want to check your work. Each green number is a Delta difference quotient and then each blue number is obtained by solving the equation of its node. Remember, from Lecture 2 ---------------------------------------------------------------------- the payoff function for this option, which shows how much Gail must have in her hedging portfolio, as a function of the final stock price. That is, if you plug ---------------------------------------------------------------------- these black numbers, one by one, into ---------------------------------------------------------------------- the function f, you get ---------------------------------------------------------------------- these red numbers, which, as a group, are called the ---------------------------------------------------------------------- contingent claim. Let's move on to another derivative. Let's talk about a fellow named Irwin. Irwin happens to hold 1000 shares of XYZ, and ---------------------------------------------------------------------- he's decided to sell them three months from now, but he's unhappy with how Harry's running XYZ corporation. He's worried that the price will collapse, and he seeks a guaranteed price of ---------------------------------------------------------------------- $970. ---------------------------------------------------------------------- He wants the right to do this sale, but not the obligation. If the share price soars, he wants to be able to sell at the market price, and an option gives him that option. ---------------------------------------------------------------------- He also works with Gail, who needs to price this option. Say, once again, that the current XYZ price, called the ---------------------------------------------------------------------- spot price, is $1 per share. ---------------------------------------------------------------------- This kind of option, guaranteeing a *sale* price is a "put" option, or, more simply, a "put". ---------------------------------------------------------------------- Until now, we've done options that guarantee the holder the right to *buy* a certain promised amount of the underlying, at a predetermined price called the strike, or strike price. ---------------------------------------------------------------------- That kind of option is a "call" option, or, more simply, a "call". Until now, we've said "option", but, from now on, we'll use the more specific terms ---------------------------------------------------------------------- "call option" and ---------------------------------------------------------------------- "put option", unless the meaning is clear from context. The predetermined price is now a ---------------------------------------------------------------------- *sell* price, not a *buy* price, but we still call it ---------------------------------------------------------------------- the strike price, or strike. ---------------------------------------------------------------------- Sometimes it's called the exercise price. In this example, ---------------------------------------------------------------------- it's $970. ---------------------------------------------------------------------- The term of an option is the time from the signing of the contract to the moment when the holder has to decide whether or not to exercise. In this example, ---------------------------------------------------------------------- it's three months. In pricing an option, one of the first steps is to write down the payoff function. So, ---------------------------------------------------------------------- let g(S) denote the payoff function for this put option. That's the cost to Gail, as a function of the ---------------------------------------------------------------------- ending price S of the stock. Now, *if* Irwin decides to ---------------------------------------------------------------------- exercise the option, then he'll be insisting on unloading his ---------------------------------------------------------------------- 1000 shares on Gail, and she must pay him ---------------------------------------------------------------------- $970 for those shares. So the dollar cost to Gail, ---------------------------------------------------------------------- if he exercises, ---------------------------------------------------------------------- is that 970, but it's partially offset, because she does get the ---------------------------------------------------------------------- 1000 shares, which are worth ---------------------------------------------------------------------- S dollars per share, lowering her dollar cost by ---------------------------------------------------------------------- 1000 S. So, *if* Irwin ---------------------------------------------------------------------- exercises, she effectively pays out ---------------------------------------------------------------------- this amount to him. So, if ---------------------------------------------------------------------- this expression is negative, then, on exercising, Irwin effectively loses money to Gail, and she gets an unexpected windfall. More accurately, if ---------------------------------------------------------------------- this is negative, then Irwin would do better, *not* to exercise, and, rather, to sell his shares on the open market, where he'd make more than $970. So the dollar cost to Gail will be ---------------------------------------------------------------------- 970 - 1000 S, *if* that amount is positive, and zero otherwise. We can express that using the ---------------------------------------------------------------------- subscript plus notation, like this, as the ---------------------------------------------------------------------- positive part of ---------------------------------------------------------------------- 970 - 1000 S. Remember that the positive part of a positive number is the number itself, and the positive part of a nonpositive number is zero. Speaking of exercising, we leave it as an ---------------------------------------------------------------------- exercise for you to ---------------------------------------------------------------------- graph this function g. Let's compare g to ---------------------------------------------------------------------- the payoff function for the *call* option that Gail sold to *Harry* back in Lecture 2. That function, which we called ---------------------------------------------------------------------- f looks a lot like g, but has ---------------------------------------------------------------------- 1000 S - 970 instead of ---------------------------------------------------------------------- 970 - 1000 S. That is, the expression in parentheses for f is the negative of the one for g. This is characteristic of call and put options. The ---------------------------------------------------------------------- call payoff function is the ---------------------------------------------------------------------- positive part of the ---------------------------------------------------------------------- final price of the underlying minus ---------------------------------------------------------------------- the strike, but for a ---------------------------------------------------------------------- put option, it's the other way around. It's important that, for both, one is taking the ---------------------------------------------------------------------- positive part of *something*, which means that the option seller, Gail, will never *receive* any money at the time of exercise -- she can only pay out a positive amount or zero. This may sound bad for Gail, but it has the nice consequence that she doesn't have to worry about engaging a collection agency if Harry or Irwin should fail to show up at the time of exercise. Failing to show up to cover a debt is called "walking away from one's position", but, buyers of options never have a debtor's position from which to walk away. This is exactly and precisely because the payoff function is never negative, which is the analytic way of expressing the fact that buyers have the option of not exercising. Why *would* they exercise, if it'd lose them money? Speaking of exercising, remember that, in Lecture 2, we left it as an ---------------------------------------------------------------------- exercise for you to ---------------------------------------------------------------------- graph the function f. Okay. Another important step in the pricing of derivatives is to immerse ourselves into the risk-neutral world. Remember the ---------------------------------------------------------------------- number line with the ---------------------------------------------------------------------- downtick and uptick factors, and the ---------------------------------------------------------------------- risk-free factor which gave us the ---------------------------------------------------------------------- risk-neutral uptick and downtick probabilities which, in turn, gave ---------------------------------------------------------------------- these expected values on stock and bank. Characteristic of ---------------------------------------------------------------------- these risk-neutral probabilities is the fact that the expected return on stock and bank are ---------------------------------------------------------------------- equal. In this case, both are 2%. This tells us that, in this imaginary risk-neutral world, *any* portfolio of stock and bank will have an expected return of 2% per month. We're now used to all this, and familiarity breeds contempt, so we reduce our reverential time of admiration to a bit less than a second. Done. Moving on to our ---------------------------------------------------------------------- template, we fill in ---------------------------------------------------------------------- XYZ share prices ---------------------------------------------------------------------- from left to right ---------------------------------------------------------------------- like so. As soon as that's done, we move on to ---------------------------------------------------------------------- the value of the option on the options market, which is the same as the value of Gail's hedging portfolio. This we compute from ---------------------------------------------------------------------- right to left. We plug ---------------------------------------------------------------------- the ending share prices, one by one, ---------------------------------------------------------------------- into the payoff function g, obtaining ---------------------------------------------------------------------- the contingent claim. Jumping to ---------------------------------------------------------------------- the risk-neutral world, and working out discounted expected values ---------------------------------------------------------------------- right to left, we find ---------------------------------------------------------------------- these option values, and we see that the option's price, to two decimals, is ---------------------------------------------------------------------- 22 cents. Let's get to work on the hedging parameters. To hedge, we'll soon see that Gail will have to hold a negative amount of the stock. That is, instead of *using up* dollars to buy stock, she'll borrow stock from people who have it, and then sell it, to *bring in* dollars. Later, instead of selling the stock to *bring in* dollars, she'll have to *use up* dollars to buy back the stock to repay her loan. As with Euros, this is called ---------------------------------------------------------------------- short selling, and we say that Gail will have to short XYZ stock to set up her hedging portfolio. To keep our lives simple, we'll assume that there are no fees or transaction costs involved in short selling -- Gail has lots of friends, with lots of XYZ stock, and they're happy to loan it to her for free. Perhaps the real world is not so accommodating. So we'll be caculating ---------------------------------------------------------------------- the number of shorted shares at each node. We can do hedging parameters in any order we like. Let's start ---------------------------------------------------------------------- here, after two downticks, and let's fill in ---------------------------------------------------------------------- this box. The Delta, at this point, is the difference quotient obtained by dividing the difference of ---------------------------------------------------------------------- these two derivative values by the difference of ---------------------------------------------------------------------- these two underlying values, which gives ---------------------------------------------------------------------- this difference quotient, and which calulates out to ---------------------------------------------------------------------- this *negative* number. The Delta of a derivative is the amount of the underlying that the derivative seller, Gail, should *hold* in her hedge in order to get the hedging portfolio's volatility to match that of the derivative. In this case, Gail should hold ---------------------------------------------------------------------- a negative number of XYZ shares, which means she should ---------------------------------------------------------------------- short the corresponding ---------------------------------------------------------------------- positive number of shares. It's an exercise for you now to fill in ---------------------------------------------------------------------- these numbers. To hedge a put option, one shorts the underlying and invests money *into* a risk-free asset, like a bank account. Holding a negative amount of an asset is called shorting it, or taking a ---------------------------------------------------------------------- short position on it. Holding a positive amount of an asset is called holding it, or taking a ---------------------------------------------------------------------- long position on it. The hoity-toity academic finance professor asserts that to hedge a put, the option seller takes a short position in the underlying, and a long position in the risk-free asset. That is, she shorts the underlying, and holds the risk-free. To sound professional, the academic finance professor typically uses a riskless ---------------------------------------------------------------------- bond market as the risk-free asset, but we've always had our option sellers running to a nearby bank. In the real world, most traders do, in fact, use bonds, but, for this lecture, we'll continue to stick with the ---------------------------------------------------------------------- bank, for the pedagogical reason that a down-to-earth presentation is, perhaps, more easily understood, and most of us have had more experience with bank than bonds. All that aside, we do need to keep in mind, that, to hedge a *put*, a long position is needed on the risk-free side of the portfolio. That is, one does not borrow *from* the bank, but invests money *into* it, in a ---------------------------------------------------------------------- savings account. Again, let's start by working out the blue hedging parameter ---------------------------------------------------------------------- here, after two downticks. We leave it for you, by solving the equation of ---------------------------------------------------------------------- this node, to verify that ---------------------------------------------------------------------- this number of dollars in the bank, combined with ---------------------------------------------------------------------- this many shorted shares, at ---------------------------------------------------------------------- this price per share, will, altogether, give us a portfolio worth ---------------------------------------------------------------------- this many dollars. Remember that ---------------------------------------------------------------------- this number represents a ---------------------------------------------------------------------- short position, and so contributes ---------------------------------------------------------------------- negatively to the portfolio while ---------------------------------------------------------------------- this number represents a ---------------------------------------------------------------------- long position, and so contributes positively. Also, don't forget to ---------------------------------------------------------------------- work out these numbers. As part of our training for real world quant jobs, let me say that we needed all these numbers yesterday, so let's get to work, people! Next on the docket is a fellow named Jack. Like Harry, ---------------------------------------------------------------------- Jack wants the right to buy 1000 shares of XYZ for $970 three months from now, but *un*like him, Jack also accepts the ---------------------------------------------------------------------- obligation to make this purchase. ---------------------------------------------------------------------- Gail will be the seller here, too. Again, let's assume that ---------------------------------------------------------------------- the spot price is $1 per share. ---------------------------------------------------------------------- This is an unusual derivative, that doesn't have a name like call or put, so far as I know. A bit later in this lecture, we'll be talking about forwards and, in fact, ---------------------------------------------------------------------- this derivative is something like a forward. Let's call it ---------------------------------------------------------------------- an obligation derivative, in honor of the fact that it involves an ---------------------------------------------------------------------- obligation. As long as we're making up terminology, let's call the ---------------------------------------------------------------------- 1000 shares of XYZ the ---------------------------------------------------------------------- promised underlying. Note that this is a stonger promise than with an option. Gail is promising Jack the 1000 shares no matter what, whereas, with Harry, Gail only promised the 1000 shares *if* Harry decided to exercise. Let's refer to ---------------------------------------------------------------------- 970 as the ---------------------------------------------------------------------- receivable dollars. One might also call it the ---------------------------------------------------------------------- strike, but remember that it's also a strong promise from Jack to Gail; Jack cannot choose not to exercise. ---------------------------------------------------------------------- Up to now we've studied options, which come in ---------------------------------------------------------------------- two flavors: ---------------------------------------------------------------------- Calls which give the right, but not the obligation, to buy and ---------------------------------------------------------------------- puts which give the right, but not the obligation, to sell. The basic pricing principles will work with this new obligation derivative, just as for calls and puts. We begin by identifying the ---------------------------------------------------------------------- payoff function. Let's call it ---------------------------------------------------------------------- h(S), where S is the final stock price, at the end of the ---------------------------------------------------------------------- three month term. h(S) is the dollar cost to Gail to meet her contractual requirement to provide Jack with ---------------------------------------------------------------------- 1000 shares for ---------------------------------------------------------------------- $970. The 1000 shares at S dollars per share cost her ---------------------------------------------------------------------- 1000 S dollars, but that's offset by the ---------------------------------------------------------------------- $970 she receives. As with all payoff functions, we ---------------------------------------------------------------------- ask that you ---------------------------------------------------------------------- graph h. Note, looking at your graph, that h can take on negative values, and, in fact, does so whenever S is below point 97. So, in some scenarios, Gail will actually receive money at the end, which is nice for her, in principle, but, again, it does mean that she has to trust Jack not to walk away from his position, if the stock tanks. Next, we make a ---------------------------------------------------------------------- template, and fill in the ---------------------------------------------------------------------- stock price ---------------------------------------------------------------------- moving left to right, ---------------------------------------------------------------------- like so. Once that's done, we'll fill in the obligation ---------------------------------------------------------------------- derivative's value, which is the same as the value of Gail's hedging portfolio. We move from ---------------------------------------------------------------------- right to left, beginning by plugging ---------------------------------------------------------------------- the final stock prices, one by one, into ---------------------------------------------------------------------- the payoff function, which yields ---------------------------------------------------------------------- the contingent claim. Using the same 80-20 risk neutral probabilities, we calculate discounted expected values, and, working from ---------------------------------------------------------------------- right to left, we ---------------------------------------------------------------------- fill in all the derivative values. Next, using Delta-hedging, we can, at each node, in any order, compute the ---------------------------------------------------------------------- number of shares of XYZ in the hedge, and we get ---------------------------------------------------------------------- these numbers. Note something interesting: Up to rounding error, ---------------------------------------------------------------------- these three numbers are all 1000, and ---------------------------------------------------------------------- these three numbers are 1000 as well. Let's make a note that ---------------------------------------------------------------------- the number of shares in the hedge is 1000 throughout the term of this derivative. The other hedging parameter is ---------------------------------------------------------------------- the bank loan, and, at each node, in any order, we can solve the equation of that node, yielding ---------------------------------------------------------------------- these numbers. Note something interesting: ---------------------------------------------------------------------- The bank loan is constant in columns. For example, up to rounding error, ---------------------------------------------------------------------- these two numbers are equal, ---------------------------------------------------------------------- as are these three numbers. Each column has only one blue number, repeated. ---------------------------------------------------------------------- Let's clear some room, and ---------------------------------------------------------------------- then clear some more. ---------------------------------------------------------------------- This number ---------------------------------------------------------------------- times 1.02 ---------------------------------------------------------------------- gives this, which is the same as ---------------------------------------------------------------------- this. ---------------------------------------------------------------------- This number ---------------------------------------------------------------------- times 1.02 ---------------------------------------------------------------------- gives this, which, up to rounding, is the same as ---------------------------------------------------------------------- this. ---------------------------------------------------------------------- This number ---------------------------------------------------------------------- times 1.02 is ---------------------------------------------------------------------- 970, which you'll recall is ---------------------------------------------------------------------- the receivable dollars on this obligation derivative. So ---------------------------------------------------------------------- 970 divided by 1.02 three times gives ---------------------------------------------------------------------- this number. ---------------------------------------------------------------------- Let's write that down. ---------------------------------------------------------------------- This number is the initial ---------------------------------------------------------------------- bank loan, so we've now found that the dollar ---------------------------------------------------------------------- amount borrowed at the start is ---------------------------------------------------------------------- 970 divided by 1.02 cubed. Now, 1.02 is the risk-free factor, so ---------------------------------------------------------------------- this is just the present value of 970 dollars, that is, the ---------------------------------------------------------------------- present value of the receivable dollars. ---------------------------------------------------------------------- Let's now clear room again, and ---------------------------------------------------------------------- fill back in all the values at the leftmost node, and ---------------------------------------------------------------------- remember that the 1000 shares in the hedge are constant through the three month term of the derivative. We now see how Gail's hedging ---------------------------------------------------------------------- strategy actually works. First, ---------------------------------------------------------------------- Gail buys and holds 1000 shares of XYZ for the full three month term. Second, ---------------------------------------------------------------------- she takes out a loan that grows to $970 in three months. Note that she makes ---------------------------------------------------------------------- no adjustments to the portfolio. That is, she never sells some stock to reduce her bank loan, nor does she increase her bank loan to buy some additional stock. She simply holds a constant 1000 shares, and watches while the bank loan grows to $970 at the end of the three month term. Remembember that, at the end of the three months, the ---------------------------------------------------------------------- derivative involves two promises: First there's the ---------------------------------------------------------------------- promised 1000 shares that Gail must give Jack and, second, there's the ---------------------------------------------------------------------- promised $970 that Jack must give Gail. So the hedging ---------------------------------------------------------------------- strategy obviously does work. Gail will have 1000 shares to give to Jack, and she's been holding them since the beginning of the three month term. Moreover, the $970 Jack pays her goes to pay off her bank loan. The point of all this is that Gail ---------------------------------------------------------------------- doesn't need to go to all the hassle of working through a template to price an obligation derivative. If she's smart enough to think of a strategy that works, she'll come up with the same one as the template would have given. One concept worth introducing now is that of a ---------------------------------------------------------------------- complete market. By definition, a complete market is one in which ---------------------------------------------------------------------- each contingent claim has a unique hedging strategy. In the market models we've studied so far, the price of the underlying changes to two different possible prices with each passing tick of time. It turns out that *that* kind of market is always complete, and so there's only one ---------------------------------------------------------------------- hedging strategy for ---------------------------------------------------------------------- each contingent claim. So, if Gail can figure out a strategy that works, without going through the template, she can rest assured that *that* strategy is the only one that works, so it's the same as the one that the template would have given. Okay. The ---------------------------------------------------------------------- cost of this obligation derivative is the initial value of the hedging portfolio, which is ---------------------------------------------------------------------- this red number. However, this number is equal to the dollar cost of the ---------------------------------------------------------------------- thousand shares minus the ---------------------------------------------------------------------- initial amount of the loan. The dollar cost of ---------------------------------------------------------------------- 1000 shares at ---------------------------------------------------------------------- $1 per share is ---------------------------------------------------------------------- 1000 times 1, and the ---------------------------------------------------------------------- initial amount of the loan is ---------------------------------------------------------------------- the present value of the receivable dollars, ---------------------------------------------------------------------- which we subtract ---------------------------------------------------------------------- to get the cost. You may wish to check ---------------------------------------------------------------------- this equation, with a calculator. It shows that the price of the obligation derivative has two parts, corresponding to the ---------------------------------------------------------------------- two promises. There's the ---------------------------------------------------------------------- initial price of the promised underlying, and ---------------------------------------------------------------------- the present value of the receivable dollars. The price of the derivative is just ---------------------------------------------------------------------- the difference between these two numbers. Let's say this again for emphasis. ---------------------------------------------------------------------- The price of the obligation derivative is ---------------------------------------------------------------------- the initial price of the promised underlying ---------------------------------------------------------------------- minus ---------------------------------------------------------------------- the present value of the receivable dollars. This is because we can hedge the two promises separately. ---------------------------------------------------------------------- The promised underlying is hedged by buying it up-front, at time zero, on the spot market, and ---------------------------------------------------------------------- the receivable dollars are "hedged" by borrowing their present value. We put ---------------------------------------------------------------------- "hedged" in quotes here because one usually imagines hedging against a loss, not against a gain. The receivable dollars, to be received by Gail, represent an anticipated gain. Our altruistic traders always avoid losses *and* gains, and strive to keep their cash flows at zero at all times. So, in that sense, Gail must hedge against her anticipated gain of 970 dollars, and she does so by borrowing their present value. Okay. In the ---------------------------------------------------------------------- remainder of this lecture, we'll learn about ---------------------------------------------------------------------- put call parity, and then about two new kinds of derivatives, called ---------------------------------------------------------------------- forward and futures contracts. Then, finally, we'll ---------------------------------------------------------------------- set up an option whose hedging strategy involves millions of adjustments. For now, though, we call for a second ---------------------------------------------------------------------- intermission. ---------------------------------------------------------------------- Welcome to the third and final act of Lecture 3 of Notes on Financial Mathematics, by Scot Adams and Fernando Reitich. Let's set aside derivative pricing for a moment. Choose some number x, and calculate ---------------------------------------------------------------------- the positive part of x and ---------------------------------------------------------------------- the positive part of its negative, and then ---------------------------------------------------------------------- subtract. I claim you'll always end up with ---------------------------------------------------------------------- x itself. We'll call this the ---------------------------------------------------------------------- reproducing equation, because it shows how to reproduce x out of its ---------------------------------------------------------------------- positive part and the ---------------------------------------------------------------------- positive part of its negative. Let's verify this reproducing equation for ---------------------------------------------------------------------- x = 7. We take ---------------------------------------------------------------------- the positive part of 7 and ---------------------------------------------------------------------- the positive part of its negative, and then ---------------------------------------------------------------------- subtract. This gives ---------------------------------------------------------------------- 7 minus 0, which reproduces the number ---------------------------------------------------------------------- 7, so it works. Let's verify the reproducing equation for ---------------------------------------------------------------------- x = -4. We take ---------------------------------------------------------------------- the positive part of -4 and ---------------------------------------------------------------------- the positive part of its negative, and then ---------------------------------------------------------------------- subtract. This gives ---------------------------------------------------------------------- 0 minus 4, which reproduces the number ---------------------------------------------------------------------- -4, so it works again. Verifification of the ---------------------------------------------------------------------- reproducing equation for all x is left as an exercise. Writing out the equation for x equal to ---------------------------------------------------------------------- this expression, we see that if we take ---------------------------------------------------------------------- the positive part of the expression and ---------------------------------------------------------------------- the positive part of its negative, and then ---------------------------------------------------------------------- subtract, then we reproduce ---------------------------------------------------------------------- the expression. Now remembering ---------------------------------------------------------------------- f, g and h, we see that if we take ---------------------------------------------------------------------- f(S) and ---------------------------------------------------------------------- g(S) and ---------------------------------------------------------------------- subtract, then we get ---------------------------------------------------------------------- h(S). For another ---------------------------------------------------------------------- exercise, we ask you to find your graphs of f, g and h and ---------------------------------------------------------------------- verify that f minus g is h using those graphs. ---------------------------------------------------------------------- Gail sold Jack an obligation derivative with payoff function ---------------------------------------------------------------------- h and which cost ---------------------------------------------------------------------- this red amount. We show ---------------------------------------------------------------------- here the leftmost node of the template. Remember that the blue number represents a bank ---------------------------------------------------------------------- loan, and so should be considered as a negative number. ---------------------------------------------------------------------- Gail sold *Harry* a *call* with payoff function ---------------------------------------------------------------------- f and which cost ---------------------------------------------------------------------- this red amount. We show ---------------------------------------------------------------------- here the leftmost node of the template. Remember that the blue number represents a bank ---------------------------------------------------------------------- loan, and so should be considered as a negative number. ---------------------------------------------------------------------- Gail sold Irwin a put with payoff function ---------------------------------------------------------------------- g and which cost ---------------------------------------------------------------------- this red amount. We show ---------------------------------------------------------------------- here the leftmost node of the template. Remember that the green number represents a ---------------------------------------------------------------------- short position in XYZ stock, and so should be considered as a negative number. If you take ---------------------------------------------------------------------- the cost of f and ---------------------------------------------------------------------- the cost of g and ---------------------------------------------------------------------- subtract, then it's not hard to imagine that you'll end up with ---------------------------------------------------------------------- the cost of h. This is, in fact, true, and we leave it to you to do the arithmetic to verify ---------------------------------------------------------------------- this equation. A similar equation holds for the ---------------------------------------------------------------------- green hedging parameters, though you should remember that ---------------------------------------------------------------------- one of them is negative. A similar equation also holds for the ---------------------------------------------------------------------- blue hedging parameters, though you should remember that ---------------------------------------------------------------------- two of them are negative. ---------------------------------------------------------------------- This equation tells us that, if you take ---------------------------------------------------------------------- the price of the call and ---------------------------------------------------------------------- the price of the put and ---------------------------------------------------------------------- subtact, then you'll get ---------------------------------------------------------------------- the price of the obligation derivative. Let's say that again, for emphasis. ---------------------------------------------------------------------- The price of the call ---------------------------------------------------------------------- minus ---------------------------------------------------------------------- the price of the put ---------------------------------------------------------------------- is equal to the price of the ---------------------------------------------------------------------- obligation derivative. Remember, though, that the obligation derivative is priceable without template, simply by hedging separately the two promises: the promised underlying and the receivable dollars. We get ---------------------------------------------------------------------- the initial price of the promised underlying ---------------------------------------------------------------------- minus ---------------------------------------------------------------------- the present value of the receivable dollars. Remember that, for an obligation derivative, there is no option not to exercise, and the ---------------------------------------------------------------------- promised underlying is a *strong* promise -- Gail *will* give 1000 shares to Jack in return for $970, whether or not Jack might do better buying them on the open market. For ---------------------------------------------------------------------- this equality to hold, we needed to ---------------------------------------------------------------------- assume, on the ---------------------------------------------------------------------- call and put, that ---------------------------------------------------------------------- the terms were equal, namely ---------------------------------------------------------------------- three months, and that ---------------------------------------------------------------------- the promised underlying of call and put were equal, ---------------------------------------------------------------------- namely 1000 shares of XYZ. For options, this underlying is only promised if the option is exercised, so it's a *weak* promise, compared to that of an obligation derivative. Finally, we needed to assume that ---------------------------------------------------------------------- the strike prices were equal, namely ---------------------------------------------------------------------- $970. On the ---------------------------------------------------------------------- obligation derivative, we also needed to make some assumptions. We needed to assume that its term was the same as the common ---------------------------------------------------------------------- term of the two options, and that ---------------------------------------------------------------------- the promised underlying was the same as ---------------------------------------------------------------------- the common promised underlying of the two options, namely 1000 shares. We also needed to assume that the number, 970, of ---------------------------------------------------------------------- receivable dollars was equal ---------------------------------------------------------------------- to the common strike price of the two options. We can therefore replace the words ---------------------------------------------------------------------- receivable dollars with ---------------------------------------------------------------------- strike price. Written this way, the equation is called ---------------------------------------------------------------------- put-call parity. For future reference, let's call ---------------------------------------------------------------------- these three assumptions ---------------------------------------------------------------------- the three equalities. So put-call parity holds, provided the three equalities do. In a moment, we'll show that ---------------------------------------------------------------------- failure of put-call parity ---------------------------------------------------------------------- yields an immediate arbitrage opportunity. This is one of the simplest ways to demonstrate that, if traders try to price derivatives with*out* careful mathematical analysis, they'll almost certainly introduce arbitrage into the economy. Those with the quantitative skills to find this arbitrage will have no trouble sending their kids to college, or even buying small apartments in New York city, not to mention the occasional coffee and chocolate mint. ---------------------------------------------------------------------- Let's clear some room, and then ---------------------------------------------------------------------- move this equation over a bit to make room on the right. Say we run to our trading desks, and come across an example where put-call parity ---------------------------------------------------------------------- fails. Say, for example, that we find a call priced at ---------------------------------------------------------------------- $8 and a put priced at ---------------------------------------------------------------------- $7, and suppose that the intial price of the promised underlying is ---------------------------------------------------------------------- $5, and that the present value of the strike is ---------------------------------------------------------------------- $1. It doesn't matter whether the underlying is corn or Microsoft shares or whatever, nor do the term or the strike matter. We only have to verify that our put and call satisfy the three equalities. Suppose they do. Note that ---------------------------------------------------------------------- 8 minus 7 is *not* equal to ---------------------------------------------------------------------- 5 minus 1. In fact, it's ---------------------------------------------------------------------- less. Our goal is now to find arbitrage. The first thing to do is to move all four numbers to ---------------------------------------------------------------------- the larger side of the inequality, ---------------------------------------------------------------------- like this. Now I'll describe for you an ---------------------------------------------------------------------- arbitrage portfolio with four steps involved in setting it up. For each step, we'll discuss the resulting ---------------------------------------------------------------------- initial cash flow. The first step is to ---------------------------------------------------------------------- short $5 worth of the underlying. That is, we borrow $5 worth of the underlying from someone, and sell it on the open market. This brings in ---------------------------------------------------------------------- $5 initially. We'll pay off this loan at the end of the term, but, initially, we get $5. Next, we ---------------------------------------------------------------------- invest $1 in the bank, which ---------------------------------------------------------------------- intially costs us $1. We'll close out the bank account at the end of the term, and get back the dollar with interest, but, initially, we lose a dollar to the bank. Next, we ---------------------------------------------------------------------- buy the call, which is priced at ---------------------------------------------------------------------- $8, so this costs us ---------------------------------------------------------------------- $8. At the end of the term, the call may have positive payoff, so we may make some money later, but, initially, we lose $8. Last, we ---------------------------------------------------------------------- short the put. That is, we borrow the put option from someone who holds it, and sell it on the options market. The put is worth ---------------------------------------------------------------------- $7, so this brings in ---------------------------------------------------------------------- $7. We'll pay off the loan at the end of the term, but, initially, we gain $7. ---------------------------------------------------------------------- This inequality tells us that ---------------------------------------------------------------------- our initial cash flow totals to a positive number, but we do have to worry about how much this is all going to cost us, at the end of the term. Here's another ---------------------------------------------------------------------- exercise. ---------------------------------------------------------------------- Show that this strategy's payoff function is 0. That is, compute the payoff functions of ---------------------------------------------------------------------- each of the four steps, remembering that the payoff function of a short position is the negative of the payoff function of the corresponding long position. Add the four payoff functions. If you do this correctly, and use the reproducing equation, you'll get the constant function ---------------------------------------------------------------------- zero. You might also graph each of the four payoff functions, and add the graphs, and see that you get ---------------------------------------------------------------------- zero. So the portfolio brings in a positive ---------------------------------------------------------------------- initial cash flow at ---------------------------------------------------------------------- no risk of future cost. That's called arbitrage. Most valuable of all, at no point, in this put-call parity analysis, did we need to make *any* assumptions about the volatility of the underlying. This arbitrage isn't dependent on such assumptions. It only depends on the three equalities, which are all objectively verifiable, by looking at numbers on our trading screens. Okay. Next, we talk about another kind of derivative called a ---------------------------------------------------------------------- forward. It may help to start out by saying that a forward is nothing more than an obligation derivative whose initial cost to the buyer is zero. Let's parse what that means. ---------------------------------------------------------------------- A forward contract specifies that ---------------------------------------------------------------------- at some time in the future, for example, one year, ---------------------------------------------------------------------- the buyer of the forward will pay the seller some amount in dollars. This ----------------------------------------- amount is specified in the contract. In this example, let's say that ---------------------------------------------------------------------- you and I are the sellers of the forward. Typically this ---------------------------------------------------------------------- amount is called the price of the forward, but in this lecture, we'll instead refer to it as the receivable dollars, to stress that it's a price that's paid ---------------------------------------------------------------------- in the future. A forward contract also specifies that the buyer ---------------------------------------------------------------------- will receive some amount of some asset from us. That asset is called the underlying, and let's say it's 100 shares of some stock. A key point to ---------------------------------------------------------------------- note is that, in executing a forward contract, ---------------------------------------------------------------------- no money changes hands at the time when the contract is signed, which we call ---------------------------------------------------------------------- time zero. In other words, a forward contract has *no* up-front cost to the buyer. The cost comes at the *end* of the term, *not* the beginning. Here's our ---------------------------------------------------------------------- problem: ---------------------------------------------------------------------- Find the amount of receivable dollars taking care not to introduce arbitrage. Let's ---------------------------------------------------------------------- assume that, in the risk-free market, money is earning two percent per year, so the ---------------------------------------------------------------------- one-year risk free factor is 1.02. Let's also assume that ---------------------------------------------------------------------- the spot price of the underlying stock is one dollar per share. Finally, let's ---------------------------------------------------------------------- define x to be the receivable dollars, which is the amount which we're trying to compute. The hedging ---------------------------------------------------------------------- strategy for an obligation derivative does *not* require a template. It involves hedging the promised underlying by buying it up-front, at time zero, and hedging the receivable dollars by taking out a loan for their present value. So we ---------------------------------------------------------------------- buy the hundred shares, at ---------------------------------------------------------------------- one dollar per share, and ---------------------------------------------------------------------- take out a loan for the present value of the x receivable dollars. The initial price of this hedging strategy is ---------------------------------------------------------------------- the difference, so we should charge our buyer this amount up-front, at time zero. However, remember that ---------------------------------------------------------------------- no money changes hands at time zero, so the cost of the hedging strategy must be ---------------------------------------------------------------------- equal to zero. Solving, we find that ---------------------------------------------------------------------- x is 102. Now suppose that ---------------------------------------------------------------------- a competitor decides to offer the same forward, but with a different receivable dollars, say ---------------------------------------------------------------------- x prime. If x' is larger than our 102, then she'll be out of luck, since we're offering a better contract. On the other hand, we leave it as an ---------------------------------------------------------------------- exercise to show that if x' is less than 102, then we can arbitrage her. Here's a hint: If x' is low, then we should follow the general precept that one buys low, which, in this case, tells us to buy her forward contract. That's Step 1. For Step 2, we negate our own hedging ---------------------------------------------------------------------- strategy. Your job is to show that, these two steps will involve no initial cost, but will, without fail, bring in some guaranteed positive number D of dollars at the end of the year. For Step 3, at time zero, we borrow the present value of D dollars from the bank, and use those guaranteed D dollars to pay off that loan one year later. These three steps give us a positive initial cash flow, at no later cost. In other words, they gives us arbitrage. Next, let's talk about a ---------------------------------------------------------------------- worry that occurs when we sell a forward. With any obligation derivative, a seller has to worry that the ---------------------------------------------------------------------- buyer may walk away from his position, particularly if the price of the underlying declines to the point where honoring the forward contract is significantly more expensive than a purchase on the open market. For that reason, a seller often wants some kind of mechanism to ensure that the buyer will honor his contract. The typical mechanism used is something called a ---------------------------------------------------------------------- margin requirement, together with something called ---------------------------------------------------------------------- margin calls. These are best described in detail in some other lecture. Without going into those details, we point out that a ---------------------------------------------------------------------- futures contract is nothing more than ---------------------------------------------------------------------- a forward contract combined with some kind of enforcement mechanism. This enforcement costs money to implement, but it's typically small compared to the cost of the promised underlying, and, in our idealization of reality, the enforcement cost is zero. So, for us, there's little difference between a forward contract and a futures contract. Last of all, let's talk about a fellow named Kyle. ---------------------------------------------------------------------- Kyle wants the right, but not the obligation to buy ---------------------------------------------------------------------- 5000 shares of ABC stock for $5000 ---------------------------------------------------------------------- thirty days from now. The right but not obligation to *buy* is a ---------------------------------------------------------------------- call option. Let's ---------------------------------------------------------------------- assume that the spot price is $1 per share. Let's also assume that, ---------------------------------------------------------------------- each second, the price changes, and ---------------------------------------------------------------------- the uptick and downtick factors are ---------------------------------------------------------------------- as given. This assumption is called a ---------------------------------------------------------------------- volatility assumption, because it gives us a sense of how volatile the stock price is. Finally, let's assume that we find a risk-free market like a bank or riskless bond market, and that it allows us to borrow and invest money in such a way that interest accrues once every second. Let's assume that ---------------------------------------------------------------------- the one-second risk-free factor is ---------------------------------------------------------------------- as given. Our ---------------------------------------------------------------------- goal is to find the right price for this option, by which we mean ---------------------------------------------------------------------- the price that doesn't introduce arbitrage. This is a typical ---------------------------------------------------------------------- call option, and can be priced in the usual way, by filling in a template. However, ---------------------------------------------------------------------- the difficulty we face is that, for *this* option, each tick is one second long, so there's one adjustment to the hedging portfolio every second. So the number of ---------------------------------------------------------------------- adjustments that we'll have to track isn't one or two or three anymore, but, rather, is the ---------------------------------------------------------------------- number of seconds in the ---------------------------------------------------------------------- 30 day term. This amount of computation might seem daunting, but our butts are ---------------------------------------------------------------------- saved by ---------------------------------------------------------------------- the Central Limit Theorem, which is the topic of our next lecture. See you there! ----------------------------------------------------------------------