---------------------------------------------------------------------- SLIDE NUMBER 1 Welcome to Lecture 1 of Notes on Financial Mathematics by Scot Adams and Fernando Reitich. Let's talk about a fellow named Dan. ---------------------------------------------------------------------- SLIDE NUMBER 2 Dan's just graduating from college and, as a graduation present, ... -------------- ... his parents have bought him an all-expense-paid trip to Europe leaving one month from today. Even though his expenses are paid, Dan would like to have a little *extra* spending money on his trip, and ... -------------- ... he wants to bring along 100 Euros. He's working at a job, and will receive $100 in pay, but not until one month from today, just before he leaves. Right now, Dan, never good at saving, ... -------------- ... has only *$3* to his name. So, again ... ---------------------------------------------------------------------- SLIDE NUMBER 3 ... Dan wants to buy 100 Euros for $100, one month from today. -------------- Let's say that the current exchange rate is $1 for 1 Euro, so, if Dan could get an *advance* on his pay, he could buy the Euros now, and have them all ready for his trip. However, his boss, acquainted with Dan's spendthrift ways, long ago told him that ... -------------- ... he could no longer get advances. Now, in this lecture, we'll be assuming, for simplicity, that we live in a universe where we know that the dollars per Euro exchange rate varies by exactly 5% each month, so that ... -------------- ... the current rate of $1 per Euro will change either to $1.05 or to 95 cents one month from now, but we *don't know* which. If Dan's lucky, the rate'll go down, ... -------------- ... and he'll buy the 100 Euros that he needs for only $95, but he'd like somehow to buy some kind of insurance against a ... -------------- ... rise in the exchange rate. That's exactly what a particular financial instrument called an "option" will do for him, as we'll see. ---------------------------------------------------------------------- SLIDE NUMBER 4 Again, we assume that, each month, the Euro exchange rate either rises or declines 5%. We'll represent that graphically this way ... -------------- So, if E is the current dollars per Euro exchange rate, then, one month from now, it'll be 1.05 E or .95 E, but we *don't know* which. Note that both changes are multiplicative, with factors 1.05 and .95, and the fact that we use multiplication is reinforced in the graphic by the small ... -------------- ... "times" sign appearing inbetween the two arrows. -------------- In this lecture, we'll also be assuming that the bank loan rate is quite high: 1% per month, so that a loan will increase by 1% each month it's not paid off. We'll represent *that* graphically this way ... -------------- Note that the bank simply doesn't care whether the Euro rises or falls; in either case, the amount owed will increase by a factor of 1.01. (pause) Okay. So these are the parameters. Dan gets the bright idea that he can simply take out a $100 loan from the bank to buy the Euros now, and can pay off the loan, with interest, in a month, using his $100 pay together with one of the three dollars he has now. However, being merely a college student with a poor credit history, ... ---------------------------------------------------------------------- SLIDE NUMBER 5 Dan's given the bum's rush by the bank's bouncer. Next, Dan is somehow introduced to Alice, who tells him that ... -------------- ... she'll guarantee him the right to buy 100 Euros for $100, one month from now, but that she'll charge him $2.97, now, for that guarantee. She also explains to him that this kind of deal is called an ... -------------- ... "option", because, at the end of the month, he has the option of buying the 100 Euros for $100, or of buying them on the open market, if the price is lower there. If, at the end of the month, Dan opts to get the Euros from Alice for $100, then we say that he "exercises" his option. Alice's $2.97 fee is three cents less than Dan's net worth in the world, and he gladly pays it. Again we make the point that, under this agreement with Alice, ... -------------- ... Dan has the right, but *not* the obligation to exercise the option, that is, to buy the 100 Euros for $100, one month from now. If the price declines to 95 cents per Euro, then Dan's free to buy them on the open market, for only $95. A contract that would *oblige* Dan to buy 100 Euros for a fixed price, one month from now, is called a "futures contract", not an "option". *That* kind of contract requires enforcement, and ... -------------- ... probably shouldn't be issued to someone like Dan, with a poor credit history. Alice, on the other hand, doesn't need to worry about Dan's creditworthiness, because an option is *structured* so that Dan needn't even show up, unless he wants to. Not showing up is one of Dan's talents, but it doesn't violate the terms of his option contract. (pause) Next, we'll describe Alice's approach to minimizing her risk. This "approach" is sometimes called a ... ---------------------------------------------------------------------- SLIDE NUMBER 6 ... "hedging strategy". We'll explain her calculations later, but here's what she does: -------------- Today, she buys 50 Euros for $50. Alice, who has *good* credit, also ... -------------- ... takes out a loan of $47.03 from the bank ... -------------- ... and, as mentioned, charges Dan $2.97. So, in dollars, she ... -------------- ... loses 50, ... -------------- ... gets 47.03 and ... -------------- ... gets 2.97. All this balances and ... -------------- ... her dollar cash flow today is 0. (pause) ---------------------------------------------------------------------- SLIDE NUMBER 7 A month passes, and one of two possible scenarios will occur. ---------------------------------------------------------------------- SLIDE NUMBER 8 In the first scenario, the exchange rate rises to $1.05. Alice then has cash flow from three sources: -------------- Dan, the bank and her Euro holdings. In this scenario, ... -------------- ... *Dan* exercises his option. That is, he asks Alice to make good on her promise to provide him with 100 Euros for $100. So Alice must give to Dan $105 worth of Euros for only $100, and she loses $5 on that transaction. Incidentally, if Dan should somehow fail to exercise his option, then Alice makes 5 extra dollars. -------------- Also, she pays off her loan to the bank, which has grown from $47.03, by 1%, to $47.50. -------------- Finally, she takes the 50 Euros she bought earlier, and trades them back for dollars. The Euros are now worth $1.05 each, and Alice gets $52.50. -------------- This again balances, and her cash flow is 0. ---------------------------------------------------------------------- SLIDE NUMBER 9 In the second scenario, the exchange rate falls to 95 cents. Again, Alice has cash flow from ... -------------- ... the same three sources. In this scenario, Dan will *not* exercise his option, since that'd cost him $100, whereas he can get the hundred Euros he needs on the open market for only $95. So, Dan will simply not show up, and Alice's cash flow from him is ... -------------- ... 0. Incidentally, if Dan should somehow decide to exercise his option, then Alice makes 5 extra dollars. Alice pays off her bank loan, costing her ... -------------- ... $47.50. She holds 50 Euros, at 95 cents each, providing her ... -------------- ... $47.50. So, in this scenario as well, ... -------------- ... her cash flow balances at 0. Note that Alice, like many option traders, loves what she does, and can't imagine making money off such an enjoyable activity. Another of our standing simplifying assumptions in this lecture will be that option traders seek no profit. (pause) Okay. Now, let's review and summarize, with different wording. ---------------------------------------------------------------------- SLIDE NUMBER 10 Alice's hedging strategy was to hold a portfolio of 50 Euros, and a $47.03 bank loan. -------------- The 50 Euros are now worth $1 each, but may rise to $1.05 or fall to 95 cents. -------------- The $47.03 now owed to the bank will grow by a factor of 1.01. Note again that 1.01 is represented on the up arrow *and* the down arrow, as the bank simply doesn't care about the viscissitudes of the Euro market. So this portfolio of Euros and a bank loan today costs ... -------------- ... 50 times 1 ... -------------- ... minus 47.03 times 1, which is equal to ... -------------- ... $2.97. Alice is able to buy this portfolio by charging Dan that exact amount. The value of the portfolio will change .... -------------- ... to one of two possible numbers in one month, and we can calculate those two numbers: The top number is ... -------------- ... 50 times 1.05 ... -------------- ... minus 47.03 x 1.01, ... -------------- ... or $5. The bottom number is ... -------------- ... 50 times .95 ... -------------- ... minus 47.03 times 1.01, ... -------------- ... or 0. So, one month from now, Alice's portfolio will be worth either ... -------------- ... $5 or ... -------------- ... 0, depending on whether the Euro rises or falls. This is perfect: If it rises, ... -------------- ... she'll have $5, which combines with the $100 Dan gives her to exercise his option, and that $105 will pay for the 100 Euros she must give *him* as part of the deal. If the Euro rate falls, ... -------------- ... her portfolio becomes worthless, but she won't need any money, because Dan won't show up to exercise his option. (pause) Next, let's talk about how Alice figures out these three numbers: ... -------------- ... 50, 47.03 and 2.97. ---------------------------------------------------------------------- SLIDE NUMBER 11 Remember that Alice needs to have $5 in the event of an uptick in the Euro exchange rate, and 0 in the event of a downtick. -------------- In this lecture, a "tick" is a one-month period of time, which is consonant with the relaxed and easygoing world of option trading. Anyway, from this point of view,... -------------- ... Dan asks Alice to make, for him, a financial instrument (which happens to be called an "option") that's worth $5 on an uptick and 0 on a downtick. Let's consider the current price of that option to be an unknown, represented by ... -------------- ... a question mark. Dan wants to know how much Alice will charge him today for the option. That is, ... -------------- ... he asks her to compute ? Let's say that Alice knows that she can pay for this option out of a portfolio of Euros and a bank loan. ---------------------------------------------------------------------- SLIDE NUMBER 12 Let's say ... -------------- ... x ... -------------- ... Euros ... -------------- ... minus y ... -------------- ... from the bank ... -------------- ... pays for the option. Here, x, y and ? are all three unknowns. (pause) -------------- From here, we can get three equations in the three unknowns. -------------- First, x times 1.05 ... -------------- ... minus y times 1.01 is equal to ... -------------- ... 5. -------------- Let's write that down. -------------- Second, x times .95 ... -------------- ... minus y times 1.01 is equal to ... -------------- ... 0. -------------- Let's write that down. -------------- Third, x times 1 ... -------------- ... minus y times 1 is equal to ... -------------- ... ?. -------------- Let's write that down. So this gives three equations in three unknowns. ---------------------------------------------------------------------- SLIDE NUMBER 13 Here they are again. We leave it as an exercise for you to solve this system, but the answers you should get are ... -------------- ... 50, 47.03 and 2.97. Do these three numbers sound familiar? (pause) We next describe a nice little labor-saving trick, which will actually become indispensible, as we get to pricing more and more complicated options. ---------------------------------------------------------------------- SLIDE NUMBER 14 Remember that three numbers that were *given* to us at the start are ... -------------- ... the factor change on a downtick of the exchange rate ... -------------- ... the factor change on an uptick ... -------------- ... and the factor change for a bank loan. -------------- Note that 1.01 is located 60% of the way from .95 to 1.05. Now I ask you to humor me for a moment, and imagine a coin-flipping ... -------------- ... game involving one player and a biased coin that comes up ... -------------- ... heads 60% of the time and ... -------------- ... tails 40% of the time. In this game, the coin is flipped ... -------------- ... one month from now, and the player receives ... -------------- ... $5 for heads and ... -------------- ... 0 for tails. What's the expected payoff? To compute it, one takes ... -------------- ... 60% of ... -------------- ... 5 dollars plus ... -------------- ... 40% of ... -------------- ... 0, which gives $3. ---------------------------------------------------------------------- SLIDE NUMBER 15 That is, the game's expected payoff, one month from now, is $3. Given that money grows 1% each month, if we discount back one month to get the ... -------------- ... *present* value, we obtain ... -------------- ... $3 over 1.01, or $2.97. -------------- Note that this is exactly the price that Alice charged for her option. -------------- Is this just a coincidence? No way! This little trick allows us to price options without solving systems of equations, and, as I said before, will be most useful later, when we get to more complicated kinds of options. We defer an explanation of why the trick works to a future lecture. ---------------------------------------------------------------------- SLIDE NUMBER 16 The title of that lecture is "The Risk-Neutral World", but we can also refer to it, colloquially, ... -------------- ... as "Coin-flippers got price". The basic theme is that, if you know how to compute time-discounted expected returns on coin-flipping games, then you can use that to find option prices. ---------------------------------------------------------------------- SLIDE NUMBER 17 Remember *this* graphic that contained the three equations in three unknowns? We just described a tricky way of computing ... --------------- ... question mark, via a coin-flipping game, although we haven't yet explained *why* that trick worked. Next, we talk about a tricky way of computing x and y. --------------- The two unknowns x and y are called "hedge parameters", and remember that they turned out to be ... -------------- ... 50 and 47.03. Imagine that we're now ... -------------- ... pricers, so we know how to compute the 2.97 price, but imagine also that we're *not* ... -------------- ... hedgers, so we do *not* know the values of x and y. On the other hand, as pricers, we do know that ... --------------- ... 2.97 is equal to ? --------------- which is equal to x - y. So, if we can figure out either x or y, then we can figure out the other one just by solving a single equation, not a system. We'll show a trick to compute x. ---------------------------------------------------------------------- SLIDE NUMBER 18 Remember that the Euro is now at $1, and will change to $1.05 or 95 cents one month from now. -------------- The difference between 1.05 and .95 is .1. -------------- The *option* is worth ? dollars now, and will be worth 5 dollars or 0 one month from now. -------------- The difference between 5 and 0 is 5. As option pricers, we know that ? is 2.97, but, since we are *not* hedgers, ... -------------- ... we don't know that x is 50. Note, though, that ... -------------- ... if we divide the Option difference by the Euro difference, we get 5 over 0.1 or 50. That is, the Option difference over the Euro difference turned out to be the number of Euros in the hedging portfolio. -------------- Is this just a coincidence? No way! This little trick allows pricers to figure out hedge parameters without solving systems of equations. Again, we defer an explanation of why the trick works to a future lecture. ---------------------------------------------------------------------- SLIDE NUMBER 19 The title of *that* lecture is "Delta-Hedging", but we can also refer to it, colloquially, ... -------------- ... as "Pricers got hedge". The basic theme is that, if you know how to price options, then you can compute hedging portfolios. (pause) Next, let's get some more practice with pricing and hedging, using the tricks we've learned. ---------------------------------------------------------------------- SLIDE NUMBER 20 Let's say that our friend Earl, like Dan, wants to buy 100 Euros for $100 one month from now, but, instead of assuming $1 per Euro, let's assume that, ... -------------- ... today, a Euro costs 1.05 divided by .95 dollars. To four decimals, that's ... -------------- ... 1.1053 dollars. We'll continue to assume that the bank charges 1% per month, and that the exchange rate goes up or down 5% each month. Then ... -------------- ... this graphic shows the two possible exchange rates one month from now. To four decimals, it might go up to ... -------------- ... a dollar sixteen point oh five. Alternatively, it might go down to exactly ... -------------- ... a dollar five. Let's say that, while *Alice* is our expert option trader when the exchange rate is at *one* dollar per Euro, ... ---------------------------------------------------------------------- SLIDE NUMBER 21 ... we use *Beth* at this new exchange rate of 1.1053. -------------- Our assumptions about the bank rate and the Euro uptick and downtick factors are unchanged, so we end up with 60% and 40%, just as before. -------------- As we just said, to four decimals, the exchange rate might go up to a dollar 16.05. If it does, then, one month from now, ... -------------- ... Beth'll have to cough up $16.05, which combines with the $100 Earl puts in, to give the one sixteen oh five needed to buy 100 Euros. -------------- Alternatively, the exchange rate might go down to exactly $1.05, ... -------------- ... in which case Beth'll need to pay out a net of exactly $5. The corresponding coin-flipping game involves, again, a 60-40 biased coin, but ... ---------------------------------------------------------------------- SLIDE NUMBER 22 ... this time, the game pays $16.05 on heads and exactly $5 on tails. -------------- As before, it pays one month from now. -------------- The expected payoff now appears in the red box, and we discount back one month, dividing by ... -------------- ... 1.01, and, to two decimals, arrive at ... -------------- ... $11.52. This is Beth's price. Now that we've reviewed *pricing*, we leave it as an ... ---------------------------------------------------------------------- SLIDE NUMBER 23 ... exercise for you to review hedging, and to use the trick from before to figure out Beth's hedging portfolio. Remember that the trick was that ... -------------- ... the Option difference over the Euro difference is the number of Euros in the hedging portfolio. (pause) Last, let's do a slightly more complicated option. ---------------------------------------------------------------------- SLIDE NUMBER 24 Let's say that Fred wants to buy 100 Euros for $100, but, instead of *one* month from now, *two* months from now. -------------- Let's say that the current exchange rate is 1 divided by .95 dollars. -------------- To four decimals, that's 1.0526 dollars. Let's say that ... -------------- ... *Cathy* is our expert at *this* exchange rate. We'll continue to assume that the bank charges 1% per month, and that the exchange rate goes up or down 5% each month. ---------------------------------------------------------------------- SLIDE NUMBER 25 This graphic then shows the possible exchange rates after one month, and after two months. Here's Cathy's strategy: If, after one month, there's a ... -------------- ... downtick, then the exchange rate will end up being *exactly* the one in which *Alice* specializes, namely ... -------------- ... $1 per Euro. Remember that Alice's price was ... -------------- ... $2.97. In this scenario, Cathy simply hands off Fred to Alice, ... -------------- ... paying the $2.97 fee, and washes her hands of him. On the other hand, if there's a *up*tick, ... -------------- ... then the exchange rate is Beth's, and Cathy hands Fred off to Beth, ... -------------- ... paying Beth's fee of $11.52. So, from Cathy's point of view, this option is *exactly* the same as a *one*-month option in which she must pay out $11.52 or $2.97 depending on whether the exchange rate goes up or down in that month. The corresponding coin-flipping game still involves a 60-40 coin, but ... ---------------------------------------------------------------------- SLIDE NUMBER 26 ... now pays $11.52 or $2.97, depending on heads or tails, ... -------------- ... one month from now. -------------- The expected payoff now appears in the numerator, and we discount back one month, dividing by 1.01, and, to two decimals, arrive at ... -------------- ... $8.02. This is what Cathy charges Fred. ---------------------------------------------------------------------- SLIDE NUMBER 27 Let's now go back to our graphic illustrating the possible evolution of the exchange rate over the two months. -------------- Note that, if there are two upticks ... -------------- ... the option must pay out $16.05 which combines with $100 to buy 100 Euros at ... -------------- ... one sixteen oh five per Euro. -------------- If there's an uptick then downtick ... -------------- ... or downtick then uptick, ... -------------- ... the option pays a net of exactly $5. -------------- If there are two downticks, Fred will *not* exercise the option, and will instead buy his Euros on the open market, so the option pays ... -------------- ... nothing. If, at the beginning of some month, the exchange rate is ... -------------- ... one dollar, and, at the end of that month, we pay ... ... $5 on an uptick and 0 on a downtick, then Alice calculated the option price to be ... -------------- .... $2.97. At Beth's exchange rate of ... -------------- ...1.05 over .95 and payouts of ... -------------- ... $16.05 and $5, Beth calculated the option price to be ... -------------- ... $11.52. Finally, at Cathy's exchange rate of ... -------------- ... 1 over .95 and payouts of ... -------------- ... $11.52 and $2.97, Cathy calculated the option price to be ... -------------- ... $8.02. Note how, to get this 8.02 price, we worked backward from ... -------------- ... the final payouts to Alice and Beth's prices, and then ... -------------- ... from those to Cathy's price. This process of calculating option values working backward in time is something we'll see again and again. By the way, also note that, in fact, the exchange rates weren't really used -- only the factor of change on uptick and downtick. -------------- Here are the payouts. ---------------------------------------------------------------------- SLIDE NUMBER 28 Here they are again. Now, let's imagine a ... -------------- ... coin-flipping game, still with a 60-40 coin, but now we make *two* flips of the coin, and the payoff is ... -------------- ... *two* months from now, as follows: If the flips come out heads then heads, we pay ... -------------- ... $16.05. If heads then tails, or tails then heads, we pay ... -------------- ... $5. If tails then tails, we pay ... -------------- ... 0. Let's discuss the ... -------------- ... expected payoff for *this* game. Note that ... -------------- ... 60% times 60% is 36%, so we multiply ... -------------- ... $16.05 by 36%. Note that ... -------------- ... 60% times 40% is 24%, but there are two ways of getting to $5, namely, heads then tails, or ... -------------- ... tails then heads. So we multiply ... -------------- ...$5 by twice 24%, or 48%. Finally, ... -------------- ... 40% times 40% is 16%, so we multiply ... -------------- ... 0 by 16%. Summing these results, the expected payoff is then ... -------------- ... 8 dollars and 18 cents. If we discount back *two* months, ... -------------- ... dividing by 1.01 twice, we obtain 8 dollars and 2 cents. -------------- Remember that Cathy's price was exactly that: 8 dollars and 2 cents. (pause) ---------------------------------------------------------------------- SLIDE NUMBER 29 Is this just a coincidence? No way! Once again, we see that coin-flippers got price, and, as mentioned earlier, we'll explain all this in a future lecture. For now, we leave it as an ... -------------- ... exercise for you to review why pricers got hedge and to use the trick from before to figure out Cathy's hedging portfolio. Remember that the trick was that ... -------------- ... the Option difference over the Euro difference is the number of Euros in the hedging portfolio. (pause) Suppose that just after selling the option to Fred, Cathy has a falling out with Alice and Beth, and, in a fit of pique, decides *not* to send any business their way. ---------------------------------------------------------------------- SLIDE NUMBER 30 Note that she doesn't really *need* them, because she can compute their hedging strategies for herself. So, undaunted, she proceeds, and sets up her own hedging portfolio. Computing *that* portfolio was left, just a moment ago, as an exercise for you, but, if you worked it out, you should have come up with ... -------------- ... 81.23 Euros, and a $77.49 bank loan. If there's a ... -------------- ... downtick in the Euro over the first month, then you can check that, up to a one penny rounding error, Cathy's portfolio will become worth $2.97, and so she can trade that portfolio in for a ... -------------- ... *new* portfolio of 50 Euros and a $47.03 bank loan. Recall that *that* was Alice's portfolio. This ... -------------- ... change of portfolios costs nothing because both are worth $2.97. More concisely, Cathy simply asks herself, "What would Alice do?", and does the same thing on her own, at no expense. -------------- On an uptick, she asks herself, "What would Beth do?" That is, she trades in her portfolio for Beth's, the computation of which we left earlier as an ... -------------- ...exercise for you. *This* ... -------------- ... change of portfolios *also* costs nothing, as you can verify, with a little work. Now that she's working on her own, we say that ... -------------- ... Cathy "adjusts her hedge" after the first month. The exact adjustment she makes depends on which way the exchange rate goes, but in case either of an ... -------------- ... uptick *or* of a -------------- ... downtick, her strategy is "self-financing", meaning that Cathy neither puts any money into the portfolio, nor takes any out, except at the beginning, when she charges Fred, and sets up ... -------------- ... the initial portfolio, and -------------- ... at the end of the two month period, when she closes out her portfolio for dollars, and settles with Fred. You can calculate that, in all possible scenarios ... -------------- ... at the end, the portfolio will be worth just what Cathy needs to pay Fred. (pause) Finally, let's talk a little bit about our assumption that the exchange rate either goes up or down by *exactly* 5% each month. This is, of course, quite unrealistic, but, given that unrealistic assumption, ... ---------------------------------------------------------------------- SLIDE NUMBER 31 ... it's logical that Cathy splits up the 2 month time-to-exercise period into two 1 month sub-periods. More commonly, though ... -------------- ... option traders will split the time period of an option into a larger number, say N, of sub-periods. One hopes that the hedging strategy will be more effective, the more closely the exchange rate is watched, and the more often the hedge is adjusted. If there are N sub-periods, then the corresponding coin-flipping game will involve N coin flips, so, since N is a large number, ... -------------- ... pricing, in this situation, involves understanding "coin-flipping in the large", and the main result one can use here is a theorem called "The Central Limit Theorem". We therefore propose one more future lecture. ---------------------------------------------------------------------- SLIDE NUMBER 32 The title of the lecture is "The Central Limit Theorem", but we can also refer to it, colloquially, ... -------------- ... as "Mathematicians got coin-flipping". The basic theme is that there are mathematical tools for figuring expected returns in coin-flipping games, even when the number of flips is large. To summarize our future lectures ... ---------------------------------------------------------------------- SLIDE NUMBER 33 ... the titles will be "The Central Limit Theorem", "The Risk-Neutral World" and "Delta-Hedging". More colloquially, we have ... -------------- ... "Mathematicians got coin-flipping", "Coin-flippers got price" and "Pricers got hedge". So, clearly, mathematicians got it all! ----------------------------------------------------------------------