---------------------------------------------------------------------------- Welcome to Lecture 2 of Notes on Financial Mathematics, by Scot Adams and Fernando Reitich. Let's talk again about our prodigal peripatetic pupil, Dan. ---------------------------------------------------------------------------- Remember, from Lecture 1, that Dan wants 100 Euros one month from now, and will receive $100 in pay at that time. ------------ Say the *current* exchange rate is 1 dollar per Euro. Remember that ... ------------ ... a bank loan goes up 1% each month. Remember that ... ------------ ... the exchange rate goes up or down 5% each month. If it goes ... ------------ ... *up* to $1.05 per Euro, then Dan's desired ... ------------ ... hundred Euros'll cost him $105, but he'll only have ... ------------ ... $100, so he'll be ... ------------ ... $5 short. Finally, remember that Dan meets an option trader named Alice, and tells her that, on *top* ... ------------ ... of his $100, he *may* need ... ------------ ... 5 *extra* dollars to buy ... ------------ ... 100 Euros, but *only* if the exchange rate ... ------------ ... goes *up*. She therefore writes a particular type of contract, called an ... ------------ ... option, which, ... ------------ ... at the end of the month, promises Dan a net of ... ------------ ... $5 on an uptick in the exchange rate, or nothing on a downtick. She needs to write a *price* into the contract. To begin, she simply calls that price an unknown, ... ------------ ... denoted by a question mark. So ... ------------ ... at the start of the month, Alice charges Dan ? dollars. Also, at the start of the month, she *uses* those dollars ... ............................................................................ ............................................................................ ---------------------------------------------------------------------------- ... to set up a hedging portfolio consisting of x Euros and a y dollar bank loan. This portfolio *starts out* financed ... ------------ ... by the ? dollars that Dan pays. If she chooses x and y correctly, then, at the end of the month, the portfolio'll *end up* worth ... ------------ ... 5 dollars on an uptick, or ... ------------ ... nothing on a downtick. We represented all this graphically ... ------------ ... like this. Note that ... ------------ ... x ... ------------ ... Euros (pause) ... ------------ ... minus y ... ------------ ... from the bank (pause) pays ... ------------ ... for $5 on an uptick or ... ------------ ... nothing on a downtick. In Lecture 1, we obtained, from this graphic, ... ............................................................................ ............................................................................ ---------------------------------------------------------------------------- ... this system of three equations in the three unknowns x, y and ?. Solving this system isn't much work for such a simple option, but the process becomes more and more intractable for options of greater and greater complexity. In *this* lecture, it's our ... ------------ ... goal to show a method to compute ? *without* solving a system of equations. In the *next* lecture, we'll show a method for computing the hedging parameters x and y, but, for now, we focus on ?, and ... ------------ ... a key point to remember is that ? does *not* depend on the actual probability of an uptick or a downtick. In fact, *those probabilities* don't appear ... ------------ ... in the system of equations. This independence of the answer on uptick-downtick probabilities gives us great flexibility, because: Whatever those probabilities are, here in *our* universe, we can ... ------------ ... imagine *another* universe in which they're different, and we can imagine how people living in that parallel reality might compute ?. Whatever *they* get, it's the same as what *we* should get, as long as ... ------------ ... the equations determining ? are the same both there and here. ------------ The *trick* is to imagine a universe ... ------------ ... in which the uptick-downtick probabilities *somehow* make the computation of ? easy. This ... ------------ ... "other universe" is sometimes called the "risk-neutral world", and we'll describe it in a moment. For now, though, say we do a market analysis, here in *our* real world, *not* in that imaginary risk-neutral one. ---------------------------------------------------------------------------- Say this market analysis shows that, here in our world, ... ------------ ... the probability of an uptick is 70%, and of a downtick is 30%. So ... ------------ ... *our* world ... ------------ ... is a 70-30 world. We now propose ... ------------ ... a Problem: Let's find the expected value *and* expected return, ... ------------ ... here, in our 70-30 world, ... ------------ ... after one month, ... ------------ ... of one dollar invested in the bank, and ... ------------ ... of one dollar invested in Euros. For simplicity, let's ... ------------ ... assume that the bank pays one percent interest per month on savings accounts, so the interest on *accounts* is the same as on *loans*. ---------------------------------------------------------------------------- For part (a), say ... ------------ ... $1 is invested ... ------------ ... in the bank, in a savings account. There's ... ------------ ... a 70% chance that the Euro exchange rate *in*creases, and ... ------------ ... a 30% chance that it *de*creases, but the bank gives out one percent of interest in *either* case, bringing the account ... ------------ ... to $1.01. So the expected dollar *value*, at the end of the month, ... ------------ ... is 70% ... ------------ ... of one point oh one ... ------------ ... plus 30% ... ------------ ... of one point oh one, (pause) ... ------------ ... which is equal to, you guessed it, 100% ... ------------ ... of one point oh one. (pause) ------------ *This* number is 1% higher than ... ------------ ... this one, so the expected bank *return* ... ------------ ... is 1%. For part ... ------------ ... (b), say ... ------------ ... $1 is invested ... ------------ ... in the Euro market. There's ... ------------ ... a 70% chance of an uptick, bringing the exchange rate *up* 5% ... ------------ ... to a dollar five, and ... ------------ ... a 30% chance of a downtick, bringing it *down* 5% ... ------------ ... to 95 cents. So the expected dollar *value*, at the end of the month, ... ------------ ... is 70% ... ------------ ... of one point oh five ... ------------ ... plus 30% ... ------------ ... of point 95, ... ------------ ... which is equal to ... ------------ ... one point oh *two*. (pause) ------------ *This* number is *2*% higher than ... ------------ ... this one, so the expected *Euro* return ... ------------ ... is *2*%. Note that ... ------------ ... the expected *bank* value of a dollar *one* ... ------------ ... is less than ... ------------ ... the expected *Euro* value of a dollar *two*. Putting this in terms of *returns*, ... ............................................................................ ............................................................................ ---------------------------------------------------------------------------- ... 1% is less than 2%. That is, ... ------------ ... the expected *bank* return is less than the expected *Euro* return. ------------ The bank is a risk-free investment, in the sense that the one percent of interest is simply guaranteed. By contrast, ... ------------ ... Euros are risky. They *don't* have a guaranteed return. It's a general tenet ... ------------ ... of economics that *most* investors are "risk-averse", meaning they dislike risk. Of course, some investors love risk, and some don't care one way or another, but the standard assumption is that we live in a *mostly* risk-averse world. ------------ So *risky* investments must have a *higher* expected return than *risk-free* investments, or they won't sell, at least not in any significant volume. The above returns ... ------------ ... of 1% and 2% fit with this view of the world, in that the risky investment has a *higher* expected return. Remember, though, that the option price does *not* depend on the uptick-downtick probabilities. In looking for the *price*, we're free, if we wish, to adjust those 70-30 probabilities. We'll soon see that, *if* we adjust them in the right way, we'll get an ... ------------ ... *imaginary* world in which bank and Euros have the *same* expected return. Risk-averse investors wouldn't like such a place, but we ... ------------ ... imagine also that investors in this *other* world are ... ------------ ... indifferent to risk, or, in other words, ... ------------ ... risk-neutral. This imaginary world is thererfore *called* ... ------------ ... the "risk-neutral world". Bear in mind the important difference between the terms ... ------------ ... "risk-free" and ... ------------ ... "risk-neutral". "Risk-*free*" means without risk, whereas "risk-*neutral*" means indifferent to risk. So a savings account can be risk-free, but, lacking a personality, it can*not* be risk-neutral. On the other hand, in *my* experience, *people* are *never* risk-free, though there are some who enjoy risk, some who avoid it and, yes, some who are risk-neutral. We'll come back to all this in just a moment. For now, recall from Lecture 1, ... ............................................................................ ............................................................................ ---------------------------------------------------------------------------- ... the number line with the Euro ... ------------ ... downtick factor, the Euro ... ------------ ... uptick factor and ... ------------ ... the bank loan factor, also known ... ------------ ... as the risk-free factor. If you take ... ------------ ... 60% ... ------------ ... of 1.05 ... ------------ ... plus 40% ... ------------ ... of point nine five, (pause) you'll get ... ------------ ... 1.01. ------------ Let's write that down. Now remember the graphic solving ... ------------ ... part (b) of the problem we posed before. Let's change ... ------------ ... 70% to 60% and ... ------------ ... 30% to 40%, and then recalculate ... ------------ ... *this* number. Then ... ------------ ... this equation will turn ... ------------ ... into this one, and the graphic on the bottom half of the screen ... ............................................................................ ............................................................................ ---------------------------------------------------------------------------- ... changes to this. (pause) Next, let's clear some room. ---------------------------------------------------------------------------- Let's ... ------------ ... clear some more. Let's review our solution to part (a). ---------------------------------------------------------------------------- Again, let's change ... ------------ ... 70% to 60% and ... ------------ ... 30% to 40%. ---------------------------------------------------------------------------- Done! So we're now *imagining* a 60-40 world, different from our own 70-30 world. In this alternate reality, we have information ... ------------ ... about the bank and ... ------------ ... about the Euro market. Note that ... ------------ ... $1 invested in the bank will, after one month, have expected value ... ------------ ... $1.01, which is an increase of 1%, so the expected bank return ... ------------ ... is 1%. Note that ... ------------ ... $1 invested in Euros will, after one month, *also* have expected value ... ------------ ... $1.01, so, in this 60-40 universe, the expected *Euro* return ... ------------ ... is *also* 1%. Here in this imaginary world, ... ------------ ... the two expected values ... ------------ ... are *equal*, as are the two expected returns, both being 1%. Risk-averse investors *wouldn't* like this place, because they want a *higher* expected return on Euros to compensate for the higher risk. On the other hand, risk-*neutral* investors are perfectly comfortable here. This 60-40 world *is* the risk-neutral world! Let's have a moment of silence now to admire the resplendent beauty that we see before us. (pause) Did you ever see ... ------------ ... an equal sign do a victory dance? Watch closely. ------------ Next, suppose we create a portfolio ... ------------ ... of 3 dollars invested in the bank. Then we should multiply the top graphic ... ............................................................................ ............................................................................ ---------------------------------------------------------------------------- ... by 3. Note that ... ------------ ... these two numbers were *just* multiplied by three, so ... ------------ ... *this* number continues to be 1% higher ... ------------ ... than *this* one. Next, suppose we create a portfolio ... ------------ ... of 2 dollars invested in Euros. Then we should multiply the bottom graphic ... ............................................................................ ............................................................................ ---------------------------------------------------------------------------- ... by 2. Note that ... ------------ ... these two numbers were *just* multiplied by two, so ... ------------ ... *this* number continues to be 1% higher ... ------------ ... than *this* one. Next, suppose we create a portfolio of $3 in the bank ... ------------ ... *and* (pause) $2 in Euros. Then we should add together these two portfolios. ---------------------------------------------------------------------------- Done! Note that ... ------------ ... these two numbers were obtained by *adding* the corresponding numbers from the preceding slide, so ... ------------ ... *this* number continues to be 1% higher ... ------------ ... than *this* one. That is, the expected *return* ... ------------ ... on *this* portfolio ... ------------ ... is 1%. (pause) Of course, there's nothing special about the numbers ... ------------ ... 3 and 2. The same logic would work on *any* portfolio of bank and Euros. We thus make the *key observation* that, ... ------------ ... in this risk-neutral world, the expected return on *any* bank-Euro portfolio is 1% per month. *This* is the *real* beauty of the risk-neutral world, and we can *finally* get to the main point of this lecture: ---------------------------------------------------------------------------- This graphic describes ... ------------ ... Alice's hedging portfolio ... ------------ ... in our 70-30 world. Let's compute ... ------------ ... ?, with*out* solving a system of equations. ? does *not* depend on the uptick-downtick probabilities, so we might as well *change* probabilities *from* 70-30 ... ------------ ... *to* 60-40, and do our pricing in the risk-neutral world. After one month, the *expected* dollar value of the portfolio will be ... ------------ ... 60% ... ------------ ... of 5 ... ------------ ... plus 40% ... ------------ ... of 0, (pause) ... ------------ ... which is equal to ... ------------ ... 3. We're in the risk-neutral world, where *any* bank-Euro portfolio has ... ------------ ... a 1% monthly expected return, so ... ------------ ... this number is 1% higher than ... ------------ ... question mark. ------------ Let's write that down. Solving, we find that ... ------------ ... ? is 3 over 1.01 or, to two decimals, 2.97. These ... ------------ ... 60-40 probabilities are the *only* ones that make this trick work. With any *other* choice, the expected return varies from portfolio to portfolio. Next, take ... ------------ ... this expression for the number 3 and ... ------------ ... plug it in here. We get ... ------------ ... this equation. ------------ The numerator is a risk-neutral expected value, and ... ------------ ... the denominator is the risk-free factor. *This* explains why option prices can be calculated as time-discounted expected payouts in coin-flipping games. In other words, and in other worlds, ... ............................................................................ ............................................................................ ---------------------------------------------------------------------------- ... coin-flippers got price! (pause) We can also get to this same idea in a more prosaic way, as follows. ---------------------------------------------------------------------------- Remember the number line, which gave us ... ------------ ... this equation. (pause) ---------------------------------------------------------------------------- Also remember this graphic which gave us ... ------------ ... these three equations. (pause) Let's multiply the first of the three ... ............................................................................ ............................................................................ ---------------------------------------------------------------------------- ... by 60% and the second ... ------------ ... by 40% and then ... ------------ ... add the two together. The coefficient on x ... ------------ ... is 60% ... ------------ ... of 1.05 ... ------------ ... plus 40% ... ------------ ... of point 95, which, ... ------------ ... according to this, ... ------------ ... is 1.01. The coefficient on y ... ------------ ... is 60% of minus 1.01 (pause) plus 40% of minus 1.01 which is equal to, you guessed it, 100% ... ------------ ... of minus 1.01. The right hand side ... ------------ ... is 60% of 5 (pause) plus 40% of 0, ... ------------ ... which is 3. ------------ In this equation, we factor out 1.01 on the left hand side, and get ... ------------ ... this. We ... ------------ ... replace x-y by ? ... ------------ ... in this equation, and get ... ------------ ... this. Solving, ... ------------ ... we again see that ? is 3 over 1.01. The risk-neutral world, in some sense, is just a fanciful place where this kind of algebraic manipulation happens naturally, and within a probabilistic framework. Next, let's use risk-neutral calculations to price a more *complicated* option. ---------------------------------------------------------------------------- Remember that Fred wants 100 Euros for $100, not *one* month from now, but *two* months. Say that ... ------------ ... the current dollar per Euro exchange rate is 1 over point 95. Remember that ... ------------ ... Cathy sells this option. ------------ Say the exchange rate still rises or falls 5% each month, and ... ------------ ... the risk-free factor is still 1.01. ------------ Then the risk-neutral world is still a 60-40 world. (pause) ---------------------------------------------------------------------------- Here we see the possible evolution of the exchange rate over the two months. Note how every ... ------------ ... uptick multiplies by 1.05 and how every ... ------------ ... downtick multiplies by point 95. Remember that, ... ------------ ... *if* there are two upticks, then Cathy'll need ... ------------ ... to pay out a net of $16.05, which, combined with Fred's $100, gives the *one hundred* sixteen oh five needed to buy 100 Euros ... ------------ ... at this exchange rate. Similarly, if the Euro experiences ... ------------ ... uptick then downtick, or ... ------------ ... downtick then uptick, then Cathy'll need ... ------------ ... to pay out a *net* of $5. Finally, if there are ... ------------ ... two downticks, then Fred will *not* pay the hundred dollars needed to exercise, because he can buy the 100 Euros on the open market for only $95. So, ... ------------ ... in this scenario, Cathy'll pay out ... ------------ ... nothing, because Fred doesn't even show up. A moment ago, we did ... ------------ ... a risk-neutral calculation that gave $2.97. ------------ The numerator is a risk-neutral expected value, and ... ------------ ... the denominator is the risk-free factor. This calculation showed that $2.97 is the correct price of a portfolio ... ------------ ... that can finance a payout of 5 dollars on an uptick or nothing on a downtick. You may remember, from all the way back in Lecture 1, that this portfolio consists of 50 Euros and a 47.03 bank loan. In *this* lecture, we'll systematically ignore those hedging parameters of 50 and 47.03, and focus only on the *value* of the portfolio, which is $2.97. A similar ... ------------ ... risk-neutral calculation shows that $11.52 is the correct price of a portfolio ... ------------ ... that can finance a payout of $16.05 or 5. Finally, a third ... ------------ ... risk-neutral calculation shows that $8.02 is the correct price of a portfolio ... ------------ ... that can finance the $11.52 or $2.97 that *then* finances the *final* payout of $16.05, 5 or 0. To pay for that ... ------------ ... initial portfolio, Cathy'll need to charge Fred $8.02, so that's the price of the option. Note how we worked backward in time ... ------------ ... from the final payouts (pause) to the portfolio values at the one month mark, and then ... ------------ ... from those (pause) to the value at the start. So we can get the $8.02 price moving baaackward, right-to-left, staaage-by-staaage like this, but it's also possible to find it more directly. Let's see how. (pause) ------------ Here are the payouts. ---------------------------------------------------------------------------- Here they are again. Let's imagine that parallel risk-neutral universe with its ... ------------ ... 60-40 probabilities, but *this* time ... ------------ ... over a two month period. Let's use ... ------------ ... "c" to denote the initial dollar cost of the hedging portfolio. Remember, from Lecture 1, that Cathy makes an adjustment to the portfolio ... ------------ ... here at the one month mark, but she neither puts any money in, nor takes any out. That is, her hedging strategy is "self-financing". Also recall, from Lecture 1, ... ------------ ... the expected ending value. Remember that there's ... ------------ ... a 36% chance of paying ... ------------ ... $16.05. There's ... ------------ ... a 24% chance ... ------------ ... no, make thaaaaaaaat ... ------------ ... *two* 24% chances ... ------------ ... of paying 5 dollars. Finally, there's ... ------------ ... a 16% chance ... ------------ ... of paying nothing. Multiplying and adding, we get an ... ------------ ... $8.18 expected ending value. In this risk-neutral world, *any* bank-Euro portfolio that's managed in a self-financing way will grow, in *expected* value, by 1% per month. So ... ------------ ... *this* is obtained ... ------------ ... from "c" by making ... ------------ ... *two* increases of 1%. ------------ Let's write that down. Solving, we see ... ------------ ... that "c" is equal to ... ------------ ... the expected ending value ... ------------ ... divided by the *square* of the risk-free factor, ... ------------ ... and this gives 8.02. (pause) ---------------------------------------------------------------------------- Here's that same computation. We have ... ------------ ... the expected ending value divided by ... ------------ ... the *square* of the risk-free factor, ... ------------ ... and this gives 8.02. Let's compare *this* calculation to the baaackward, right-to-left, staaage-by-staaage computationssss that we did before. ------------ Here's the risk-neutral computation that gave $2.97. ------------ The numerator is a risk-neutral expected value, and ... ------------ ... the denominator is the risk-free factor. ------------ Here's the computation that gave $11.52. Finally, ... ------------ ... here's the one that went from $11.52 and $2.97 to $8.02. These ... ------------ ... two ways of arriving at Cathy's $8.02 price may *seem* different, but they're really *not*, and here's why: Take ... ------------ ... this expression for $11.52, and ... ------------ ... plug it in here. Then take ... ------------ ... this expression for $2.97, and ... ------------ ... plug it in here. We leave it as an exercise for you to expand, to find a common denominator and, eventually, ... ------------ ... to arrive at this formula up at the top. (pause) Okay. ---------------------------------------------------------------------------- READ FAST In the remainder of this lecture, we'll ... ------------ ... introduce some important terminology. This terminology'll be ... ------------ ... used in later lectures, so, if you intend to listen to them, it's important ... ------------ ... *not* to skip the rest of *this* lecture, *even if* you already understand the idea of risk-neutral pricing. SLOW DOWN Also, we'll ... ------------ ... price another option, for practice. For variety, let's make it ... ------------ ... a stock option. This is a good place for a break, so we call ... ------------ ... for an INTERMISSION!!!! (LONG PAUSE) ---------------------------------------------------------------------------- Welcome to the second and final *act* of Lecture 2 of Notes on Financial Mathematics, by Scot Adams and Fernando Reitich. Let's talk about a fellow named Harry. ---------------------------------------------------------------------------- Harry's the CEO of XYZ corporation, and he's interested in buying an option on his company's stock. Specifically, let's say that ... ------------ ... Harry wants to buy one *thousand* shares of XYZ stock, for 9 hundred 70 dollars, *three* months from now. More precisely, Harry wants ... ------------ ... the right, but *not* the obligation to do this. Should Harry turn out to be a *bad* CEO, and should the stock plummet, then Harry wants ... ------------ ... no *obligation* to buy the worthless stock at the high, high price ... ------------ ... of 9 hundred 70 dollars. On the other hand, should Harry turn out to be a *good* CEO, with a wide array nefarious accounting gimmicks, and should the stock soar, then Harry wants ... ------------ ... the right to buy the artifically inflated stock at the low, low price ... ------------ ... of 9 hundred 70 dollars. In these lectures, the only heroes are quants, ... ------------ ... like Gail, who sells this option contract. She seeks to compute ... ------------ ... the right *option* price, without a thought to profit for herself. Let's say that ... ------------ ... the current XYZ *share* price is a dollar. A *current* price is sometimes called ... ------------ ... a *spot* price. ------------ The 9 hundred 70 dollars mentioned above is called ... ------------ ... the *strike* price ... ------------ ... or *exercise* price of the option. This is the amount that Harry must *pay* to exercise. (pause) Back when Harry's company had its IPO several years ago, it, simply by *going* public, introduced a brand new public financial market, which trades ... ------------ ... shares of XYZ stock. Soon after that, other *related* markets started to appear. For example, one of them trades ... ------------ ... XYZ *forward* contracts, whatever *those* are. The market that concerns *us* in *this* lecture trades ... ------------ ... option contracts on shares of XYZ. The point I want to make *now* is that we had ... ------------ ... an underlying market, in shares of XYZ, which then gave rise to other financial marketssss. Each of these other markets was ... ------------ ... *derived* out of this underlying market, and so each one is commonly referred to ... ------------ ... as a derivative market. A good example *is* ... ------------ ... the option market on XYZ shares, but there are others, and they can get quite complicated and exotic, which is part of the reason that more good financial mathematicians and engineers are needed. ---------------------------------------------------------------------------- Remember Alice and Dan? For them, ... ------------ ... the *underlying* market was the Euro market, and they traded with each other in ... ------------ ... the derivative market of Euro options. ------------ Another good example of an underlying market would be a commodity market, like corn, wheat, soybeans, and so on. Each of these has a variety of derivatives associated to it. Let's look, for example, ... ------------ ... at wheat. One may buy, say, ... ------------ ... an option ... ------------ ... on a futures contract ... ------------ ... on wheat. So, in the beginning, there was ... ------------ ... wheat, followed by the wheat ... ------------ ... *market*, then ... ------------ ... futures on wheat, whatever *those* are, and *then* ... ------------ ... options on futures on wheat. This may *sound* involved, but, in fact, *this particular* ... ------------ ... *second* derivative market is not at all exotic, by today's standards. The world of finance *can* get *very* complicated, but we return now to our modest example ... ------------ ... of Gail and Harry. For them, ... ------------ ... the underlying ... ------------ ... consists of XYZ shares, and ... ------------ ... the derivative ... ------------ ... consists of XYZ options. (pause) In order to compute an option price, we need to make some kind of assumption about the price volatility ... ------------ ... of the underlying. So let's assume that, each month, ... ............................................................................ ............................................................................ ---------------------------------------------------------------------------- ... the price ... ------------ ... S of an XYZ share either ... ------------ ... increases 3% or ... ------------ ... decreases 2%. That'll be our volatility assumption. So the spot price of $1 per share, after the first month, will either go up to $1.03 or go down to 98 cents. Let's also assume that the venal, usurious bank managers, with their big, burly bouncers, aren't happy making only 1% per month, and raise their loanshark rate ... ------------ ... to 2% per month. We plot three numbers ... ------------ ... on a number line, namely, the XYZ ... ------------ ... downtick factor, the XYZ ... ------------ ... uptick factor, (pause) and ... ------------ ... the bank loan factor, also known as ... ------------ ... the risk-free factor. *This* time, the risk-free factor is found ... ------------ ... *80*% of the way from the downtick factor to the uptick factor. Then 80% ... ------------ ... is the risk-neutral *up*tick probability, while 20% ... ------------ ... is the risk-neutral *down*tick probability. Do note that ... ------------ ... the *down*tick probability appears on the *right*, while ... ------------ ... the *up*tick probability is over on the *left*. That might seem *backward*, but here's why it's correct: If you take ... ------------ ... 80% ... ------------ ... of 1.03 ... ------------ ... plus 20% ... ------------ ... of point 98, (pause) you'll get ... ------------ ... 1.02. Let's *imagine* a world ... ------------ ... with 80-20 probabilities here on the stock. Then, after one month, the expected *value* will be ... ------------ ... 80% ... ------------ ... of 1.03 *S*, ... ------------ ... plus 20% ... ------------ ... of point 98 *S* (pause) which is equal to... ------------ ... one point oh two *S*. That's an increase of 2%, so ... ------------ ... the expected stock *return* is 2% per month. (pause) If we also put ... ------------ ... 80-20 probabilities here on loans, then, after one month, the expected *value* will be ... ------------ ... 80% ... ------------ ... of 1.02 *L* ... ------------ ... plus 20% ... ------------ ... of 1.02 *L*, (pause) which is equal to, you guessed it, 100% ... ------------ ... of 1.02 *L*. That's *also* an increase of 2%, so ... ------------ ... the expected *bank* return is *also* 2% per month. Risk-averse investors won't like the fact that the risky stock and the risk-free bank have the same expected return, but risk-*neutral* investors won't care. These ... ------------ ... 80-20 probabilites give us our risk-neutral world, in which *any* self-financed portfolio of XYZ stock and bank has ... ------------ ... an expected return of 2% per month. We call for another moment of reverential silence, in appreciation of the beauty and simplicity before us. (pause) Now I know you're overwhelmed with emotion, but we must *somehow* carry on. ---------------------------------------------------------------------------- Here's a template for the evolution, ... ------------ ... over *three* months, ... ------------ ... of *four* market variables, and we'll describe the four in just a moment. Note how the leftmost ... ------------ ... node has four boxes, one box for each variable, and ... ------------ ... so do all the other nodes. ------------ In the top boxes, we'll, *eventually*, fill in the XYZ ... ------------ ... share price, which goes ... ------------ ... up 3% or ... ------------ ... down 2% each month, and which ... ------------ ... starts out at the spot price of one dollar. ------------ In the second boxes, we'll, *eventually*, fill in ... ------------ ... the value of Gail's hedging portfolio. Money goes into the portfolio ... ------------ ... at the start, and comes out ... ------------ ... at the end, but, ... ------------ ... inbetween, her strategy is self-financing, in the sense that, when Gail adjusts the portfolio, she keeps its value unchanged. So any gain or loss in portfolio value will come *inbetween* adjustments, and will come from changes in the XYZ share price, and from bank interest charges, but *not* because Gail added or removed any money. (pause) ------------ At the end, in all possible scenarios, ... ------------ ... the *red* value will be *exactly* the net amount that Gail *pays* to meet her obligation to Harry. ------------ At the beginning, ... ------------ ... the *red* value will be the amount she *charges* Harry, in order to set up the initial portfolio. So ... ------------ ... *this* red value *will be* the option's price. (pause) ------------ The third boxes'll have ... ------------ ... the number of XYZ shares in the hedging portfolio, and ... ------------ ... the bottom boxes'll have ... ------------ ... the bank loan in the hedging portfolio. So ... ------------ ... these are the *four* market variables that we'll track, and these ... ------------ ... last two are the hedging parameters. *They* won't be filled in until the *next* lecture, on Delta-hedging. ---------------------------------------------------------------------------- Let's get to work on the top boxes, ... ------------ ... each of which needs a share price. Let's clear some room. ---------------------------------------------------------------------------- Remember that the spot price ... ------------ ... is $1 per share. Remember ... ------------ ... the uptick and downtick factors. At the end of the first month, the share price will either ... ------------ ... go up to $1.03 or ... ------------ ... go down to 98 cents. Because ... ------------ ... 1.03 squared ... ------------ ... is 1.0609, the dollar share price might ... ------------ ... go up to 1.0609. We also have ... ------------ ... 1.03 times point 98 ... ------------ ... is 1.0094, so another possible share price *is* ... ------------ ... 1.0094 dollars. Also, ... ------------ ... point 98 squared ... ------------ ... is point 9604, ... ------------ ... and we fill in that possible dollar share price, as well. We leave it as an exercise for you to check that ... ------------ ... these are also correct, to five decimals. Again, let's clear some room. ---------------------------------------------------------------------------- Now we *move on* ... ------------ ... to the value of hedging portfolio. Remember that ... ------------ ... the strike price is 9 hundred 70 dollars. That's what Harry must pay, to exercise. We'll calculate portfolio values working ... ------------ ... backward in time, from right to left. Let's work out the portfolio value first in the case ... ------------ ... of three upticks, which puts the share price ... ------------ ... here. (pause) Gail must give Harry ... ------------ ... one thousand shares ... ------------ ...in return for 9 hundred 70 dollars, so, to two decimals, this transaction costs Gail a *net* ... ------------ ... of $122.73. So that's the dollar value that needs to be in the portfolio ... ------------ ... here, so that she can meet her obligation to Harry, in *case* of three upticks in the stock price. (pause) Next, let's assume ... ------------ ... three *down*ticks, which puts the share price ... ------------ ... here. A similar calculation ... ------------ ... is this, which gives ... ------------ ... a negative number, namely, minus 28.81. Moving ... ------------ ... 9 hundred 70 over to the *right* hand side gives us ... ------------ ... this equation, which tells us that Harry can get his ... ------------ ... thousand shares, on the open market, ... ------------ ... for $28.81 ... ------------ ... *less* (pause) than ... ------------ ... the 9 hundred 70 dollar strike price he'd have to pay to exercise. So, unless he's a financial masochist, he will *not* exercise, and Gail needs ... ------------ ... *nothing* in her portfolio, because Harry won't even show up. Speaking of exercising, we leave it as an exercise for you to do similar calculations, and to get ... ------------ ... these two numbers. At each of these ... ------------ ... two nodes, you should calculate the dollar price of 1000 shares, and then subtract 9 hundred 70. If the result comes out positive, put it in the box. Otherwise, put 0 in the box. To two decimals, ... ------------ ... here's what you should get. (pause) Next, let's clear some room. ---------------------------------------------------------------------------- Let's clear some more room. ------------ Let's define ... ------------ ... x plus, which is called ... ------------ ... the positive part of x. ------------ The positive part of x is just ... ------------ ... x, if x is positive, ... ------------ ... and zero otherwise. So, for example, ... ------------ ... the positive part of 3 is 3, but ... ------------ ... the positive part of -3 is 0. Next, define a function f ... ------------ ... by f(S) ... ------------ ... is the positive part ... ------------ ... of 1000 times S minus 9 hundred 70. Let's plug ... ------------ ... this share price ... ------------ ... into f. We get ... ------------ ... this, and you may remember, from the last slide, that, to two decimals, ... ------------ ... the expression inside the brackets ... ------------ ... is 122.73. The positive part of a positive number ... ------------ ... is itself. By an *amazing* coincidence, 122.73 is also ... ------------ ... *exactly* the net dollar amount that Gail pays Harry, in case of three *up*ticks. Similarly, if we plug ... ------------ ... this share price down here ... ------------ ... into f, ... ------------ ... we get this, and you may remember, from the last slide, that ... ------------ ... the expression inside the brackets ... ------------ ... is a negative number, namely, minus 28.81. The positive part of *any* negative number ... ------------ ... is 0. By an *amazing* coincidence, 0 is also ... ------------ ... *exactly* the net dollar amount that Gail pays Harry, in case of three *down*ticks. We leave it as an exercise for you to plug ... ------------ ... these two share prices into f, and to see that, to two decimals, you get ... ------------ ... 69.68 and 19.21. By two *amazing* coincidences, these numbers are also ... ------------ ... *exactly* the net dollar amounts that Gail pays Harry in *these* two cases. Here's another ... ------------ ... exercise: ... ------------ Graph the function f. This function ... ------------ ... f is called the *payoff function*, because it relates ... ------------ ... the final share price ... ------------ ... to the net amount Gail needs, ... ------------ ... to *pay off* Harry. So f provides a transition between ... ------------ ... the *underlying* ... ------------ ... and the *derivative*. (pause) Okay. ---------------------------------------------------------------------------- These four red numbers are the possible ending values ... ------------ ... of the portfolio. As a group, they're sometimes called ... ------------ ... a "contingent claim", because: According to the terms of the option contract, Harry has a monetary ... ------------ ... *claim*, at the end of three months, on some of Gail's assets, but the amount of that claim ... ------------ ... is *contingent* on the ending stock price. Using our volatility assumptions and the payoff function, we found that the claim could end up ... ------------ ... as *high* as $122.73, or ... ------------ ... as *low* as 0. So we have ... ------------ ... four different possible *claims* which, as a group, form ... ------------ ... the contingent claim. (pause) We'll continue moving ... ------------ ... backward from the contingent claim ... ------------ ... to the option price. First, though, let's clear some room. ---------------------------------------------------------------------------- We'll need to remember ... ------------ ... the bank's risk-free factor, which is 1.02. We'll also need to remember ... ------------ ... the risk-neutral uptick-downtick probabilities, which are 80-20. Imagine a coin-flipping game ... ------------ ... with an 80-20 coin, and a one-month delayed payoff ... ------------ ... of $122.73 or $69.68. The *expected* payoff'll be ... ------------ ... 80% ... ------------ ... of this ... ------------ ... plus 20% ... ------------ ... of this. Let's ... ------------ ... write down that amount. The payoff is delayed by one month, so, to get the *present* value, we ... ------------ ... divide by the risk-free factor. To two decimals, we end up ... ------------ ... with 109.92, and *that* is then the value of the hedging portfolio ... ------------ ... here. A well-chosen portfolio worth ... ------------ ... 109.92 can finance a payout ... ------------ ... of 122.73 on an uptick or ... ------------ ... 69.68 on a downtick. In the *next* lecture, we'll figure out ... ------------ ... the details of how to set up that well-chosen portfolio, but we're ignoring the hedging parameters in *this* lecture. We leave it as an exercise for you to verify ... ------------ ... these two numbers. In both cases, find the expected value ... ------------ ... using the risk-neutral uptick-downtick probabilities, and then discount back one month, ... ------------ ... dividing by the risk-free factor. Let's do another portfolio value. Imagine a coin-flipping game ... ............................................................................ ............................................................................ ---------------------------------------------------------------------------- ... with an 80-20 coin, and a one-month delayed payoff ... ------------ ... of 109.92 or 58.42. Then the *discounted* expected payoff ... ------------ ... is this, and, to two decimals, ... ------------ ... we get 97.67, and *that*'s then the value of the hedging portfolio ... ------------ ... here. We leave it as an exercise for you to do a similar calculation, and to get ... ------------ ... 48.77 here. Finally, let's do the leftmost portfolio value. Imagine a coin-flipping game ... ............................................................................ ............................................................................ ---------------------------------------------------------------------------- ... with an 80-20 coin, and a one-month delayed payoff ... ------------ ... of 97.67 or 48.77. Then the *discounted* expected payoff ... ------------ ... is this, and, to two decimals, ... ------------ ... we get 86.16, and *that*'s then the value of the hedging portfolio ... ------------ ... here, at the starting time. So $86.16 is the amount that Gail should charge Harry, to *finance* the purchase of the initial portfolio. (pause) *That* was a *lot* of work to get the $86.16 price. It involved many staaages and intermediate calculaaations. Let's see how we could have skipped past some of them. Let's back up. ---------------------------------------------------------------------------- This is the point where we'd only just calculated ... ------------ ... the contingent claim. To avoid clutter in the graphic, let's ... ------------ ... drop the share prices, and look *only* ... ------------ ... at portfolio values for a moment. Let's imagine the risk-neutral ... ------------ ... 80-20 world. We endeavor to go *directly* ... ------------ ... from the contingent claim ... ------------ ... to Gail's price. Remember that, in this risk-neutral world, the expected value of the portfolio grows ... ------------ ... by a factor of 1.02 each month. Then the *expected* ... ------------ ... ending value of the portfolio is 1.02 *cubed* times ... ------------ ... the initial value. In other words, the *expected* ... ------------ ... contingent claim is 1.02 *cubed* times ... ------------ ... the option price. So, to get the option price, let's calculate that expected ... ------------ ... contingent claim, and then divide by 1.02 cubed. (pause) In a moment, we'll see that we have eight possible scenarios for how the stock price evolves over the three months. These scenarios range ... ------------ ... from upticks-uptick-uptick ... ------------ ... to downtick-downtick-downtick. To find the expected contingent claim, we'll consider each of the eight scenarios and multiply the probability *of* that scenario by the amount ... ------------ ... of the claim that arises at the end of that scenario. We'll then add all those results. Okay, here we go! First, we can get ... ------------ ... to *this* claim *only* ... ------------ ... by this path, which happens with probability ... ------------ ... point 8 cubed. So *this* path contributes point 8 cubed times ... ------------ ... *this* to the expected contingent claim. Let's ... ------------ ... write down that amount. Next, we can get ... ------------ ... to *this* claim by, for *example*, ... ------------ ... this path, which happens with probability ... ------------ ... point 8 squared times ... ------------ ... point 2, so this path contributes ... ------------ ... this to the expected contingent claim. There are, in fact, *three* paths ... ------------ ... to the 69.68 claim, namely, ... ------------ ... this one, ... ------------ ... this one ... ------------ ... and this one. Each happens with probability ... ------------ ... point 8 squared times ... ------------ ... point 2, ... ------------ ... so we multiply by 3. Next, we can get ... ------------ ... to *this* claim by, for *example*, ... ------------ ... this path, which happens with probability ... ------------ ... point 8 times ... ------------ ... point 2 squared, so this path contributes ... ------------ ... this to the expected contingent claim. There are, in fact, *three* paths ... ------------ ... to the 19.21 claim, namely, ... ------------ ... this one, ... ------------ ... this one ... ------------ ... and this one. Each happens with probability ... ------------ ... point 8 times ... ------------ ... point 2 squared, ... ------------ ... so we multiply by 3. Finally, we can get ... ------------ ... to *this* claim *only* ... ------------ ... by this path, which happens with probability ... ------------ ... point 2 cubed, so this path contributes ... ------------ ... this to the expected contingent claim. We've now looked at 1+3+3+1 different scenarios, each one represented by a path. That makes eight, and that's all there are. Multiplying and ... ------------ ... adding, ... ------------ ... we get 91.44, so that's ... ------------ ... the expected contingent claim, or, in other words, the expected ... ------------ ... ending value. Here in this risk-neutral world, the expected value grows 2% each month, so ... ------------ ... the expected ending value is 1.02 *cubed* times ... ------------ ... the initial value of the portfolio. We therefore ... ------------ ... divide by 1.02 cubed, and, to two decimals, ... ------------ ... we get 86.16, which we insert ... ------------ ... here. That gives *another* method of computing Gail's 86.16 price. Does it seem any easier than the *first* method, with all its staaages and intermediate calculaaations? Maybe not, but the time period of *this* option was three months, and that period was split ... ............................................................................ ............................................................................ ---------------------------------------------------------------------------- ... into only *three* one-month *sub*-periods. If we had, say, 100 sub-periods, it'd actually become quite clear that this second method *is* easier, once we develop a system for counting the number of paths to *each* of the claims. Moreover, with a large number of sub-periods, we'll see in a later lecture, that ... ------------ ... the Central Limit Theorem gives a definite integral that's extremely close to the expected contingent claim. That'll save us lots of work. In the limit, as the number of subperiods approaches infinity, we obtain the famous ... ------------ ... Black-Scholes Option Pricing Formula. (pause) Let's do a quick summary review of the pricing of this option. ---------------------------------------------------------------------------- READ FAST NOW We started with an empty template. ------------ We added the spot price of a dollar. Using ... ------------ ... the uptick-downtick factors, and working *forward* from left to right, we ... ------------ ... populated the template with share prices. We then plugged ... ------------ ... all of the *ending* share prices ... ------------ ... into the payoff function, and obtained ... ------------ ... the contingent claim, thereby moving from black numbers to red numbers, in other words, from the underlying to the derivative. We then used the risk-neutral ... ------------ ... uptick-downtick probabilities, ... ------------ ... together with the risk-free factor, to compute discounted expected values. Working *backward*, from right to left, we filled in ... ------------ ... all the remaining portfolio values. Finally, we backed up and looked again ... ------------ ... at the contingent claim, and showed how to go *directly* from there ... ------------ ... to the option price. (pause) ---------------------------------------------------------------------------- It remains to work out the two hedging parameters, but we save *that* for our next lecture, on Delta-hedging. Now that we're pricers, ... ------------ ... let's go get hedge! ----------------------------------------------------------------------------