 So as always, what I'd like to do is give a review of what we've covered last time and then move on to the new material. So in the last lecture, we were going over all of these concepts of no arbitrage once again. The fact that in this economy where there are two underlying assets and one is risky and one is risk-free, we showed that there's no arbitrage if and only if there exists a probability measure such that the current value is equal to the future value discounted and then taken an expectation under using those probabilities. And these are not real probabilities. And the existence of the absence of arbitrage is equivalent in this model to the assets having a particular ordering. The risk-free asset does better than one state and worse than another. And this is equivalent to the fact that the little Q probabilities that we calculated are in the range 01. Then in the lecture, we talked about this idea of an incomplete market and did some examples and show that there were various bounds on the risk-neutral prices that we could find. The idea here was that since there are three possible outcomes, when you write down the risk-neutral condition, the risk-neutral expectation condition, you find that actually there are multiple choices that could happen. There are many possible measures, not just one, and the Q is not unique in that case. And because of the non-uniqueness of Q, you were able to find non-unique values for contingent claim or for other assets. And in this case, the other asset was able to take on some bound, and I guess in the example we had it was somewhere between 0 and 10 for this particular kind of option. And then we discussed the fact that, well, how, since you have a range of prices, what's a reasonable way to come up with a unique price? What would you yourself do, because no arbitrage theory is not enough, doesn't pin us down. And one idea here was to say, okay, let's take a look at all possible portfolios that you can attain by trading in the assets that already exist. And if you look at all possible portfolios that you can attain, in this situation when you have three outcomes, the attainable assets effectively span a plane. And if your claim, if your new asset doesn't lie on that plane, you can't, it's not attainable. But we thought about the idea, perhaps we can somehow get as close as possible. And as close as possible in some sense. And the sense in which we talk about close as possible is by minimizing, in this case, the L2 distance between the attainable asset, the class of attainable asset, and the asset that you're trying to replicate. Okay, so this gives you a particular projection onto this plane, and it allows you to replicate, or allows you to replicate the asset that lies on the plane. And then you claim that is in fact a value that you can assign to the new asset. But there was a key question that I left open-ended. And it's not one that I'm going to ask on a test or anything like that. But it's an interesting open-ended question for you to think about, is whether a value associated with the underlying asset, whether a value associated with an asset in this type, is a reasonable value to use. Does it lead to a no arbitrage price? It's a reasonable approach, but does it actually give you a no arbitrage price? And there's a couple of approaches that you can take to answer that question, and I'll leave it for you to discuss. And if you want to ask me, you can go ahead. And then, I believe, we then went on and discussed what to do in multi-period setting. And in the multi-period setting case, we thought of the idea that if you had multiple time steps at each point in principle, you can take on two new values. But this led to a computational problem, the computational problem being that the tree, at the number of points that you have after stepping out n number of time steps, grows exponentially. It grows as a power of two. And one way to avoid that is to force your trees to do what's called recombining. And that is, when you go up and then down, you get to the same state as when you go down and then up. So this is the idea behind a recombining tree. And the total number of nodes that you have is, in fact, only quadratic in the number of steps. Because at every time you have n, so if you sum that up, you get n squared. So that's a nice computational reduction when you go to the multi-step case. Then we were discussing, well, how do you deal with valuation and no arbitrage conditions and so on within the multi-period setting? And the idea was actually quite simple. It said the idea is to just focus in. If you're at a particular node, say you're at that node where the asset A takes on the value AU, so you're inside that yellow circle, there are only two possible outcomes for you from that node onward. And similarly for the money market account, there are only two possible values or for the secondary asset, there are only two possible values for it as well. So focusing it on just that yellow circle, it looks exactly the same as it did for the one period case. So there's no difference. You can just repeat all of your knowledge set from before and apply it in that yellow circle. And then repeat that again, apply it in the sort of turquoise circle. And then repeat that again because now you know the asset's value in the, now you know whether or not there's arbitrage or not, or you know the new asset value at the two nodes, AU and AD. And then you can repeat the analysis again for the purple circles because at every particular subset, it's just a binomial tree. There's nothing new. And in fact, I'm going to be a little bit more explicit about this later on today when we look at American options and we look at past dependent options and when we'll also talk about some simulations. So we're going to go back to this and do some more examples and a little more details on it in a bit. But that's the basic idea, is that you simply have to focus in on each part of the tree. And if you're asking a question about no arbitrage, you would make sure that all of the Q probabilities that you got from these branches are in fact, or all of these Qs that you get by imposing risk neutrality are in fact probabilities. In other words, that they are in the range 01. And then if you can find such Qs and they don't have to be the same, in general they won't, you can find such Qs then you do have an arbitrage free model and you can go on and value some new contingent claim in the same way as before. So either use a risk-neutral expectation on the new claim or the new asset or do a replicating portfolio. Again, focus in on just the two nodes or sorry, focus in on any one node which just takes on two possible values in which case it reduces to your old analysis again. Okay? So that's where we discussed for the multi-period case and that naturally led to the situation of, well, what if we were to take this model and actually try to replicate some behavior that we see in the real world? What features would you try to replicate? And at a first order analysis, you're going to try to replicate at least the mean and the variance of the returns of your underlying asset. And we discussed this fact that there are two ways that you could look at returns, either as absolute returns, so the difference divided by the initial value, or the logarithm of the ratio. And I also mentioned that these two are more or less the same, up to some small corrections. For the purpose of the analysis, it's much easier to deal with the logarithm. So we use the logarithm, logarithmic returns and what we want to impose on our tree model is we would like to impose that the expected return in our tree model is also equal to the historical return that we have observed. So this quantity, the starred quantity, the mu star minus the half sigma star squared times delta t, that's our historical quantity. And the variance, we also want our model to match this variance. This is our historical variance. And the approach that we then took was that, well, if we have a simple binomial tree which recombines, one way to do that is to have the asset grow up or down by the same factor. And then we would like to choose that factor and choose the probabilities so that we match these first two moments. So matching these first two moments are the key aspect of this analysis. So once you've matched those first two moments and you do this to order little delta t to lowest order that you can, in other words, the most important contributions are what you keep, what we found after doing some of this analysis is this very nice result. That the probability is approximately half, it's not exactly, it's a half plus a correction term that depends on the drift and the volatilities. And that correction term is also of order square root delta t. As well, the up and down ticks, if you want to call it a tick, up and down of the asset is proportional to square root delta t in log space. And the coefficient of proportionality of the up and down tick in log space is actually the volatility, the sigma star is your vol. So once we've been able, so then when we did this analysis, we then achieved our initial goal that is create a binomial model that recombines that replicates the behavior that we've seen in the market. At least the mean and the variance, it replicates that. And then we want to go on and ask, okay, what about the continuous time limit of this model? Suppose we took the time steps going to zero, what do we end up with? So we then said, okay, let's take our time, break it up into a whole bunch of pieces, write down our asset price in terms of all of these little increments of whether we went up or down, that's these little x Bernoulli random variables. And by central limit theorem, basically, we were able to say that the distribution of, we can write, sorry, the asset prices as a times the exponential of some random variable, the terminal asset price as a times the exponential of a random variable. And by central limit theorem, that random variable is normally distributed. And our goal was to figure out what's the mean and what's the variance of that normal. We then went ahead to that calculation, and the end result is right here, actually. There we go. So we found that the distribution of x under the measure p is normal with that particular mean and that variance that shows up there. Now, from the start, we had this sort of weird convexity correction term that I had written in terms of the historical mean. You might remember, let me slide back here. When I wrote down the expected value of the log returns, I didn't call that mu star delta t. I called it mu star adjusted by convexity, mu star minus the half sigma squared, and everything times delta t. And I said, just take it at fate. There's a reason that I've chosen it to be that. I could choose it to be anything, but I'm calling it this. And the reason then becomes evident once we've got this continuous time model. And we know the distribution of the underlying asset is the exponential of a normal with that mean and that variance. And then if we compute the expected value of that underlying asset, what we found was that the expected value of the underlying asset is, in fact, the initial value grown at the rate of mu star. And it only grows at the rate of mu star because we had that convexity correction. If you didn't include the convexity correction in the definition of mu star, what you'd end up with is the expected return of the asset being equal to mu star plus the half sigma star squared. So that was the underlying reason for including it in the first place. Okay? And I believe, yeah, good. And then we did one last calculation and that was take, now that we've got a model that replicates the historical behavior of the underlying asset. Under the P measure, we now want to do derivative valuation. So we want to somehow value new contingent claims or new assets. And in order for us to do that, we have to introduce the risk-neutral measure. So we then said, let's calculate the risk-neutral measure induced by such a model. We compute the Q and we have this nice exact expression. But we can play the same game as before. We're going to take delta T to zero. So let's do some asymptotics on that. You carry out the asymptotic expansion. And what you find is that the Q probability here looks exactly the same form as the P probabilities did when we did this calibration, except instead of mu star, you have R. And that's kind of a nice thing. It sort of corresponds to the intuition that we had about the risk-neutral probability. The risk-neutral probabilities tell us that the rate of return of assets in terms of Q probability is equal to R. So it's nice and comforting to see that Q, in fact, it doesn't have mu star in it at all. It only has R. It doesn't depend on the P measure in some sense. The probability P only really depends on R. It depends on the sigma stars and so on, but that really defines your state space. So the real degree of freedom in the probabilities P is R. Sorry, it's mu star. It's the rate of growth. And here we see that rate of growth, mu star getting replaced by the risk-free one. And that's something we'll see happening again, even in the continuous time setting. When we introduce Brownian motions and go through the whole continuous time dynamic hedging, you'll see this happen again. The risk-neutral measure induces something which changes anything that has to do with real-world drift, mu stars, into R. They'll all get changed into R somehow, some sort of magical thing that happens, but happens from a completely different perspective when we do the dynamic hedging approach. Okay, so we ended up finally then saying that under the Q measure, the asset still has a log-normal distribution because the form of Q is the same as it was for P. So obviously the asymptotic limits, the limits when we take A and compute the distribution will be exactly the same as what we found under P, but instead of mu, you have R. So this result is nothing, you don't have to do much work to do this. You can just quote your old result and say the asset A under Q has this distribution, and this is where we've landed. Okay, so that's a sort of synopsis of a number of things that we did last time. Do we have any questions that I can address or issues that you'd like clarified about the analysis so far? Crystal Clare, okay. Okay, so we'll begin with our new lecture today. Oh, actually there was one more thing here. Yeah, that was your quiz, okay. Well, I posted the solution, so you can go ahead and take a look at that. Okay, so what I want to do today is a number of things actually. I want to derive for you the Bloch-Scholes pricing equation, which is actually a very simple calculation, and discuss how various options payoffs behave in terms of their prices without using any mathematics, just sort of intuitively looking at them and trying to figure out how they actually behave. And then I would like to discuss American options and optimal exercise boundaries associated with them and some pass-dependent options. That'll sort of be our little outline for today. Okay, so the first part is on Bloch-Scholes. And again, I think this is something that perhaps 90% or more of you have already seen, so I'm not going to go slow. I'm just going to go quickly through it if you have questions, ask me. So the analysis that we're going to do first is let's take a call option, and what we'd like to do is we want to value a call on an underlying asset, F, and the strike is K, and the maturity is sometimes capital T. So how are we going to approach this? Well, we know that the value of this call option has to be equal to, and for now we're just going to use a constant interest rate. So think of R as the spot interest rate for maturity capital T. We know that this has to be the expected value under the measure Q of a function of that payoff. And what is it my payoff? Sorry, the expectation of the payoff itself. What is the payoff? It's this. It's just F capital T minus K, for a call option. Now we know, and this is under the measure Q, now from the analysis that we've done last time, we know that this equals in distribution the initial value of S times the exponential of a normally distributed random variable with a mean, so let me just slide back here to remind you of the, where is it? There we go. We showed this result that the underlying asset has this mean and this variance. So what I'm going to do is I'm going to write it instead of using a non-standard normal random variable. I'm going to write it in terms of a standardized normal random variable. It's always easiest and best to do that. So this becomes R minus one-half, and I'm not going to bother to put the little asterisks, the stars on things anymore. It's not necessary, right? What I was using the asterisks before was to emphasize the fact that that was coming from data, right? It's an estimate based on data, and that is my estimator of the variance. But now we have a model, so let's just get rid of the asterisks and say, I don't care if you got it from data or you just picked it out from the model. I don't care. So we know that this in distribution has to be equal to sigma squared, or that's the mean, and that's the variance. And this variable here that I'm introducing is just some standardized normal under the measure Q. And our whole goal now is really simply to compute that expectation. That's the goal of the calculation. But at this point, I could actually just say, okay, here's the answer. Just write it down for you. Because there's no more financial theory that goes into this anymore. We're done with financial theory. It's only basic math. First tier calculus. But since the first time we do it, I do it for you explicitly anyway, okay? So let's go ahead and just compute the expected value. Well, the way that I like doing this is to write this expectation in two terms. Okay, remember, what does the little subscript plus means? It means whatever is in the argument if it's positive, otherwise it's zero. So I can also write that simply as s minus k if s is bigger than k, otherwise it's zero. Remember this little, this one script by definition, this is an indicator function, right? This is zero or this is one if a is true, so if a occurs and is zero otherwise, okay? That's just my, that's the notation that I used for it. So this expectation, there are two terms in it. There's the expectation of this term and the expectation of the second term. Now, we can go ahead and just use standard calculus but we can do something even better. The second term here, what's the expected value of an event that, of the indicator of an event? What's one if the event occurs, zero if it doesn't occur? The expected value of such random variable is simply the probability that that event occurs. So this is in fact just the Q probability because this is the Q expectation that s is bigger than k. So for this one, we actually don't have any work to do. We simply write down since we know that the s at maturity is equal in distribution to that expression at the top of the screen. Therefore I can simply write this as Q. I can insert that expression in there, take logarithm, subtract out the mean and rearrange it so that I isolate the only random variable which is z. So this is the same thing as z is greater than negative log s over k plus r minus a half sigma squared t all over sigma squared t. That's just the same thing. I've just done a few lines of algebra for you all at once. And then you recognize well, z is standard normal so this is the upper tail probability which is actually the same thing as the lower tail probability of negative the point. And let me just sketch that for you. That's my normal distribution. So whatever this number is I don't know, call that thing z star. Okay. I'm looking for the probability of that event from a standard normal and of course take that same thing and just reflect it and that's identical to the probability of the reflected event because it's a standard normal. It's completely symmetric. So this is minus z star and I've drawn it as if z star was negative but that's just the way I've drawn it. So that's the same probability and therefore you can write that in terms of the standard normal CDF of negative z star. In other words of log s over k and this is the standard normal CDF. Often in this course capital phi will represent standard normal CDF. In fact it will always represent standard normal CDF. So you've got one term of that expectation now you just need the other term expected value of s times the indicator. It turns out that you can actually do this calculation as well without writing down a single integral by using some ideas of measure changes but I'm not going to dwell on that today. If you want to see that come see me during my office hours I'll explain to you how that works but we'll just do it instead by direct integral calculation. So what I do is I condition on the little z random variable and I know that s of capital T in distribution is this plus sigma square root T times little z in the event that well since I introduced that little variable z star up there the event that s is bigger than k is the same as the event that little z is bigger than z star. So I can always write it that way times the density of little z or times the density of the standard normal sorry. Any questions there? A straightforward plugin of the distributional property of s capital T and of course that truncation of the integral all that indicator does is truncate the integral and I can pull out a bunch of constants here. That part is constant and the integral goes from z star to plus infinity e to the sigma square root T z and I'll collect both exponential terms together one half one half z squared and the trick in all of these kinds of integrals if you want to call it a trick at all is just to complete the square in the exponential. It's a quadratic function of z so you can complete the square and you'll end up with a new quadratic function of z without a linear term and then you can shift it and do the integral. So let's work on just the integral z star that's e to the minus one half z minus sigma square root T all squared plus one half sigma squared T so you can convince yourself that that indeed is what happens when you complete the square. If I square that term I'll get minus a half z squared that's the term that was there before if you look at the cross term that gives me minus 2 sigma square root T z times minus a half so that's plus sigma square root T z and that's the term that we had before and then there's a quadratic term on the sigma which gives me minus a half sigma squared T and a plus half sigma squared T camp with that so this is identical and now it's a matter of shifting the integral so first of all there's a constant part that can be pulled out introduce a new variable called z prime z prime is z minus sigma square root T and the Jacobian for that transformation is one just linear coefficient one so this is z star minus sorry minus sigma square root T plus infinity e to the minus one half z prime squared z prime over square root 2 and then again you can recognize that integral as the upper tail probability of a standard normal and so you can rewrite it again in terms of the standard normal CDF of negative of that value and if you simplify if you just plug back in what z star in fact is you would get log s minus k end up with that expression and then if you put this together with your previous result you find that the final value is actually let's scroll back up again for a second one thing that you notice is this i function there's a probability but then there's this exponential factor out front and that exponential factor exactly cancels the minus a half sigma square that showed up up there on the second line you see it's s0 e to the r minus a half sigma squared T so that exponential factor down in front of the probability will cancel the exponential the convexity correction term there and so the end result is you just get phi and I'm going to write this thing I'm going to introduce a new constant called d plus it's the usual constant that you see in a black-should equation and d minus and what are they well this is just log of this ratio plus r plus or minus a half sigma squared T all over sigma squared T so this is our black-should price or the black-should formula for call option okay now you could also do this for a put option so maybe I'll put little c up there just to denote the fact that that's a call and I'll put a little p here and if you do the calculation for put you end up with with this expression has a very very similar form so you've all I think have seen this at some point and what I'd like to do is something that you may not have seen I'm not sure how much of it you've seen is looking at the behavior of this particular option as a function of a certain parameters change and then trying to understand what happens if you try to build values of other type of contingent claims that are not so obvious they're not like a standard call or a standard put they have slightly different behavior in shape how would you figure out how to sketch their prices and that's what we'll investigate next unless there's questions about this formula okay so let's address a couple of the questions or a couple of the behaviors so first of all let's look at we'll look at call and then we'll look at put and what I'm drawing here is the initial asset price or the asset price at some point in time where we're going to draw the value at some point in time now at maturity you know that this is simply the call payoff we end up with a straight line call payoff function so this is at, this is the at maturity now if we're at some point prior to maturity if you look at two weeks before maturity then this call function call price function has a shape that, sorry I'm not drawing it too well, there we go has a shape that looks like that and the question is does that asymptotically approach the payoff or not asymptotically approaches the payoff and yes I see some yeses okay the answer is no it actually approaches a different straight line it approaches let me re-label this thing this is at maturity and if there's capital T left to maturity so this is a time to maturity this red curve will asymptotically approach that straight line and that straight line it shifted right it hits the axis at K discounted that's K and over there is K e to the minus RT and the way that you can see that the way that you can think about that is from a financial perspective you can say well I know that the call option if the asset is extremely high in value the optionality embedded in there has pretty much no value if I'm very far off to the right then really all I have is a payoff is S minus K the fact that this is S minus K gets flat if it goes below zero doesn't really affect me because I'm so far deep in the money if your current asset price is way out here then all you really see is just a straight line you don't see this little protection you don't see this protection part here all you see is a straight line that continues out very very far out you can formalize that by saying while the payoff of the call is clearly bounded below by sorry I need a pen here the payoff of the call option phi call is always greater than or equal to the asset price at maturity minus K do you agree it's always greater than or equal to that that S minus K is a straight line that does that so clearly the payoff is always bigger or equal to that so if there's no arbitrage in the market the value of the call option has to be more valuable than something which will pay me S minus K and what kind of portfolio can I build that pays S minus K well if I hold one unit of the asset today and I owe K dollars in the future then in the future the value of that portfolio would be S minus K because on that date in the future I would be owing K dollars that's the minus K I have the asset in my hand I can sell it and that gives me S so right now what's the value of owning one unit of the asset and owing K dollars in the future what's the value someone tell me what's the value minus right so there's a minus there so this here is just my value of the call the value of the payoff of the call receiving that in the future is the value of the call today so I have this inequality so this tells me that the call has to be bounded below by that straight line and then arguing that asymptotically it actually approaches a straight line you just simply do that take the limits take the limit as S gets infinitely large the probability of the event that I drop below K is zero and therefore I should asymptotically approach that equality there this equality becomes strict equality becomes binding for large S because for large S there's zero chance that I would actually drop below K you can imagine that right if I have the larger and larger my current level of the asset price is much less likely the smaller the probability is of the event occurring that I fall below K and it's only those events below K which cause the payoff to be more valuable than than this little portfolio right only because only because of this part here that you have an inequality right so that's the S minus K and the yellow portion is where the call is more valuable than that little portfolio and so if S is very large the probability of getting into the yellow region gets to be very small and therefore this bound becomes binding so you know asymptotically approaches now that's not a formal rigorous proof or anything but I'm not trying to do rigorous proofs here trying to give you intuition questions about that okay well you could extend that idea then and then answer the question to the call price as maturity increases so let's say let's compare so we know that we know that the value of the call it's going to look something like that and this is approaching that straight line suppose I had another call that had twice the maturity date maturity twice in the future so this was the one year call price the red line and I wanted to draw a purple line that's the two year call price where would it lie? it should lie off to the left lie off to the left and I'll have something that goes like that and this would asymptotically approach the straight line where the discount factor is a two year discount factor the red line would be for the one year discount factor so this would be e to the minus r2 for example if we talked about the two year and this would be S minus K e to the minus r times 1 okay the discounted value of K over two years is smaller than the discounted value of K over one year so the line moves off to the left do we agree? makes sense? okay so let's say we can repeat that argument for a put option and then I'm going to try to use a formalized argument to show you this inequality with Jensen's inequality basically okay so for a put the payoff of a put looks like this and if I ask for the price at some earlier date how do I figure out should it be positive here? should it be zero? if I'm in this region out here the put matures in a year from now should clearly be positive right because the put never gives me a negative value and therefore it should always have a positive value the payoff is always positive under some events therefore I should be tamed for that that's basically the idea now the further I move off to the right the lower that value is because the less likely it is that I end up in the money so we know that here it's got to be increasing in this direction or decreasing for large increases increasing as I move towards zero now the question is there's a couple of options that we could have here one is it could look like that another is it could look like that another is it could look like that and of course it could also do that in principle those are features now what makes financial sense or I should have posted the other way which of these don't make financial sense and why? so what about this? does that one make financial sense? why not? is it decreasing in S? in S0? it's pretty much that the value is a decreasing function of S0 why? because forget about the explicit black-scholes formula the explicit black-scholes formula is a bit of a mess you look at that and then you're trying to answer oh sorry there's an S0 there you're trying to answer whether this increases or decreases with S but S appears in a few places appears here appears here and appears there which means it also appears there and there so it's not obvious from that expression but if you go back to the representation of the price as an expectation then it should become quite clear the price is the expected since we're talking about a fixed maturity the discount factor doesn't affect anything the expectation under Q of S0 e to the r minus a half plus sigma square root T times Z minus K that's that oh sorry K minus this that's the call how do I want to do that? as S increases S0 as that increases clearly this function decreases and you're taking an expectation but that doesn't change anything the expectation is a linear operator so as I increase S my payoff decreases therefore the value will decrease so it's a decreasing function very simply so you can't have this funny kink you can't decrease and then increase now it could in principle become it could have like sort of sharp edges to it and so on in principle but you can prove that that's not the case either you can use convex dearguments and you can use convexity to argue through Jensen's inequality that you get convexity as well but we won't go there you can just intuitively think the payoff is this blue shape and the expectation is what you're computing so you're kind of smoothing that blue shape out so you should get something that's kind of smooth and now we know it should be decreasing so now the question is should it be decreasing but starting above, at or below the blue curve when you get to zero asset price in other words am I above below or at that point okay let's get votes so for option number one two and three how many ones do we get one, two, three, four, five, six, seven, eight okay we got about eight or so, eight or ten something like that, okay what about option number two any takers no twos, what about number three it's like about four or five okay come on somebody from the non-participating group what do you think the behavior should be one, two or three okay so somebody from the group that thinks it should be number one why, why do you think that anybody from who thinks it's number one what's the intuition, no one wants to provide any okay so if you're before maturity then there should be some time value in the option the fact that the optionality still exists in some sense right okay so potential argument now well since we got nobody saying number two I'll skip that what about number three, someone from the group who thinks that three is correct what's the intuition okay assuming it stays at zero forever okay okay kind of in the right direction sort of but not quite so you're so the correct answer is three okay one you'll see why time value is not sufficient to make it stay above okay but the reason why three is correct it's a very simple argument look at this let's delete let's delete those two is that line a bound for my put option is it an upper bound it's clearly an upper bound right so again five for the put is less than or equal to k no matter what the asset price is at maturity current value of k if I if I have something that gives me k dollars in the future what is it that I have in my hand now I have a bond right how much does that bond cost k e to the minus r t agree if I get the put payoff in the future what do I have in my hand the value of the put the value of the put has to be less than or equal to the discounted value of k well the discounted value of k is going to be somewhere like that and in fact the way that I've drawn it there is exactly this here okay so it has the put price has to be lie below that straight line so at zero it cannot go above it for sure so the time value that's in the put is there but it's only there if the underlying asset price is large enough if it's too small then it's actually more valuable to immediately exercise the put and I will talk more about that in a second what was being said before you could also think about it from the perspective that if the asset is zero dollars right now it will remain zero because that's how our model is designed as zero equals zero then all future asset price is zero and if it's zero now and I hold it to maturity then the underlying asset price is zero and the put would pay me k dollars so in fact if the asset is zero I get k at maturity and the current value of that is k discounted so I know as well that this point has to be exactly there it's not just an upper bound in fact it's exactly there so this is kind of what you're trying to get at this replicating idea but it's not really replicating it's something more simple it's just if the asset zero is always zero at maturity the put will pay what's the value of getting k in the future the value of a bond now that's it okay so these are our two bounds that's k, k e to the minus rt okay so if I have a put that has a longer maturity date then you're going to get decreased even more over here and you're going to get increased value over here so this will still automatically go down to zero maybe I should draw that in a different color okay this is still going down to zero and they don't cross at the same point I hope it doesn't look like that in the picture the crossover point there is not the same so these are two different discount times so this may be a two-year contract and this may be a one-year contract so for a put you have an odd behavior that you get increases of prices in this direction and decreases in prices in that direction as maturity increases for small enough the asset values it decreases for larger asset values it increases and then there's some sort of bound in there in between eventually time value of money is going to wipe everything and make them zero you can imagine if this exists out to infinite maturity dates it's going to eventually come down to zero because that purple line will shift all the way down so here I should have drawn that in here for a call it just always goes up okay any questions about this behavior? okay let's do a couple more before we go for our first break let's think about interest rates let's suppose so we're at a fixed maturity date maturity is out to some fixed time one year out we're looking at a call option with one interest rate level we have that let's say that's the 5% interest rate level now let's suppose we increase the interest rate what's going to happen with the call price it's going to decrease is it going to increase and how would you figure that out I'm writing stuff down to give you hints that's the call price it's equal to this expectation and we know it's explicit formula in terms of a box sold formula but again, R appears in too many places it's difficult to tell what it does but from this expression can you tell you take this and you multiply it through it's going to multiply through this term and it will multiply through that term agree? when it multiplies through the first term look there's a positive R there there's a positive R times T there so this will cancel that you agree? so the only place that R actually shows up after pulling it through is in that second term is in the K term oh yeah that's correct so what happens as I increase R as I increase R I reduce this K discounted and that's something that's being subtracted from my asset price so I'm subtracting a smaller number that means the whole thing is getting bigger so the asset price should increase or the option price should increase so if R goes up the underlying asset price should go up okay? but I'm still asymptotically I have to approach the well not the same line but the same form so this is S minus K E to the minus RT that's my asymptotic there and here it's also but I just have two different R's one is R, let's call that R2 let's call that one R1 and R2 is bigger than R1 so you can see from this bound this asymptotic bound that we talked about before you can get the same relationship that the price has to increase what about for put can you answer the question without me going through the detail now? for put let's say this is one interest rate level if I increase the interest rate what's going to happen decrease right? it's going to decrease and in fact it should lie below at every level if you think about the same basic the same formula there except the K term and the S term are interchanged right? you're you're getting a smaller number and you're subtracting the same thing from it in every event so you're going to get something that is smaller in every event R increasing and here now volatility is a harder one to actually think about from this expression if we ask what about as volatility increases if I look at that expression it's not at all obvious is it? because it all appears in two places there's a coefficient in front of Z but then there's also that sort of little convexity correction term the minus a half sigma squared term and the only way to actually figure it out is to explicitly compute the derivative with respect to volatility and I'm not going to do that here not today we may do it at another time tell you the answer the answer is that for both puts and calls as volatility increases the price increases one way to think of that is when there's more variability in the outcome the option has more value that's sort of a generic way to think of it when there's more variability in the outcome the option actually has more value so this is one level of volatility and then we'll have another level of volatility and this time the asymptotics of both of these have to be the same because the interest rate hasn't changed the interest rate hasn't changed so they both asymptotically approach the same line and this is sigma increasing and for the put you have the same as I said already as sigma increases for a put you find that the price also increases but what about with zero asset price initial asset price it's got to be the same right when s equals zero the price doesn't depend on vol no matter how volatile it is the initial price is zero sorry so this will have to lie strictly above in fact it's got to touch there and we've got to get an increase in price okay so I think we've done all of the parameters that I'd like to talk about right now sorry well time is not really a parameter is it? time is not a parameter time is actually the flow of time and we talked about that effectively when we address the issue of what happens as maturity increases you could view time flowing forward sorry can we have the chatter down I didn't release you as yet so if we have a fixed maturity and we just let time flow forward that's the same thing as maturity getting shorter in this sort of stationary model time flowing forward is identical to thinking of an option with a shorter maturity date so if you want to understand the variability of time you just look at this graph t increasing is big t increasing is the same thing as little t getting smaller for in other words if little t gets larger I would go the opposite way as little t approaches maturity I have to eventually hit the option payoff okay and in fact you can draw a nice little 3D graph as a function of time if you wanted to of actual time you could do something like this and say this is t here this is capital T there this is the underlying asset price and this is value and what you would end up with is a surface so you know at maturity you have to end up with that and let's actually draw it with a steeper line like that and it's going to increase as I move away from here and curve on the projection onto the onto this Vs axis has to be a curve has to be this increasing curve and I'm doing something like that so you'll get some sheet that looks like this and that shouldn't that goes more or less to zero you get a sheet that looks like that so this is like a nice smooth sheet that eventually becomes kinked at the end maybe I'll actually when you come back from the break I'll throw up a 3D graph for you okay so now we can take a break I'll show you some 3D plots of the option prices as a function of time and asset price so on this graph here on the left this is a call option and it's the call option price in this axis here down here this is the asset's price the underlying asset price on the vertical axis is the price of the option and time is flowing forward here and one is the maturity date of that option so prior to one you can see the price is positive and it's pretty smooth and as you approach maturity if you look at the contours here you can see that the contours if you look at the constant color this moves in closer and closer and closer to here so this is where the zero but the contour constant price moves in and it approaches and eventually becomes zero in this case and here we can see that the smoothness that you have in this payoff becomes a kink and the way that I coded this up in such a quick way you don't really get that maturity date point showing up because there's a little bit of a singularity there so I need to correct that by uh oh that's my put I've now corrected that little singularity so you can see it's not going up to exactly one point that's a little bit less than one but it's smoothing out and it becomes eventually that kinked behavior well for the put which is on this side you can see the put's always bounded above by a discounted value of 50 and if I rotate this let me see if I can for some reason the machine is being a little bit slow and it's not responding very well to uh there but from this perspective you can see what I wanted to show you look this is close to maturity you're getting the payoff almost exactly there's not quite a kink here yet but it will get kinked if we went closer and closer to maturity what I wanted to show you was this value at a zero asset price which is along this dotted line here and at zero asset price you can see the put price decreases as we move away from maturity as you approach maturity it goes toward it so that's exactly the behavior that we were discussing here in the second graph in this one here you can see that as we went away from maturity the initial price goes down and that's what we're seeing there but as we go away from maturity here the price in this region goes up so this is uh I think you can take this home as a good snapshot of how it behaves and that's where your time value is also being affected by discounting and this is why you get these things compensating with one another alright so what I wanted to do that took a bit longer than I was expecting but I don't want to skip over what I wanted to do next so I will cover it in any case and I may have to push one of the topics to next class that I imagined to do today so what I wanted to do now is look at other payoffs that are a little bit non-standard and see if we can try to guesstimate or get an idea of how those non-standard payoffs should behave and I'll pull up for you as well an Excel spreadsheet that will allow you to view how they do actually behave so let's see let's take a payoff that will start off with something relatively simple plus one and a slope of minus one and that's a strike of K now this type of payoff I believe it's called a str- I can never remember if it's strangle or straddle they're very similar related straddle straddle and you can build this of course out of a put struk at K and a call struk at K and so here's the question how should this behave as a function of asset price? some fixed maturity, it matures in a year from now well you know at zero the asset price has to get discounted you're going to get K dollars for sure if the asset is currently zero will this be stuck at zero and the put portion of this will end in the money you'll get exactly K dollars asymptotically out here you know that the put portion as I said if I'm asymptotically large the put portion of the payoff has a probability zero of occurring and so asymptotically it should behave just the same as a call so I know asymptotically out here I should get that and I know that as well for very small asset prices it's extremely unlikely that the asset will end in the money for the call option it's very unlikely that the asset will get large enough so that the call payoff takes an effect so initially at least it has to start decreasing just like it does for a put so there's got to be some sort of decreasing portion and then eventually it's got to turn around and start looking like a call so it's sort of a blend of these two things and exactly where exactly where it turns around oh damn it exactly where it turns around is not going to be too obvious to pick it out you'd have to actually do a numerical calculation so we should find a payoff or a price that looks like that and this point here is of course k e to the minus rt okay what about let's look at a strangle so a strangle looks very much like a straddle except it's got two strikes and how should those behave what should the price function look like for this more or less the same but it should be a little more flat in the middle right should be a little bit more flat it should behave more like kind of like the put and then start to turn around and behave like the call and asymptotically here it would be k2 and that point is k1 discounted okay let's try one more what about this guy this is called a beer spread how should that behave well if I'm very far away over here I know that the value should go down to 0 right asymptotically and since this is also bounded above by whatever level that is let's call that level L okay then I know that if the asset price is 0 this also has to be the discounted version of L initially and it's got to decrease the question is does it decrease does it do that this kind of behavior does it do that kind of behavior what should happen if you think about the expectation operators more or less smoothing the payoff that's what you should expect you more or less get a smoothing of this payoff function so it actually changes curvature it's not always convex convex and it's concave one region is convex the other region is concave so you end up with these kinds of behaviors now on the website I've uploaded a number of things for you by the way there's some old problem sets that these used to be required for handing in at one point but we've stopped that so you can take a look at some old problems as well as some old term test why am I pointing that out let's pull up a term test and I don't have a connection it lost hopefully it will reconnect in a second yeah this wire is not very it doesn't have a click okay well we'll wait until that connects the point is that the term test I always ask the question of this kind always ask you a sketching question from option payoff I'll give you a payoff and I'll ask you sketch it's behavior okay it looks like it's back up there we go this is your term test there's some true faults as before so here's a question that's asking about the option price as a function of the current spot level for maturities zero, one month and one year the exact times are irrelevant obviously you're not going to know what is the exact one month the point is I just simply want you to plot or sketch for three different maturities maturity, the actual maturity date it's one year out to maturity and it's one year out to maturity so that you tell me you can show me the behavior and it increases or decreases and in this case it's asking you for something called a digital call and it tells you what that is it pays one if S is bigger than K and zero otherwise and you're to draw these on the same graph okay so let's address that question so there's K so at maturity says it's one if you're bigger than K or zero otherwise so the payoff has to look like that if you're say one month from maturity it's going to be relatively close to the payoff right and if you're one year from maturity it should be even more smooth and the question is well okay it's more smooth does it asymptotically approach one this is one right according to the thing it said one or zero does it asymptotically approach one no what should it asymptotically approach discounted value of one right it's bounded above by one a bond bounds it so therefore it cannot be larger than the discounted version of one dollar so you've got to draw on another dotted line there and just say something like e to the minus r times one one year contract the next question is another sketch portfolio four long puts one long call both struck at one so it's telling me the strike is one four long puts one long call how does that look so that's going to be one what's this point you have four puts so that should be four right if it was one put it would be it would be one and one long call so the slope is one quarter that slope and then pretty much you just have to sketch it out so close to maturity again you'll be close to that and this is asymptotically going to approach s minus k e to the negative so one month one month is one twelve times r and then this is more or less the same thing as the straddle so of course when you're sketching these I urge you don't do it in pen do it in pencil and then once you've got your once you're confident that you like the shape of it then you could highlight it over and with a pen and indicate which is which there you go so pretty easy questions I would say but if you if you didn't try one on your own you would kind of fumble and there's a bunch of exercises in the problem set here's a bunch of them that you can try as well to help you out if you want to gain more intuition about the behavior there is a file which I don't see here okay I will upload it I think I might have deleted it by mistake I'll upload this for you it's an Excel file that you can play with that allows you to build portfolios out of whatever underlying assets you want so here you put in the risk-free rate we didn't talk about dividends and I'm not even going to bother and those of you who've seen it before you know all you do basically you just modify by a little bit of a discount factor right it's not a huge issue you put in the volatility you put in the maximum minimum spot price you want to see on this you put in the there's a you can put a stock a bond or a call or a put that's your basic building block your maturity date your strike how many of them you're going to hold and then you get plots of the price and a bunch of other things a bunch of other Greeks that we will talk about later on delta gamma vega row etc so you can also play with the maturity date here and you can see that this in fact it's one of the questions on the problem right when maturity is zero you get that payoff which was one of the questions on the problem and you can see what happens as you increase maturity smoothen things out exactly in the way that I told you you basically take your payoff and kind of smoothen it out but you have to think about whether it stays above or below a certain line right and that's where you these bounding arguments and if you increase volatility you can see what the value of an option is you can keep the scale all the same so increasing volatility in this case you can see does not always increase the value of an option unlike what you normally think for basic puts and calls it does but here's a clear example where I'm increasing vol and you can see the value increasing in one region but decreasing in another and similarly if we let's just pick a moderate vol here if we change interest rate increasing in one I can't here we go in this case actually in this case it's always decreasing because this is a bounded payoff so you're okay for interest rate behavior so I urge you to play with this and get comfortable with you know put in different types of portfolios that you'd like to play with and get a sense of how they behave with the parameters you can only build that intuition by playing with a sheet like this okay I'll post it up on the site alright I think that's about all I would like to say about these behaviors any questions nope alright so now I want to go back to a little bit of theory and then we'll do some example again so the theory that I want to talk about is going back to this key result that we found in our last lecture about the branching probabilities in this CRR model so we found that if you take if you calibrate to historical data you have these probabilities are approximately half plus some correction terms and under P and under Q you have two different probabilities showing up okay now what I want to do is I want to really convince you that when you generate sample paths the sample paths mean start the asset at a particular value flip a coin that has this probability of going up and then decide whether it went up or not but at the same time use sort of the same fundamental coin the same uniform random variable to generate whether you went up or down with the P probability and the Q probability and if you generate a sample paths in this way what you're going to see is there's some sort of strong relationship between the P paths and the Q paths and one of the relationships that we know has to come out is that the variance of those paths have to be the same that's one key result that we showed variance did not change under P and under Q you had the same variance but the drifts do change and I want to show you how that happens in this kind of continuous time setting so let's go here and I think this is my file that I'd like to open so let's generate let me actually step through this code with you so that you get a sense of what it is that I'm actually doing so we've got a few parameters set up here for our model we're going to do 10 years out we're going to take 100 steps these are the sizes of my steps this is the drift that we're going to put in for the P measure this is the volatility that's our risk that's our risk free rate and then I'm calling this function called simcrr so let's go into there what this does is first of all it calculates the P and the Q exactly as in the note okay and then what I'm going to do is I'm going to flip I'm going to generate a uniform random number between 0 and 1 now if that uniform random number suppose we wanted to generate an outcome from the P measure how would I generate that outcome I generate a uniform random number and then I simply check to see whether that uniform is smaller than P smaller than my uptake probability if it is then that event is an event that I would say happened that event happened I should go up now if the uniform is bigger than P then the event doesn't happen that means I go down so that's how I'm picking out whether I go up or down that gives me these little Bernoulli random variables here so I just check to see is that uniform less than P if it is that number is 0 1 if it is it's 1 and if it's not it's 0 so to generate something that is plus or minus 1 to tell me whether I go up or down I multiply by 2 and I subtract 1 I do the same thing for the Q path so it's using the same uniform number and this is because I want the same line probability space I just want the outcomes to be different or I should say I have the same outcome the same path but I want them to occur with their appropriate probability P probability and Q probability and then once I've got that all I do is you know sum up all of those all of those paths all of those X's for P and they're multiplied by sigma squared delta T so that's just the relationship or where's the previous slide there that's just using this A capital T is my initial value times the exponential of sigma squared delta T times the sum of the little X's so that's what this little piece of code is doing for you that's just doing that and it's going to sum it up it's going to take the first element add it to the next and then add it out and you're going to get a whole vector that's the sequential sum and then I also do that for the Q path that's all it is so something pretty simple so let's run that and we should have gotten there we go so here's an outcome from this tree so behind all of this there's a big CRR tree embedded in there there's a giant tree underlying there and these little up or down ticks are representing the choice that happened when it flipped the uniform coin so it drew the uniform random numbers 100 of them and then it decided for example over here they both the P and the Q probabilities are actually the P and the Q probabilities are fairly close to one another actually let's take a let's actually see where are they P and Q so P is about 0.52 and Q is whoops Q is about 0.49 let me put P back there Q is 0.49 say 0.5 and P is 0.52 so it's clear that it's more likely under the P probabilities that you're going to go up because the P probability is higher so it's more likely that you're going to go up and that's exactly what you see happening here this blue path these are the P measure paths okay and there are more times in which it goes up than the red path and you can see that they kind of match up at first but then there's a point in which the probability the uniform was generated in between 0.52 and 0.5 so one went up and the other one went down okay there's a 2% chance that you get one going up and the other one going down because again the difference between this and this is 2% so it's only if you generate something in between the two that you'll get the P path go up and the Q path go down if it's smaller than this they both go down if it's bigger than this they both go up so it's only 2% chances that you actually have a deviation in the path but but you can see even though there are some of these deviations that happen there the variability the general variability of the paths are very similar you see that they're very similar I mean if you were to look at one path versus the other you'd say yeah they have pretty much the same kind of variability and that is reflecting the fact that the volatility along the P paths and the volatility along the Q paths are identical now if we increase the number of points that we would like to generate here so let's do a thousand the keyboard there we go so now you see what happens of course these little structures that we saw before these very jagged edge structures become finer and finer and this limiting procedure in fact is what's going to generate a Brownian motion at the end of the day we haven't really introduced a Brownian motion yet but that will be what our underlying process behaves like the important point that I want you to see is that of course the general drift of the Q path is lower than the P path the P path drift is mu the Q path drift is R so this is not just a fictitious thing it's not just some sort of mathematical calculation that we did but actually if you generate paths using those probabilities you really do get the drift that the theory is telling you about now the other thing we can do is we can run this over so I can generate say thousands of these scenarios and then look at the end points of those scenarios take its logarithm and then compute the standard deviation and the mean of that or look at that distribution and ask about the mean and standard deviation and what should I find okay let's that's done in this piece of actually let's execute that and then here I do 5000 scenarios so they're done and which this is Q so on the left you have the P histogram like I said I've done this 5,000 times and we've picked off just the end point where the underlying asset actually ended up and you can clearly see that the P distribution has a larger mean than the Q distribution the mean of the Q distribution is somewhere around I don't know two or something like that this is actually not taking the logs yet the underlying asset price and I think if we look at this here it's going to tell us yeah so here are the the first order or the first and second order statistics that we have the P mean turned out to be about 0.8 the P standard deviation 0.63 the Q mean negative 0.1 and the Q standard deviation 0.63 how can I check if this is corresponding to my theory what should theory say is taking log of the terminal value and taking the mean and expected value taking the mean and the variance what should the P mean be theoretically what's the distribution of the underlying asset according to theory when we take the infinite limit log normal not mu star remember convex decorrection it's the exponential of a normal with mean mu star minus a half sigma star squared so if I took the log I'm going to end up with something that has a mean of mu minus a half sigma squared so let's just check that mu minus 0.5 times sigma to be squared times why is that taking forever to do oh I know why it's lost the connection and matlab is annoying in that it makes sure that you have the license from the license file so there we go 0.8 what was the P mean 0.798 so extremely close so we basically matched it and what about the Q mean what should that be again this is the Q mean of the logarithm of the underlying asset so it should be r minus a half sigma squared minus 0.1 and again that's pretty darn close simulation error if I did this 10 million times I would get exactly minus 0.1 what about the standard deviation well we know that it should be sigma squared times t 0.4 sorry square root of that because I think I have standard deviation reported so 0.6325 what would we get 0.63 0.63 so it actually does work out that's really just the point yeah this all does work out nicely and really to hammer home the fact that when you take a sample path from and you do get paths on a sample by sample basis that are drifted upwards relative to the Q path okay so the next topic that I'd like to cover here okay so far what we've been discussing are assets that just take on some value in a tree and so this is some giant tree and we've so far been viewing it as if okay I tell you what the value is at the end and then you can go ahead and figure out the values earlier on by doing discounted expectations we now to take continuous time limit so we know we can represent this in terms of an expectation of a random variable that's measurable or it's a random variable that we know it's distribution of and you can go ahead and use your standard calculus rules to compute that expectation so these are really all of European style contingent claims right the fact that the option can mature only at maturity so how do you deal with American style contingent claim that's what I'd like to discuss and I think we all more or less know what you do operationally right so what's the difference between an American style contingent claim versus European style what's the key difference American style is exercisable at any time right okay and there in fact may be various clauses in American style contingent claims you may have regions in which you're allowed to exercise and other regions in which you're not so one example classic example is an employee stock option so if you are you're an employee or you're some sort of manager in a firm often they will issue you a warrant for equity in the firm and you cannot exercise that warrant the same as a call option you cannot exercise that within the next two years but after that you can so there may be blackout period the standard American option is you can exercise at any time but in principle you can have blackout period so again this simply means periods in which you're not allowed to exercise so let's just go through the procedure very quickly because again I think you've all seen this before at some point or another some arbitrary point in the middle of your tree suppose we've already figured out everything else and we're there and we want to figure out how do we value this American option so we already know the value in those two states of the world and what we want is we want to know this value at that point well it's a very simple thing if you're sitting here at this point here then there's really two things in front of you two options in front of you either you hold the option and you just if you hold the option you then have the potential of receiving either Cu or Cd in one time step from now or you exercise the option those are my two options there's only two things you can do so therefore it makes rational sense to do whatever is better so the actual value so let's call this holding value H0 and this exercise value E0 so C0 is a maximum of hold or exercise alright that's all it is very simple you do whatever is better okay well what is the exercise value equal to so if we think of an American put let's be very specific and think of an American put option in that case this is just K minus S0 plus right if I immediately exercise I'll get the value of the put now if I hold the option what should the value be so if I'm holding onto the option I'm not exercising it that means that what I have in front of me are the two possibilities to get Cu or Cd in one time step from now so that claim that option of holding actually is itself an option right it's an option to receive Cu or Cd but we've already dealt with valuations of options of that kind that is in fact just a standard one period claim and so therefore H0 has to be the value the discounted expected value of C one time step from now so in other words it will be if we do this in a tree and then we probably want to take small steps eventually we'd use a discount factor like that and we'd have the Cu corresponding to that node in the CRR model we already know Cu is constant everywhere it's the same Cu in general models it won't be but in the CRR model it's just constant so we have this and then one minus Cu Cd and you're taking the maximum of those two things and then you choose and that's it then you're done because once you've figured out what C0 is every other part of the tree looks exactly the same if I think of building this say a two step tree for example and I've already gotten the value here by taking that maximum I use the same procedure I get the value there the exact same procedure I decide on which is better hold their exercise whatever is better I compute both of them I get a value and then I finally get my value here as well and I can just repeat this through the entire tree operationally it's trivial but even though operationally it's trivial there's an important concept embedded in there and that is when there are going to be some nodes in which this is optimal and some nodes in which this is optimal and if you label if you go through your tree and let's I'm going to just draw a whole bunch of nodes here I'm going to try to do it so suppose we went through this tree and what I'm going to do is I'm going to color wherever it's optimal to I'm going to color it blue and wherever it was optimal to exercise I'm going to color it red okay well if you're an American putt you know that it will be optimal to exercise at maturity whenever you are below K K or below so there's going to be some set of nodes here below which they'll be red and above which they will be blue so if you go out you can show that the pattern that you will receive that you will get by doing this always looks like that the red where you exercise always lie below the blue where you exercise it's lying below the blue and there's this obviously creates sort of two regions in space this sort of region here that's separated from the other region and that region this is called your exercise region and the other region is a continuation region and when we get to continuous time settings you'll see how this changes into partial differential equations with three boundaries there's a point where these two regions sort of meet this layer here that layer is the optimal exercise so why is that thing called an optimal exercise curve how does it operationally work well if you're following an asset price dynamic so you're starting here you're going to go through this tree the very first time that you hit that optimal exercise curve it's actually best to exercise then you will never never get to this node in an optimal way without exercising in other words this is an upper bound of that entire region so once you hit it you make sure you exercise now the way I've drawn it there is not how it typically looks in these many steps you'll get much more reasonable behavior so you'll find that for put options the optimal exercise curves trace out something that looks like that so this is s this is time that's maturity that's k the yellow region this exercise region is somewhere below a nice smooth curve and the continuation region is up there okay that's what you generally find now you can it's easy to do these computations just one step at a time and so on but I want you to see how this evolves when you take multiple steps so let's see what I'd like to do is pull up a matlab file for you but maybe before I do that is this concept clear how people have done this before who has not seen this before would you like to see an example yes so the example is best done in excel rather than me writing down numbers and then I'll send you the sheet or I'll post the sheet so we're going to have the underlying asset price, the risk-free rate the volatility and the step size so let's say this is 100 the risk-free rate is say 2% the volatility is 20% let's take steps of 1 month 1 over 12 so the way I will do this this is this is f0 this is my risk-free rate this is my volatility this is my dt so to do that I also need to know my up and down ticks so this is the exponential of sigma times square root of dt that's my up and down up and down value so this is f0 here that's time 0 step 1, step 2, step 3, step 4 step 5, step 6 let's do 6 steps so it's a 6 month contract so rather than by the way, rather than looking at the tree this way up down rotate the tree 45 degrees rotated 45 degrees that way going across really means going up and going this way means going down okay so this is just the same tree rotated 45 degrees so that means when I go across here it equals this times u sorry, times the up value that's my outcomes if I went up, up, up and over here I'm going down so it's my previous value divided by the up value because it's either the negative sigma square root of dt can you see this or is this too small can you see it, it's fine speak up otherwise I don't know so we can just copy this across and did I get them all 1, 2, 3, 4, 5, 6, 7 good, yeah that's correct I need 7 states by the end of a 6 step tree every step I have one more so I've got them all so you can see if I went up and then that's down this would be up and then going there would be a down step and I got back to 100 I go up, I go back down that's the 100 if I go down, down I get to there sorry, that's wrong down, down start there, go down go down, go up and go back up again I went down twice, up twice, I got back to 100 so the tree is correct so this is my asset price tree and then we'll have the value of the option tree option value tree so let's put a strike, let's say 100 so what do I do? we go out here to the last step of the tree and the option payoff for an American option is the same as the European counterpart that's the only day on which you can exercise so you decide whether you do so it's the maximum of the strike, which I didn't give a name here maximum strike minus the corresponding value in that tree the maximum of that and zero they get all 1, 2, 3, 4, 5, 6, 7 okay, there we go, so that's my payoff and now all I do is I use the discounted expectation stuff well, it might as well compute Q and keep it over here because we use it everywhere, right? the Q probability so what is it? it's the exponential of RF times DT minus the down tick divided by the up tick minus 1 over the down tick you can just go ahead and calculate and check that that's the case and I called it up, not U okay, so it's almost exactly 5 by coincidence here so perhaps if I change this volatility I guess, yeah, there we go get something a little bit bigger maybe 0.3, that's 2.3 okay, so what value goes here? well in principle, I mean I can just do the calculation I can say discounted expectation the whole value what's the whole value? discounted expectation, that's my whole value I have two outcomes of 0 so discounted expected value is 0, clearly and what's the exercise value? it's also 0, right? it's out of the money at this node that node corresponds to that node, right? so it's actually 0 it's out of the money so both cases are 0 whenever you have an equality you usually just choose the whole value you just don't exercise the option you just hold it so, in formulaically, we can do it like this we can take the maximum of discounted value of, oh, I need Q okay, I'll introduce a variable called Q up in a second times pay off there plus 1 minus Q up times the payoff going down close bracket that versus maximum strike minus the corresponding spot value okay, and yes, it's giving me an error because I didn't call this thing anything Q up okay okay, so this is this is just implementing exactly this that formula there, maximum of H0 and E0 whole value versus exercise value and all I have to do is copy this formula over okay and that tells me the value of the option so using this method I can't really tell which one where the exercise points were, can I? because it doesn't store that for me but I can create another cell here that does that I can say equals if, for example this is bigger sorry, if it's smaller than the immediate exercise value which is maximum of strike minus the spot then let's just put an exclamation mark or let's put a number sign, actually then it'll put that, otherwise it'll put nothing I hope that works one, two, three, four, five, six, seven it's not giving me anything at maturity because I put strict equality but that's strange okay, I think I have the formulas copied in all of the cells that need to be but these little number signs are showing me where I should be exercising okay, so it's saying I should exercise the node corresponding to this one and not this node here I could replace it with putting another symbol there so the exercise boundary in this case is nice and simple in terms of this tree, it's just a straight line here as soon as I hit this cell I would exercise and if you wanted to get fancy you can make it color these cells to tell you when to exercise so according to this it's using those cells as the exercise points if the asset ever hits that okay, so that's a simple example I'll save that for you later and let's take a look at how this actually plays out if we do this in multiple in multiple steps okay, plot time, comma boundary okay, so this is a setup to do 100 steps and now you're starting to see that shape that I was drawing for you, right? remember I was claiming that once you do this in continuous time you're going to get this thing that eventually becomes a nice downward sloping curve and if we increase the number of points even more I actually don't remember what parameter represents the number of points here let me see, it's the last one okay, good so this is a thousand steps and you can see it's getting finer and finer and we could actually just stop there and just ask some questions is that behavior reasonable? so does it make sense, first of all that that optimal exercise boundary which splits the exercise region from the no exercise region is it reasonable that it approaches 100? it is because at maturity you know you should exercise at 100 now it turns out that there's a very funny thing with call options call options, it turns out when there are no dividends it's never optimal to exercise early but if there is a dividend it also turns out that the call option the shape for call options are kind of reflected from the put so call optimal exercise boundary kind of looks like this looks like that but it doesn't always hit K for a call option you may in fact find situations where the boundary doesn't go down below K stays above it and this depends on the number of the dividends so this is for call this is for put and only with dividends I don't think I want to get into the details as to why exactly that happened and how there's a little bit of analysis necessary for you to show that but that's the behavior what is kind of interesting and what we can what we can try to assess right now is how should for the put option let's just stick with the put option how should this boundary behave as we change some of the parameters so suppose you were to increase volatility what do you think should happen to this optimal exercise boundary remember the idea of the optimal exercise boundary if the asset price is moving around let's copy this figure and paste it over here oh where'd it go okay so if the asset price is moving the instant that you touch it that's when you should exercise if that was the underlying asset price dynamic if it actually evolved that way that's the optimal exercise point this is why it's called an optimal exercise curve so how should that curve behave increase volatility what's your intuition more convex and what do you mean by more convex so it goes like this like this so you think it's here okay so why what's the intuition for that there is a good intuition for it volatility increases my paths become much wilder right so the actual probability of exercise doesn't change much with the lower path if I kept the same exercise curve suppose I kept that exercise curve but volatility increases then it's more likely I'm going to hit that original exercise curve because my paths are more volatile and that turns out to be sub-optimal to do it's more optimal to wait longer and have the opportunity that the asset drops deeper in the money because that is a likely path the asset price dropping is actually a likely path when vol is high if vol is low the asset value dropping is not a likely path so my exercise region would be much tighter does that make sense so the curves generally do this as vol increases and let's see if we can let's see if we can illustrate that by just running that code again with a different volatility let's increase volatility let's go to 0.3 and what's my range before this range was down to 70 so well you can immediately see that it's clearly going lower sorry I want to put it side by side oh it doesn't want to do that maybe I can do this there you can see that this is dropping below 65 and this state above 75 so it's clearly dropping below it does that so that's one parameter now what about interest rates what do you think should happen for interest rates so let's go back to 0.2 volatility but let's decrease interest rates what's going to happen this one's a little surprising it also went down in fact it's going down even more if interest rates are 0 it's in fact never optimal to exercise this is all just computational noise this is just computational noise so you get a similar behavior with r decreasing the curves move down with a decreasing r and you can argue why that's the case or at least argue why it's sub-optimal when there's zero interest rates by using Jensen's inequality so do you all know Jensen's inequality you think you can prove this with Jensen's inequality okay here's what I'm going to do I'm going to give you Jensen's inequality and then that's going to be your little quiz for today