In the last couple of modules we introduced the volatility surface. We saw how to construct the volatility surface, and we also discussed the skew and why we might see a skew in practice. In this module we're going to discuss what the volatility surface tells us. We will see that the volatitlity surface gives us the marginal risk neutral distributions of the stock price. It does not tell us anything about the joint risk neutral distributions of the stock price at various times. So that is the key part of the volatility surface. It is very important to appreciate it. It only tells us the marginal risk neutral distributions of the stock price at a given fixed time. It tells us nothing about the joint risk mutual distributions. And we will emphasize that in this module and indeed in later modules. So recall again this is an example of implied volatility surface and just remind yourselves again to make sure we don't forget. It is constructed as follows, we see a set of strike expiration pairs in the market place. So we have k1 t1 up to k n t n. We see the option prices in the marketplace for all of these. So, we actually see the caller put price, let's say call price c subscript mkt for market of kiti, and that's true for i equals 1 to n. So, we see these prices in the marketplace, and what we do is, we set these prices equal to the Black-Scholes price with S, r, K, T, C, and sigma of kt. So we set this market price equal to the Black Shoal's price. We know the left hand inside, we know the Black Shoal's formula, we know s r kt. We can estimate C, and so there's just one unknown in this equation, and we can actually back out of this unknown for sigma k i t i. And that will give us the implied volatility at the strike k i and expiration t i. So that will give us a number of points on our surface here. And then we fill in the rest of the surface using some sort of interpolation or extrapolation procedure. I didn't really discuss how we would do this interpretation or extrapolation, but one has to be careful when doing it. So we're going to continue to assume that the volatility surface has been constructed from European option prices. We're going to discuss now what the volatility surface tells us and what we can use it for. Certainly we can use it for risk management purposes. I've mentioned that already. We can do scenario analysis. We can actually stress the volatility surface by moving it up or down, or moving parts of the volatility surface up or down. Recomputing the value of a portfolio, computing the pnl, and so on. So, the volatility surface is certainly used for risk management purposes. What we're going to discuss in the next couple of slides, is what can it tell us in terms of being able to price derivative securities beyond call and put options. So, to answer this question, let's first of all consider, a butterfly strategy. Now a butterfly strategy centered at k, does the following. It buys a call option with strike k minus Delta K. It buys a call option with strike k plus Delta K. And then it sells two call options with strike k. The value of the butterfly, B0, we'll say, at time t equals 0, is therefore given to us by this expression here. Where C is the call option price at the strike K minus Delta K, and maturity T, and so on. And in fact, in practice what we'll be doing is, doing this, using the market prices, so if you like you can assume that these are market prices. MKT, being shorthand for market. let's also see what the payoff of this butterfly strategy is at maturity. So maturity is capital T. Let's get an idea of what this looks like at maturity. So, let's draw a plot. So we will call this the payoff, and we will call it B capital T, for the payoff of the butterfly strategy at maturity. And along the x axis we will have the underlying security price, which is st, at maturity capital T. And, let's, mark off k. k minus Delta K. And k, plus Delta K. Well, it's pretty straight forward, to see, that this strategy earns nothing, if the stock price at time capital T is less than k minus Delta K. It also earns nothing if the stock price at time capital t is greater than k plus k Delta K. Moreover, is easy to see that the maximum payoff of this strategy is equal to k and it occurs if the stock price itself at maturity is equal to k. And it grows linearly for values of stp low k and then it decreases down to zero, at k plus Delta K. So, in fact this is the payoff of the butterfly strategy at maturity as a function of st. Now, a couple of things to keep in mind, the maximum payoff is k. And, if you like, if you're inside this interval where you do get a payoff at time capital t, the average payoff will be k over 2. So the average payoff, if you're paid off, will be k over 2. Alright another thing to keep in mind, we know from risk mutual pricing that the fair value of this payoff. This is a payoff at maturity. So the fair value of this payoff at maturity is the current value of the butterfly today which is B0. We know B0 is equal to the right inside of 6. But from risk mutual pricing this is also equal to the expected value of time 0. Using risk mutual probabilities e to the minus r times t times the payoff. And the payoff we will call B capital T. Now I know I've used b in the past to refer to the cash account. Here it's referring to the butterfly payoff here, and this is our butterfly payoff. So keeping this in mind we're going to get an alternative expression for b0. We have one expression for B0 here in equation 6. On the next line we're going to get an alternative expression for B0 using this representation here. So what we can say is that B0 is equal to e to minus rt or rather is. It is approximately equal to e to the minus rt. Times the risk neutral probability of st being between k minus Delta K, and k plus Delta K times Delta K over 2. Now, where does that came from? Well, if you think about it, it comes from this idea here. So the payoff occurs if st is in k minus Delta K. Up to k plus Delta K. So the risk neutral probability of that is q of k minus Delta K being less than or equal to st, being less than or equal k plus Delta K. So, that's the probability that s t is inside this interval here. Now, were imagining Delta K being small, by the way. In fact, soon we're going to let Delta K go to zero. So we can imagine Delta K is very small. So, this is the probability, the risk neutral probability that st is inside this interval here. We already explained that if s t is in this interval then the pay off you expect to get is k over 2. And indeed that is why we multiply by the k over 2 here. So we have our e to minus r t term, the probability that st is inside this interval, times the average payoff in this interval. And so that's how we get this first line here. It's an approximation, but it is a very good approximation for small Delta K. Now, if you recall something about density functions, then you will understand why we're letting q. The risk neutral probability that st is in this interval is equal to the risk neutral density times the width of the interval. So we're saying the risk neutral probability that st is between k minus Delta K and k plus Delta k. That is approximately equal to the density, risk neutral density evaluated at k times the width of the interval to Delta K. And that just follows from a property of PDFs. We actually explain this in one of the additional modules on, on probability that we also recorded, they're also available on the, on the plat, course platform. So, remember, if you've got a PDF, in general. So if this is our PDF, f of x. And suppose we want to compute the interval that the random variable x is inside x0 to x 0 plus Delta X. Well, the density satisfies that the probability, that the random variable x is in, x0 plus Delta x where Delta x small. That's approximately equal to f of x 0 times the width of the interval which is Delta X. This is a standard property of probability density function and that's all we're using here. So we're saying the portability that st is inside this interval here is equal to the density which is ft, times the width of the interval which is 2 Delta K. So therefore we're going to get B0 is approximately equal e to the minus rt. Times ft of k, times Delta K squared. We have a two here, but that counts as with a two there, and we get a Delta K times Delta K, which is delta k squared. So what we've done now is we've come up with 2 expressions for the value of the butterfly strategy. We have this expression here in equation 6 which is exact. I'm here with this expression here which is an approximation. But as Delta K goes to zero, this approximation also becomes exact. So, what we're going to do is, we're going to equate equation 7 with equation 6 and then solve for ft of k. Or in other words, bring ft of k over to the left hand side. So we will see ft of k, is approximately equal to e to the rt, times this expression on the right hand side of 6 here, divided by Delta K squared. If we now let Delta K go to 0 in 8. Well, if you recall your, your calculus, you'll see that all you're doing when you do this is actually computing the second partial derivative of the call option price with respect to the strike. And so what we're seeing is that by constructing a butterfly strategy where Delta K goes to 0. We're actually able to come up with a risk neutral probability density function for st, f t is equal to e to the r t, Delta 2 c, Delta k squared. And so. The volatility surface gives us the marginal risk-neutral distribution of the stock price, st, for any fixed time, t. So this is a really interesting observation. We see option prices for finite number of strikes and maturities. We compute those implied volatilities. We then actually fit the volatility surface to those finite number of points. I mentioned earlier that we need to fit the volatility surface very carefully. And the reason is if we want to be able to compute something like f t of k, then we're computing partial derivatives and second partial derivatives. So we need to make sure we do things, we fit things very smoothly. This means that, given the implied volatility surface, sigma kt. We can compute the price p0 of any derivative security whose payoff f only depends on the underlying stock price of the single and fixed time capital T. Now, maybe I've chosen f. Unfortunately here because I used f subscript t. F subscript t to denote the risk neutral PDF of st. Here, f, this f here has got nothing to do with this. This is the risk neutral PDF of st. F, here, is just some arbitrary function representing the payoff of some derivative security. So just to be clear, the f that I'm using here has nothing to do with the f subscript t on the previous slide. Which was the risk neutral probability density function of st. Why is it we can compute this quantity here? Well, if I know the risk neutral density of st, I can just perform and integration against that risk neutral density to compute this quantity here. So therefore I can compute the price of any derivative security if the payoff of that derivative security only depends on the stock price at a single and fixed time t. And that's because I will know f subscript t, the risk neutral density for that stock price. And therefore, I can evaluate this expectation on the right hand side. However, knowing the volatility surface tells us nothing about the joint distribution of the stock price at multiple times t1 up to tn. And this is not surprising since the volatility surface is constructed from European option prices. And European option prices only depend on the marginal distributions of st. Just to be clear, the joint distribution of the stock price at multiple times t1, tn, what I'm referring to there is the following. It would be this distribution, t1, up to tn, of s, t1, up to stn. So this is the risk neutral joint distribution of the stock price at times t1 up to tn. And what we're saying here is that we don't know anything about this joint distribution. The only thing that we can learn from the volatility surface is the marginal distribution for each individual time t1 up to tn. Here's an example. Suppose we wish to compute the price of a knockout put option with time t payoff given to us here. So, it's this piece here is like a regular put option. So this is a like a regular European put option, the maximum of k minus s, t, and 0. However we only get that payoff if the minimum stock price over the interval of 0 to capital T is greater than or equal to B. So B here represents a barrier. If the stock price ever falls below B, then this indicator function here becomes 0, and so we get nothing. Remember the indicator function, this indicator function, can take on two possible values. It takes on the value 1, or 0. It takes on the value 1, if the minimum of st is greater than or equal to B. And that's the minimum over 0 less than or equal to little t, less than or equal to capital T and it takes on the value 0 otherwise. So this is an example of a knockout put option. And the point I'm trying to make here is that, we cannot compute the process of this option just using the implied volatility surface. The implied volatility surface will only give us the marginal distribution, marginal risk neutral distribution of the stock price. It doesn't give us the joint distribution. And in order to evaluate this, I would need to compute the following. So if the value of the security is p0. It will be equal to the expected value using risk-neutral probabilities e to the minus or t times the maximum of k minus st and 0 times the indicator function of st being greater than or equal to B. And the point I'm trying to make is, in order to compute this expectation. I would need to know the joint risk neutral distribution for the stock price at all times between 0 and capital T. But I don't know that. I can not compute that risk neutral distribution from the, from the implied volatility surface. And so in practice and we'll come back to this soon. We would need to use some sort of model, some arbitrage free model to estimate the price of this quantity.