We're now going to see how we can use the volatility surface to see how we can price different types of derivative securities. Obviously, we can use the volatility surface to price European call and put options. After all, we actually constructed the volatility surface using European call and put option prices. But we will see that there are other derivative securities that can also be priced. These are derivative securities whose value only depends On the marginal risk-neutral distributions of the stock price. So, we're going to see some examples. In particular we will see how to price a digital option. And we will also see how to price a so-called range accrual using the information in the implied volatility surface. Suppose we wish to price a digital option, which pays $1 if the time t stock price, ST, is greater than K, and 0 otherwise. [INAUDIBLE] . We actually know that we can price the security given the implied volatility surface. Now why is that? Well, the reason for that, as we saw in one of the more recent modules, that if you know the implied volatility surface, then you know the marginal risk-neutral distribution of The security processor on a fixed time 'T', so the pay off of this option is going to be equal to the maximum of '0' and the indicative function '1' which pays off '1' is 'ST' is greater than 'K'. So this is the pay off of the digital option. So the risk neutral distribution of this option only depends on the marginal risk neutral distribution of st. And so we know from the volatility surface that we can calculate this marginal risk neutral distribution. And therefore, we can evaluate. This quantity here. Which is the initial value of this digital option. So lets see how we can actually go ahead and price this. It is easy to see. That the digital price. Were going to call it dkt. K is the strike. T is the maturity. Is given by the following. So dkt is equal to, well we have this limit as delta k goes to 0, of the s, the market price of a call option with strike k maturity t, minus the market price for a call option with strike k plus delta k and maturity t, all divided by delta k. Now why is that? Well, it's easy to see this if we draw a picture and that's what we'll do. So we will draw a picture. This is going to be our payoff of this strategy, so buying 1 over delta K times and option of strike K maturity T and selling 1 over delta K times the call option strike K times delat K and maturity T T. So this is represents S T, the stock price at maturity. We have this value here, whchi is K. And we have the value here K plus delta K. Now it is easy to check. That this option, this strategy here, of going long option and shortest option gives a payoff of 0 up as far as K, and then it grows linearly up to a value of 1. At k plus delta k. And thereafter, gives a constant value one. And you can see this easily. So you see that if the terminal stock price is greater than k plus delta k. Then you're going to make a profit of delta k from these two positions. Divide that by the delta k here, and you get a profit of 1. So, this is the payoff. And so it should be clear that, as we let delta k go to zero. Then this line here is becoming more and more vertical. And we're getting closer and closer to the payoff of a digital option which pays 1 dollar, only if st is greater than k. And so that's why we get this limiting argument here. So dkt is equal to the limit as delta k goes to 0 with this. We're just going to multiply through by minus 1, take the minus outside here and we get it's the limit is delta k goes to 0 of this quantity here. And we just recognize this as being the partial derivative of the market price of a call option with strike k maturity t with respect to the strike k. So it's very straightforward to see that the price of the digital option is given to us by this. Now recall how we defined the implied volatility of an option. What we did is, we equate the market price of the option with the Black Scholes price. And we figure out what is the implied volatility parameter, sigma of k and t. Which makes the Black-Scholes price match the market price. Just to simplify notation, I haven't bothered to include the other parameters that I often include here, R C and S 0. So we know that sigma K, T is the volatility parameter that must go into the Black-Scholes formula so that we get a price on the right-hand side that is equal to the market price of the option. As I said before, sometimes this is likened to Plugging the wrong number into the wrong formula to get the right price. So now, if you recall from the previous slide. What we need to do is, we need to compute this partial derivative here with respect to k. Well, we know that the partial derivative of this with respect to k is the partial derivative of the right hand side with respect to k. And there are actually two terms that come into this. We see, the first argument. K appears here. But also in the sigma argument. K also appears in there. So were actually going to have two terms corresponding to, to, to k here. So were going to get that the partial derivative, with respect of k. Is equal to the partial derivative of the Black Scholes formula with respect to the strike. Plus the Black-Scholes formula with respect to Sigma. Well, that's our vega times delta sigma, delta K. And that's actually what we will call the skew. Of course, we have a minus, because we have in both terms. Because we had a minus outside here as well. So, we're going to get D cave T is equal to minus delta CBS, delta K, minus the vega. Times the skew. And so the skew we're just going to refer to this as being delta sigma delta k. And remember that for a fixed time maturity. Maturity in the Acuity market we'll see a skew like this. So this will be s t or if you like k. Either one. And this is sigma kt. So, we can actually calculate delta cbs, delta k, and the vega from the Black Scholes formula. These are straightforward to compute, because we know the Black Scholes formula. And there, we can compute these derivatives. The skew can be estimated from the implied volatility surface. So we will have calculated our, our volatility surface, and we'll be able to estimate the sku. In other words, we'll be able to compute what delta sigma, delta k is. For example, suppose this is the strike here. Well, this, therefore, has. A value of, let's call it Sigma K. Maybe you go up to K plus Delta K. This has a value, Sigma K plus Delta K, and so we can estimate the partial derivative, the skew, or Delta Sigma Delta K... As being approximately equal to sigma of k plus delta k minus sigma k divided by k plus delta k minus k, which is delta k. And of course as I let delta k go to zero, this approximation becomes better and better. So we can approximate. Delta sigma delta k, or if you like the skew we're calling it, from the implied volatility surface that we will have available to us. So this is an example of how the Black-Scholes terminology, or technology, is used in practice. Even though the Black Scholes model is known to be wrong. We can still compute option prices with, with the, with the Black Scholes terminology. In this case, we're u-, computing option prices from the implied volatility surface. The implied volatility surface, if you recall, has been set up, so that, by construction. Call and put options will match the prices of call and put options in the marketplace. And we're going to be able to use this volatility surface to compute other types of options as well. And in this case, we're going to compute the price of a digital option. As I also mentioned before, we can use the volatility surface to price any security Who's payoff only depends on the stock price at a given fix time tee. That is because we know the marginal retribution distribution once we know[UNKNOWN] and we know the marginal retribution then we can compute the derivative who's payoff only depends on the stock price and a fix time T So here's an example. This example is taken from the book, the Volatility Surface, by Jim Gatheral. It's an advanced text. So I wouldn't necessarily advise anyone to go out and look at it. It's more of a doctoral text on financial mathematics and financial engineering. But there's a nice example in that text that we'll go through here. So what we're going to do, is, we're going to, we're going to be pricing a digital option. The digital option is, has a strike of k equals 100. The current stock price is 100. So the digital is at the money. Remember, just to be clear, the payoff of this digital time t is equal to. The maximum of zero. And the indicator function of st being greater than or equal to k. And the fact if you stop for a second, you can see you don't really need the maximum here because this is simply the indicator function of st being greater than or equal to k. So, if you recall the vega from the Black-Scholes formula, well it is as follows. We know that vega, I'll write it here. We know that vega is equal to e. To the minus c T times S square root capital T times five of D one. Where D one was equal to the, the log of S zero divided by K. Plus R minus C plus sigma squared over two times T divide, all divided by sigma square root T. Now, in this situation, in this example, we're going to assume r equals c equals zero. T equals 1 year. And s zero equals k. So in that case, d is equal to, it turns out to be simply sigma over 2. And it also implies that the vega, in this case. Is equal to. While c is zero. S is 100. T is one. So it's five of sigma over two. So what were actually going to do. Is were going to get this as equal to. S0 times 5 of sigma over 2. So now we can go to the To the task at hand, which is to compute the price of this digital option. We know this price is given to us by this quantity here. So we can actually calculate delta cbs, delta k from the Black Scholes formula. You can check. But it actually turns out to be this quantity here. The vega is given to us by s 0 phi of sigma atm over 2. Sigma atm is the octamum implied volatility. And we're told it's 25 percent. Finally, how about the skew? Well, we are told that the skew is 2.5 percent per 10 percent change in strike. So a 10 percent change in strike is equal to 10 percent of s zero. Because the strike is equal to s zero. And it's 2.5% per 10% change. So it's going to be 0.025 divided by .1 s0. This is the skew. However, we also have a minus sign here. Now the minus sign is, we're not explicitly told there's a minus sign here. But we know that there must be, because we know the equity markets we see askew like this. So clearly as, so this is K. Clearly as K increases, the implied volatility falls. So this number here, which if you recall, is delta sigma delta K, that's going to be negative. And so I implicitly understand that the skew here presents the negative 2.5% per 10% change in strike, and so that's how I get this quantity here. The S zero accounts with the S zero her. I can evaluate this quantity using, I can do it simply in Excel, and I get a digital price of 0.55... What's interesting is if we ignored the skew component. In other words, if I just took the market price, see market price of the call option. To be equal to just the black[UNKNOWN] price, as a function of KT and sigma at the money. And ignore the fact that sigma is also a funcion of K as we see in this skew here. If I ignore that, then I would only get a delta cbs delta k term appearing. Remember when we take partial derivatives here with respect to k, we get a term from the k argument. But we also get a term from the implied volatility argument. Because the implied volatility's a function of k. And that's why I get this second term here. But if I ignored the second term, pretend that signal was a constant, that would be the case of the Black-Scholes model held, then I would only get the 0.45 value here. And, so in fact, by taking the skew into account correctly, I see the price of the digital option as 0.55 and not 0.45, and actually this is significant. This represents ten cents extra. On .54 dollars. So consider now a 3 month range accrual on the S & P 500 index with range of 1,500 to 1,550. After 3 months the product pays X% of notional where x equals the percentage of days over the 3 months that the index is inside the range. So, for example, the notional is $10M and the index is inside the range 70% of the time; then the payoff will be $7M. The question is, is it possible to calculate the price of this range accrual using the volatility surface? The answer is, yes. Consider a portfolio consisting of a pair of digital's For each date between now and the expiration. So, actually lets expand on this answer and see how indeed we can use the volatility surface to price this range accrual. So lets assume that there are N Trading days in the three month period in question. Then the payoff at time t Let's write it here. The payoff at time T, that's called a P subscript T, will be equal to 10,000,000 times the summation from I equals 1 to N of the indicator function. That the underlying security price, we'll call it si, is inside the range in question and this range is 1,500 less than or equal to si, less than or equal to 1,550. We have to divide by n because its the percentage of days that were inside the range. And so we must divide. So this summation is the total number of days that were inside the range that the security price, the underlined security price, in this case the S&P 500, is inside the range. So the percentage of days that were inside the range is the summation divided by M. So this is the payoff at time, capital T. So how might we price this security? Well, we know, let's write it here. From Risk-neutral pricing that the initial value of the security is equal to the expected value under the Risk- neutral probability distribution of E to the minus R capital T. Capital T is assumed to be the maturities of To three months. Of times pt. So let's expand on that. This is equal to 10 divided by n. So I'm going to omit the m, for millionths. So now my units are in millionths. So it's 10 over n. Times the summation from i equals 1 to n of e to the minus r times t minus ti. Times the expected value of e to the minus, or ti, times this quantity here. Now, what can we do with this. We'll notice by the way, that I have a minus minus TI which is a plus TI and that counts as with a minus TI inside here. So let's look at this expression here. So we have the expected value of e to the minus rti, times the indicator function. That 1,500 is less than or equal to si. Is less than or equal to 1,550. Where S I is the underlining security price on day I. Well I can write this as the following, this is equal to, E to the minus R T I TImes the indicator function of si being greater than or equal to 1,500 minus the indicator function of si being greater than or equal to 1,550. So it's quite straightforward to see that this indicator function here. One on the event s size between 1,500 and 1,550 is equal to the difference of these 2 indicator functions here. And if you think about it, what you will see is that this is equal to. Well, using our earlier notation for the price of the digital option. This is equal to D 1500 on Date TI minus D 1550 on Date TI. And so actually what we've managed to do is we've managed to break down the range accrual into a strip of digital pairs... A different pair for each[UNKNOWN] TI, and so therefore the price of the rang accrual is equal to 10 million dollars divided by N, the number of days, times the summation... From i equals 1 to n of e to the minus r times capital T maturity minus t little i. Times the digital option. Would strike 1500 maturity ti minus the digital option would strike 1550 in maturity ti. And indeed, we can compute these digital option prices from the implied volatility surface as we saw a short while ago.
