We're now going to see how we can use the volatility surface to price certain 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 one dollar if the time T stock price ST is greater than K and zero otherwise. We actually know that we can price this 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 price at any fixed time T. So the payoff of this option is going to be equal to the maximum of zero and the indicator function one which pays off one if ST is greater than K. So this is the payoff of a digital option. So the risk-neutral distribution of this option only depends on the marginal risk-neutral distribution of ST. So we know from the volatility surface that we can calculate this marginal risk-neutral distribution. Therefore we can evaluate this quantity here which is the initial value of this digital option. So let's see how we can actually go ahead and price this. It is easy to see that the digital price we're going to call it DKT, K is the strike, T the maturity, is given by the following. So DKT is equal to, well, we have the limit as Delta K goes to zero of the market price of a call option with strike K and 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 an option with strike K and maturity T, and selling 1 over Delta K times a call option with strike K plus Delta K and maturity T. So this represents ST, the stock price at maturity. We have this value here which is K, and we have the value here K plus Delta K. Now, it is easy to check that this strategy here of going along this option ensures this option gives a payoff of zero up as far as K. Then, it grows linearly up to a value of one at K plus Delta K and thereafter gives a constant value one. You can see this easily. So you see that at 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 got a profit of one. 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 one dollar only if ST is greater than K. So that's why we get this limiting argument here. So DKT is equal to the limit as Delta K goes to zero of this. We're just going to multiply through by minus 1, take the minus outside here and we get it's the limit as Delta K goes to zero of this quantity here. We just recognize this as being the partial derivative of the market price of a call option with strike K and 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, RC and S0. So we know that Sigma KT is the volatility parameter that must go into the Black-Scholes formula so that we get a price on the right hand side that's 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. 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 we're actually going to have two terms corresponding to K here. So we're going to get that the partial derivative with respect to 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. 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 DK of T is equal to minus Delta CBS Delta K, minus the vega times the skew. So the skew, we're just going to refer to this as being Delta Sigma Delta K. Remember that for a fixed time to maturity generally in the equity markets would see a skew like this. So this will be ST 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 of our volatility surface and we'd be able to estimate the skew. 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 let's call it Sigma K. Maybe we go up to K plus Delta K. This has a value Sigma K plus Delta K. 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. 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 the Black-Scholes terminology. In this case we computing option prices from the implied volatility surface. The implied volatility surface if you recall has been setup so that by construction call and put options will match the prices of call and put options in the marketplace. We're going to be able to use this volatility surface to compute other types of options as well. 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 whose payoff only depends on the stock price at a given fixed time T. That is because we know the marginal risk-neutral distribution once we know the implied volatility surface. If we know the marginal risk-neutral distribution, then we can compute the price of any derivatives whose payoff only depends on the stock price at a fixed time T.
