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 be pricing a digital option. The digital option 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 at time T is equal to the maximum of zero and the indicator function of S_T being greater than or equal to K. In 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 S_T 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 CT times S square root capital T times Phi of d_1, where d_1 was equal to the log of S_0 divided by K plus or minus C plus Sigma squared over 2 times t, all divided by Sigma square root t. Now, in this situation, in this example, we're going to assume r equals c equals 0, T equals one year, and S_0 equals K. So in that case, d is equal to, it turns out to be simply Sigma over 2. It also implies that the vega in this case is equal to, well, C is zero, S is 100, T is one, and so it's Phi of Sigma over 2. So what we're actually going to do is we're going to get this is equal to S_0 times Phi of Sigma over 2. So now, we can go 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 C_BS, 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. So sigma_atm is the ATM 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 a 10 percent of S_0 because the strike is equal to S_0, and it's 2.5 percent per 10 percent change. So it's going to be 0.025 divided by 0.1S_0. This is the skew. However, we also have a minus sign here. Now, the minus sign is, we're not explicitly told that there's a minus sign here, but we know that there must be, because we know in the the equity markets we see a skew like this. 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. So I implicitly understand that the skew here represents a negative 2.5 percent per 10 percent change in strike. So that's how I get this quantity here. The S_0 will cancel with the S_0 here. 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 skewed component, in other words, if I just took the market price, C_market price of the call option to be equal to just the Black Scholes price as a function of K, T, and Sigma at the money, and ignored the fact that Sigma is also a function of K, as we see in the skew here. If I ignore that, then I would only get a Delta C_BS, 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 implied volatility is a function of K, and that's where I get the second term here. But if I ignored the second term, pretend that sigma was a constant as would be the case that the Black-Scholes model held, then I would only get the 0.45 value here. So in fact, by taking the skew into account correctly, I see the price of the digital option is 0.55 and not 0.45. Actually, this is significant. This represents $0.10 extra on $0.45.