Consider now a three-month range accrual on the S&P 500 index with range 1500 to 1550. After three months, the product pays X percent of notional where X equals the percentage of days over the three months that the index is inside the range. So for example if the notional is $10 million and the index is inside the range 70 percent of the time, then the payoff will be $7 million. 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, that's expounded on this answer and see how indeed we can use the volatility surface to price this range accrual. So let's 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 let's call it, P_T will be equal to 10 million times the summation from i equals one to N of the indicator function. That the underlying security price, we'll call it S_i is inside the range. The question of this range is 1500 less than or equal to S_i, less than or equal to 1550. We have to divide by N because it's the percentage of days that were inside the range. So we must divide. So this summation is the total number of days that were inside the range of the security price, the underlying 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 N. 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 rT. Capital T is assumed to be the maturity of three months of times P_T. So let's expand on that. This is equal to 10 divided by N. So I'm going to omit the M for millions. So now my units are in millions so it's 10 over N times the summation from i equals one to N of e to the minus r times T minus ti times the expected value of e to the minus rt_i times this quantity here. Now, what can we do with this? Well, notice by the way that I have a minus minus ti which is a plus ti and that cancels with the minus ti inside here. So let's look at this expression here. So we have the expected value of e to the minus rt_i times the indicator function. The 1500 is less than or equal to S_i, is less than or equal to 1550 where S_i is the underlying security price on the i. Well, I can write this as the following. This is equal to e to the minus rt_i times the indicator function of S_i being greater than or equal to 1,500 minus the indicator function of S_i 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_i is between 1,500 and 1,550, is equal to the difference of these two indicator functions here. If you think about it, what you will see is that this is equal to, well, using our earlier notation for the price of a digital option, this is equal to D, 1,500 on date t_i minus D, 1,550 on date t_i. 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 date t_i. So therefore, the price of the range accrual is equal to $10 million divided by N, the number of days, times the summation from i equals one to N of e to the minus r times capital T, maturity, minus t_i, times the digital option with strike 1,500 and maturity t_i minus the digital option with strike 1,550 and maturity t_i. Indeed, we can compute these digital option prices from the implied volatility surface as we saw a short while ago.