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.