So keeping this in mind we're going to get an alternative expression for b0.

We have one expression for B0 here in equation 6.

On the next line we're going to get an alternative expression for B0 using this

representation here. So what we can say is that B0 is equal to

e to minus rt or rather is. It is approximately equal to e to the

minus rt. Times the risk neutral probability of st

being between k minus Delta K, and k plus Delta K times Delta K over 2.

Now, where does that came from? Well, if you think about it, it comes

from this idea here. So the payoff occurs if st is in k minus

Delta K. Up to k plus Delta K.

So the risk neutral probability of that is q of k minus Delta K being less than

or equal to st, being less than or equal k plus Delta K.

So, that's the probability that s t is inside this interval here.

Now, were imagining Delta K being small, by the way.

In fact, soon we're going to let Delta K go to zero.

So we can imagine Delta K is very small. So, this is the probability, the risk

neutral probability that st is inside this interval here.

We already explained that if s t is in this interval then the pay off you expect

to get is k over 2. And indeed that is why we multiply by the

k over 2 here. So we have our e to minus r t term, the

probability that st is inside this interval, times the average payoff in

this interval. And so that's how we get this first line

here. It's an approximation, but it is a very

good approximation for small Delta K. Now, if you recall something about

density functions, then you will understand why we're letting q.

The risk neutral probability that st is in this interval is equal to the risk

neutral density times the width of the interval.

So we're saying the risk neutral probability that st is between k minus

Delta K and k plus Delta k. That is approximately equal to the

density, risk neutral density evaluated at k times the width of the interval to

Delta K. And that just follows from a property of

PDFs. We actually explain this in one of the

additional modules on, on probability that we also recorded, they're also

available on the, on the plat, course platform.

So, remember, if you've got a PDF, in general.

So if this is our PDF, f of x. And suppose we want to compute the

interval that the random variable x is inside x0 to x 0 plus Delta X.

Well, the density satisfies that the probability, that the random variable x

is in, x0 plus Delta x where Delta x small.

That's approximately equal to f of x 0 times the width of the interval which is

Delta X. This is a standard property of

probability density function and that's all we're using here.

So we're saying the portability that st is inside this interval here is equal to

the density which is ft, times the width of the interval which is 2 Delta K.

So therefore we're going to get B0 is approximately equal e to the minus rt.

Times ft of k, times Delta K squared. We have a two here, but that counts as

with a two there, and we get a Delta K times Delta K, which is delta k squared.

So what we've done now is we've come up with 2 expressions for the value of the

butterfly strategy. We have this expression here in equation

6 which is exact. I'm here with this expression here which

is an approximation. But as Delta K goes to zero, this

approximation also becomes exact. So, what we're going to do is, we're

going to equate equation 7 with equation 6 and then solve for ft of k.

Or in other words, bring ft of k over to the left hand side.

So we will see ft of k, is approximately equal to e to the rt, times this

expression on the right hand side of 6 here, divided by Delta K squared.