A cap consists of a sequence of caplets, all of which are the same strike, so these
caplets would all have different maturities but all have the same strike.
And a floor consists of a series of floorlets, again all of them having the
same strike, but different materials. So here's our familiar short rate lattice.
This is the lattice that we have been working with throughout this section on
fixing[UNKNOWN] derivitive securities. We start off with 6% for the short rate,
and it grows by a factor of u equals 1.25, or falls by a factor of d equals 0.9 in
each period. What we're going to do is we're going to
price a caplet with this model. So we're going to assume the expiration of
the caplet is t equals six. In other words, the payment actually takes
place at t equals six. We'll assume a strike of 2% for the
caplet, and now we're going to go ahead and price it.
But there's one thing to keep in mind here, the caplet, the pay off of the
caplet takes place in arrears. So for example, suppose we're at this node
down here 0.28, now in reality, the payment that is determined at, at that
node will take place at time t equals 6. But it is somewhat awkward to record the
payment here or here, because if we were to record the payment at this point, well,
we're not sure, actually, which node is responsible for this payment.
Is it this node, 0.28, or is it this node, 0.045?
So, that makes the accounting just a little bit awkward.
So instead what will do is we will actually record the cash flow, the final
cash flow at the node of which it is determined.
So that is at t equals 5, but we will discount it appropriately to reflect the
fact that payment does not take place until t equals 6 and that is what we have
done here. So if we look at this point 0.015, 0.015
is equal to the payoff of the caplet af, determined at t equals 5, so this is the
maximum of zero and the short rate that prevails at that period, which is 3.54%
minus the[INAUDIBLE] 0.2. So this is the payoff, of the capital,
which is determined at this note, however it is not paid unto type t equals six, and
so that is why we must discount it by 1 plus 3.54% to reflect the fact that the
payment is made in the rears. So we do that for all of the notes at time
t equals five. These are all of the payments of the
caplet discounted by 1 period. Once we do that we can work backwards in
the lattice using our usual risk neutral pricing.
And just to remind ourselves our usual risk neutral pricing takes the following
form s t is the price of whatever security we want to price and it's equal to the
expected value at conditional time t information using the risk neutral
probabilities of St plus 1 divided by 1 plus rt, where rt of course is the
short-rate prevailing at the node your at time t.
So this is risk neutral pricing for a security, that does not pay any
intermediate cash flows or coupons. If the security did pay a coupon at time t
plus 1 or some intermediate cash flow time t plus 1 we would of course inject that
cash flow into this point here. And here we can see the lattice with all
of the caplet prices displayed, for example, the value of 0.021 is calculated
as follows. It is simply one over one plus 3.94%,
remember that is the short rate prevailing at this note here, 3.94%.
So we get simply 1 over one plus 3.94% times the expect, expected value of the
cap at one period ahead, using the risk neutral probabilities which are a half and
a half. One period ahead it's either worth point
0.28, which we have here, or 0.15, which is what we have down here.
So we get the value of the caplet at this node.
So again all we're doing is using what we had on the previous slide, we're using the
fact that s t equals the expected value, of s t plus 1 over 1 plus r t.
So we start off at time t equals 5. At time t equals 5, that is where t plus 1
equals 5. We know the value of the caplets; It's
this value, these values here, and we just work backwards one period at a time to get
to time zero and we see a caplet value of 0.042.
Now just to be clear, this is the value of the security that pays off r 5 Minus 2%
the maximum of this and it's paid at time t equal to 6.
Well this 0.42 is how much this is worth, but in practice of course, you might
actually not just buy just one of these, you might buy a million of these.
So a million dollars, that would be a notion of 1 million dollars, that price of
that would therefore be 0.042 times 1 million dollars.
So this value here, is the value for a notional of one dollar.
We're multiplying this pay off by if you like one dollar here.
In general, in practice, you would be actually buying many of these, or selling
many of these, and so you would need to multiply the price you obtain here by the
notional of the counters. And here we see our familiar spreadsheet,
that I hope you've opened in front of you while you go through these calculations.
Here we have our parameters for a short-rate lattice or starting off with 6%
which neutral probabilities of a half and a half, u equals 1.25, d equals 9.
So we have a familiar short-rate lattice that we're using throughout these
examples, down here we see how to price the caplet.
The .0149 is the payoff of the caplet at the time as determined the time equals 5.
So that is the maximum of 0, and the short rate which is g 16, which is 3.5%, minus
the strike 2%. But of course that payment does not take
place until time t equals six so we have to discount by the one plus the short rate
prevailing at that node which is again in g 16.
So we compute the value of the caplet at time t equals 5, and then we just use risk
mutual pricing to work backwards. And I've highlighted the, the cell here in
bold because it is this cell which contains the formula, the risk mutual
pricing formula for the value of the caplet at this cell.
We can then copy and drag this cell formula throughout lattice to determine
the prices at every node. And again, we see the value of the caplet
is 0.0420.
