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In this module we're going to see how to price caplets and floorlets, and

Â particularly we'll focus on caplets but floorlets are priced the same way.

Â Caplets and floorlets in more generally caps and floors are very liquidly traded

Â fixed income derivatives securities that are found in the market place.

Â So it's important that we understand how to price these securities and understand

Â the mechanics behind these securities. A caplet is similar to a European call

Â option on the short-rate r t. It is usually settled in arrears, but a

Â caplet may also be settled in advance. If the maturity is time tau and the strike

Â is c, then the payoff of a caplet is r tau minus 1 minus e the positive part, or if

Â you like, this is the maximum of 0, and or, tau minus 1 minus c.

Â This payment occurs at time tau, so this is what we mean by being settled in

Â arrears. So, you can think of the caplet as being a

Â call option on the short rate prevailing at time tau minus 1, but settled at time

Â tau. A floorlet is the same as a caplet, except

Â the payoff is c minus r tau minus 1 positive part, or as we've been writing,

Â the maximum of zero and c minus or tau minus 1.

Â But again, this payoff would typically take place at time tau.

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A cap consists of a sequence of caplets, all of which are the same strike, so these

Â caplets would all have different maturities but all have the same strike.

Â And a floor consists of a series of floorlets, again all of them having the

Â same strike, but different materials. So here's our familiar short rate lattice.

Â This is the lattice that we have been working with throughout this section on

Â fixing[UNKNOWN] derivitive securities. We start off with 6% for the short rate,

Â and it grows by a factor of u equals 1.25, or falls by a factor of d equals 0.9 in

Â each period. What we're going to do is we're going to

Â price a caplet with this model. So we're going to assume the expiration of

Â the caplet is t equals six. In other words, the payment actually takes

Â place at t equals six. We'll assume a strike of 2% for the

Â caplet, and now we're going to go ahead and price it.

Â But there's one thing to keep in mind here, the caplet, the pay off of the

Â caplet takes place in arrears. So for example, suppose we're at this node

Â down here 0.28, now in reality, the payment that is determined at, at that

Â node will take place at time t equals 6. But it is somewhat awkward to record the

Â payment here or here, because if we were to record the payment at this point, well,

Â we're not sure, actually, which node is responsible for this payment.

Â Is it this node, 0.28, or is it this node, 0.045?

Â So, that makes the accounting just a little bit awkward.

Â So instead what will do is we will actually record the cash flow, the final

Â cash flow at the node of which it is determined.

Â So that is at t equals 5, but we will discount it appropriately to reflect the

Â fact that payment does not take place until t equals 6 and that is what we have

Â done here. So if we look at this point 0.015, 0.015

Â is equal to the payoff of the caplet af, determined at t equals 5, so this is the

Â maximum of zero and the short rate that prevails at that period, which is 3.54%

Â minus the[INAUDIBLE] 0.2. So this is the payoff, of the capital,

Â which is determined at this note, however it is not paid unto type t equals six, and

Â so that is why we must discount it by 1 plus 3.54% to reflect the fact that the

Â payment is made in the rears. So we do that for all of the notes at time

Â t equals five. These are all of the payments of the

Â caplet discounted by 1 period. Once we do that we can work backwards in

Â the lattice using our usual risk neutral pricing.

Â And just to remind ourselves our usual risk neutral pricing takes the following

Â form s t is the price of whatever security we want to price and it's equal to the

Â expected value at conditional time t information using the risk neutral

Â probabilities of St plus 1 divided by 1 plus rt, where rt of course is the

Â short-rate prevailing at the node your at time t.

Â So this is risk neutral pricing for a security, that does not pay any

Â intermediate cash flows or coupons. If the security did pay a coupon at time t

Â plus 1 or some intermediate cash flow time t plus 1 we would of course inject that

Â cash flow into this point here. And here we can see the lattice with all

Â of the caplet prices displayed, for example, the value of 0.021 is calculated

Â as follows. It is simply one over one plus 3.94%,

Â remember that is the short rate prevailing at this note here, 3.94%.

Â So we get simply 1 over one plus 3.94% times the expect, expected value of the

Â cap at one period ahead, using the risk neutral probabilities which are a half and

Â a half. One period ahead it's either worth point

Â 0.28, which we have here, or 0.15, which is what we have down here.

Â So we get the value of the caplet at this node.

Â So again all we're doing is using what we had on the previous slide, we're using the

Â fact that s t equals the expected value, of s t plus 1 over 1 plus r t.

Â So we start off at time t equals 5. At time t equals 5, that is where t plus 1

Â equals 5. We know the value of the caplets; It's

Â this value, these values here, and we just work backwards one period at a time to get

Â to time zero and we see a caplet value of 0.042.

Â Now just to be clear, this is the value of the security that pays off r 5 Minus 2%

Â the maximum of this and it's paid at time t equal to 6.

Â Well this 0.42 is how much this is worth, but in practice of course, you might

Â actually not just buy just one of these, you might buy a million of these.

Â So a million dollars, that would be a notion of 1 million dollars, that price of

Â that would therefore be 0.042 times 1 million dollars.

Â So this value here, is the value for a notional of one dollar.

Â We're multiplying this pay off by if you like one dollar here.

Â In general, in practice, you would be actually buying many of these, or selling

Â many of these, and so you would need to multiply the price you obtain here by the

Â notional of the counters. And here we see our familiar spreadsheet,

Â that I hope you've opened in front of you while you go through these calculations.

Â Here we have our parameters for a short-rate lattice or starting off with 6%

Â which neutral probabilities of a half and a half, u equals 1.25, d equals 9.

Â So we have a familiar short-rate lattice that we're using throughout these

Â examples, down here we see how to price the caplet.

Â The .0149 is the payoff of the caplet at the time as determined the time equals 5.

Â So that is the maximum of 0, and the short rate which is g 16, which is 3.5%, minus

Â the strike 2%. But of course that payment does not take

Â place until time t equals six so we have to discount by the one plus the short rate

Â prevailing at that node which is again in g 16.

Â So we compute the value of the caplet at time t equals 5, and then we just use risk

Â mutual pricing to work backwards. And I've highlighted the, the cell here in

Â bold because it is this cell which contains the formula, the risk mutual

Â pricing formula for the value of the caplet at this cell.

Â We can then copy and drag this cell formula throughout lattice to determine

Â the prices at every node. And again, we see the value of the caplet

Â is 0.0420.

Â