# Caps floors and collars

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**Caps, Floors and Collars****Caplet****Floorlet****Collar****The Black Model and Option on LIBOR****Applying the Black Model to Option on LIBOR**

### Caps, Floors and Collars

In this section, we introduce caps, floors and collars all of which are combinations of calls and puts on LIBOR that can be used to control interest rate risk. We begin the analysis by considering a single future period for a loan and how the cost of the loan over that period can be constrained with a single call on LIBOR or a single put on LIBOR. As we will see, these applications involve the same logic as forward put-call parity.

### A Caplet

Assume that a firm needs to borrow $10 million for a three-month period beginning four months from now. Assume also that today the three-month forward LIBOR for a period starting four months from now is 8 percent. In terms of our notation, FRA_{4,7} = 0.08. Given this financing need there are at least two ways that the firm could arrange now to borrow these funds at a fixed rate of 8 percent.

(1) With FRA_{4,7} = 0.08 percent, the firm could enter a loan agreement at a fixed rate of 8 percent for the period in question.

(2) With FRA_{4,7} = 0.08, the firm could enter a floating rate loan by purchasing a call and selling a put for the same period, both with a strike rate of 8 percent. From our discussion of forward put-call parity, the price of the call and put would be equal, so the long call/short put combination would be costless and the borrowing cost would be fixed at 8 percent.

Instead of pursuing either of these alternatives, the firm might merely wait to see what rate prevails for three-month LIBOR in four months. If the firm does this, it might result in paying a higher or lower rate than the 8 percent that the firm might reasonably expect to pay for its loan based on the prevailing forward rate. Whatever rate prevails, the actual interest cost of the loan will be:

Observed LIBOR x 1/4 x $10,000,000

where in our example, the Observed LIBOR is the three-month rate prevailing at month four, and the interest payment will be due at month seven.

Rather than bearing the full interest rate risk associated with its financing need, the firm decides that it would like to limit the maximum interest rate it has to pay to 8.5 percent. Given the prevailing forward rate of 8 percent, it should be possible to transact to guarantee that the actual loan rate will not exceed the desired 8.5 percent. To place an upper bound on the actual cost of the loan at 8.5 percent the firm purchases a call on three-month LIBOR with a notional principal of $10 million to expire in four months, which is the same time that the loan commences. The pay-off on this call will occur at month 7 and will be:

MAX {0, (Observed LIBOR - 0.085) x 1/4 x $10,000,000}

We now evaluate the total financial result on the loan and the call on LIBOR. Assume that the Observed LIBOR at month four for a three-month maturity is 9 percent. The interest cost of the loan will be:

.09 x 1/4 x $10,000,000 = $225,000

and the payoff on the call will be

(.09 - .085) x 1/4 x $10,000,00 =$12,500

Thinking of the loan cost of $225,000 diminished by the $12,500 payoff on the call, the outcome is the same as if the firm entered a loan at 8.5 percent, for which the interest would have been:

.085 x 1/4 x $10,000,000 = $212,500

From this, we can see that the firm has put an upper bond on the cost of the loan at 8.5 percent through the purchase of the option. In market parlance, we can say that the firm has capped the loan at 8.5 percent through the purchase of the call on LIBOR. A caplet is a call on LIBOR purchased in conjunction with a floating rate loan to cover a single period. An interest rate cap is a sequence of calls on LIBOR organized to correspond to a multi-period floating rate loan based on LIBOR. Note that the 8.5 percent rate on the loan does not include the cost of the call in the caplet or the sequence of calls in the cap.

To review our example, consider the following facts. The firm initiated a floating rate loan for a three-month period to begin in four months. At the time the firm initiated the loan, the relevant FRA rate was 8 percent, FRA_{4,7} = .08. In conjunction with the loan, the firm purchased a call on three-month LIBOR to cover the same period with a strike rate of 8.5 percent. As we have seen, combining the floating rate loan with the purchase of the call effectively capped the loan rate at the strike rate of the call, or 8.75 percent, plus the purchase price of the call. This capped loan also allows the possibility for the loan cost to fall below the anticipated cost of 8 percent. If the observed three-month LIBOR at month four is 7 percent, for example, the loan cost will be the 7 percent market rate, plus the cost of the call. To evaluate the cost of the call, let us assume that the yield curve is flat at 8 percent, and that the annualized standard deviation of the three-month LIBOR is 0.25. According to Equation 19.12 (see bottom of page), with these data, the call will cost $6,342.45.

### Floorlet

Let us now assume that the firm also decides to sell a put on LIBOR to go with the capped loan, choosing a strike rate of 7.75 percent. As we will see, this effectively places a lower bound, or floor, on the interest cost at 7.75 percent. A floorlet is a put on LIBOR sold in conjunction with a floating rate loan to cover a single period. An interest rate floor is a sequence of puts on LIBOR organized to correspond to a multi-period floating rate loan based on LIBOR. For the firm of our example, the effective rate of interest cannot fall below the floor level of 7.75 percent, plus the effect of the options. For example, assume that the three-month Observed LIBOR in four months is 7.0 percent. The call expires worthless. The firm then pays the 7 percent rate on the loan:

0.07 x 1/4 x $10,000,000 = $175,000

The put is exercised against the firm, for an outlay of

MAX {0, (0.0775 - 0.07) x 1/4 x $10,000,000} = $18,750

The total effective cost of the loan, not counting the options, is $193,750. This is equivalent to an interest rate of 7.75 percent.

0.0075 x 1/4 x $10,000,000 = $193,750

Given the flat term structure at 8 percent and a volatility of 0.25 as the standard deviation of the three-month forward LIBOR on an annualized basis, Equation 19.12 gives the value of the put as $8087.73.

### Collar

The purchase of the call and the sale of the put, together, place bounds on the cost of the loan. The upper bound is the strike rate of the call, and the lower bound is the strike rate on the put. The portfolio of a cap and a floor used in this way is called a collar. A collar may be defined as a combination of a call and a put on LIBOR, with the same expiration and underlying good, with a long position in one instrument and a short position in the other. Thus, if the firm adds a collar to its floating rate loan, the rate on our example loan is collared between 7.75 percent and 8.5 percent. A collar is referred to as a floor rate - cap rate percent collar, so our example loan has a 7.75 - 8.50 percent collar. For any observed LIBOR of 7.75 percent or below, the rate on the collared loan will be the floor rate of 7.75 percent. For any observed LIBOR rate from 7.75 to 8.50 percent, the loan rate will be the same as the observed LIBOR rate. For any observed LIBOR rate greater than 8.50 percent, the loan rate will be capped at 8.50 percent.

Our discussion to this point do not take into account the cost of buying the call nor the proceeds from selling the put that form the collar. We may view the addition of a collar to the loan as part of the final cost of the loan. To review, our collared loan consists of three parts: the determination to borrow $10 million for three months at the three-month LIBOR rate prevailing in four months: the purchase of a call (caplet) to parallel this loan at a cost of $6,342.45; and the sale of a put (floorlet) to parallel this loan for an inflow of $8,087.73. The purchase of the collar (purchase of a call for $6,342.45 and sale of a put for $8,087.73) actually generates a cash flow of $1,7245.28 when the option positions are instituted. We may think of this inflow from the collar as reducing the rate on the loan. To evaluate all of the different loan opportunities in similar terms, the future value of this inflow at the date when the interest must be repaid in seven months is:

= exp.08x(7/12) x 1,745.28 = $1,828.66

The addition of the collar to the loan terms reduces the resulting rate by 7.3 basis points, because

$10,000,000 x 0.25 x .00073146 = $1,828.66

This reduction will arise no matter what the actual interest rate on the loan is. Therefore, the final interest rate of the loan would be depicted if the interest rates shifted down to 7.3 basis points, so the effective cost of the loan runs from 7.677 (=0.0775-0.00073) to 8.427 (=0.085-0.00073) percent, depending on the rate observed in four months for the three-month LIBOR.

For our example, we have considered only some of the many possible ways to cap the loan rate, to put a floor under the loan rate, or to collar the loan rate. The upper panel of Table 19.7 presents the Black model prices for a variety of options that might be combined with the floating rate loan of our example. All have a four-month expiration and a $10 million notional principal, pertain to a three-month loan, are based on a flat yield curve at 8 percent, and assume that the forward rate has a standard deviation of 0.25 per year. The lower panel presents various financing alternatives, including capped rates and rates with a floor, as well as several collars.

As a first comment on Table 19.7, we may note that the borrower could secure a floating rate note at the three-month LIBOR rate that prevails in four months, or a fixed rate loan at an interest rate of 8 percent. These can be achieved without using options. The borrower could also obtain a capped rate of 8 percent by buying a call on LIBOR at 8 percent for a dollar outlay of $10,981.33 (corresponding to a future value of $11,505.94). This call purchase would add 46 basis points (=11505.94/250) to the interest rate, because each basis point on the loan is worth $250 ($10,000,000 x 1/4 x .0001 = $250):

For the loan capped at 8 percent, the true cost would be the observed LIBOR plus 46 basis points to a maximum of 8.46 percent. All of the other financing alternatives are presented in similar terms. Notice, that it is cheap to cap the loan at 10 percent or to establish a 6 percent floor rate. This reflects the slight chance that the observed LIBOR would exceed 10 percent or be less than 6 percent. While Table 19.7 presents 11 different financing possibilities, the real range of possibilities is endless given all the possible cap rates, floor rates, and collars that could be devised. Finally, note that no one opportunity is absolutely superior to others. They are all consistent with the existing term structure and other information. Therefore, the financing choice depends on preferences and beliefs about the future course of interest rates.

So far, we have considered a caplet, a floorlet, and a collar on a single-period loan. Caps and floors really pertain to multi-period loan agreements. For example, one might enter a five-year floating rate loan to pay three-month LIBOR. This loan would have 20 payments each dependent on an observation of three-month LIBOR on different dates. It is possible to cap this loan, to put a floor rate under it, or to put a collar on it. For example, to cap this loan, one would need a portfolio of 20 call options, with maturities that match each of the observation dates for LIBOR. Each option would be priced using the Black model, and the total cost of the option would be combined into a single fee that would cap the rate at the selected common strike price for the option. The analysis for the multi-period case is identical to our extended example of the caplet and floorlet. It must simply be repeated for each of the payment dates on the multi-period loan. It is common for lenders, such as commercial banks, to offer capped, floored, or collared loans for a specified rate plus a single fee. The fee represents the value of the portfolio of options that constitute the cap, floor, or collar, and would be paid at the inception of the loan. Note also that it is possible to offer capped, floored, or collared loans without an explicit fee. This could be done by offering terms that are somewhat above the market rate, with the excess rate on the loan covering the cost of the associated options.

Table 19.7 (a) Option values and (b) financing alternatives for the three-month loan example (a) Black model option prices for various strikes rates ($10 million notional principal; four month expiration; three-month loan; 0.25 standard deviation on loan rate; yield curve flat at 8 percent)

### Table 19.7 (a) Option values

Strike Rate | Call Price | Put price |

0.0600 | $47,926.27 | $205.87 |

0.0700 | 26,327.13 | 2,466.93 |

0.0725 | 21,752.05 | 3,856.89 |

0.0750 | 17,647.13 | 5,717.03 |

0.0775 | 14,052.27 | 8,087.22 |

0.0800 | 10,981.33 | 10,981.33 |

0.0825 | 8,422.45 | 14,387.50 |

0.0850 | 6,341.95 | 18,272.05 |

0.0875 | 4,690.31 | 22,585.46 |

0.0900 | 3,408.96 | 27,269.16 |

0.1000 | 810.51 | 48,530.91 |

### Table 19.7(b) Financing alternatives and the effect of options on the resulting loan rate

Description | Call cash flow | Put cash flow | Option effect on loan rate in basis points* | Resulting loan rate (to the nearest basis point) |

Floating | 0 | 0 | 0 | Observed LIBOR |

Fixed at 8% | 0 | 0 | 0 | 8% fixed |

Cap at 8% | -10981.33 | 0 | 46.02 | Observed LIBOR + 46 b.p. up to maximum of 8.46% |

Cap at 10% | -810.51 | 0 | 3.4 | Observed LIBOR + 3 b.p. up to maximum of 10.03% |

Floor at 6% | 0 | 205.87 | -0.86 | Observed LIBOR -1 b.p. down to minimum of 5.99% |

Floor at 7% | 0 | 2,466.93 | -10.34 | Observed LIBOR -10 b.p. down to minimum of 6.90% |

Floor at 8% | 0 | 10,981.33 | -46.02 | Observed LIBOR -46 b.p. down to minimum of 7.54% |

7.75% - 8.5% collar | -6341.95 | 8,087.22 | -7.31 | Observed LIBOR -7 b.p. from minimum of 7.68% to maximum of 8.43% |

6%-10% collar | -810.51 | 205.87 | 2.53 | Observed LIBOR 3 b.p. from minimum of 6.03% to maximum of 10.03% |

7% - 9% collar | -3,408.96 | 2466.93 | 3.95 | Observed LIBOR 4 b.p. from minimum of 7.04% to maximum of 9.04% |

8% - 8% collar | -10981.33 | 10981.33 | 0 | 8% fixed |

*The option effect in basis points is the future value of the option cash flows at the termination of the loan, divided by the value of one basis point = $250. For the 7.75 - 8.5 percent collar, the net cost of the option is $1,745.27. The loan terminates in seven months, and the yield curve is flat at 8 percent. The future value is $1,828.65. Because the option cash flow is a net inflow, the effect on the interest rate is a reduction of the rate by $1,828.65/$250 = 7.31 basis points.

### The Black Model and Option on LIBOR

The Black model applies quite directly to calls and puts on LIBOR. The currently observed forward LIBOR (FLIBOR) plays the role of the futures price. Specifically, FLIBOR, is observed at time t, when the option is being valued. It is the forward LIBOR with a maturity corresponding to the maturity of the option and it is the forward rate with a time horizon that matches the expiration date of the option. Notice that FLIBOR, should be identical to the FRA rates for the same period in the future.

Using FLIBOR, in the Black model is tantamount to assuming that the forward LIBOR is log-normally distributed. The strike rate plays the role of the exercise price, while the standard deviation of the LIBOR forward rate (FLIBOR) of the requisite maturity is the volatility measure for the model:

Equation (19.12)

C_{t}^{FLIBOR} = NP x FRAC x exp^{-r(T + FRAC - t)}[FLIBOR_{t} x N(d_{1}^{FLIBOR}) - SR N(d_{2}^{FLIBOR})]

P_{t}^{FLIBOR} = NP x FRAC x exp^{-r(T + FRAC - t)}[SR x N(-d_{2}^{FLIBOR}) - FLIBOR_{t} x N(-d_{1}^{FLIBOR})]

d_{1}^{FLIBOR} = ln (FLIBOR_{t}/SR) + 0.5σ^{2} / σ√T-t

d_{2}^{FLIBOR} = d_{1}^{FLIBOR} - σ√T-t

In this equation we have the Black model with the following substitutions, FLIBOR, takes the role of the futures price and SR substitutes for the exercise price. The payoff on each option is discounted for a longer period, the time to expiration plus FRAC to account for the delayed payment, and the values of each option are multiplied by NP (FRAC) to convert the values to dollar amounts.

The following equation is identical to Equation 19.12, except that it uses the zero-coupon factor instead of the continuously compounded rate to discount the payoff on the option from the pay off date (the expiration of the option plus FRAC) to the present:

Equation (19.13)

C_{t}^{FLIBOR} = NP x FRAC x 1/Z_{t,T+FRAC} x [FLIBOR_{t} x N(d_{1}^{FLIBOR}) - SR N(d_{2}^{FLIBOR})]

P_{t}^{FLIBOR} = NP x FRAC x 1/Z_{t,T+FRAC} x [SR x N(-d_{2}^{FLIBOR}) - FLIBOR_{t} x N(-d_{1}^{FLIBOR})]

d_{1}^{FLIBOR} = ln (FLIBOR_{t}/SR) + 0.5σ^{2} / σ√T-t

d_{2}^{FLIBOR} = d_{1}^{FLIBOR} - σ√T-t

### Applying the Black Model to Option on LIBOR

As an example, consider a call and put on one-month LIBOR that both expires in eight months. The strike rate on the call and put is 7 percent. The notional principal is $10 million. From the FRA market, the historical standard deviation of the one-month LIBOR rate has been 0.23, which is our estimate of the standard deviation of LIBOR. The yield curve is downward sloping, as illustrated in the last two columns of Table 19.6, which assumes monthly compounding. The one-month forward rate for a period from month eight to month nine is the value we need for FLIBOR. From Table 19.6, we see that the eight- and nine-month zero-coupon factors are 1.0527 and 1.0594. The one-month forward rate is, therefore, 1.0594/1.0527 - 1 = 0.006365. With monthly compounding, the annualized rate is (1 + .006365)^{12} - 1 = 0.079111.

The one-month maturity corresponds to 0.0833 years (= 1/12). Using our zero-coupon factor of 1.0594 to cover the nine months until payment on the options would be received, the option values according to Equation 19.13 are as follows:

We first find the d_{1}^{FLIBOR} and d_{2}^{FLIBOR} values

1/Z_{t,T+FRAC} = 9-month zero-coupon rate

√T-t = 8/12 = .6666667

FRAC = 1/12

d_{1}^{FLIBOR} = ln (FLIBOR_{t}/SR) + 0.5σ^{2} / σ√T-t

=( ln (0.079111/0.0700) + 0.5 x (0.23)^{2} x 0.6667) / (0.23 x √1/12) = .745448434

d_{2}^{FLIBOR} = d_{1}^{FLIBOR} - σ√T-t

= .7454484 - 0.187794 = 1.575072

Finding the cumulative normal values for the put and call, we have,

N(d_{1}F) = N(0.745104) = 0.745104

N(d_{2}F) = N(0.55731) = .711342178

N(-d_{1}F) = N(-0.745104) = 0.228104426

N(-d_{2}F) = N(-0.55731) = 0.288657822

The cost of the call and put are as follows:

C_{t}^{FLIBOR} = NP x FRAC x 1/Z_{t,T+FRAC}[FLIBOR_{t} x N(d_{1}^{FLIBOR}) - SR N(d_{2}^{FLIBOR)}]

= 10,000,000 x .0833x 1/1.0594 *(.07910 x .771896 - .07 x .711342)

= $8,863.33

P_{t}^{FLIBOR} = NP x FRAC x 1/Z_{t,T+FRAC} x [SR x N(-d_{2}^{FLIBOR}) - FLIBOR_{t} x N(-d_{1}^{FLIBOR})]

= 10,000,000 x .0833x 1/1.0594 *(.07 x .288658 - .079106 x .228104)

= $1,700.83

Therefore, the call is worth $8,859.69 and the put is worth $1,699.70.

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