Caps and Floors Defined.
Caps and floors trade over the counter, and are tools that companies can use to manage interest rate exposure,
and that fixed income managers can use to make trading profits and manage their portfolio
A cap can be associated with any underlying interest rate. For the most part, they are linked to LIBOR.
A cap must specify a cap rate--which is the strike price of the option, a notional principle, used to compute the cash flows, and
a tenor, which indicates the term of the rate as well as the frequency of resets and payments.
Example:
Cap on three-month LIBOR with three-month tenor
Notional Principal of $10 million
Cap Rate 4.78821%
Current Date 12/14/05
Valuation Date 12/16/05
Maturity: 12/16/10
This cap is shown on the Bloomberg screen.
The cash flows on this cap follow:
Date Reset Amt (R(T)) Cash flow
+
3/16/2006 (L(3) - X) P (dc) -
+
6/16/2006 (L(3) - X) P (dc) R(T-1)
+
9/16/2006 (L(3) - X) P (dc) R(T-1)
+
12/16/2006 (L(3) - X) P (dc) R(T-1)
+
3/16/2007 (L(3) - X) P (dc) R(T-1)
+
6/16/2007 (L(3) - X) P (dc) R(T-1)
+
9/16/2007 (L(3) - X) P (dc) R(T-1)
+
12/16/2007 (L(3) - X) P (dc) R(T-1)
+
3/16/2008 (L(3) - X) P (dc) R(T-1)
+
6/16/2008 (L(3) - X) P (dc) R(T-1)
+
9/16/2008 (L(3) - X) P (dc) R(T-1)
+
12/16/2008 (L(3) - X) P (dc) R(T-1)
+
3/16/2009 (L(3) - X) P (dc) R(T-1)
+
6/16/2009 (L(3) - X) P (dc) R(T-1)
+
9/16/2009 (L(3) - X) P (dc) R(T-1)
+
12/16/2009 (L(3) - X) P (dc) R(T-1)
+
3/16/2010 (L(3) - X) P (dc) R(T-1)
+
6/16/2010 (L(3) - X) P (dc) R(T-1)
+
9/16/2010 (L(3) - X) P (dc) R(T-1)
+
12/16/2010 (L(3) - X) P (dc) R(T-1)
here dc is the percentage of the year between the reset date and the payment date.
In this particular case, we use the actual number of days divided by 360.
So if on any of the reset dates, (Spot) 3-Mo LIBOR is less than the cap rate, the cash flow 3 months later is 0.
Suppose that on 9/16/06, spot 3-month LIBOR is 5.2%.
In this case, the cash flow on 12/16/06 will be calculated as follows:
Reset Date: 9/16/2006
Pay Date 12/16/2006
Not Princ 10000000
Cap Rate 0.0478821
Days between: 91
dc 0.252777778
Spot LIBOR 0.052
Cash Flow: $10,409.14
We know exactly how to value a certain cash flow that occurs on this future date. We would use the discount factor.
This is the discount rate from the Bloomberg "Curves" screen:
Note that the one-year discount factor is 0.953072
Spot Rate 0.00483 4
0.995184545 This is how the discount is computed, but we can't replicate
Bloomberg because of the day count conventions.
Note that each of the Reset/Payment events is a caplet.
As such, a cap is a portfolio of caplets.
Bloomberg's Calculation of Forward Rates
Variables:
d1 number of days from settlement date until the start date of the forward period (I.e., the
d2 number of days from settlement date until the end date of the forward period (I.e., the n
r1 spot rate for d1 days.
r2 spot rate for d2 days.
FV = future value: FV = (1 + [(r2*d2)/360]) / (1 + [(r1*d1) / 360]))
f = forward rate = [(FV - 1) / (d2-d1)] * 360
So, for example for the caplet that is reset on 9/14/06, 12/14/2005
r1 0.0477
r2 0.0483
d1 274 9/14/2006
d2 365 12/14/2006
FV 1.0122221 0.048970833 0.036305
f 0.0483512 Note that this corresponds to the Reset Rate used to value the
erest rate exposure,
heir portfolio
ed to compute the cash flows, and
and payments.
months later is 0.
use the discount factor.
ate of the forward period (I.e., the Reset Date).
ate of the forward period (I.e., the next Reset Date).
r1*d1) / 360]))
the Reset Rate used to value the caplet that resets on 9/14/06. (See the next worksheet.)
Once we understand the way the cash flows are determined on a cap we can think about how we would come up with a value o
Black's Model is widely used by Fixed Income Traders to characterize the value of caps.
Black's model was originally designed to ascertain the value of an option on a futures contract.
In the Black-Scholes model, (or any finance model where there is no arbitrage), we simply take the expected value of the contra
For convenience sake, we do this in the "Equivalent risk-neutral World" since only in this (equivalent) world do we know the disc
(And that is the risk-free rate).
In Black-Scholes, the numeraire is $1 today.
To use Black's model, the numeraire is $1 at the time of delivery. This trick allows us to handle the problem of
having the bond price be a random variable at a future date, but assuming that the risk-free rate between now and
then is constant. Thus we not only do the valuation in our equivalent risk-neutral world, but we move to an equivalent
risk-neutral forward world.
In this world, the expected future spot rate is the forward rate, and its standard deviation is the same as in the "physical world."
Black's Model assumes that the future spot rate is lognormally distributed.
So turn to the 9/14/06 settlement. We see on the attached Bloomberg screen that the corresponding
forward rate is 4.83516% We also see that the volatility (used in describing the caplet value) is 11.8%.
This means that for valuing this caplet, the probability description of the 3-month LIBOR spot rate, 9 months from
now is lognormal, with mean and standard deviation computed as follows:
Mean rate (= forward rate): 0.0483516
std dev (1-Yr) 0.118 Var * T 0.010443
Term (Years) 0.75 (Again approximate)
Mean of log rate: -3.034477466
Var of log rate: 0.010443
std dev of log rate: 0.102190998
Notional Principal 10000000
So the cumulative density looks like this:
CDF
1.2
1
Cumulative Probability
0.8
0.6
0.4
Cumulative Prob
0.2
0
0 0.01 0.02 0.03 0.04 0.05 0.06
90-Day LIBOR
And the probability density function looks like this:
4.5
4
3.5
3
Probability
2.5
2
1.5
1
0.5
0
0 0.01 0.02 0.03 0.04 0.05 0.06
90-Day LIBOR
Let me stress that this is not the market's distribution of the 90-Day spot LIBOR in 9 months. Instead it is that as it relates to va
Black's Model values each caplet in a distinct forward risk-neutral world.
As is the case in the Black-Scholes Model, we take the discounted expected cash flows from the option. The difference is that
whereas Black's model uses the forward risk-neutral world corresponding to each caplet.
I wrote a VB macro to evaluate a caplet using Black's model, and the type of input from Bloomberg:
Cap Rate: 0.0478821 Discount 0.95307
#NAME? Bloomberg gives 5326.85
Bloomberg's Intrinsic PV:
The intuition behind this value is as follows:
1) What is the equivalent risk-neutral probability that we would exercise this caplet?
The strike rate is 0.0478821 which is less than the expected 3-month rate (in the f
We look at the pdf graph and ask for the probability that 90-LIBOR will exceed the strike rate.
#NAME? So in this case, there's a 51.8% probability that this caplet will be in the money when
2) What is the expected value under the truncated distribution--where we only look at the portion above the exercise
#NAME? is the probability associated with
So what we expect to get if we exercise: #NAME?
what this costs: #NAME?
Expected $ value (in 1 Year) #NAME?
3) Bring this expected value back to today. #NAME?
For the most part, the market does not use Black's model to value caps and floors, so much as it uses this model as a tool to c
In particular, prices are often expressed as implied volatilities. Notice on the Bloomberg valuation screen that the implied volati
as we move out in time.
As in the valuation screen there is a (potentially) different implied vol for each caplet. This is a method sometimes called spot v
We could, of course solve for a cap value by forcing all caplet vols to be the same. Such a situation is called flat volatilities.
Floors work analogously to caps. We can think of a caplet as a call option on the future spot rate, the floorlet is an analogous p
are identical to caps'.
#NAME? The delta of a floorlet is -N(-d1):
Furthermore, there is a put-call-parity relationship between caps, floors, and swaps.
Consider a trader who is long the caplet and short the floorlet.
If 90-Day LIBOR, 9 months from now is above the cap rate, then this trader will receive the tenor-adjusted differenc
If 90-Day LIBOR, 9 months from now is below the cap (floor) rate, then this trader will pay the tenor-adjusted differe
These are exactly the same state-dependent cash flows that this date's position in a receiver (of floating rate) swap,
fixed rate (equal to the cap/floor rate), and receive 90-Day LIBOR (90 days hence).
So in this case, the value of such a swap is: #NAME?
The "Swap Rate" is that rate which such a swap holder pays fixed makes the value of the swap 0.
As such one way to solve for the swap rate would be to use Solver to identify such a strike price:
Numerical computation of derivatives and hedging.
Derivatives can be calculated numerically by looking at the effect of a small change in a function's input on the value of the func
Example: The Caplet's Delta:
Start with the current value of the function: #NAME?
Identify the small change in the input: 0.000000001
Evaluate the function by adding this small change: #NAME?
Change in the function: #NAME? This corresponds to the effect of a small change in the
Chg in F'n / Chg in Input: #NAME? it equals N(d1) * P* tenor * disct
Units: #NAME? Numerical approximation to N(D1).
Hedge Portfolio: Current Change in F New
Long in the FRA: 1131.095534 0.0001 1372.01045
Sell 1/Delta units of the cap: #NAME? 0.0001 #NAME?
Portfolio #NAME? #NAME?
DV01
Start with the current value of the function: #NAME?
Identify the small change in the imput: 0.0001
Evaluate the function by adding this small change: #NAME?
Discount Effect: Discount: 0.95307
T 0.755555556
-LN(disct)/T 0.06361799
add 1 bp: 0.06371799
New Discount: 0.952997993
#NAME?
DV01 #NAME?
Vega:
Start with the current value of the function: #NAME?
Identify the small change in the input: 0.000000001
Evaluate the function by adding this small change: #NAME?
Change in the function: #NAME?
Chg in F'n / Chg in Input: #NAME?
Units: #NAME?
ow we would come up with a value of these cash flows.
Constructing the forward rate:
ake the expected value of the contract at expiration and discount this back to today. forward:
quivalent) world do we know the discount factor without doing a lot of work.
ndle the problem of
rate between now and
we move to an equivalent
he same as in the "physical world."
e caplet value) is 11.8%.
t rate, 9 months from
today 12/16/2005 38702
reset date 9/14/2006 38974
pay date 12/18/2006 39069
next expiry 12/14/2006 39065
#NAME?
CDF
0.06 0.07 0.08
Series1
0.06 0.07 0.08
. Instead it is that as it relates to valuing an option (the forward risk neutral world).
m the option. The difference is that Black-Scholes uses the equivalent risk-neutral world,
Bloomberg's Intrinsic PV: 1131.07
Our,reconciliation: 0.00047 1186.896
1131.195
the expected 3-month rate (in the forward risk-neutral world)
will exceed the strike rate.
is caplet will be in the money when it expires. (This is N(d2))
ok at the portion above the exercise price.
is the probability associated with this truncated mean. (This is N(d1))
Again, ignoring day-count issues.
h as it uses this model as a tool to characterize the prices.
uation screen that the implied volatilities are getting higher
s a method sometimes called spot volatilities.
situation is called flat volatilities.
t rate, the floorlet is an analogous put option. Its institutional features
#NAME?
eceive the tenor-adjusted difference times the notional principal.
er will pay the tenor-adjusted difference times the notional principal.
in a receiver (of floating rate) swap, where the swap makes its owner pay the
90 days hence).
#NAME? We find the rate to be 4.835%.
--which, of course is the forward rate.
ction's input on the value of the function.
o the effect of a small change in the forward rate on the Cap Value.
P* tenor * disct
ximation to N(D1).
12/14/2005 360
ing the forward rate: 3-Mo Spot 0.0449125 3/14/2006 90 0.25 1.011228
6-Mo Spot 0.046625 6/14/2006 182 0.505556 1.023572
1.012206348 FV = (1 + [(r2*d2)/360]) / (1 + [(r1*d1)/360]))
0.048825393
0.043
0.044
0.045
0.046
0.047
0.048
0.049
0.05
0.051
0.052
0.053
0.054
0.055
0.056
0.057
0.058
Grid used to describe the probability of the future spot rate:
Grid Pts CDF pdf Use the Normdist function which allows us
0.001 0 to get the exact pdf:
0.002 6.6E-213 6.6E-213
0.003 1.2E-162 1.2E-162
0.004 4E-131 4E-131
0.005 4.9E-109 4.9E-109
0.006 1.58E-92 1.58E-92
0.007 1.21E-79 1.21E-79
0.008 2.78E-69 2.78E-69
0.009 9.38E-61 9.38E-61
0.01 1.3E-53 1.3E-53
0.011 1.5E-47 1.5E-47
0.012 2.42E-42 2.42E-42
0.013 7.91E-38 7.91E-38
0.014 6.94E-34 6.94E-34
0.015 2.03E-30 2.03E-30
0.016 2.36E-27 2.36E-27
0.017 1.25E-24 1.25E-24
0.018 3.35E-22 3.33E-22
0.019 4.99E-20 4.95E-20
0.02 4.45E-18 4.4E-18
0.021 2.53E-16 2.49E-16
0.022 9.69E-15 9.44E-15
0.023 2.61E-13 2.51E-13
0.024 5.12E-12 4.86E-12
0.025 7.58E-11 7.07E-11
0.026 8.73E-10 7.97E-10
0.027 7.99E-09 7.12E-09
0.028 5.96E-08 5.16E-08
0.029 3.69E-07 3.09E-07
0.03 1.92E-06 1.55E-06
0.031 8.59E-06 6.67E-06
0.032 3.33E-05 2.47E-05
0.033 0.000113 8.02E-05
0.034 0.000343 0.00023
0.035 0.000932 0.000588
0.036 0.002288 0.001356
0.037 0.005124 0.002836
0.038 0.010545 0.005421
0.039 0.020075 0.009531
0.04 0.035581 0.015505
0.041 0.059048 0.023467
0.042 0.092253 0.033205
0.043 0.136376 0.044123
0.044 0.191664 0.055288
0.045 0.257242 0.065578
0.046 0.33113 0.073888
0.047 0.410467 0.079337
0.048 0.491893 0.081426
0.049 0.571993 0.0801
0.05 0.64771 0.075717
0.051 0.716652 0.068943
0.052 0.777251 0.060599
0.053 0.828776 0.051525
0.054 0.871236 0.04246
0.055 0.905208 0.033972
0.056 0.931642 0.026434
0.057 0.951677 0.020035
0.058 0.966489 0.014812
0.059 0.977185 0.010696
0.06 0.98474 0.007555
0.061 0.989965 0.005225
0.062 0.993507 0.003542
0.063 0.995863 0.002357
0.064 0.997404 0.00154
0.065 0.998393 0.00099
0.066 0.999019 0.000626
0.067 0.999409 0.00039
0.068 0.999648 0.000239
0.069 0.999793 0.000145
0.07 0.99988 8.66E-05
0.071 0.999931 5.11E-05
0.076 0.999996 #REF!
cdf pdf
0 0
6.6E-213 2E-210
1.2E-162 3.2E-160
4E-131 9.6E-129
4.9E-109 1.1E-106
1.58E-92 3.15E-90
1.21E-79 2.23E-77
2.78E-69 4.8E-67
9.38E-61 1.51E-58
1.3E-53 1.96E-51
1.5E-47 2.13E-45
2.42E-42 3.24E-40
7.91E-38 9.97E-36
6.94E-34 8.25E-32
2.03E-30 2.29E-28
2.36E-27 2.51E-25
1.25E-24 1.25E-22
3.35E-22 3.18E-20
4.99E-20 4.49E-18
4.45E-18 3.79E-16
2.53E-16 2.04E-14
9.69E-15 7.38E-13
2.61E-13 1.88E-11
5.12E-12 3.48E-10
7.58E-11 4.86E-09
8.73E-10 5.27E-08
7.99E-09 4.55E-07
5.96E-08 3.19E-06
3.69E-07 1.85E-05
1.92E-06 9.07E-05
8.59E-06 0.000379
3.33E-05 0.001374
0.000113 0.004362
0.000343 0.012273
0.000932 0.030881
0.002288 0.070091
0.005124 0.144617
0.010545 0.273151
0.020075 0.475311
0.035581 0.766406
0.059048 1.151167
0.092253 1.618499
0.136376 2.139444
0.191664 2.669716
0.257242 3.156631
0.33113 3.54869
0.410467 3.805158
0.491893 3.903083
0.571993 3.840149
0.64771 3.633142
0.716652 3.312991
0.777251 2.918098
0.828776 2.487676
0.871236 2.056433
0.905208 1.651275
0.931642 1.290077
0.951677 0.982118
0.966489 0.729595
0.977185 0.5296
0.98474 0.376102
0.989965 0.261614
0.993507 0.178441
0.995863 0.119468
0.997404 0.078588
0.998393 0.05084
0.999019 0.032373
0.999409 0.020306
0.999648 0.012556
0.999793 0.00766
0.99988 0.004613
0.999931 0.002745
0.999996 0.000174
The third application of Black's model is to swaptions. A swaption is an option to enter a swap agreement.
We will first review swaps.
An interest rate swap is a portfolio of forward rate agreements.
A forward rate agreement (FRA) entails the exchange of a pre-determined interest rate for the market rate at a pre-
A FRA is valued by assuming that the current market forward rate will be realized at at swap date.
FRAs and swaps are priced to have 0 value at origination. The "price" is the pre-specified fixed rate
In the previous sheet, we saw that the one-period swap (I.e., a FRA) will have a price equal to the rel
Thus, swap valuation entails the following steps:
1) Ascertain the relevant LIBOR forward rates,
2) Calculate the cash flows that will accrue -- assuming that the realized future floating rates equal relevant the forw
3) The value of the swap is the PV of these cash flows. (So at origination, the fixed "swap rate" will be the rate tha
Example:
Here's an already existing swap where the holder pays 6-month LIBOR in exchange for 8% (fixed) (semi-annual com
Notional Principal: $100,000,000
6-Mo LIBOR at last reset date: 0.102 Semi-annual compounding)
Continuously Compd LIBOR - 3-Mo: 0.1
Continuously Compd LIBOR - 9-Mo: 0.105
Continuously Compd LIBOR - 15-Mo: 0.11
Fixed Rate: 0.08
Valuation:
Risk-Neutral
Expected
Time forward Sem-Ann Frwd term cash flow disct
0.25 0.102 0.102 0.5 -$1,100,000 0.97530991
0.75 0.1075 0.110441528 0.5 -$1,522,076 0.92427096
1.25 0.1175 0.12102016 0.5 -$2,051,008 0.87153435
Value of Swap:
Swaptions
In the spirit of Black's model, we will value swaptions by assuming that the swap rate at the maturity of the option is
Example: Consider an option to enter a 3-Year swap in 5 years, with semiannual payments and notional principal of:
Scenario:
LIBOR yield curve is flat at 6% per annum (continuous compounding).
Vol at 5 years : 0.2 T: 5
IdentifyCash flows:
Years ahead Discount tenor $ Multiplier
5.5 0.718924 0.5 0.359461867
6 0.697676 0.5 0.348838163
6.5 0.677057 0.5 0.338528437 Black's Model uses this annuity as the
7 0.657047 0.5 0.32852341 numeraire in valuing the swaption.
7.5 0.637628 0.5 0.318814076
8 0.618783 0.5 0.309391696
2.003557649 Total Value discount.
1) Payer Swaption (I.e., right to pay fixed over the swap's life). Strike:
d1 0.183911397 N(d1) 0.572958518 f N(d1)
d2 -0.263302199 N(d2) 0.396158832 X N(d2)
Option Value: 2.070981704
2) Receiver Swaption (I.e., Right to receive fixed (pay floating LIBOR) over the swap's life). (A Put option on the sw
N(-d2) 0.603841168 X N(-d2)
N(-d1) 0.427041482 f N(-d1)
Option Value: 2.289556238
-0.218574534 Verification of Put-Call
(Long a Payer Swaption and Writ
ap agreement.
st rate for the market rate at a pre-sepcified date, on a pre-specified notional principal.
d at at swap date.
rice" is the pre-specified fixed rate that will be exchanged for the market rate in the future.
A) will have a price equal to the relevant forward rate.
loating rates equal relevant the forward rates.
xed "swap rate" will be the rate that makes the PV of these flows equal to 0.)
nge for 8% (fixed) (semi-annual compounding applies).
Semi-annual compounding)
PV
-$1,072,841
-$1,406,811
-$1,787,524
-$4,267,176
rate at the maturity of the option is a lognormally distributed random variable.
s and notional principal of: 100
0.06 (translates into: 0.060909 on a semi-annual compounding basis.)
Since we have a flat curve, this is the forward swap rate.
Payer Swap Value
-0.03921
-0.03806
es this annuity as the -0.03693
uing the swaption. -0.03584
-0.03478
-0.03375
Total Swap Value: -0.21857
0.062 (A Call option on the swap rate -- valued in the forward risk neutral world.)
0.034898369 (The Expected Future Swap Rate equals
0.024561848 the forward swap rate.)
wap's life). (A Put option on the swap rate).
0.037438152
0.026010699
Verification of Put-Call Parity
(Long a Payer Swaption and Write a Receiver Swaption is equivalent to entering into the forward swap.
is the forward swap rate.