Transcript Document 7417256

Interest Rate
Derivative
Pricing
IRD Valuation
Caps, Floors and Collars
 Swaptions

Caps and Floors

Application of B-S model as modified by
Black (1976):

d 
 Notional Value *

Basis  * e-rT *  FN (d )  XN (d )  ,
Caplet  
1
2




 d 
1
+
F





Basis






where, d = days in period, Basis is number of days per year, F = that caplet's Forward rate ,
Notional Value is principal basis of transaction, and

F
ln
d1 
X





T

2
2
T

 , while,
d 2  d1  
T
Then, Cap   Capletsi
all i
Caps and Floors

Application of Put-Call Parity by
Black (1976):

d 
 Notional Value *

Basis  * e-rT *  XN ( d )  FN ( d ) 
Floorlet  
2
1



 d 


1 + F  Basis  





Then,
Floor   Floorletsi
all i
Caps & Floors Example
Assume you are concerned with rising
rates on a $100m, variable debt your
company owes in 1 year.
 Currently the variable rate is 6.5%,
and you would like to fix it for no
charge. The current forward rate is
6.65%, the riskless rate is 4.35%, and
the rate volatility is 15%.
 (Note: days = Actual/360)

Cap (Caplet)






D1= .227 N(D1)=.5898
D2 = .077 N(D2)=.5307
Black-76 = .004525 or 45 ¼ BPs
Adjustment for $1 Notional Value =
0.93765
Cap = .004243 or a bit less than 42 ½ BPs
On $100m NP = $424,284…expensive!
Floor (Floorlet)






D1= .227 N(-D1)=.4102
D2 = .077 N(-D2)=.4693
Black-76 = .003089 or 31 BPs
Adjustment for $1 Notional Value =
0.93765
Floor = .002896 or a bit less than 29 BPs
On $100m NP = $289,624
Collar
Collar
 Collar
= Buy Cap and Sell Floor
= - Cap + Floor
= - 424,284 + 289,624
= $134,660 Payment
 As F > X, Collar in-the-money.
 Fix rate at 6.5%, no higher, but none
of the benefit if lower.
 If set strike rate at 6.65%,
zero-cost collar.

Swaptions

Also usage of Black (76) extension

Payer (Call) swaption:



The right (but not the obligation) to pay the fixed rate,
and receive the floating rate in a swap of prespecified term and rate.
The right to be the swap buyer.
Receiver (Put) swaption:


The right (but not the obligation) to receive the fixed
rate, and pay the floating rate in a swap of prespecified term and rate.
The right to be the swap seller.
Payer (Call) Swaption


1 

C






1
 
1+
F
m
F
S *m






* e-rT *  FN ( d1 )  XN ( d 2 )  ,
where, m = number of periods of compunding in each year, S = term of swap,
F = Fixed rate of swap that starts at end of option life (assumes Euro-option),
and,

F
ln
d1 
X





T

2
2
T

 , while,
d 2  d1  
T
Receiver (Put) Swaption


1 

P 






1
 
1+
F
m
F
S *m






* e-rT *  XN (d 2 )  FN (d1 ) 
Swaption Example

2 year call (put) swaption on a 4 year
swap (semi-annual resets) that has a
pay fixed rate of 7%. The call strike is
7.5%, the riskless rate is 6% and rate
volatility is 20%.
Swaption Example
D1= -.1025
N(D1)=.4592
 D2 = -.3854
N(D2)=.3500
 Black-76 Call = .0052 or 52 BPs
 Adjustment for $1 Notional Value =
3.4370


Call Swaption =.01796 or a bit less than 180 BPs
Swaption Example
D1= -.1025
N(D1)=.4592
 D2 = -.3854
N(D2)=.3500
 Black-76 Put = .0097 or 97 BPs
 Adjustment for $1 Notional Value =
3.4370



Put Swaption =.03221 or a bit more than 322 BPs
Note: Put more expensive as F < X, so put (not call)
in-the-money.