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Interest Rate Derivative Pricing

IRD Valuation Caps, Floors and Collars Swaptions

Caps and Floors Application of B-S model as modified by Black (1976):

Caps and Floors Application of Put-Call Parity by Black (1976):

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) D 1 =.227N(D 1 )=.5898 D 2 =.077N(D 2 )=.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) D 1 =.227N(-D 1 )=.4102 D 2 =.077N(-D 2 )=.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 = Buy Cap and Sell Floor Collar = - 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 pre- specified 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 pre- specified term and rate. The right to be the swap seller.

Payer (Call) Swaption

Receiver (Put) Swaption

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 D 1 = -.1025N(D 1 )=.4592 D 2 = -.3854N(D 2 )=.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 D 1 = -.1025N(D 1 )=.4592 D 2 = -.3854N(D 2 )=.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.

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