Function DefinitionsTODO: define some functions such as min, max, bound, etc.
Interest Rate Cap, Floor and StraddleAn interest rate cap is a derivative in which the buyer receives money at the end of each cash flow period in which the specified floating rate
R exceeds the agreed strike price
K. An example of a cap would be an agreement to receive money for each month the LIBOR rate exceeds 2.5%.
The interest rate cap can be analyzed as a series of European call options or caplets which exists for each cash flow period the cap agreement is in existence. In formulas, the caplet rate that will be applied on the notional for a cash flow period is:
R_{payoff} = max(R - K, 0)Where
R is the underlying floating rate, and
K is the cap strike.
An interest rate floor is a series of European put options or floorlets on a specified floating rate
R. The buyer of the floor receives money if on the maturity of any of the floorlets, the fixed floating rate
R is below the agreed strike
K of the floor. In formulas, the floorlet rate that will be applied on the notional for a cash flow period is:
R_{payoff} = max(K - R, 0)An interest rate straddle is a portfolio of an interest rate cap and an interest rate floor on the same floating rate
R with the same strike
K. The payoff rate of a straddle cash flow (a straddle-let) is:
R_{payoff} = max(R - K, 0) + max(K - R, 0) = abs(R - K)An interest rate straddle can be replicated by a portfolio of an interest rate cap and an interest rate floor.
Leveraged Cap/FloorA capped/floored floating rate is a floating rate
R with an upper bound
U and a lower bound
L. The payoff rate of the capped/floored floating rate is:
R_{payoff} = bound(R, L, U)
= U (if R > U), or
R (if L <= R <= U), or
L (if R < L)
We can re-write the formula above as the following:
R_{payoff} = R + max(L - R, 0) - max(R - U, 0)So, the capped/floored payoff rate may be replicated using a portfolio of:
-
+1 floating rate
R,
-
+1 floor on the floating rate
R with a strike of
L,
-
-1 cap on the floating rate
R with a strike of
U.
In a capped/floored floating rate, if we replace the underlying floating rate
R by a structured rate
(L * R + fix), where the leverage
L and the fixed rate
fix are constants, we can obtain a leveraged cap/floor. The payoff rate is:
R_{payoff} = bound(L * R + fix, L, U)The payoff rate may be replicated using a portfolio of:
-
+1 leveraged floating rate
R', where
R' = L * R + fix,
-
+1 floor on the leveraged floating rate
R' with a strike of
L,
-
-1 cap on the leveraged floating rate
R' with a strike of
U.
Leveraged Spread Cap/FloorIn a capped/floored floating rate, if we replace the underlying floating rate
R by a leveraged spread
(L * (L1 * R1 - L2 * R2) + fix), where
R1 and
R2 are two floating rates (such as 3-month LIBOR and 6-month LIBOR), and the leverages
L,
L1,
L2, and the fixed rate
fix are constants, we can obtain a leveraged spread cap/floor. The payoff rate is:
R_{payoff} = bound(L * (L1 * R1 - L2 * R2) + fix, L, U)Digit Cap/FloorThe payoff rates of a digit cap and a digit floor are defined by the following formulas:
For digit cap:
R_{payoff} = fix (if R > K), or
0 (otherwise)
For digit floor:
R_{payoff} = fix (if R < K), or
0 (otherwise)
By using a proper epsilon
e which is small enough, we can approximately replicate a digit cap using:
-
+(fix/2e) cap on the floating rate
R with a strike of
(K - e),
-
-(fix/2e) cap on the floating rate
R with a strike of
(K + e).
We can approximately replicate a digit floor the same way, using:
-
+(fix/2e) floor on the floating rate
R with a strike of
(K + e),
-
-(fix/2e) floor on the floating rate
R with a strike of
(K - e).
Barrier Cap/FloorThere are four types of barrier cap/floor: cap-up-and-in, cap-up-and-out, floor-down-and-in, and floor-down-and-out. Their payoff rates are
defined by the following furmulas:
For cap-up-and-in:
R_{payoff} = max(R - K, 0) (if R < B), or
0 (otherwise)
For cap-up-and-out:
R_{payoff} = max(R - K, 0) (if R > B), or
0 (otherwise)
For floor-down-and-in:
R_{payoff} = max(K - R, 0) (if R < B), or
0 (otherwise)
For floor-down-and-out:
R_{payoff} = max(K - R, 0) (if R > B), or
0 (otherwise)
Where
R is the floating rate,
K is the strike, and
B is the barrier. In a barrier cap, the barrier should be greater than the strike
(B > K). In a barrier floor, the barrier should be less than the strike
(B < K).
We can use the following portfolio to replicate a cap-up-and-in:
-
+1 cap on the floating rate
R with a strike of
K,
-
-1 digit cap on the floating rate
R with a strike of
B and a fixed rate of
(B - K),
-
-1 cap on the floating rate
R with a strike of
B.
Since a portfolio of a cap-up-and-in and a cap-up-and-out on the same floating rate
R with the same strike
K and the same barrier
B gives the payoff of a cap on the floating rate
R with the strike
K, we can replicate a cap-up-and-out using the following portfolio:
-
+1 cap on the floating rate
R with a strike of
K,
-
-1 cap-up-and-in on the floating rate
R with a strike of
K and a barrier of
B.
We can use the following portfolio to replicate a floor-down-and-out:
-
+1 floor on the floating rate
R with a strike of
K,
-
-1 digit floor on the floating rate
R with a strike of
B and a fixed rate of
(K - B),
-
-1 floor on the floating rate
R with a strike of
B.
And the following portfolio to replicate a floor-down-and-in:
-
+1 floor on the floating rate
R with a strike of
K,
-
-1 floor-down-and-out on the floating rate
R with a strike of
K and a barrier of
B.
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