Risk Latte - Interest Rate Swaps and the Gamma Problem

Interest Rate Swaps and the Gamma Problem

Rahul Bhattacharya
Feb 23, 2005

Do swaps in general and Interest Rate (IR) swaps in particular have gamma?

Take an interest rate swap for example where there is floating rate payment depending on the level of floating interest rate (say, 3 month LIBOR) and fixed rate F, which is the swap rate. The payoff of the swap is:

The delta is the first partial derivative of this equation with respect to F. And clearly delta is non-zero. It will be positive or negative depending on whether the dealer is a fixed rate payer or fixed rate receiver. All swaps have deltas. Mathematically, it can be written as:

The gamma is the second partial derivative with respect to F, which in this case will be zero because F, the fixed rate has a power of only one in the above equation. Gamma is the derivative of delta with respect to F and since the equation for delta will have no F in it the value of the derivative of delta with respect to F will be zero. Remember, the derivative of a constant is zero. Mathematically, this can be written as:

Thus given the above variables the swaps can be said to have zero gamma.

In fact if we look at the partial derivatives of the NPV with respect to both the floating rate, LIBOR and the discount rate the swap will have both delta as well as the gamma. In practice, delta and gamma are calculated by shifting both the fixed and the floating rate by one and two basis points and then calculating the sensitivities. A one basis point shift gives the delta of the payoff and a two basis point shift helps us to calculate the change in the delta or in other words the gamma. A one basis point shift in the interest rate, either just one rate or the entire curve, is mathematically equivalent to the delta of the swap as stated above.

With respect to the floating rate the delta is:

and the gamma is:

Therefore, swaps can have gamma, depending on how we are calculating it. This gamma is almost always a very small number and most traders and risk managers ignore it. Generally speaking, swaps are considered linear instruments but they are actually quasi-linear.

Another way to look at this is that a swap payoff is a sum (with appropriate sign adjustment for long and short) of the payoff for a floating rate bond and a fixed rate bonds. If we look at the first equation above then we see that it is simply the sum of the payoffs of two bonds on with a floating rate x and the other with a fixed rate F. And we know that bond payoff has a second mathematical derivative with respect to the short rate, which can be seen as the discount rate in the above equation for the NPV of the swaps. Bonds have first mathematical derivative which is the duration of the bond (analogous to delta) and the second mathematical derivative which is the convexity or the curvature (analogous to gamma). There since bonds have gamma or a gamma-type measure swaps which is a combination of bond payoffs should also have gamma.

As mentioned above IR Swaps have gamma, which can be calculated using shifting the short rate curve but usually this gamma is ignored because it is very small.

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