# Synthetic Relationship Between Swaps and Derivatives

The unifying theory of interest rate risk management rests upon the basis that all hedging instruments (swaps, caps, floors, collars) can be created directly or synthetically from each other.

The relationship between option contracts and swap transactions operates at an important level.

The Relationship between Option Contracts and Interest Rate Swaps

A call option can be replicated by continuously adjusting or dynamically managing a portfolio of securities on the underlying asset and cash. As the price of the asset rises, the call option equivalent portfolio would contain an increasing proportion of the assets .As the financial price of the asset decreases, the call option equivalent portfolio would reduce the holding of the assets.

More directly, options may have a direct relationship to interest rate swaps insofar as an interest rate swap can be characterized as a portfolio of purchased and sold options. Swap instruments, such as caps, floors and collars are in effect a series of option contracts.

The following will demonstrate the synthetic concept using simple mathematics.

A seller can provide an interest rate cap directly or synthetically.

Synthetically:

A cap =  Purchase floor + Swap (pay fixed rate & receive floating rate)

A quick look at the cap payoff rate will bear this relationship out. Suppose a 10% cap was sold directly. At an 11% reset, the cap would make a payment of 11%-10% =1%

At 9% reset, the cap would make no payment.

Alternatively, suppose the cap provider created the cap synthetically by purchasing a 10% floor and swapping cash flows. This synthetic 10% cap would have the following payoff rate:

At an 11% reset, the floor would be worth zero (no payment made) and the swap would pay a fixed 10% and receive a floating 11% for a net gain of 1%.

At a 9% reset, the floor would be worth  1% (a payment of11%-10% would be made) and the swap would pay a fixed 10% and receive a floating for a net result of 1%-1% =0.

Comparing the two payoff’s results in the following matrix:

 Type of Cap Reset at 9% Reset at 11% Direct 0% 1% Synthetic 0% 1%

The results are identical and show that the provider of the floor has more than one way to sell this option.

A fixed /floating swap can be provided directly or synthetically by the seller.

Synthetically:

A swap = Purchase cap and sell floor at same strike prices

Using the same numbers from the previous example, the two payoff’s result in the following matrix:

 Type of Cap Reset at 11% Reset at 9% Direct 1% -1% Synthetic 1% -1%