Valuation and Hedging of Differential Swaps (original) (raw)
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Pricing and Hedging of Swaptions
2003
Interest rate swaps (IRS) have been widely used by the larger corporates for some time as an efficient method to manage interest rate exposure. In this way, a floating-rate borrower who expects a rise in interest rates can swap his floating rate obligation to a fixed rate obligation, thus locking in his future cost. Should he subsequently decide that rates have peaked, and that the trend is reversing, the interest obligation could be swapped back to a floating rate basis, thereby gaining advantage from the anticipated fall in rates. Swaptions first came into vogue in the mid-1980s in the US on the back of structured bonds tagged with a callable option issued by borrowers [13]. With a callable bond, a borrower issues a fixed-rate bond which he may call at par from the investor at a specific date(s) in the future. In return for the issuer having the right to call the bond issue at par, investors are offered an enhanced yield. Bond issuers often issue an IRS in conjunction with the bon...
Interest Rate Swaptions: A Review and Derivation of Swaption Pricing Formulae
In this paper we outline the European interest rate swaption pricing formula from first principles using the Martingale Representation Theorem and the annuity measure. This leads to an expression that allows us to apply the generalized Black-Scholes result. We show that a swaption pricing formula is nothing more than the Black-76 formula scaled by the underlying swap annuity factor. Firstly, we review the Martingale Representation Theorem for pricing options, which allows us to price options under a numeraire of our choice. We also highlight and consider European call and put option pricing payoffs. Next, we discuss how to evaluate and price an interest swap, which is the swaption underlying instrument. We proceed to examine how to price interest rate swaptions using the martingale representation theorem with the annuity measure to simplify the calculation. Finally, applying the Radon-Nikodym derivative to change measure from the annuity measure to the savings account measure we arrive at the swaption pricing formula expressed in terms of the Black-76 formula. We also provide a full derivation of the generalized Black-Scholes formula for completeness.
In this tutorial article, the strategies available to hedge market risks arising from different financing instruments are explained. Financial derivatives, whether futures or options have been widely applied in companies to mitigate or eliminate potential losses due to the uncertainty in interest or foreign exchange currency rates. However, the mathematical complexity of derivatives has sometimes been a barrier to non-highly specialised financial managers in understanding their foundations, advantages and ways to apply them in exposure reduction strategies. To address this issue, a practical approach to the use of derivatives is presented in this article. The swap valuation concepts and foundations of pricing are dealt with in the text, but their formal valuation techniques are described separately in the appendices. Explanations will be provided to calculate option premiums with the extensively used and free downloadable software of John Hull (Derivagem), but the mathematics involving the models used in option valuation will not be shown as they are outside the scope of this paper.