Swaption

Swaptions are options that allow the buyer, in return for a one-off premium, to enter an interest rate swap at a certain point in time (European swaption), up to a certain point in time (American swaption, extremely rare) or at specified successive points in time (Bermuda swaption) to enter. The swap is fixed in terms of its term and interest rate.

Types of swaptions

A distinction is made between payer swaptions and receiver swaptions:

• Receiver swaption (rarely also called: put swaption ): The buyer of a receiver swaption has the right to enter into a swap in which he receives a fixed interest rate and pays a variable interest rate. The receiver swaption is a hedge against falling interest rates.

• Payer swaption (seldom also called call swaption ): The buyer of a payer swaption has the right to enter into a swap in which he pays a fixed interest rate and receives a variable interest rate. The payer swaption is a hedge against rising interest rates.

The terms “call” and “put” swaption are rather uncommon in practice and their use in the literature is inconsistent.

Exercise of swaptions

With swap settlement (or physical settlement), the buyer and seller of the swaption enter into an interest rate swap when exercised.

With cash settlement, the seller pays the buyer the current cash value of the interest rate swap. This results from discounting the difference between the agreed fixed rate ( strike of the swaption) and the fixed rate currently traded on the swap market ( current swap rate ). The prevailing convention in the euro area is to discount using the current swap rate, i. H. no discounting on the yield curve.

Option price calculation

One of the most widespread models for evaluating European swaptions is the classic model by Fischer Black from 1976. It is still very popular today precisely because of its ease of understanding and simple implementation. The model was originally developed by Fischer Black and Myron Samuel Scholes in the Black-Scholes model named after them in 1973 for the valuation of stock options and assumes logarithmic-normally distributed stock prices. By applying this model to the valuation of swaptions, the assumption of the log-normal distribution of share prices is transferred to changes in the swap rate.

The price for a payer swaption with cash settlement is as follows: ${\displaystyle c}$

${\displaystyle c=a\cdot e^{-rT}\left[FN(d_{1})-XN(d_{2})\right]}$

The price for a receiver swaption with cash settlement is as follows: ${\displaystyle p}$

${\displaystyle p=a\cdot e^{-rT}\left[XN(-d_{2})-FN(-d_{1})\right]}$

With:

${\displaystyle d_{1}={\frac {\ln \left({\frac {F}{X}}\right)+\left({\frac {\sigma ^{2}}{2}}\right)T}{\sigma {\sqrt {T}}}}\quad {\text{und}}\quad d_{2}=d_{1}-\sigma {\sqrt {T}}}$

${\displaystyle a={\frac {1-\left(1+{\frac {F}{m}}\right)^{-t\cdot m}}{F}}}$

• ${\displaystyle t}$ = Term of the swap (in years)
• ${\displaystyle F}$= Forward rate of the swap
• ${\displaystyle X}$= the strike, d. H. the contractually agreed fixed rate of the swap
• ${\displaystyle r}$ = risk-free interest rate
• ${\displaystyle T}$ = Term of the option (in years until exercise)
• ${\displaystyle \sigma }$= Implied volatility of the swap's forward rate
• ${\displaystyle m}$ = Payment frequency of the fixed rate (1 = annually, 2 = every six months)

Individual evidence

1. ↑ Rolf Beike, Andreas Barckow: Risk management in financial derivatives. Control of interest rate and currency risks. 3rd updated and expanded edition. Oldenbourg, Munich et al. 2002, ISBN 3-486-25848-6 ( textbooks and manuals on money, the stock exchange, banking and insurance ).
2. ^ Atsuo Konishi, Ravi E. Dattatreya (Ed.): The Handbook of Derivative Instruments. Investment Research, Analysis and Portfolio Applications. Probus Publishing Co., Chicago IL 1991, ISBN 1-55738-154-2 ( An Institutional Investor Publication ).
3. ^ Fischer Black : The Pricing of Commodity Contracts. In: Journal of Financial Economics. 3, 1/2, 1976, ISSN  0304-405X , pp. 167-179.
4. ^ Fischer Black, Myron Scholes : The Pricing of Options and Corporate Liabilities. In: Journal of Political Economy. 81, 3, 1973, ISSN  0022-3808 , pp. 637-654.