Endpräsentation Diplomarbeit

1. Endpräsentation Diplomarbeit Analysis and valuation of interest rate swap options Betreuer: Prof. Dr. Günther Pöll

2. Themes • Introduction • Market for fixed income and interest rate swaps • Basic valuation methods for fixed income assets • Basics of options and swaptions • Valuation of interest rate swap options • Conclusion

3. Introduction • The basics of fixed income assets • coupon rate • maturity date, • issued amount, • outstanding amount, • issuer, • issue date • market price, • market yield, • Contractual features and • Credit-rating category • Interest rate Swaps: Exchange of a fixed interest rate with a floating rate • Option on Interest rate Swaps: swaption • Swaptions are derivatives of swaps

4. Market for fixed income and interest swaps • Market for fixed income assets • Primary • Secondary • Participants • Issuers • Intermediaries • Investors • Key players • Governments • Central banks • Corporations • Banks • Financial institutions and dealers • Households

5. Market for fixed income and interest swaps

6. Market for fixed income and interest swaps

7. Basic valuation methods for fixed income assets • Value of continously compounded fixed deposit: • Zero-Coupon bond countinously compounded: • Yield curve given a set of bond prices

8. Basic valuation methods for fixed income assets • Forward interest rate: • For Instantenous fr, fr and yield curve are given by:

9. Basics of options and swaptions • Option gives buyer the right (not the obligation to buy (call option) or sell (put option) an aggreed quantity n of a predetermined underlying S at a specific price, the strike X at maturity T. • 3 kind of options: • European options • American options • Bermudan options • 3 price points: • at-the-money • in-the-money • out-of-the-money

10. Basics of options and swaptions • Black-Scholes-Merton model • Following example for a ۲ by T European payer swaption with fixed coupon rate c. FSR(0, ۲,T) is the forward swap rate and using A(t, ۲,T) as the numeraire leads to the following solution • Practical usage with following discount factors: D(0,1y) = 0.95, D(0,1.5y) = 0.925, D(0,2y) = 0.9, D(0,2.5y) = 0.875, D(0,3y) = 0.85 and the implied volatility is 18.5%. First step for calculating a ATM forward payer swaption is to calculate the 2-year par swap rate at 1 year foward with semiannual payment:

11. Basics of options and swaptions • Strike K equals the forward swap rate, K = 5,663. The maturity of the option is 1 year (T0 = 1) and the volatility is σ = 0.185. Plugging in Blacks formula and testing for expected value. • Final price of the swaption:

12. Valuation of interest rate swap options-factor models • Modelling yield curve and term structure • how interest rates of a given maturity evolve over time • All prices develop under the assumption of no arbitrage • Forward rates do not have to be lognormally distributed like in Black‘s formula

13. Valuation of interest rate swap options-factor models • The Vasicek model • Developement of short term interest rate r as simple mean reverting process • The Cox-Ingersoll-Ross model • Similiar like Vasicek and volatility depends of the level of r

14. Valuation of interest rate swap options-factor models • The Heath-Jarrow-Morton model • Drift term and white noise process • Forward rate is driven by the white noise process • Shock at t from R(t) influences all future rates

15. Valuation of interest rate swap options-market models • Market models are directly based on market data • Parameters set from historical data • Libor market model • Uses Libor rates as input • Swap market model • Uses swap rates as input • String market model • Interprets every distinct point at the term structure as random variable

16. Valuation of interest rate swap options-market models

17. Conclusion • Massive increase of the volume of interest rate derivatives since 2000 • Higher debt levels are the main reason for the volume increase in interest rate derivatives and swaptions • Market models with 3-4 factors are best for describing term structure