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

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

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

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

Market for fixed income and interest swaps

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

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

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

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:

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 = Plugging in Blacks formula and testing for expected value.  Final price of the swaption:

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

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

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

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

Valuation of interest rate swap options-market models

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