Endpräsentation Diplomarbeit

Analysis and valuation of interest rate swap optionsBetreuer: 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

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 = 0.185. 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