Browsing by Subject "options"

This thesis covers martingale based theory of interest modelling. Short-rate models and their characteristics are introduced to deal with the pricing of zero-coupon bonds, including a defaultable bond. In order to back the results concerning the characteristics of the short-rate models, there are few simulations done with Matlab. The thesis covers a formulation for a necessary condition for Heath-Jarrow-Morton-model (HJM) to have a martingale property. As a practical example of zero-coupon bonds, a self-financing hedging strategy for the bond call option is presented. The last part of the thesis handles the derivatives of Secured Overnight Financing Rate (SOFR) and The London Inter-Bank Offered Rate (LIBOR). It is shown that the Gaussian cumulative distribution may be used in forming the arbitrage free price of SOFR and LIBOR derivatives. Also a hedging strategy for the swaption is introduced. The receipt to manage these merely complex looking equations is quite simple. The effective use of martingale representation theorem together with Girsanov-Meyer’s theorem and forward measure is the key. The first theorem guarantees that the square-integrable martingale Mt admits to representation dMt =θtdBt, where Bt is a Brownian motion and θt is unique adapted square-integrable stochastic process. The second one guarantees us a way to find Brownian motion process under the new measure. It is of the form dB∗ t = Htdt+dBt, In addition to the concepts of mathematical finance there are results governing the probability theory. For example, the construction of Wiener and Itô integrals, Lèvy’s characterization theorem and already mentioned Martingale represantation theorem with proofs are covered.