A lattice model for pricing interest rate derivatives
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Keywords
Interest-rate derivatives ; Short-rate models ; Term structure of interest rates ; Binomial tree ; Trinomial tree ; Hull-White model ; Zero-coupon curve ; Lattice calibration ; Bond options ; Backward inductionAbstract
This thesis investigates lattice methods for pricing interest rate derivatives when the short rate follows a one-factor arbitrage-free term structure model. Two recombining trees are considered in the analysis. The first one is a symmetric binomial lattice and the second one is a mean-reverting trinomial lattice. Both models are calibrated to the AAA euro area government bond yield curve on 31 January 2024.
The calibration is performed in order to match the observed zero-coupon term structure exactly. This is achieved by calculating recursively the sequence of shift parameters 𝑎! at each time step of the tree. After the calibration stage, the models are used to price European call options on zero-coupon bonds. The valuation is done using backward induction through the lattice.
The numerical results show that both models reproduce the market discount factors with negligible differences. However, when the trinomial tree is constructed without restrictions, some transition probabilities become negative or greater than one. To avoid this issue, a truncated recombining trinomial tree is implemented together with boundary adjustments, while the exact fit of the initial curve is maintained.
The empirical findings show that the binomial tree produces consistently higher option prices than the truncated trinomial tree.