Fries, Christian P.; Rott, Marius G.: Cross Currency and Hybrid Markov Functional Models (PDF, 282 KB) (Version 1.5.1 - 04.05.2004).
An earlier version of this paper is available on SSRN.
| 1 | Introduction | 2 |
| 2 | One-factor single-currency models | 3 |
| 2.1 | Markov-functional model under the terminal measure | 3 |
| 2.1.1 | The LIBOR model | 4 |
| 2.1.2 | Calibration of the Markov-functional model under terminal measure | 5 |
| 2.1.3 | Backward induction step | 5 |
| 2.2 | Markov-functional model under the spot measure | 5 |
| 2.2.1 | Calibration of the Markov-functional model under spot measure | 6 |
| 2.2.2 | Forward induction step | 7 |
| 2.2.3 | Dealing with the path-dependency of the numéraire | 7 |
| 2.2.4 | Efficient calculation of the LIBOR functional from given market prices | 7 |
| 2.3 | Remark on the implementation | 8 |
| 2.3.1 | Fast calculation of price functionals | 9 |
| 2.3.2 | Discussion on the implementation of the Markov-functional model under terminal and spot measure | 9 |
| 2.4 | Change of numéraire in a Markov-Functional Model | 10 |
| 3 | The two-factor cross-currency model (stochastic FX rates) | 12 |
| 3.1 | Discrete approximations of the driving processes | 12 |
| 3.1.1 | A note on the importance of 'early' discretization | 12 |
| 3.2 | The functional form and the derivation of the drift | 13 |
| 3.2.1 | A note on the functional form | 14 |
| 3.3 | Examples for analytical drift calculation for special functional forms | 14 |
| 3.3.1 | Linear functional form | 14 |
| 3.3.2 | Exponential functional form | 14 |
| 3.3.3 | Other functional forms | 15 |
| 3.4 | Calibration to FX option prices | 15 |
| 4 | The three-factor cross-currency model | 17 |
| 5 | Other hybrid Markov-functional models | 19 |
| A | Auxiliary calculations | 20 |
In this paper we consider cross currency Markov functional models and their calibration under the spot measure. Hunt, Kennedy and Pelsser [1] introduced a single currency Markov functional interest rate model in the terminal measure and showed how to efficiently calibrate it to LIBOR or swaprate options. Building upon their work we will present a multi-factor cross currency LIBOR model under different measures. We view zero correlated FX spot and LIBOR rates as a natural starting point. Under this assumption we don't need a change of numéraire drift correction and the functionals of the foreign currency rates under the domestic numéraire are identical to the functionals of the foreign currency model under its (foreign) spot measure. This provides the motivation for first deriving a spot measure version of single currency Markov functional model. We will show that in the spot measure it is possible to formulate and implement a very efficient calibration procedure comparable to that provided by Hunt, Kennedy and Pelsser [1] for the terminal measure.
Combining single currency Markov functional interest rate models with a Markov functional FX spot model we build two and three factor cross currency models. Relaxing the zero correlation assumption is technically quite simple, but it entails considerable additional computational costs, mainly for the calibration of the model to FX options. To circumvent this problem we suggest a more efficient approximate procedure, which seems to work quite well for low correlations.
[1] Hunt, Phil J.; Kennedy, Joanne E.; Pelsser, Antoon: Markov-Functional Interest Rate Models.