Numerical Methods for the Markov Functional Model

Simon Johnson proposes some numerical methods for efficient implementation of the 1- and 2-factor Markov Functional models of interest rate derivatives. These methods allow a sufficiently rapid implementation of the standard calibration method, that joint calibration to caplets and swaptions becomes possible within reasonable CPU time. Prices for Bermudan swaptions generated within the Markov Functional model are found to be very close to market consensus prices. Bermudans are therefore a good example of a product ideally suited to this model.

The Libor Market Model of Brace Gatarek and Musiela (BGM) (1997) is the market standard model for pricing and hedging exotic interest rate derivatives. Its advantages include model parameters which are easy to interpret in terms of financial variables, ability to define realistic correlation dynamics, and the ability to price essentially any callable Libor exotics by means of Monte Carlo valuation.

There are two main difficulties in practical implementation of the BGM Model. Firstly the drift is strongly state-dependent and cannot be reduced to a low-dimensional Markovian form. Whilst simulations with long time-steps can still be performed using a suitable differencing scheme (Hunter et al., 2001; Joshi, 2005), this does mean that Monte Carlo simulation is the only practical method for pricing. One must work hard to achieve acceptable convergence, particularly when computing hedge ratios of callable Libor exotics Piterbarg (2004). Secondly, although calibration to market prices of caplets is of course trivial, there is a great deal of debate surrounding the best way to perform global calibration of a BGM model to market prices of all at-the-money swaptions.

The choice of calibration method becomes even richer when extensions to BGM, including displaced diffusion, local volatility or stochastic volatility are considered Rebonato (2004). However when the model is used in a production environment, it is by no means simple to ensure that a small change in market quotes give rise to a correspondingly small change in model parameters and hence stable hedges are obtained.

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