# MFM

**The Markov-functional Approach**

The class of Markov-functional models (MFMs) provides a framework that can be used to define interest-rate models of any finite dimension that can be calibrated to any arbitrage-free formula for caplet or swaption prices. Low-dimensional MFMs are of particular interest as they allow for efficient implementation on a low-dimensional grid. As these models provide significant advantages over other market models they are widely accepted and used in the banking industry.

**One-dimensional Markov-functional models**

A one-dimensional Markov-functional model is specified by a driving process which is Markovian under the equivalent martingale measure corresponding to the chosen numeraire and the arbitrage-free pricing formulae of the chosen calibrating vanilla instruments. LIBOR- and swap-based versions of the model with a one-dimensional Gaussian driver were first introduced in 2000 by Hunt, Kennedy and Pelsser. These versions of the models are still widely used in practice. When using the models to price Bermudan swaptions the driving process should be parametrized by time rather than by maturity (Kennedy and Pham (2013)) so that the models have suitable hedging properties. Efficient algorithms that can be used to implement an MFM driven by a not-necessarily-Gaussian driver under terminal and spot measure are described in Gogala and Kennedy (2019). In principle all one-dimensional interest-rate models (for example quasi-Gaussian and Affine LIBOR models) can be viewed and implemented from the Markov-functional perspective. With the separation of choice of driver and marginal distributions of the calibrating instruments the Markov-functional approach offers much modelling flexibility. In principle observed smiles can be reproduced exactly and, through the choice of driving process, the joint distribution of market rates at their setting dates can be chosen. It is this joint distribution that specifies the final form of the model. Care must be taken to ensure a suitable evolution of the implied volatility surface is achieved for the particular combination of driver and marginal distributions chosen. A copula approach (Gogala and Kennedy (2019)) enables us to determine when two Markov-functional models are equivalent and provides insight into how to express the driving process in a form that facilitates implementation.

**Prior model and multi-dimensional Markov-functional models**

Multi-dimensional Markov-functional models can be specified by introducing the idea of a prior model (Kennedy (2010)). The prior model expresses each market rate as a function of the driving process which is now of dimension greater than one. The prior model is chosen to capture the basic dynamics of the market but may admit arbitrage. When the functional fitting is done to fit the chosen implied marginal distributions, referred to as the Markov-functional sweep, no-arbitrage conditions are enforced. If the prior model was well chosen the Markov-functional sweep will be a small perturbation of the prior model to enforce no arbitrage. The idea of a prior model can be illustrated by considering the SDEs which specify LIBOR or swap market models. If we assume a market model is driven by two correlated Brownian motions, the volatility structure is separable and if we ignore the drift in the SDE for each market rate, the resulting equations specify a prior model which is a function of a two dimensional driving process. This prior model will admit arbitrage. If we now carry out the Markov-functional sweep and fit implied distributions corresponding to Black’s formula for caplets in the LIBOR case or swaptions in the swap case we will have an arbitrage-free Markov-functional model which is efficient to implement. (Details for the implementation of a two-factor LIBOR Markov-functional model can in Bennett and Kennedy (2005).)

**Link between Market models and Markov-functional models**

One- and two-factor separable LIBOR and swap market models are very close numerically to the analogous Markov-functional models (Bennett and Kennedy (2005) for the lognormal case, Gogala and Kennedy (2017) for the local volatility case). The link between separable market models and the analogous Markov-functional models has proved strong enough to base a calibration routine around, even in the case when stochastic volatility is included (Guo and Kennedy (2017)). This link between market models and Markov-functional models provides the practitioner with a well-understood starting point when developing a tailor-made Markov-functional model for a particular pricing problem. The assumption of separability though a restriction on the flexibility of a model is not as restrictive as it might first appear (Gogala and Kennedy (2017)). Conversely it should be noted that in addition to offering much more flexibility in modelling choice the Markov-functional approach offers a model very close to a high dimensional separable market model but which can be implemented efficiently.

**Higher dimensional Markov-functional models**

A high-dimensional Markov-functional model which combines the Markov-functional approach with the Monte-Carlo method and allows for a separate Gaussian driver for each market rate of interest is described in Kaisajuntti and Kennedy (2013). This model is relatively efficient for products which depend on LIBORs at their setting dates. A three-factor cross-currency model suitable for modelling long-dated FX products (for example a Power reverse dual currency swap) has been developed in Gogala and Kennedy (2020) and code is available here. The development of this model is based on the prototype algorithm formulated in Gogala and Kennedy (2019).

**References:**

**References:**

* MFM with a Non-Gaussian Driver*

1. Gogala, J and Kennedy J (2020) A Cross-currency Markov-functional model with FX volatility skew, (working paper). [pdf], [code]

2. Gogala, J and Kennedy, J (2019) One-dimensional Markov-functional models driven by a non-Gaussian driver, Journal of Computational Finance, Vol. 23, Issue 3 (pp 1-39). [pdf]

*Multidimensional MFM*

3. Kennedy J.E (2010) Markov-functional models and Numeraire approach, Encyclopedia of Quantitative Finance edited by Rama Cont, Wiley.

4. Kaisajuntti, L and Kennedy, J (2013) An n-dimensional Markov-functional Interest Rate Model, Journal of Computational Finance, Volume 17, Issue 1.[pdf]

5. Gogala, J and Kennedy, J (2017) Classification of two-and three-factor time-homogeneous separable LMMs, Int. J. Theor. Appl. Finan. Volume 20, No. 02.[pdf]

6. Guo,C and Kennedy J.E.(2017) A Markov-Functional Model with stochastic volatility, (working paper)

7.Bennett, Michael N. and Kennedy, J. E. (2005) Two-factor LIBOR Markov-functional model (working paper, code available on request)

*One-factor MFM*

8. Hunt, P.J. , Kennedy, J.E. Kennedy and Pelsser, A.J. (2000) Markov-Functional Interest Rate Models, Finance and Stochastics volume 4, pages391–408 [pdf]

9. Bennett, Michael N. and Kennedy, J. E. (2005) A comparison of Markov-functional and market models : the one-dimensional case. The Journal of Derivatives, Vol.13 (No.2). pp. 22-43. [pdf]

10. Kennedy, J and Pham, D (2013) Implications for Hedging of the choice of driving process for one-factor Markov-functional models, Int. J. Theor. Appl. Finan. Volume 16, No. 05.[pdf]