Fries, Christian P.: Markov Functional Modeling of Equity, Commodity and other Assets. Version 0.8.1. March 31, 2006. [276 KB].

1 | Introduction | 3 |

2 | Markov Functional Model with Asset Numéraire | 3 |

2.1 | Markov Functional Assumption | 3 |

3 | Example: The Black-Scholes Model | 4 |

4 | Numerical Calibration to a Full Two Dimensional European Option Smile Surface | 5 |

4.1 | Market Price | 5 |

4.2 | Model Price | 5 |

4.3 | Solving for the Functional | 6 |

5 | Interest Rates | 6 |

5.1 | A Note on Interest Rates and the No-Arbitrage Requirement | 6 |

5.2 | Where are the Interest Rates? | 6 |

6 | Model Dynamics | 7 |

6.1 | Introduction | 7 |

6.1.1 | Time Copula | 8 |

6.1.2 | Time-Discrete Markovian Driver | 8 |

6.2 | Interest Rate Dynamics | 8 |

6.2.1 | Example: Black-Scholes Model with a Term Structure of Volatility | 8 |

6.2.2 | Calibration to arbitrary Interest Rate Dynamics | 10 |

6.3 | Forward Volatility | 10 |

6.3.1 | Example: Black-Scholes Model with a Term Structure of Volatility | 10 |

6.3.2 | Example: Exponential Decaying Instantaneous Volatility | 11 |

7 | Implementation | 11 |

7.1 | Calibration of the Functional Forms | 11 |

8 | Conclusion | 13 |

List of Symbols | 14 | |

References | 15 | |

Notes | 15 |

In this short note we show how to setup a one dimensional single asset model, e.g. equity model, which calibrates to a full (two dimensional) implied volatility surface. We show that the efficient calibration procedure used in LIBOR Markov functional models may be applied here too. In a addition to the calibration to a full volatility surface the model allows the calibration of the joint asset-interest rate movement (i.e. local interest rates) and forward volatility. The latter allows the calibration of compound or Bermudan options.

The Markov functional modeling approach consists of a Markovian driver process *x* and a mapping functional representing the asset states *S(t)* as a function of *x(t)*. It was originally developed in the context of interest rate models, see [Hunt Kennedy Pelsser 2000]. Our approach however is similar to the setup of the hybrid Markov functional model in spot measure, as considered in [Fries Rott 2004].

For equity models it is common to use a deterministic Numéraire, e.g. the bank account with deterministic interest rates. In our approach we will choose the asset itself as Numéraire. This is a subtle, but crucial difference to other approaches considering Markov functional modeling. Choosing the asset itself as Numéraire will allow for a very efficient numerically calibration procedure. As a consequence interest rates have to be allowed to be stochastic, namely as a functional of *x* too. The Black-Scholes model with deterministic interest rates is a special case of such a Markov functional model.

The most general form of this modeling approach will allow for a simultaneous calibration to a full two dimensional volatility smile, a prescribed joint movement of interest rates and a given forward volatility structure.

- MSC-class: 65C05 (Primary), 68U20, 60H35 (Secondary).
- ACM-class: G.3; I.6.8.
- JEL-class: C15, G13.
- Keywords: Markov functional model, implied modeling, equity, commodity, lattice model.

Comments and suggestions welcomed at email@christian-fries.de.