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.