Extended Libor Market Models with Affine and Quadratic Volatility

The market model of interest rates specifies simple forward or Libor rates as lognormally distributed, their stochastic dynamics has a linear volatility function. In this paper, the model is extended to quadratic volatility functions which are the product of a quadratic polynomial and a level-independent covariance matrix. The extended Libor market models allow for closed form cap pricing formulae, the implied volatilities of the new formulae are smiles and frowns. We give examples for the possible shapes of implied volatilities. Furthermore, we derive a new approximative swaption pricing formula and discuss its properties. The model is calibrated to market prices, it turns out that no extended model specification outperforms the others. The criteria for model choice should thus be theoretical properties and computational efficiency.