Extended Libor Market Models with Affine and Quadratic Volatility

The market model of interest rates specifies simple forward or Libor rates as lognormaly distributed, their stochastic dynamics has a linear volatility function. This model is extended to quadratic volatility which is the product of a quadratic polynomial and a level-independent covariance matrix. This extension of the Libor market model allows for closed form cap pricing formulae and I give examples for the possible shapes of implied volatilities. For affine specification I derive a new approximative swaption pricing formula and discuss the properties of the approximation. 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.