Paul Kruehner: On a term structure approach for stock options

A usual practice for modeling stock markets is to specify a model class for the underlying and calibrate the class to the market data, i.e. one chooses the parameters in such a way that the real market option prices coincide with the theoretic option prices. However, the calibrated parameters for the different time points usually don’t coincide. That means that the chosen model class cannot explain the joint behavior of real market securities. In the spirit of [Carmona 09] we try to specify the dynamics for the options. However, special care has to be taken since arbitrage-free option prices observed at one time point have awkward restrains, see [Davis Hobson 07]. To avoid this constraints we map the possible option price configurations into a more suitable space, namely a convex cone, and specify their dynamics in this cone. Finally, we characterize absence of arbitrage and existence of an underlying dynamic generating the option price processes.

References:

[Carmona 09], Carmona, R., HJM: A Unified Approach to Dynamic Models for Fixed Income, Credit and Equity Markets, Lecture Notes in Mathematics, vol. 1919, 2009.

[Davis Hobson 07], Davis, M. and Hobson, D., The Range of Traded Option Prices, Mathematical Finance, vol. 17, 2007.