Final Report Summary - RIVAL (Risk and Valuation of Financial Assets: A Robust Approach)
The present research is motivated by a basic problem in Mathematical Finance. A central concept in this theory is the notion of volatility. Mathematically speaking, this is the quadratic variation of the price process of the underlying asset. In particular, the paths of the price processes are modelled by functions with a critical Hölder exponent of ½. The no arbitrage theory of pricing and hedging of derivative securities is fundamentally based on this assumption. In fact, it was shown in 1994 by F. Delbaen and the PI that, for continuous price processes, the no arbitrage paradigm is necessarily violated if the critical Hölder coefficient is different from ½.
On the other hand there is some empirical evidence that the “true” critical Hölder exponent, or rather its statistical estimate based on real world data, is slightly higher than ½ (but, of course, still far away from 1 which corresponds to smooth functions). The assumption of a critical Hölder exponent different from ½ leads to different models, such as the so-called “fractional Brownian motion” which was proposed more than 50 years ago by B. Mandelbrot. But the crux is that these models always allow for arbitrage, as mentioned above, and therefore do not fit into the economically quite convincing no arbitrage theory.
In the present project we develop a way out of this deadlock and try to reconcile these two approaches. The idea is to introduce transaction costs. It has been shown by the PI and several co-authors that the assumption of (arbitrarily small) transaction costs makes models such as fractional Brownian motion free of arbitrage.
While the idea of replication, which is central in the Black-Scholes theory, does not make economic sense under the assumption of transaction costs (there are mathematical theorems which make this statement precise), the concept of portfolio optimization does make good sense from a mathematical as well as from an economic point of view.
A main result of the present project was established in a series of four papers of the PI jointly with the project PostDoc C. Czichowsky (one of these papers is also joint with the project PreDoc J. Yang). Under natural assumptions there is a “shadow price” associated to a portfolio optimization problem under transaction costs. While the original price process may fail to have a “volatility”, i.e. a non-trivial quadratic variation, the shadow price always does so. The optimal portfolio for the original price process under transaction costs then coincides with the optimal portfolio for the shadow price process without transaction costs. This result establishes a bridge between the above sketched two approaches.
We also mention a purely mathematical consequence of the above result: For a given fractional Brownian motion one may find a diffusion process which, with high probability, touches the paths of the fractional Brownian motion in a one-sided way. The mathematically surprising aspect is that there is no reflection or local time involved. We consider this result as a prime example of a purely mathematical insight originating from a very applied problem.
On the other hand there is some empirical evidence that the “true” critical Hölder exponent, or rather its statistical estimate based on real world data, is slightly higher than ½ (but, of course, still far away from 1 which corresponds to smooth functions). The assumption of a critical Hölder exponent different from ½ leads to different models, such as the so-called “fractional Brownian motion” which was proposed more than 50 years ago by B. Mandelbrot. But the crux is that these models always allow for arbitrage, as mentioned above, and therefore do not fit into the economically quite convincing no arbitrage theory.
In the present project we develop a way out of this deadlock and try to reconcile these two approaches. The idea is to introduce transaction costs. It has been shown by the PI and several co-authors that the assumption of (arbitrarily small) transaction costs makes models such as fractional Brownian motion free of arbitrage.
While the idea of replication, which is central in the Black-Scholes theory, does not make economic sense under the assumption of transaction costs (there are mathematical theorems which make this statement precise), the concept of portfolio optimization does make good sense from a mathematical as well as from an economic point of view.
A main result of the present project was established in a series of four papers of the PI jointly with the project PostDoc C. Czichowsky (one of these papers is also joint with the project PreDoc J. Yang). Under natural assumptions there is a “shadow price” associated to a portfolio optimization problem under transaction costs. While the original price process may fail to have a “volatility”, i.e. a non-trivial quadratic variation, the shadow price always does so. The optimal portfolio for the original price process under transaction costs then coincides with the optimal portfolio for the shadow price process without transaction costs. This result establishes a bridge between the above sketched two approaches.
We also mention a purely mathematical consequence of the above result: For a given fractional Brownian motion one may find a diffusion process which, with high probability, touches the paths of the fractional Brownian motion in a one-sided way. The mathematically surprising aspect is that there is no reflection or local time involved. We consider this result as a prime example of a purely mathematical insight originating from a very applied problem.