## Financial asset risk and valuation methods

The recent financial crisis brought to the foreground the importance of a correct and robust valuation of financial assets and their underlying risk. The Black-Scholes theory on current valuation methods implies that the risk involved in a financial asset can be eliminated through a proper dynamic hedging strategy.

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The project's central concept was the notion of volatility, which is the quadratic variation of the price process of the underlying asset. The paths of the price processes are modelled by functions with a critical Hölder exponent of 1/2, on which the arbitrage theory of pricing and hedging of derivative securities is based on.

Empirical evidence has also suggested that the 'true' critical exponent, or rather its statistical estimate, is slightly higher than 1/2 but also significantly lower than 1. The assumption of a critical Hölder exponent different from 1/2 leads to different models, such as the so-called fractional Brownian motion, which was proposed more than 50 years ago by B. Mandelbrot. However, since these models allow for arbitrage, they don't fit into the economically quite convincing no arbitrage theory.

RIVAL (Risk and valuation of financial assets: A robust approach) developed a way to reconcile both approaches by introducing transaction costs. It has been already shown that the assumption of arbitrarily small transaction costs make models such as fractional Brownian motion free of arbitrage.

Although the idea of replication makes no economic sense under the assumption of transaction costs, the concept of portfolio optimisation is, however, more appropriate under a mathematical and economic point of view.

It is assumed that there is a shadow price associated with a portfolio optimisation problem under transaction costs. Though the original price process may fail to have volatility, the shadow price always does. Additionally, the optimal portfolio for the original price process under transaction costs coincides with the optimal portfolio for the shadow price process without transaction costs, establishing a connection between the two approaches.

There is also a mathematical consequence of the above result where, for a given fractional Brownian motion, a diffusion process touching the paths of the fractional Brownian motion in a one-sided way can be observed. A mathematical, surprising aspect is that there is no reflection or local time involved. This result is considered a prime example of a purely mathematical insight that originates from a very applied problem.

Project outcomes were jointly published with another postdoctoral project in a series of four papers, one of which was also jointly published with a third pre-doctoral project.

Empirical evidence has also suggested that the 'true' critical exponent, or rather its statistical estimate, is slightly higher than 1/2 but also significantly lower than 1. The assumption of a critical Hölder exponent different from 1/2 leads to different models, such as the so-called fractional Brownian motion, which was proposed more than 50 years ago by B. Mandelbrot. However, since these models allow for arbitrage, they don't fit into the economically quite convincing no arbitrage theory.

RIVAL (Risk and valuation of financial assets: A robust approach) developed a way to reconcile both approaches by introducing transaction costs. It has been already shown that the assumption of arbitrarily small transaction costs make models such as fractional Brownian motion free of arbitrage.

Although the idea of replication makes no economic sense under the assumption of transaction costs, the concept of portfolio optimisation is, however, more appropriate under a mathematical and economic point of view.

It is assumed that there is a shadow price associated with a portfolio optimisation problem under transaction costs. Though the original price process may fail to have volatility, the shadow price always does. Additionally, the optimal portfolio for the original price process under transaction costs coincides with the optimal portfolio for the shadow price process without transaction costs, establishing a connection between the two approaches.

There is also a mathematical consequence of the above result where, for a given fractional Brownian motion, a diffusion process touching the paths of the fractional Brownian motion in a one-sided way can be observed. A mathematical, surprising aspect is that there is no reflection or local time involved. This result is considered a prime example of a purely mathematical insight that originates from a very applied problem.

Project outcomes were jointly published with another postdoctoral project in a series of four papers, one of which was also jointly published with a third pre-doctoral project.