Stefan Rostek

More than You Ever Wanted to Know About the VIX: Bringing Together Serial Correlation and Volatility Clustering

In this paper, I show that the volatility index VIX is not model-free as soon as the diff usion term is not Brownian motion even when correcting for jumps. In a stock index model that allows for temporary periods of under- or overreaction, such as a multifractional model, a wrong annualization procedure associated with the VIX, may lead to severe mispricings as the three concepts of the VIX, the volatility per annum and (the square-root of) realized variance do not coincide any longer.As a byproduct, this article proves that multifractionality is a parsimonious and economically sound explanation for the observable phenomenon of volatility clustering as it suggests that clusters of abnormally high (low) volatility can be attributed to periods of underreaction (overreaction). In this way, the paper brings together two observable patterns in fi nancial markets: the transitory existence of periods with serially correlated asset returns and the phenomenon of volatility clustering.

Explaining the Volatility Surface: A Closed-Form Solution to Option Pricing in a Fractional Jump-Diffusion Market

This paper prices European options in a framework that captures both non-normality of returns and serial correlation within financial time series. The underlying security dynamics are driven by a jump-diffusion process where the diffusion part is fractional Brownian motion while jumps exhibit a double-exponential distribution. These model characteristics suffice to overcome most of the evident drawbacks of the classical Black-Scholes setting, while the parsimony of my model still ensures analytical tractability. Due to market incompleteness, I suggest an equilibrium model a la Brennan (1979). I derive a closed-from solution to the problem, which contains the Black-Scholes pricing formulae and the formulae of Kou (2002) as limit cases. As an intuitive illustration of the models power, I choose the phenomenon of volatility surfaces: I show that the derived formulae are able to reflect observable patterns of real market data as the model entails a smile over moneyness as well as a non-flat term structure of implied Black-Scholes volatilities.

Estimating LGD with Stochastic Collateral

This article addresses to the appropriate modeling of loss given default (LGD) for the retail business sector. We assume small or mid-size loans that are assigned in a standardized way and collateralized by residential or commercial property. The focus on this specific type of loans entails two major advantages: Firstly, reduction of complexity is followed by easier-to-grasp methodology and increased handiness of results when comparing with other recent approaches in the field. Secondly, the focussing allows to take into account the characteristic properties of the housing market and its underlying uncertainty and so choose a tailor-made modeling for the collateral. The choice of an exponential Ornstein-Uhlenbeck diffusion as the stochastic process of the collateral combines the desirable features with the charm of analytical solvability which seems to be of advantage as regards to acceptance among practitioners. Further key improvements of this approach are the explicit consideration of loan ranking, the disentanglement of the time of default and the time of liquidation as well as the introduction of liquidation cost.

The Valuation of M&A Targets by Relative Indifference Pricing

In this paper, we promote the approach of relative indifference pricing as a conceptual enhancement of real options theory whenever incompleteness comes into play. As until now the discussion of this concept is limited to the field of mathematical finance, our goal is to stimulate its diffusion amongst business researchers. In a simplified example, we derive M&A target prices that differ for the sell and buy side even if both sides are equally risk-averse and share identical information. Moreover, our approach reveals investors current level of engagement in the very business field to be another key determinant of the price.

Fractional Brownian Motion and the Inherent Exclusion of Arbitrage

Over the last years, the usage of fractional Brownian motion for financial models was stuck. The favorable time-series properties of fractional Brownian motion exhibiting long-range dependence came along with an apparently insuperable shortcoming: the existence of arbitrage. In this article, we solve this dilemma: We provide a fractional analogue to the work of Sethi and Lehoczky (1981) thereby confirming that fractional Brownian motion and continuous tradability are incompatible. In the light of a market microstructure perspective to fractional Brownian motion, it becomes clear that the correct usage of fractional Brownian motion inherently implies dynamic market incompleteness.