Nils Detering

Pricing and hedging Asian-style options on energy

We solve the problem of pricing and hedging Asian-style options on energy with a quadratic risk criterion when trading in the underlying future is restricted. Liquid trading in the future is only possible up to the start of a so-called delivery period. After the start of the delivery period, the hedge positions cannot be adjusted any more until maturity. This reflects the trading situation at the Nordic energy market Nord Pool, for example. We show that there exists a unique solution to this combined continuous–discrete quadratic hedging problem if the future price process is a special semimartingale with bounded mean–variance tradeoff. Additionally, under the assumption that the future price process is a local martingale, the hedge positions before the averaging period are inherited from the market specification without trading restriction. As an application, we consider three models and derive their quadratic hedge positions in explicit form: a simple Black–Scholes model with time-dependent volatility, the stochastic volatility model of Barndorff-Nielsen and Shephard, and an exponential additive model. Based on an exponential spot price model driven by two NIG Lévy processes, we determine an exponential additive model for the future price by moment matching techniques. We calculate hedge positions and determine the quadratic hedging error in a simulation study.

Model Risk of Contingent Claims

Paralleling regulatory developments, we devise value-at-risk and expected shortfall type risk measures for the potential losses arising from using misspecified models when pricing and hedging contingent claims. Essentially, P&L from model risk corresponds to P&L realized on a perfectly hedged position. Model uncertainty is expressed by a set of pricing models, each of which represents alternative asset price dynamics to the model used for pricing. P&L from model risk is determined relative to each of these models. Using market data, a unified loss distribution is attained by weighing models according to a likelihood criterion involving both calibration quality and model parsimony. Examples demonstrate the magnitude of model risk and corresponding capital buffers necessary to sufficiently protect trading book positions against unexpected losses from model risk. A further application of the model risk framework demonstrates the calculation of gap risk of a barrier option when employing a semi-static hedging strategy.

Return distributions of equity-linked retirement plans

In the recent years an increasing demand for capital guaranteed equity-linked life insurance products and retirement plans has emerged. In Germany, a retirement plan, called Riester-Rente, is supported by the state with cash payments and tax benefits. Those retirement plans have to preserve the invested capital. The company offering a Riester-Rente has to ensure that at the end of the saving period at least all cash inflows are available. Due to the investors demand for high returns, banks and insurance companies are not only offering saving plans investing in riskless bonds but also in products with a high equity proportion. For companies offering an equity-linked Riester-Rente the guarantee to pay out at least the invested capital is a big challenge. Due to the long maturities of the contracts of more than 30 years it is not possible to just buy a protective put. Many different concepts are used by banks and insurance companies to generate this guarantee or to reduce the remaining risk for the company. They vary from simple Stop Loss strategies to complex dynamic hedging strategies. In our work we analyze the return distribution generated by some of these strategies.

Model Risk in Incomplete Markets with Jumps

We are concerned with determining the model risk of contingent claims when markets are incomplete. Contrary to existing measures of model risk, typically based on price discrepancies between models, we develop value-at-risk and expected shortfall measures based on realized P&L from model risk, resp. model risk and some residual market risk. This is motivated, e.g., by financial regulators’ plans to introduce extra capital charges for model risk. In an incomplete market setting, we also investigate the question of hedge quality when using hedging strategies from a (deliberately) misspecified model, for example, because the misspecified model is a simplified model where hedges are easily determined. An application to energy markets demonstrates the degree of model error.

Measuring the Model Risk of Quadratic Risk Minimizing Hedging Strategies with an Application to Energy Markets

Measures of model risk based on the residual error from hedging in a misspecified model were recently proposed in (Detering and Packham, 2013). These measures rely on the assumption that the model used for hedging represents a complete financial market. We show that under certain conditions, in a diffusion setup, markets can be completed to derive measures of model risk for the original market. If the market can not be completed, as it is the case in most market models that allow for jumps, we derive measures that are applicable in a more general setup. In a case study we measure the model risk that is present when hedging options on energy futures with a simplified model compared to a model that better fits the empirical returns observed in the market.

A stochastic approach to serotonergic fibers in mental disorders

Virtually all brain circuits are physically embedded in a three-dimensional matrix of fibers that release 5-hydroxytryptamine (5-HT, serotonin). The density of this matrix varies across brain regions and cortical laminae, and it is altered in some mental disorders, including Major Depressive Disorder and Autism Spectrum Disorder. We investigate how the regional structure of the serotonergic matrix depends on the stochastic behavior of individual serotonergic fibers and introduce a new framework for the quantitative analysis of this behavior. In particular, we show that a step-wise random walk, based on the von Mises-Fisher probability distribution, can provide a realistic and mathematically concise description of these fibers. We also consider other stochastic models, including the fractional Brownian motion. The proposed approach seeks to advance the current understanding of the ascending reticular activating system (ARAS) and may also support future theory-guided therapeutic approaches.