Nicole Branger

Why is the Index Smile So Steep

Empirical evidence shows that the implied volatility smiles for index options are significantly steeper than those for individual options. We propose a model setup where we start from the joint dynamics of the stocks and where the index value is a weighted sum of individual stock prices. Then the differences between the index smile and the smiles for individual stocks are entirely determined by the dependence structure among the stocks. We illustrate our idea in a jump-diffusion framework where both the diffusion and the jumps are decomposed into common and idiosyncratic components. Empirical data for options on the German stock index DAX and on Deutsche Bank are used to show that the model can explain the stylized facts on implied volatility smiles.(This abstract was borrowed from another version of this item.)

The Optimal Demand for Retail Derivatives

It has been shown that investors can benefit from including derivatives into their portfolios. For retail investors, however, a direct investment in derivatives is often too complicated. Investment certificates offer a potential solution to this problem. We analyze if retail investors who buy and hold their portfolio for one year can indeed benefit from an investment in these certificates. We use a model with stochastic volatility and jumps calibrated to the German stock market index DAX. We find that the benefit of investing in typical retail products is equivalent to an annualized risk-free excess return of at most 35 basis points for a CRRA investor with a low risk aversion. If we take transaction costs into account, this number reduces to at most 14~bp. In terms of the types of contracts, we find that discount certificates perform best, while more sophisticated certificates, in particular those with knock-in or knock-out features, should often not be held by investors at all. Therefore, standard preferences cannot explain the large observed demand for investment certificates.

Discrete-time implementation of continuous-time portfolio strategies

Optimal portfolio strategies are easy to compute in continuous-time models. In reality trading is discrete, so that these optimal strategies cannot be implemented properly. When the investor follows a naive discretization strategy, i.e. when he implements the optimal continuous-time strategy in discrete time, he will suffer a utility loss. For a variety of models, we analyze this discretization error in a simulation study. We find that time discreteness can be neglected when only the stock and the money market account are traded, even in models with stochastic volatility and jumps. On the other hand, when derivatives are traded the utility loss due to discrete trading can be much larger than the utility gain from having access to derivatives. To benefit from derivatives, the investor has to rebalance his portfolio at least daily.

Pricing Two Heterogeneous Trees

We consider a Lucas-type exchange economy with two heterogeneous stocks (trees) and a representative investor with constant relative risk aversion. The dividend process for one stock follows a geometric Brownian motion with constant and known parameters. The expected dividend growth rate for the other tree is stochastic and in general unobservable, although there may be a signal from which the investor can learn about its current value. We find that the equilibrium quantities in our model significantly depend on the information structure and on the level of risk aversion. While an observable stochastic drift mainly makes the economy more risky, a latent expected growth rate process with learning significantly changes the equilibrium price-dividend ratios, price reactions to dividend and drift innovations, expected returns, volatilities, correlations, and differences between the stocks. These effects are the more pronounced the more risk averse the representative investor.

Pricing electricity derivatives on an hourly basis

The purpose of this paper is to develop a framework for pricing electricity derivatives on an hourly basis. We do not in contrast to most current approaches focus exclusively on spot models which primarily reflect empirical spot price dynamics, but also ensure a straightforward applicability to the valuation of electricity derivatives. We show that a model with a jump and a spike component can be calibrated to both the time-series of hourly spot prices and the cross-section of futures prices, once we allow for time-dependent jump and spike parameters. Furthermore, we illustrate the importance of derivative pricing in electricity markets and present some examples of options on futures and hourly spot-options, such as operating reserves and physical transmission rights.

Optimal Derivative Strategies with Discrete Rebalancing

Continuous-time derivatives models are based on the assumption that a hedged portfolio can be constructed and adjusted continuously over time. But practically speaking, of course, continuous rebalancing is impossible. This means that a hedged option position will still be exposed to risk. It also means that the delta hedge from the model, which an investor would choose if continuous rebalancing were possible, may not be optimal when the hedge is only adjusted at discrete intervals. In this article, Branger, Breuer, and Schlag simulate hedging strategies in a world with periodic rebalancing and stochastic volatility. They consider hedging an option with the underlying alone, with the underlying and another option, and with the underlying and a contract based on realized variance. Interestingly, the optimal hedge changes sharply in some cases when rebalancing is only at discrete intervals, but a large fraction of the theoretically possible utility increase from continuous hedging can be achieved by trading continuous-time derivatives models are based on the assumption that a hedged portfolio can be constructed and adjusted continuously over time. But practically speaking, of course, continuous rebalancing is impossible. This means that a hedged option position will still be exposed to risk. It also means that the delta hedge from the model, which an investor would choose if continuous rebalancing were possible, may not be optimal when the hedge is only adjusted at discrete intervals. In this article, Branger, Breuer, and Schlag simulate hedging strategies in a world with periodic rebalancing and stochastic volatility. They consider hedging an option with the underlying alone, with the underlying and another option, and with the underlying and a contract bas at much longer intervals.

When Do Jumps Matter for Portfolio Optimization

We consider the continuous-time portfolio optimization problem of an investor with constant relative risk aversion who maximizes expected utility of terminal wealth. The risky asset follows a jump-diffusion model with a diffusion state variable. We propose an approximation method that replaces the jumps by a diffusion and solve the resulting problem analytically. Furthermore, we provide explicit bounds on the true optimal strategy and the relative wealth equivalent loss that do not rely on quantities known only in the true model. We apply our method to a calibrated affine model. Our findings are threefold: Jumps matter more, i.e. our approximation is less accurate, if (i) the expected jump size or (ii) the jump intensity is large. Fixing the average impact of jumps, we find that (iii) rare, but severe jumps matter more than frequent, but small jumps.

Why is Portfolio Insurance Attractive to Investors

This paper examines whether and how the popularity of portfolio insurance strategies can be justified theoretically. The analysis employs three different return generating processes with and without stochastic volatility and jumps. We find that an investor with constant relative risk aversion does not profit from portfolio insurance. For a cumulative prospect theory investor, on the other hand, the investment performance, measured by the certainty equivalent return, doubles from around 5% for buy-and-hold to around 10% due to portfolio insurance. Across all models, both loss aversion and probability weighting turn out to be crucial to explaining the attractiveness of portfolio insurance.

The Case for Herding is Stronger than You Think

In case of herding, investors follow each other, prices move together more than they normally do, and the cross-sectional dispersion of returns decreases. Chang, Cheng, and Khorana (2000) suggest to test for herding by regressing the cross-sectional absolute deviation on the absolute and squared excess market return. They argue that there is evidence for herding in case of large market movements when the coeffcient of the squared excess market return is signifcantly smaller than zero. We show that the true coeffcient of the squared excess market return is positive under the null hypothesis of no herding. The test of Chang, Cheng and Khorana is thus biased against finding evidence for herding. We find that this bias matters. For the S&P 500, the test of Chang, Cheng and Khorana signals that there is no herding over the period from 2008 to 2013, while the modified test based on the correct null hypothesis provides clear evidence for herding.

The Dynamics of Crises and the Equity Premium

This paper analyzes the equilibrium pricing implications of contagion risk in a Lucastree economy with recursive preferences and jumps. We introduce a new economic channel allowing for the possibility that endowment shocks simultaneously trigger a regime shift to a bad economic state. We document that these contagious jumps have far-reaching asset pricing implications. The risk premium for such shocks is superadditive, i.e. it is 2.5% larger than the sum of the risk premia for pure endowment shocks and regime switches. Moreover, contagion risk reduces the risk-free rate by around 0.5%. We also derive semiclosed-form solutions for the wealth-consumption ratio and the price-dividend ratios in an economy with two Lucas trees and analyze cross-sectional effects of contagion risk qualitatively. We find that heterogeneity among the assets with respect to contagion risk can increase risk premia disproportionately. In particular, big assets with a large exposure to contagious shocks carry significantly higher risk premia.It is a major challenge for asset pricing models to generate a high equity premium and a low risk-free rate while imposing realistic consumption dynamics. To address this issue, our paper proposes a novel pricing channel: we allow for consumption drops that can spark an economic crisis. This new feature generates a large equity premium even if possible consumption drops are of moderate size. In turn, our model also matches the consumption data of 42 countries along several dimensions. In particular, our approach generates a realistic number of crises that have realistic durations and involve clustering of moderate consumption drops. Received October 17, 2014; accepted August 18, 2015 by Editor Pietro Veronesi.