Christian Schlag

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.)

Price discovery in international equity trading

This study addresses two questions: where does price discovery occur for internationally-traded firms and how do international stock prices adjust to an exchange rate shock ? These questions are answered by analyzing quotes originating in New York and Frankfurt for three large German firms, DaimlerChrysler, Deutsche Telekom, and SAP, during overlapping trading hours. A high-frequency sample of quotes from both locations along with the dollar/euro exchange rate yields evidence of one cointegrating relation among the three variables. Vector error correction models are estimated for each firm and the associated vector moving average representations are utilized to infer the share of price discovery coming from the exchange rate, New York, and Frankfurt quotes. The evidence suggests a structure of the international equity market that has the home-market largely determining the random walk component of the international value of a firm along with an independent role for exchange rate shocks to affect prices in the U.S. markets. However, there is a significant information share for New York in the case of DaimlerChrysler and an even bigger role for New York with respect to SAP. Following a shock to the exchange rate, we find that almost all of the adjustment comes through the New York price.

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.

Has There Always Been Underpricing and Long-Run Underperformance? - IPOs in Germany Before World War I

This paper provides empirical evidence on initial public offerings (IPOs) by investigating the pricing and long-run performance of IPOs using a unique data set collected on the German capital market before World War I. Our findings indicate that underpricing of IPOs has existed, but has significantly decreased over time in our sample. Employing a mixture of distributions approach we also find evidence of price stabilization of IPOs. Concerning long-run performance, investors who bought their shares in the early after-market and held them for more than three years experienced significantly lower returns than the respective industry as a whole..

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.

Volatility-of-Volatility Risk

We show that time-varying volatility of volatility is a significant risk factor which affects the cross-section and the time-series of index and VIX option returns, beyond volatility risk itself. Volatility and volatility-of-volatility measures, identified model-free from options data as the VIX and VVIX indices, respectively, are only weakly related to each other. Delta-hedged index and VIX option returns are negative on average, and more negative for strategies more exposed to volatility and volatility-of-volatility risks. Volatility and volatility of volatility significantly and negatively predict future delta-hedged option payoffs. The evidence is consistent with a no-arbitrage model featuring time-varying volatility and volatility-of-volatility factors, which are negatively priced by investors.

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.

An empirical examination of the effect of dividend taxation on asset pricing and returns in Germany

Abstract This research finds evidence that required pretax returns on German stocks are unchanged as a result of the enactment of a law in Germany providing shareholders with tax credits for dividends received. In the most recent time interval, higher risk-adjusted pretax returns are discovered on high-yielding German stocks. These findings imply that the effect of the tax credits has been more than offset by other factors.

The Dynamics of Risk-Neutral Implied Moments: Evidence from Individual Options

We study the estimation, the dynamics, and the predictability of option-implied risk-neutral moments (variance, skewness, and kurtosis) for individual stocks from various perspectives. We first show that it is in the estimation of the higher moments essential to use an interpolation with a narrow grid as well as a wide interval. We show that implied moments are well explained cross-sectionally by a number of firm characteristics. We use the characteristics that have been shown to exhibit correlation with expected returns (like size and the market-to-book ratio of equity). In a next step, we investigate the joint dynamics of the three moments in a vector autoregressive model. We find that the moments are significantly linked to each other over time. Finally, adding exogenous variables to the vector autoregression improves the explanatory power of our model even further. Granger causality tests show significant differences between the three implied moments.

The German Equity Market: Risk, Return, and Liquidity

Within the international research community knowledge about German capital markets is not widespread. This may be due to the absence of a central data base and the fact that empirical results in German journals cannot be easily accessed by the English speaking majority. However, data bases on stocks, bonds, warrants and all derivative products of the German options and futures exchange are now available to researchers.