A Relationship Between Regression Tests and Volatility Tests of Market ncy

Volatility tests are an alternative to regression tests for evaluating the joint null hypothesis of market efficiency and risk neutrality. Acomparison of the power of the two kinds of tests depends on what the alternative hypothesis is taken to be. By considering tests based on conditional volatility bounds, we show that if the alternative is that one could"beat the market" using a linear combination of known variables, then the regression tests are at least as powerful as the conditional volatility tests.If the application is to spot and forward markets, then the most powerful conditional volatility test turns out to be equivalent to the analogous regression test in terms of asymptotic power. In other applications,the volatility test will be less powerful than regression tests against our chosen alternative. However, these results are not inconsistent with the observation that volatility tests may be more powerful against other alternative hypoth-eses, such as that risk-averse investors are rationally maximizing the present discounted utility of future consumption,with a time-varying discount rate.

Utility indifference pricing with market incompleteness

Utility indifference pricing and hedging theory is presented, showing how it leads to linear or to non-linear pricing rules for contingent claims. Convex duality is first used to derive probabilistic representations for exponential utility-based prices, in a general setting with locally bounded semi-martingale price processes. The indifference price for a finite number of claims gives a non-linear pricing rule, which reduces to a linear pricing rule as the number of claims tends to zero, resulting in the so-called marginal utility-based price of the claim. Applications to basis risk models with lognormal price processes, under full and partial information scenarios are then worked out in detail. In the full information case, a claim on a non-traded asset is priced and hedged using a correlated traded asset. The resulting hedge requires knowledge of the drift parameters of the asset price processes, which are very difficult to estimate with any precision. This leads naturally to a further application, a partial information problem, with the drift parameters assumed to be random variables whose values are revealed to the hedger in a Bayesian fashion via a filtering algorithm. The indifference price is given by the solution to a non-linear PDE, reducing to a linear PDE for the marginal price when the number of claims becomes infinitesimally small

From Smile Asymptotics to Market Risk Measures

The left tail of the implied volatility skew, coming from quotes on out-of-the-money put options, can be thought to reflect the market's assessment of the risk of a huge drop in stock prices. We analyze how this market information can be integrated into the theoretical framework of convex monetary measures of risk. In particular, we make use of indifference pricing by dynamic convex risk measures, which are given as solutions of backward stochastic differential equations (BSDEs), to establish a link between these two approaches to risk measurement. We derive a characterization of the implied volatility in terms of the solution of a nonlinear PDE and provide a small time-to-maturity expansion and numerical solutions. This procedure allows to choose convex risk measures in a conveniently parametrized class, distorted entropic dynamic risk measures, which we introduce here, such that the asymptotic volatility skew under indifference pricing can be matched with the market skew. We demonstrate this in a calibration exercise to market implied volatility data.Comment: 24 pages, 4 figure

An expansion in the model space in the context of utility maximization

In the framework of an incomplete financial market where the stock price dynamics are modeled by a continuous semimartingale (not necessarily Markovian) an explicit second-order expansion formula for the power investor's value function - seen as a function of the underlying market price of risk process - is provided. This allows us to provide first-order approximations of the optimal primal and dual controls. Two specific calibrated numerical examples illustrating the accuracy of the method are also given

Expert Opinions and Logarithmic Utility Maximization in a Market with Gaussian Drift

This paper investigates optimal portfolio strategies in a financial market where the drift of the stock returns is driven by an unobserved Gaussian mean reverting process. Information on this process is obtained from observing stock returns and expert opinions. The latter provide at discrete time points an unbiased estimate of the current state of the drift. Nevertheless, the drift can only be observed partially and the best estimate is given by the conditional expectation given the available information, i.e., by the filter. We provide the filter equations in the model with expert opinion and derive in detail properties of the conditional variance. For an investor who maximizes expected logarithmic utility of his portfolio, we derive the optimal strategy explicitly in different settings for the available information. The optimal expected utility, the value function of the control problem, depends on the conditional variance. The bounds and asymptotic results for the conditional variances are used to derive bounds and asymptotic properties for the value functions. The results are illustrated with numerical examples.Comment: 21 page

Pricing swing options and other electricity derivatives

The deregulation of regional electricity markets has led to more competitive prices but also higher uncertainty in the future electricity price development. Most markets exhibit high volatilities and occasional distinctive price spikes, which results in demand for derivative products which protect the holder against high prices. A good understanding of the stochastic price dynamics is required for the purposes of risk management and pricing derivatives. In this thesis we examine a simple spot price model which is the exponential of the sum of an Ornstein-Uhlenbeck and an independent pure jump process. We derive the moment generating function as well as various approximations to the probability density function of the logarithm of this spot price process at maturity T. With some restrictions on the set of possible martingale measures we show that the risk neutral dynamics remains within the class of considered models and hence we are able to calibrate the model to the observed forward curve and present semi-analytic formulas for premia of path-independent options as well as approximations to call and put options on forward contracts with and without a delivery period. In order to price path-dependent options with multiple exercise rights like swing contracts a grid method is utilised which in turn uses approximations to the conditional density of the spot process. Further contributions of this thesis include a short discussion of interpolation methods to generate a continuous forward curve based on the forward contracts with delivery periods observed in the market, and an investigation into optimal martingale measures in incomplete markets. In particular we present known results of q-optimal martingale measures in the setting of a stochastic volatility model and give a first indication of how to determine the q-optimal measure for q=0 in an exponential Ornstein-Uhlenbeck model consistent with a given forward curve

Malliavin calculus method for asymptotic expansion of dual control problems

We develop a technique based on Malliavin-Bismut calculus ideas, for asymptotic expansion of dual control problems arising in connection with exponential indifference valuation of claims, and with minimisation of relative entropy, in incomplete markets. The problems involve optimisation of a functional of Brownian paths on Wiener space, with the paths perturbed by a drift involving the control. In addition there is a penalty term in which the control features quadratically. The drift perturbation is interpreted as a measure change using the Girsanov theorem, leading to a form of the integration by parts formula in which a directional derivative on Wiener space is computed. This allows for asymptotic analysis of the control problem. Applications to incomplete It\^o process markets are given, in which indifference prices are approximated in the low risk aversion limit. We also give an application to identifying the minimal entropy martingale measure as a perturbation to the minimal martingale measure in stochastic volatility models