Tradable measure of risk

The main idea of this paper is to introduce Tradeable Measures of Risk as an objective and model independent way of measuring risk. The present methods of risk measurement, such as the standard Value-at-Risk supported by BASEL II, are based on subjective assumptions of future returns. Therefore two different models applied to the same portfolio can lead to different values of a risk measure. In order to achieve an objective measurement of risk, we introduce a concept of {\em Realized Risk} which we define as a directly observable function of realized returns. Predictive assessment of the future risk is given by {\em Tradeable Measure of Risk} -- the price of a financial contract which pays its holder the Realized Risk for a certain period. Our definition of the Realized Risk payoff involves a Weighted Average of Ordered Returns, with the following special cases: the worst return, the empirical Value-at-Risk, and the empirical mean shortfall. When Tradeable Measures of Risk of this type are priced and quoted by the market (even of an experimental type), one does not need a model to calculate values of a risk measure since it will be observed directly from the market. We use an option pricing approach to obtain dynamic pricing formulas for these contracts, where we make an assumption about the distribution of the returns. We also discuss the connection between Tradeable Measures of Risk and the axiomatic definition of Coherent Measures of Risk

Dynamic Scoring: Probabilistic Model Selection Based on Utility Maximization

We propose a novel approach of model selection for probability estimates that may be applied in time evolving setting. Specifically, we show that any discrepancy between different probability estimates opens a possibility to compare them by trading on a hypothetical betting market that trades probabilities. We describe the mechanism of such a market, where agents maximize some utility function which determines the optimal trading volume for given odds. This procedure produces supply and demand functions, that determine the size of the bet as a function of a trading probability. These functions are closed form for the choice of logarithmic and exponential utility functions. Having two probability estimates and the corresponding supply and demand functions, the trade matching these estimates happens at the intersection of the supply and demand functions. We show that an agent using correct probabilities will realize a profit in expectation when trading against any other set of probabilities. The expected profit realized by the correct view of the market probabilities can be used as a measure of information in terms of statistical divergence

Drawdowns preceding rallies in the Brownian motion model

We study drawdowns and rallies of Brownian motion. A rally is defined as the difference of the present value of the Brownian motion and its historical minimum, while the drawdown is defined as the difference of the historical maximum and its present value. This paper determines the probability that a drawdown of a units precedes a rally of b units. We apply this result to examine stock market crashes and rallies in the geometric Brownian motion model.Drawdowns, Brownian motion model, Rallies,

Options on a traded account: Vacation calls, vacation puts and passport options

In this article we study options on a traded account. In terms of the actions available to the buyer, the options we study are more general than a class of options known as {\em passport options}; in terms of the model of the underlying asset they are more restrictive. Using probabilistic techniques, we find the value of these options, the optimal strategy of the buyer, and the hedging strategy the seller should use in response to a (not necessarily optimal) strategy by the buyer.Passport options, Vacation options, Stochastic control, Hamilton-Jacobi-Bellman equation, Comparison theorem, Put-call parity, Hedging