Value at Risk (VaR) has emerged in recent years as a standard tool to measure and control the risk of trading portfolios. Yet, existing theoretical analyses of the optimal behavior of a trader subject to VaR limits have produced a negative view of VaR as a risk-control tool. In particular, VaR limits have been found to induce increased risk exposure in some states and an increased probability of extreme losses. However, these conclusions are based on models that are either static or dynamically inconsistent. In this paper we formulate a dynamically consistent model of optimal portfolio choice subject to VaR limits and show that the conclusions of earlier papers are incorrect if, consistently with common practice, the VaR is reevaluated dynamically making full use of conditioning information. In particular, we find that the risk exposure of a trader subject to a VaR limit is always lower than that of an unconstrained trader and that the probability of extreme losses is also lower. We also consider the Tail Conditional Expectation (TCE), a coherent risk measure often advocated as an alternative to VaR, and show that in our dynamic setting it is always possible to transform a TCE limit into an equivalent VaR limit, and conversely.

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We extend the theory of super-replicating a European option by relaxing its two main assumptions: we take into account the constraints on trading the option and allow it to be traded inter-temporally. The first extension has a dramatic effect on the price of a portfolio hedging the option, while the second has a dramatic effect on finding arbitrage opportunities in the market. We introduce a new approach for identifying the best arbitrage opportunities in the market with friction.Derivative pricing models, Derivatives hedging, Asset pricing, Risk management,