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This article considers the pricing and hedging of a call option when liquidity matters, that is, either for a large nominal or for an illiquid underlying asset. In practice, as opposed to the classical assumptions of a price-taking agent in a frictionless market, traders cannot be perfectly hedged because of execution costs and market impact. They indeed face a trade-off between hedging errors and costs that can be solved by using stochastic optimal control. Our modelling framework, which is inspired by the recent literature on optimal execution, makes it possible to account for both execution costs and the lasting market impact of trades. Prices are obtained through the indifference pricing approach. Numerical examples are provided, along with comparisons to standard methods.

In this paper, we consider the optimal execution problem associated to accelerated share repurchase (ASR) contracts. When firms want to repurchase their own shares, they often enter such a contract with a bank. The bank buys the shares for the firm and is paid the average market price over the execution period, the length of the period being decided upon by the bank during the buying process. Mathematically, the problem is new and related to both option pricing (Asian and Bermudan options) and optimal execution. We provide a model, along with associated numerical methods, to determine the optimal stopping time and the optimal buying strategy of the bank.

When stricken by a terrorist attack, a war, or a natural disaster, an economic unit or a critical infrastructure may suffer significant loss of productivity. More importantly, due to interdependency or interconnectedness, this initial loss may propagate into other systems and eventually lead to much greater derivative loss. This belongs to what is known as a cascading effect. It is demonstrated in this article that the cascading effect and the derivative loss can be significantly reduced by effective risk management. This is accomplished by deliberately distributing the initial inoperability to other systems so that the total loss (or inoperability) is minimized. The optimal distribution strategy is found by a linear programming technique. The same risk management can also be applied to situations where objectives need to be prioritized. A case study featuring 12 economic sectors illustrates the theory. The result suggests that using the same amount of resources, minimizing risk (inoperability) of infrastructures will generally give rise to highest payoff, whereas overlooking it may result in greatest total loss. The framework developed in this work uses a steady-state approach that applies primarily to managing situations where the attack is catastrophic resulting in very long recovery time.

This paper presents several models addressing optimal portfolio choice, optimal portfolio liquidation, and optimal portfolio transition issues, in which the expected returns of risky assets are unknown. Our approach is based on a coupling between Bayesian learning and dynamic programming techniques that leads to partial differential equations. It enables to recover the well-known results of Karatzas and Zhao in a framework à la Merton, but also to deal with cases where martingale methods are no longer available. In particular, we address optimal portfolio choice, portfolio liquidation, and portfolio transition problems in a framework à la Almgren-Chriss, and we build therefore a model in which the agent takes into account in his decision process both the liquidity of assets and the uncertainty with respect to their expected return.