Derivative pricing, Valuation and hedging, Incomplete markets, Dynamic shortfall risk, Average value-at-risk, Utility indifference pricing, Convex measure of risk, Coherent risk measure, Risk-efficient options, Semimartingale models, Risk indifference pricing, C60, D46, G13,

Abstract. In this article we study arithmetic Asian options when the underlying stock is driven by special semimartingale processes. We show that the inherently path dependent problem of pricing Asian options can be transformed into a problem without path dependency in the payoff function. We also show that the price satisfies a simpler integro-differential equation in the case the stock price is driven by a process with independent increments, Lévy process being a special case.

Abstract. In this article we study arithmetic Asian options when the underlying stock is driven by special semimartingale processes. We show that the inherently path dependent problem of pricing Asian options can be transformed into a problem without path dependency in the payoff function. We also show that the price satisfies a simpler integro-differential equation in the case the stock price is driven by a process with independent increments, Lévy process being a special case.

Value-at-risk (VaR) and conditional value-at-risk (CVaR) are popular risk measures from academic, industrial and regulatory perspectives. The problem of minimizing CVaR is theoretically known to be of a Neyman–Pearson type binary solution. We add a constraint on expected return to investigate the mean-CVaR portfolio selection problem in a dynamic setting: the investor is faced with a Markowitz type of risk reward problem at the final horizon, where variance as a measure of risk is replaced by CVaR. Based on the complete market assumption, we give an analytical solution in general. The novelty of our solution is that it is no longer the Neyman–Pearson type, in which the final optimal portfolio takes only two values. Instead, in the case in which the portfolio value is required to be bounded from above, the optimal solution takes three values; while in the case in which there is no upper bound, the optimal investment portfolio does not exist, though a three-level portfolio still provides a sub-optimal solution

In this paper, we show that the Mean Comparison Theorem which is valid for Brownian motion, cannot be extended to Poisson process. A counter example in the Poisson case, for which the Mean Comparison Theorem does not hold, is provided.