Trade Duration Risk in Subdiffusive Financial Models

Subdiffusive processes can be used in finance to explicitly accommodate the presence of random waiting times between trades or "duration", which in turn allows the modelling of price staleness effects. Option pricing models based on subdiffusions are incomplete, as they naturally account for the presence of a market risk of trade duration. However, when it comes to pricing this risk matters are quite subtle, since the subdiffusive Levy structure is not maintained under equivalent martingale measure changes unless the price of this risk is set to zero. We argue that this shortcoming can be resolved by introducing the broader class of tempered subdiffusive models. We highlight some additional features of tempered models that are consistent with economic stylized facts, and in particular explain the role of the stability and tempering parameters in capturing the time multiscaling properties of equity prices. Finally, we show that option pricing can be performed using standard integral representations