*Article* **Pricing, Risk and Volatility in Subordinated Market Models**

#### **Jean-Philippe Aguilar 1,\*, Justin Lars Kirkby 2 and Jan Korbel 3,4,5,6**


Received: 26 October 2020; Accepted: 13 November 2020; Published: 17 November 2020

**Abstract:** We consider several market models, where time is subordinated to a stochastic process. These models are based on various time changes in the Lévy processes driving asset returns, or on fractional extensions of the diffusion equation; they were introduced to capture complex phenomena such as volatility clustering or long memory. After recalling recent results on option pricing in subordinated market models, we establish several analytical formulas for market sensitivities and portfolio performance in this class of models, and discuss some useful approximations when options are not far from the money. We also provide some tools for volatility modelling and delta hedging, as well as comparisons with numerical Fourier techniques.

**Keywords:** Lévy process; subordination; option pricing; risk sensitivity; stochastic volatility; Greeks; time-change
