Pricing formulas for American perpetual knock-out and callable volatility options

This study investigates challenges associated with the valuation of knockout and callable American put options on volatility, where volatility dynamics are described as a mean-reverting 3/2 volatility process. We propose pricing formulas for the perpetual American-style up-and-out volatility put option and perpetual American-style down-and-out volatility put option. Furthermore, we obtain the corresponding pricing formula for the perpetual American callable volatility put option under regularity conditions. We also conduct sensitivity analyses for various model parameters via simulations. Finally, we compare our model results to an empirical analysis.