A Stochastic-Volatility Model for Pricing Power Variants of Exchange Options

Power‐type derivatives for rough volatility with jumps

“…The payoff structure (44) grants the option investor a leveraged view on the average‐forward volatility. Since $${I}_{T}(\Delta )$$ takes values within the unit interval under normal conditions, $${p}_{1}\gt 1$$ actually reduces the option investor's risk exposure, other things equal, while $$0\le {p}_{1}\lt 1$$ expands it, which is the exact opposite of the case of equity options (see Xia, 2019, p. 119). For any fixed $${p}_{2}\ge 0$$, $${P}_{T}^{(1,{p}_{2},({\rm{a}}))}$$ corresponds to the terminal payoff of the standard volatility put option, while $${P}_{T}^{(2,{p}_{2},({\rm{a}}))}$$ represents that of a standard put option on the average‐forward variance.…”

“…As in the case of equity options, the volatility option investor's risk exposure can also be adjusted by directly forcing a mutual power on the standard option payoff (similar to Raible, 2000, Sect. 3.4; Xia, 2019, p. 120). This way of generalization does not build any direct connection between options on the average‐forward volatility and the corresponding forward variance and is hence considered less important from the viewpoint of this paper's motivation.…”

“…On the implementation side, our study objects are a more general class of derivatives raising the underlying volatility or the standard option payoff to a nonnegative power. Derivatives with power payoff functions written on equity have been thoroughly examined in the literature; see Tompkins (1999), Raible (2000), Macovschi and Quittard-Pinon (2006), and Xia (2017) on single-asset options and Blenman and Clark (2005), Wang (2016), and Xia (2019) on exchange options. Similar exchange options on zero-coupon bonds have recently been studied in Blenman et al (2020).…”