Displaced Lognormal and Displaced Heston Volatility Skews: Analysis and Applications to Stochastic Volatility Simulations

We analyze the displaced (anti- )lognormal (DL) and displaced (anti- ) Heston (DH) volatility skew. In particular, for the displaced lognormal, we prove the global monotonicity of the implied volatility, and an at-the-money bound on the steepness of the downward volatility skews, which therefore cannot reproduce some features observed in the equity market. A variant, the displaced anti-lognormal, overcomes this steepness constraint, but its state space is bounded above and unbounded below. We prove the global monotonicity of its implied volatility too. For the displaced Heston dynamics, we show that the at-the-money slope has the same sign as the displacement. What's more, we give an explicit formula for the DL and DH's short-expiry limiting volatility skew, which allows direct calibration of their parameters to volatility skews implied by market data or by other models. In the end, we analyze the large-expiry limiting volatility of the displaced lognormal and give an asymptotic formula of it in the region of large-strike and fixed-strike respectively.