Displaced lognormal volatility skews: analysis and applications to stochastic volatility simulations

We analyze the implied volatility skews generated by displaced lognormal diffusions. In particular, we prove the global monotonicity of implied volatility, and an at-the-money bound on the steepness of downward volatility skews, under displaced lognormal dynamics, which therefore cannot reproduce some features observed in equity markets. A variant, the displaced anti-lognormal, overcomes the steepness constraint, but its state space is bounded above and unbounded below. In light of these limitations on what features the displaced (anti-)lognormal (DL) can model, we exploit the DL, not as a model, but as a control variate, to reduce variance in Monte Carlo simulations of the CEV and SABR local/stochastic volatility models. For either use—as model, or as control variate—the DL’s parameters require estimation. We find an explicit formula for the DL’s short-expiry limiting volatility skew, which allows direct calibration of its parameters to volatility skews implied by market data or by other models.