Displaced Diffusion Model

The standard Black-Scholes model assumes an asset $S_t$ follows geometric Brownian motion, which forces it to stay strictly positive and gives it lognormal returns. That's a poor fit whenever the underlying can plausibly go to zero or negative (interest rates in a low- or negative-rate environment, or commodity forwards, which briefly went negative in 2020) and it also can't reproduce the volatility smile shapes seen in many option markets on its own. Displaced diffusion fixes both by shifting the process.

The model applies ordinary GBM not to $S_t$ directly, but to $S_t$ plus a constant displacement $\beta$:

$$d(S_t + \beta) = \sigma (S_t + \beta) \, dW_t,$$

so $S_t + \beta$ is lognormal — meaning $S_t$ itself is now allowed to go as low as $-\beta$ before the process would need to hit zero. In plain English: instead of forcing the asset itself to be lognormal, you force a shifted version of it to be lognormal, which lets the asset dip below zero by up to $\beta$ while keeping the whole model as tractable as Black-Scholes (there's a closed-form option pricing formula, just applied to $S_t+\beta$).

The displacement also controls the shape of the resulting volatility skew: $\beta = 0$ recovers ordinary lognormal Black-Scholes, while larger $\beta$ pushes the model closer to a normal (Bachelier-style) process, which produces a flatter, more symmetric implied-volatility smile — giving practitioners one tunable parameter to match observed skew shapes without a fully separate stochastic-volatility model.