Displaced diffusion from zero
1/5Shift the origin, get a smile
Black-Scholes assumes the spot price diffuses lognormally from its current level. Displaced diffusion changes one thing: it shifts the origin. The diffusion is still lognormal, but the axis it lives on has moved.
The SDE is dead simple:
That is the entire model. One extra parameter, d, added to standard BS. The diffusion coefficient is proportional to (S + d) instead of just S. That shift breaks the symmetry of the lognormal smile and creates skew.
Why does shifting the origin produce skew? Because the percentage volatility of the shifted variable is σ, but the percentage volatility of S itself varies with the level. When S is low, S + d is relatively large compared to S, so the effective vol in percentage terms is higher. When S is high, the displacement d matters less, and you approach the BS case.
Imagine you are measuring from a different zero point. Instead of measuring from 0, you measure from −d. The underlying has not changed, but the measuring stick has. That change in reference frame is enough to produce a tilted smile.
The displacement parameter
The displacement d is the only knob you get. It controls the direction and magnitude of the skew. Understanding what it does is understanding the whole model.
d > 0 (positive displacement): The origin shifts right. For a given σ, low prices see a bigger effective vol (because S + d is large relative to S), while high prices see a smaller one. Result: the implied vol curve slopes downward from left to right. This is negative skew -- the same direction as equity and crypto markets.
d < 0 (negative displacement): The origin shifts left. Now high prices see proportionally more vol. Result: positive skew. This is uncommon but can model markets where vol rises with price (some commodities, for instance).
d = 0: No shift. You are back in Black-Scholes. Flat smile.
Drag the slider above. Notice how the smile tilts progressively as you increase d. There is no curvature in the displaced diffusion smile -- it is nearly linear in the wings. This is the fundamental limitation: DD can produce tilt but not the U-shape you see in real markets.
Displaced diffusion = shifted Black-Scholes
Here is the operational insight that makes DD so useful: you do not need a new pricing formula. You run standard Black-Scholes with shifted inputs. Replace S with (S + d) and K with (K + d). Done.
The logic is straightforward. Define S̃ = S + d. Then the SDE becomes dS̃ = σ·S̃·dW, which is just geometric Brownian motion for the shifted variable. Standard BS applies to S̃ with strike K̃ = K + d.
This is why DD was adopted so quickly by rates desks in the negative-rate era. They did not need new software. They added a shift to their inputs and kept their entire Black-Scholes infrastructure running. The shift was usually calibrated once per day from the ATM vol and one additional point.
The Greeks also shift. Delta is the BS delta of the shifted option. Gamma is the BS gamma. Vega is the BS vega. The only subtlety is that you need to adjust sensitivities back to the original (unshifted) coordinates when computing hedges.
Connection to CEV and SABR
Displaced diffusion is the linearized version of the CEV model. SABR with β = 1 and a shift parameter is approximately displaced diffusion. Understanding this connection tells you exactly where DD sits in the model hierarchy.
CEV (constant elasticity of variance) uses dS = σ·Sᵝ·dW where β is the elasticity. When β = 1, it is BS. When β < 1, the vol is higher at low S and lower at high S -- the same qualitative behavior as DD.
The connection: a first-order Taylor expansion of Sᵝ around S = F gives approximately (S + d) for a particular d that depends onβ and F. So DD is the linearized approximation of CEV around the forward. They produce nearly identical smiles near ATM and diverge in the far wings.
Notice how the two curves overlap near ATM but diverge in the wings. DD produces a smile that is nearly linear in strike. CEV produces curvature because the power-law backbone bends. For most practical purposes within a few strikes of ATM, they are interchangeable.
SABR connection: The SABR model with β = 1 is lognormal SABR. Adding a shift to the forward (shifted SABR) gives you SABR(β = 1) on the displaced variable. In the zero-vol-of-vol limit (ν = 0), this collapses exactly to displaced diffusion. So DD is a degenerate case of shifted SABR -- the simplest possible member of that family.
This is why DD is called the simplest way to add skew to BS. You get one extra parameter, one direction of tilt, and exact compatibility with existing BS infrastructure. If you need curvature, wings, or stochastic dynamics, you graduate to CEV, SABR, or Heston.
When it is enough
DD is a single-parameter extension of Black-Scholes. That is both its strength and its limitation. Know when to use it and when to move on.
Use DD when:
1. You need a quick skew adjustment and do not need a full model. Quoting a rough skew for a desk conversation, sanity-checking a more complex model, or pricing a vanilla book where tilt matters more than wings.
2. Your underlying can go to zero or negative (rates, spreads). The displacement keeps the shifted variable positive even when the original crosses zero. This is the canonical use case -- rates desks in the negative-rate era lived on shifted lognormal.
3. You want to keep existing BS infrastructure intact. No new numerical methods, no Monte Carlo, no Fourier inversion. Just shift the inputs.
Move past DD when:
1. You need smile curvature. DD produces a nearly linear skew. Real markets have U-shaped smiles with convexity in both wings. DD cannot capture that.
2. You need dynamic smile behavior. DD is a static model -- the displacement is fixed. It says nothing about how the smile moves when spot moves. For dynamic hedging, you need SABR, Heston, or SLV.
3. You are pricing exotics. Path-dependent options need a model that describes the dynamics of vol, not just a snapshot. DD has no vol dynamics.
For crypto specifically, DD is too simple. Crypto smiles are steep, curved, and dynamic. DD can give you a rough first tilt, but any production surface will use SVI, SABR, or a more sophisticated model.
Think of the model hierarchy as a ladder: Black-Scholes (flat smile) → displaced diffusion (tilted smile) →CEV/SABR (curved smile with dynamics) →Heston/SLV (stochastic vol with rich structure). Each step adds complexity but also explanatory power. DD is the first rung above BS. It is worth knowing even if you never use it in production, because it teaches you that skew is fundamentally about how volatility scales with the underlying level.
Where to go next:
CEV Model -- the nonlinear cousin of DD, with curved smiles
SABR Model -- stochastic vol on top of a backbone, the production standard
SVI Parameterization -- direct smile fitting, the crypto standard