Skip to main content

Displaced Diffusion

Displaced diffusion (also called the shifted lognormal model) takes Black-Scholes and shifts the price axis. Instead of modeling the forward price FF directly, you model F+dF + d as lognormal, where dd is the displacement. This creates skew without any stochastic vol -- just a coordinate shift.

💡
A coordinate shift creates skew

Negative displacement allows the underlying to go below zero (useful for rates). Positive displacement shifts the smile right. The shift breaks Black-Scholes symmetry and creates skew. ATM level stays the same; OTM options reprice.

Explore the Parameters​

Move the displacement slider to see how shifting the price axis creates asymmetry. The vol slider controls overall level. The dashed blue line shows the un-shifted (Black-Scholes) case.

Displaced Diffusion Explorer

Zero displacement. Pure lognormal -- flat smile, no skew.
37%44%51%758595ATM105115125StrikeImplied Vol (%)
Vol level40%
Base volatility of the shifted process
Displacement (d)0
Negative = allows negative prices, Positive = right-shifted

Move the displacement slider to see how shifting the price axis creates skew. The dashed blue line shows the un-shifted smile for reference.

What each parameter does​

  • sigma (vol level): The implied volatility applied to the shifted forward. Higher sigma = everything costs more.
  • displacement (d): How far you shift the price axis. Negative d creates put skew (vol rises as price drops). Positive d creates mild call skew. Zero displacement is standard Black-Scholes.

Strengths and Limitations​

Strength
What it means for you
Handles negative values
With negative displacement, the model allows negative underlying prices. This was crucial when interest rates went negative.
Closed-form pricing
It is literally Black-Scholes with shifted inputs. Every BS formula, every Greek -- they all carry over exactly.
Two parameters
Vol level and displacement. Simple to calibrate, hard to overfit.
Limitation
What it means for you
No smile curvature
Like CEV, displaced diffusion produces skew (tilt) but not smile (curvature). It cannot fit a market smile that curves up on both wings.
Linear skew only
The skew it produces is nearly linear across strikes. Real market skew has curvature, especially for short-dated options.
Displacement is arbitrary
There is no economic reason for a particular displacement value. It is a fitting knob, not a model insight.
💡
Quickest path from Black-Scholes to skew

Displaced diffusion is the quickest way to add skew to Black-Scholes. Good starting point, but real markets need more parameters. For proper delta and vega hedging across the term structure, you need a richer model.

Equation Explorer​

Convert between implied vol, total variance, log-moneyness, and option prices.

Equation Explorer

w = σ2 × Ttotal variance = IV2 × time
%
The implied volatility
days
Calendar days to expiration
Total Variance (w)
0.022225
Annualized Variance (σ²)
0.2704
Round-trip IV
52.00%
Total variance is what SVI and other models fit. It scales with time, so a 50% vol for 30 days has less total variance than 50% vol for 90 days.

Test your understanding before moving on.

Q: What does a negative displacement do to the vol smile?
Q: Why was displaced diffusion popular in rates markets around 2014-2016?

💡 Tip: Try answering each question yourself before revealing the answer.

Building mathematical intuition​

Learn displaced diffusion from scratchInteractive lesson · no prerequisites

This lesson explains the shifted-axis trick in plain English, shows how the displacement parameter changes the smile, and connects the model back to Black-Scholes intuition.

See also:

Last updated on