A displaced diffusion makes a shifted price lognormal. See how the shift changes option values, volatility skew and the lower bound, with an exact pricing example.
Displaced diffusion model
A displaced diffusion model makes a shifted underlying value lognormal: the random quantity is proportional to the underlying plus a displacement. It preserves a simple European option formula while producing a volatility skew when prices are quoted as ordinary Black implied volatilities. The shift also moves the model's lower bound, so choosing it changes both pricing and the values the underlying can reach.
Specify what is being shifted
Use a zero-carry pricing example: the interest rate and proportional dividend yield are both zero, r=q=0. For a price-like variable St, set
dSt=σ(St+a)dWt,S0+a>0,σ>0.
Here Wt is Brownian motion under the risk-neutral measure, a is a constant displacement in the same currency units as S, and time is in years. The coefficient σ is annualised percentage volatility of the shifted coordinate. This is the driftless displaced-diffusion form in Fries, Nigbur and Seeger, §3.1.
Define Yt=St+a. Then
dYt=σYtdWt,St=(S0+a)exp(−21σ2t+σWt)−a.
Thus Y follows geometric Brownian motion, and S has a shifted lognormal distribution. With a>0, its support is (−a,∞): negative S values are possible, although the boundary −a is not reached in finite time from an interior starting point. This can suit a signed financial variable; it does not preserve the nonnegative price constraint of limited-liability equity.
Carry needs its own convention. If instead a spot model has drift (r−q)St and the same diffusion coefficient, the fixed shift gives
dYt=(r−q)(Yt−a)dt+σYtdWt.
Its drift is not generally proportional to Yt, so the shift alone does not make it geometric Brownian motion. One alternative, for constant r and no dividends, uses a time-dependent shift at=a0ert and diffusion σ(St+at); then Yt=St+at has drift rYt. Carr and Wu, §2.1, describe the equivalent construction with the opposite displacement sign. Forward-rate implementations instead specify the relevant pricing measure and discount/payoff factors; see the related shifted-lognormal rate model. A spot shift, a forward shift and their drift conventions must not be interchanged silently.
Shift the strike too
For a European call with expiry T, the payoff identity is exact:
(ST−K)+=(YT−(K+a))+.
Under the zero-carry convention above, when K+a>0,
Here Φ is the standard normal cumulative distribution function. This is the zero-rate Black–Scholes formula applied to both shifted inputs. If K+a≤0, the call is always in the money because YT>0; its zero-carry price is S0−K, and the logarithmic formula is unnecessary.
Take a synthetic contract with S0=K=100 currency units, a=50, σ=0.20 and T=1 year. Then Y0=150 and the transformed strike is also 150, giving d1=0.1 and d2=−0.1:
C0=150[Φ(0.1)−Φ(−0.1)]=11.948351….
Without the shift, keeping the coefficient at 20% gives 7.965567…. The shifted call is more expensive because the same coefficient now multiplies 150 at inception rather than 100. These are different uncertainty assumptions, even though both parameter lists contain “20%”.
The instantaneous absolute diffusion coefficient is σ(S+a), in currency units per square-root year. At a positive state S, the corresponding local return volatility is
σloc(S)=σSS+a.
In the example, the shifted-coordinate coefficient is 20%, the initial local return volatility is 30%, and the ordinary unshifted Black implied volatility of the one-year ATM call is approximately 30.0629%. The last number solves the unshifted formula at spot and strike 100 for the price 11.948351. It is neither automatically 20% nor exactly 30%.
Synthetic zero-carry comparison: at S100 the absolute diffusion coefficient rises from20 to30 with displacement50; ordinary Black implied volatility from shifted prices is31.262% at strike80,30.063% at100 and29.188% at120, versus20% throughout without displacement.
Original theoretical comparison with S0=100, T=1, r=q=0 and σ=20%; no market calibration. The left panel measures absolute instantaneous uncertainty, while the right reports ordinary Black implied volatility from European call prices.
For a>0 and S>0, local percentage volatility decreases as S rises. The figure shows the resulting downward implied-volatility skew for this scenario. These are separate curves: local volatility is indexed by underlying state; implied volatility is indexed by option strike. Brigo's slides 2–5 explain the procedure of generating option prices from alternative dynamics and then inverting Black–Scholes to obtain model-implied quotes.
At S=0, the absolute coefficient can remain finite while the percentage expression is undefined. For negative S, percentage-return volatility is no longer the useful positive-price interpretation above. The shifted-coordinate dynamics remain well defined on their stated support.
The normal limit requires rescaling
Setting a=0 recovers the driftless lognormal model. A normal, or Bachelier, limit requires more than making a large while holding σ fixed; that would make the initial absolute volatility grow without bound.
To keep the scale comparable, fix S0>0 and an annualised reference coefficient v, and write
dSt=v[βSt+(1−β)S0]dWt,0≤β≤1.
At β=1 this is lognormal; at β=0 it is arithmetic Brownian motion with constant absolute volatility vS0. For β>0, the equivalent displaced parameters are
σ=vβ,a=β1−βS0.
As β tends to zero, a grows while σ shrinks, with σa tending to vS0. This makes the scaling behind the normal limit explicit. A single constant displacement remains a restrictive skew model: it cannot fit an arbitrary smile, and a fit to vanilla prices does not establish realistic future smile dynamics or exotic-option hedges.
