Differential Geometry, (m,λ)-SABR and a Formula by Pierre-Henry Labordère

How differential geometry turns a stochastic volatility model into a tractable implied volatility formula

Options desks price volatility smiles every day using models that, under the hood, require translating a stochastic process into a usable number: the Black-Scholes implied volatility at each strike and expiry. For models with stochastic volatility, that translation is not straightforward. Numerical solvers are slow, and calibration requires evaluating thousands of prices per second. What practitioners need is a closed-form approximation that is accurate, fast, and grounded in the model's actual dynamics. This paper by Dr. Ingo Fahrner, circulated on SSRN (2009), delivers exactly that for a broad class of stochastic volatility models, including a unified generalization of SABR, by using differential geometry to connect the model's probability structure to the shape of the volatility smile.

The geometry of a diffusion. The central idea is that any diffusion model defines a curved space. For a local volatility model df_t = C(f) dW_t, the natural metric on the forward space is g(f) = 2 / C²(f). The transition density, the probability of moving from one price level to another in time t, is then approximated by the heat kernel expansion: a Gaussian centered on the geodesic distance d(f₀, K) between current price and strike, multiplied by curvature correction terms. This is not a numerical approximation; it is an asymptotic expansion in t, accurate to first order in time-to-expiry, that exploits the geometry of the model rather than discretizing it.

Implied volatility from local volatility. Plugging the heat kernel density into the call price formula yields an explicit approximation for the Black-Scholes implied volatility σ_BS(K, T) for any local volatility model. The leading term is σ₀ = ln(K/f₀) / ∫ dx / C(x), the log-moneyness divided by the integrated inverse local vol between spot and strike. The time correction adds terms proportional to C''(f_av)/C(f_av) and (C'(f_av)/C(f_av))² at the midpoint f_av = (f₀ + K)/2, capturing the curvature and slope of the local vol surface. The result is a closed-form smile formula for any model in this class, including CEV and displaced diffusion as special cases.

Bridging stochastic volatility to local volatility. To extend this framework to stochastic volatility models, the paper applies a key equivalence result: any stochastic volatility model has an equivalent local volatility model with identical one-dimensional marginal distributions, and therefore identical vanilla option prices. The equivalent local vol is found by conditioning the instantaneous variance on the forward level. For the (m, λ)-SABR model, df_t = a_tC(f_t) dW_t with C(f) = mf^β + (1 − m)L and da_t = λ(θ − a_t)dt + νa_t dZ_t, dW_t dZ_t = ρ dt, this conditioning integral is evaluated via a saddle-point approximation in the volatility dimension.

The hyperbolic geometry of SABR. A structurally important result is that after a coordinate transformation, the two-dimensional metric of the (m, λ)-SABR model becomes the hyperbolic metric on the Poincaré half-plane. The geodesic distance between two points z₁ and z₂ is d = Arcosh(1 + |z₁ − z₂|² / (2y₁y₂)). This hyperbolic structure encodes the correlation ρ between the forward and its stochastic volatility. The saddle-point minimizer in the volatility dimension has the closed form a_m(f) = √(a₀² + 2a₀νρ q(f) + ν²q(f)²), where q(f) = ∫ dx / C(x) is the time-changed distance from spot to forward. This clean expression is what makes the final smile formula tractable.

What practitioners get. The paper delivers three concrete outputs for anyone building or calibrating options models. First, an explicit σ_BS(K, T) formula for the (m, λ)-SABR model that covers CEV (m = 1), displaced diffusion (β = 1), and normal SABR (β = 0, m = 1) as special cases, unifying them in a single parameterization. Second, an approximation to the full transition density of the forward: not just the smile shape, but the probability distribution itself, which is the input to exotic pricing and risk scenario generation. Third, Monte Carlo simulation schemes for both the mean-reverting volatility process and, in closed form for the displaced diffusion case, the forward process itself. The geometric framework turns a complex stochastic model into a set of formulas that can be implemented directly on a trading desk.