3.13 Local-Stochastic Volatility - The Leverage Function, Markovian Projection & the Particle Method

Basic Prerequisites: Advanced Volatility - Heston, SABR & Stochastic-Vol Dynamics and Volatility Surfaces & Smiles. (these are the folder-level prerequisites for pages 02–06; page 01 states its own, smaller, entry requirements)

1. Intuition & Practical Objective

Local volatility fits today’s smile exactly and then gets the dynamics wrong - it has no forward skew. Stochastic volatility gets the dynamics right and then fails to fit today’s smile - it has four parameters and a rigid two-dimensional surface. Local-stochastic volatility (LSV) is the attempt to buy both, and it is bought with exactly one object: a deterministic function of time and spot, the leverage function $\sigma (t,S)$, that rescales the stochastic-volatility driver so the model’s one-dimensional marginals are the market’s.

The one-sentence essence:

Take a stochastic-volatility model $\sqrt{v_t}$ with the dynamics you want, multiply it by a deterministic leverage $\sigma (t,S)$, and choose $\sigma$ so that the product’s conditional second moment reproduces the Dupire local variance: σ²(t,S) = σ_loc²(t,S) / E[v_t|S_t=S]. That single scalar equation forces the model to match every vanilla in the market while leaving the joint $(S,v)$ law - hence the forward skew, the skew stickiness and the vol-of-vol term structure - governed by the stochastic-volatility driver you started from.

Three consequences, and they are the whole folder:

1. The leverage function is not a free fit. Vanilla prices constrain only the product σ(t,S)² E[v_t|S_t=S]. Once the SV driver is fixed, the split is fixed too, so the leverage is determined, not calibrated to vanillas. It is a function on a two-dimensional domain whatever the number of volatility factors, which is why it scales to multi-factor and rough drivers.
2. The calibration is a nonlinear (McKean–Vlasov) fixed point. E[v_t|S_t=S] is an expectation under the model being calibrated, and the model depends on σ. The particle method (Guyon–Henry-Labordère 2012) solves this by Monte-Carlo binning - simulate, bin by spot, re-estimate the conditional variance, update σ, repeat - and the deterministic alternative is a Fokker–Planck / forward-PDE solve for the joint density.
3. “LSV” is not automatically a usable model. Bergomi’s admissibility condition is that the pricing function must not depend on the SV state variables for fixed hedge instruments. A forward-variance-driven LSV passes; a Heston-driven LSV generally fails. Quoting Bergomi: “most local-stochastic volatility models are not usable models.”

This folder is a hub: (a) the fast formula lookup below, and (b) six sub-pages that walk from the raw intuition and the leverage identity, through Markovian projection and Dupire’s local vol, the particle method and the Fokker–Planck/McKean–Vlasov route, into the practice (proxy bias, binning noise, gauge freedom) and the frontier (LSVJ, multi-factor and rough drivers, LSV-LMM, neural calibration).

2. Mathematical Ground Truth & Derivations

Notation. $S$ spot, $F_T=S_0{e}^{(r-q)T}$ forward, $K$ strike, $T$ maturity, $t$ calendar time, $k=\ln (K/F_T)$ log-moneyness, $w(k,T)={\sigma }_{BS}(k,T{)}^{2}T$ total implied variance, $v_t$ instantaneous variance of the stochastic-volatility driver, $m(t,S):=\mathbb{E}[v_t|S_t=S]$ its conditional expectation, $\sigma (t,S)$ the leverage function, ${\sigma }_{loc}(t,S)$ the Dupire local volatility, $\rho$ spot/variance correlation, ${\alpha }_t$ the instantaneous volatility of a generic Itô process, $\mathcal{L}[v](t,y)=\mathbb{E}[v_t|X_t=y]$ the Markovian-projection (conditional-expectation) operator.

Quick-Reference Lookup (job #1). Every formula is transcribed from the verified corpus (Guyon–Henry-Labordère 2012; Bergomi, Stochastic Volatility Modeling, ch 6–12 and his Risk article “Local-stochastic volatility: models and non-models”; Gyöngy 1986; Dupire 1994; Gatheral, The Volatility Surface) and every number in the check column was re-executed in §3 and the sub-pages of this folder.

Critical caveat (flagged in this folder). The leverage formula is conditional: $\mathbb{E}[v_t|S_t=S]$ is an expectation under the LSV model’s own law, which itself depends on $\sigma$. Substituting the unconditional forward variance ${\xi }_0^t$ for $m(t,S)$ - the natural first guess, and the one most implementations start from - is not a small approximation when $\rho \ne 0$: on the real Heston-LSV test of §03 the true $m(1,S)$ runs from $0.32\times$ to $5.6\times$ the unconditional ${\xi }_0^1$, and the “proxy” leverage produces a worse smile than no leverage at all ($24.4\%/13.6\%$ across the $K/S_0=0.75/1.30$ wings against a flat-$20\%$ target). The fixed-point iteration, not the proxy, is what calibrates.

3. Computational Implementation - the LSV engine

Stdlib only (math, cmath) - no numpy/scipy. Two primitives carry the whole folder: Dupire’s local variance (in both forms, used as a cross-check) and the leverage function. Every number in §2 and in the sub-pages was produced by code of this shape.

What the engine establishes.

• (a) The Dupire formula is right - on a flat surface the total-variance form returns $0.040000000$ to nine decimals at every strike. This is the non-negotiable first test of any Dupire implementation.
• (b) Two independent Dupire forms agree, and the ATM local variance heads to $v_0=0.0174$ as $T\to 0$ ($0.017478$ at $T=0.02$, $0.4\%$ of which is the finite-difference error, not model error). The ratio of local to implied variance at the money rises with maturity ($1.0046\to 1.1597$) - local volatility is not implied volatility, and the gap is what the leverage function has to absorb.
• (c) The leverage at $t=0$ is exactly computable: $\mathbb{E}[v_0|S_0]=v_0$ because the variance is known at $t=0$. To bend the Heston driver onto a flat-$20\%$ surface the instantaneous volatility must be scaled up by $51.6\%$ at the origin; to reproduce Heston’s own surface the leverage is $1$ (here $1.00197$, the residual being the $T=0.02$ finite-difference stand-in for $T\to 0$). Those two numbers - $1.5162$ and $1.0$ - are the two ends of the LSV programme.
• (d) The naive proxy $\sigma (t)={\sigma }_{loc}/\sqrt{{\xi }_0^t}$ uses the unconditional forward-variance curve. It is exact only in the LV limit and, as §03 shows, wrong by a factor of several in the wings once $\rho \ne 0$.

4. Failure Modes & First-Principles Breakdowns

Hub signposts - the folder’s failure-mode analysis lives in 05 · Failure Modes & Practice. In one line each:

1. The forward-variance proxy. Replacing $\mathbb{E}[v_t|S_t=S]$ by ${\xi }_0^t$ is the single most common LSV bug. On the Heston test of §03 it produces a worse smile than the un-levered model ($24.4\%/13.6\%$ wings against a flat-$20\%$ target). The conditional variance is not the unconditional one whenever $\rho \ne 0$.
2. The leverage is not a free calibration knob. Vanillas fix only ${\sigma }^{2}m$; the split is set by the SV driver. Fitting $\sigma$ to vanillas and then choosing the driver is circular, and grid-based leverage surfaces with more parameters than quotes are unidentifiable.
3. Binning noise propagates as $\delta \ln \sigma =-\frac{1}{2}\delta \ln m$. In the wings of the spot distribution the bins are thin and $m$ is estimated from few paths: with $500$ paths per bin the conditional-variance error is $5.7\%$ and the local-vol error $\approx 0.57$ vol points - a systematic-looking wing distortion, not white noise.
4. Gauge freedom is real. ${\zeta }^u\to {\phi }^u{\zeta }^u$ with $\sigma \to \sigma /\sqrt{{\phi }^u}$ leaves the spot process invariant, so a calibrated leverage is only defined up to this transformation. Two implementations can report “different” leverage functions and identical prices.
5. “Usable” is a theorem, not a taste. Bergomi’s condition - $\partial P/\partial {\lambda }_k{|}_{S,\{O_i\}}=0$ for the SV state variables - is what separates a model from a non-model. The forward-variance LSV satisfies it; a Heston-LSV generally does not, and its hedged P&L leaks.
6. Jumps break the naive Dupire reading. For a jump-diffusion, Dupire’s local variance is not the instantaneous quadratic-variation rate (§06: no-jump control returns $0.022500$ exactly; with jumps it returns $0.024213$ at $T=0.005$ against a QV rate of $0.035633$). “Strip the jump QV and divide by $m$” is a first-principles error.

5. References

• Guyon, J. & Henry-Labordère, P. (2012), Being particular about calibration, Risk 25(1), 91–107
• Gyöngy, I. (1986), Mimicking the one-dimensional marginal distributions of processes having an Itô differential, Probability Theory and Related Fields 71(4), 501–516
• Dupire, B. (1994), Pricing with a smile, Risk 7(1), 18–20
• Bergomi, L., Stochastic Volatility Modeling (CRC, 2016)
• Hagan, P. S., Kumar, D., Lesniewski, A. & Woodward, D. (2002), Managing smile risk, Wilmott Magazine, 84–108
• Gatheral, J., The Volatility Surface (Wiley, 2006)

6. Connected Graph Bridges

• Foundational base: Stochastic Calculus & Itô’s Lemma · Probability & Measure Theory · Econometrics & Time Series
• Sibling topics: Advanced Volatility - Heston, SABR & Stochastic-Vol Dynamics (the SV drivers this folder leverages) · Volatility Surfaces & Smiles (Dupire local vol and the empirical object) · Calibration & Market Practice (the calibration discipline this folder lives inside)
• Related flat notes: Implied vs Local Vol (the local-vol object being inverted) · SV Dynamics & the Bergomi–Guyon Expansion (the dynamics LSV is built to preserve) · The Greeks & Dynamic Hedging (why the leverage changes delta)
• Sub-pages (in-folder): 01 From Zero · 02 Markovian Projection & the Leverage Function · 03 The Particle Method · 04 The Fokker–Planck / McKean–Vlasov Route · 05 Failure Modes & Practice · 06 Advanced Extensions

Beginner: start at 01 · Practitioner: start at 05