The Leverage Effect and the Retarded Volatility Model
Does a fall in price mechanically predict a rise in later volatility?
Measure how later volatility changes after positive and negative returns of comparable size, then repeat across dated subperiods. The retarded-volatility model explains an observed asymmetry through past-price response in studied samples; it is a model interpretation, not automatic causation.
Observed index and stock differences are sample-bounded; retarded-volatility explanation is a model interpretation.
Evidence map
| Aspect | Finding |
|---|---|
| What it is | The negative correlation between past returns and future volatility: a fall in price raises subsequent volatility more than a rise of the same size lowers it. |
| Key result / formula | The leverage correlation function is L(τ) = ⟨r_t · r²_(t+τ)⟩ / ⟨r²⟩², negative for τ > 0 and zero within error for τ < 0 — the asymmetry in time is itself a finding: volatility does not predict the sign of returns. |
| Why it matters for backtesting | The asymmetry in τ is a free test of look-ahead in a volatility feature: if something built from volatility appears to predict the sign of the next return, the leverage function shows that the arrow runs the other way and the feature is probably leaking the return it claims to forecast. |
What it is
Bouchaud, Matacz & Potters (2001) measured its amplitude and time profile on individual stocks and on indices, found the two qualitatively different, and proposed a model in which the difference is the signature of two different mechanisms.
Key result / formula
On a panel of US individual stocks the effect is moderate and decays over roughly 50 trading days; on a set of major stock indices it is much stronger in amplitude and decays much faster. For single stocks the amplitude is rationalised by the "retarded" model: volatility responds to a moving average of past prices, so the process interpolates between a purely additive walk (a fixed dollar volatility, hence a higher relative volatility after a fall) and a purely multiplicative one — which reproduces the measured magnitude with no behavioural ingredient. For indices the retarded mechanism is too weak by construction, and a separate "market panic" effect, a rise in volatility after collective losses, is needed to match the amplitude.
Why it matters for backtesting
The stock/index difference is a warning against pooling: a volatility model fitted on index data and applied to single names overstates the immediate response and understates the persistence. And the retarded model is the null that stops the behavioural story: for individual stocks the measured leverage is consistent with mechanical dollar-versus-percent volatility, so a claim that stock-level leverage reveals fear or forced selling needs an amplitude beyond what the retarded model already gives. What makes the measurement itself fragile is the fourth moment: L(τ) involves r² and is dominated by the largest days, so confidence bands from a block bootstrap belong next to it.
Source
Bouchaud, Matacz & Potters, "Leverage Effect in Financial Markets: The Retarded Volatility Model", Physical Review Letters 87(22), 2001, 228701. Primary source