Mean Reversion

Mean reversion is the tendency of a variable to revert to a long-term level, which can also be called an unconditional mean. Examples are:

• Fixed-coupon bonds are mean-reverting because they are pulled-to-par
• Interest rates are generally mean reverting

_{$ S_{t} $ – $ S_{t-1} $= a (μs – St-1)Δt+ $ σ_{s} $ ε√Δt}

Where:
St: price at time t
St-1: price at the previous point in time − 1
α: degree of mean reversion, also called mean reversion rate or gravity, 0 ≤ ≤ 1
μ_s: long-term mean of S
σ_s: volatility of S
ε: random drawing from a standardised normal distribution at time t, (t): n ~ (0,1)

Example of Mean Reversion:

The long-term mean of the correlation data is 34.83%. In February 2012, the average correlation of the 30 × 30 Dow correlation matrices was 26.15%. From the regression function from 1972 to 2012, we find that the average mean reversion is 77.51%. What is the expected correlation for March 2012
S_t – S_t-1 = a (μ_s – S_t-1)?

Solving equation for S(t), St – S_t-1 = a (s – S_t-1) such that: S(t) =0.7751 * (0.3483 – 0.2615) + 0.2615 = 0.3288.