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# 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√ΔtWhere: 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 μs: long-term mean of S σs: 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 St – St-1 = a (μs – St-1)?Solving equation for S(t), St – St-1 = a (s – St-1) such that: S(t) =0.7751 * (0.3483 – 0.2615) + 0.2615 = 0.3288.
Owais Siddiqui