Financial
Ornstein-Uhlenbeck conditional mean, half-life and uncertainty formulas
Calculate conditional expectation and dispersion under a constant-parameter mean-reverting model, with consistent time units and validated parameters.
A conditional stochastic model, not a guaranteed rate path
Assume dr=kappa*(theta-r)*dt+sigma*dW with constant kappa>0 and sigma>=0. The conditional mean is theta+(r0-theta)*exp(-kappa*T). Actual paths fluctuate around it and need not move monotonically toward theta. This assumption must be tested and calibrated; it is not a universal description of rates, volatility or trading spreads.
Use years consistently here: kappa is per year, T is years, and sigma is rate units per square-root year. The Gaussian model allows negative rates and has unbounded support. A finite stationary standard deviation does not bound possible outcomes.
Half-life measures decay of the conditional mean deviation, not the time by which a path must cross theta or a recommended holding period. Real-world forecasting and risk-neutral pricing parameters differ; these inputs are fictional.
| Vasicek / OU Metric | Exact Closed-Form Equation | Inputs: r0=5.5%, theta=3.5%, kappa=0.50 | Calculated Output |
|---|---|---|---|
| 2-Year Expected Rate E[r_2] | B3 + (B2 - B3) * EXP(-B4 * B5) | 3.5% + 2.0% * EXP(-1.0) | 4.24% (0.0424) |
| Mean-Reversion Half-Life | LN(2) / B4 | 0.69315 / 0.50 yr^-1 | 1.39 years |
| Stationary Long-Run Vol | B6 / SQRT(2 * B4) | 1.20% / SQRT(1.0) | 1.20% (0.0120) |
| 20-Year Asymptote (T = 20) | B5 set to 20 years | EXP(-10) -> 0.000045 | 0.035 after four-decimal rounding; not exact equality at finite T |
Vasicek Conditional Expected Rate at Horizon T: theta + (r0 - theta) * EXP(-kappa * T)
Smoothly interpolates between today's spot rate r0 (at T = 0) and long-run equilibrium theta (as T -> infinity).
=IF(COUNT(B2:B5)<>4,"Invalid inputs",IF(OR(B4<=0,B5<0),"Invalid inputs",ROUND(B3 + (B2 - B3) * EXP(-B4 * B5), 4)))Conditional-mean deviation half-life
LN(2)/kappa is expressed in the same time unit used to calibrate kappa. The model restricts kappa to positive values; a zero-speed diffusion has no finite mean-reversion half-life.
=IF(COUNT(B4)<>1,"Invalid inputs",IF(B4<=0,"Invalid inputs",ROUND(LN(2) / B4, 2)))Stationary standard deviation
sigma/sqrt(2*kappa) is the limiting standard deviation of the state distribution. It is not the annualized instantaneous diffusion parameter sigma.
=IF(COUNT(B4,B6)<>2,"Invalid inputs",IF(OR(B4<=0,B6<0),"Invalid inputs",ROUND(B6 / SQRT(2 * B4), 4)))Finite-horizon conditional standard deviation
Starting from a known initial rate, dispersion is zero at T=0 and approaches the stationary value as T grows. This describes model uncertainty, not a confidence guarantee for an unvalidated market forecast.
=IF(COUNT(B4:B6)<>3,"Invalid inputs",IF(OR(B4<=0,B5<0,B6<0),"Invalid inputs",ROUND(B6*SQRT((1-EXP(-2*B4*B5))/(2*B4)),6)))Try the example
Use the original values or change them to test your understanding.