Mean-reverting processes OU and CIR

Skill positionModule 4 of 5 · Topic 1 of 3

4.1 Mean-reverting processes OU and CIR

—Lab score— / 100Total XP—Streak—

Prices that get pulled back

A Brownian motion wanders away for ever. Many things in finance do not: interest rates, volatility, the gap between two related stocks. They are pulled back toward a long-run level. The Ornstein-Uhlenbeck (OU) process does exactly that. In $dX=\kappa(\theta-X)\,dt+\sigma\,dW$ the number $\theta$ is the level it returns to, $\kappa$ is the speed of the pull and $\sigma$ is the size of the random shocks. The further $X$ is above $\theta$ , the harder it is pulled down.

dX=\kappa(\theta-X)\,dt+\sigma\,dW,\quad E[X_t]=\theta+(X_0-\theta)e^{-\kappa t},\quad t_{1/2}=\frac{\ln2}{\kappa},\quad \mathrm{sd}(X_\infty)=\frac{\sigma}{\sqrt{2\kappa}}

(1)

Three numbers describe an OU process. The average distance to $\theta$ halves every $\ln2/\kappa$ time units, the half-life. In the long run $X$ settles into a stable range with standard deviation $\sigma/\sqrt{2\kappa}$ . And $X$ can go negative, which is fine for a spread but wrong for an interest rate or a variance. The CIR model fixes this by making the shocks smaller near zero: $dr=\kappa(\theta-r)\,dt+\sigma\sqrt r\,dW$ . The rate stays above zero when $2\kappa\theta\ge\sigma^{2}$ , called the Feller condition.

A spread that returnsA spread with

\kappa=0.35

per day has a half-life of

\ln2/0.35\approx1.98

days, so a pairs trade should last a few days and not weeks. For a rate that starts at

0.08

with

\theta=0.04

and

\kappa=0.5

per year, the average after two years is

0.04+0.04\,e^{-1}\approx0.0547

: it has covered about

63\%

of the way back.

OU process: mean reversion with shocks of constant size. It can go negative.
Half-life $\ln2/\kappa$ and long-run spread $\sigma/\sqrt{2\kappa}$ .
CIR process: shocks shrink near zero, so the rate stays positive when $2\kappa\theta\ge\sigma^{2}$ .

Lab · what's graded

The half-life15 pts
The expected level15 pts
The long-run spread15 pts
Which rates stay positive15 pts
Match the vocabulary15 pts
Fitting an OU model to data10 pts
Why a spread suits an OU model15 pts

0/ 7 answered

1Numeric answer15 ptsNot answered

A spread follows an OU process with $\kappa=0.35$ per day. What is its half-life $\ln2/\kappa$ in days? Give two decimals.

h =

accepted: ± 0.01

2Numeric answer15 ptsNot answered

A rate starts at $X_0=0.08$ and follows an OU process with $\theta=0.04$ and $\kappa=0.5$ per year. What is $E[X_2]=\theta+(X_0-\theta)e^{-\kappa t}$ after $t=2$ years? Give four decimals.

m =

accepted: ± 0.0003

3Numeric answer15 ptsNot answered

An OU process has $\sigma=0.06$ and $\kappa=0.18$ . What is its long-run standard deviation $\sigma/\sqrt{2\kappa}$ ?

s =

accepted: ± 0.002

4Choice15 ptsNot answered

For a CIR rate, which parameter sets satisfy the Feller condition $2\kappa\theta\ge\sigma^{2}$ and so stay positive? Pick all that apply.

5Match15 ptsNot answered

Match each item with its meaning.

Pick the partner of each item on the left.

OU process

CIR process

\kappa

\theta

6Order the steps10 ptsNot answered

Put the steps in order.

Drag to reorder, or focus an item, press Space to lift it, the arrow keys to move it and Space to drop it.

1Compute the half-life
2Regress each change on the previous level
3Take the series and its changes
4Read the speed and the level from the fit

7Explain in words15 ptsNot answered

In one or two sentences, explain why the gap between two related stocks is modelled with an OU process and how a trader can use that.

Graded on:The gap is pulled back toward a long-run levelSo buy it when low and sell it when high0 / 400 characters

Math modeKeyboard or LaTeX inputTask 1 of 7 · Ctrl ⏎ to check

Mean-reverting processes OU and CIR

Skill positionModule 4 of 5 · Topic 1 of 3

4.1 Mean-reverting processes OU and CIR

—Lab score— / 100Total XP—Streak—

Prices that get pulled back

dX=\kappa(\theta-X)\,dt+\sigma\,dW,\quad E[X_t]=\theta+(X_0-\theta)e^{-\kappa t},\quad t_{1/2}=\frac{\ln2}{\kappa},\quad \mathrm{sd}(X_\infty)=\frac{\sigma}{\sqrt{2\kappa}}

(1)

A spread that returnsA spread with

\kappa=0.35

per day has a half-life of

\ln2/0.35\approx1.98

days, so a pairs trade should last a few days and not weeks. For a rate that starts at

0.08

with

\theta=0.04

and

\kappa=0.5

per year, the average after two years is

0.04+0.04\,e^{-1}\approx0.0547

: it has covered about

63\%

of the way back.

OU process: mean reversion with shocks of constant size. It can go negative.
Half-life $\ln2/\kappa$ and long-run spread $\sigma/\sqrt{2\kappa}$ .
CIR process: shocks shrink near zero, so the rate stays positive when $2\kappa\theta\ge\sigma^{2}$ .

Lab · what's graded

The half-life15 pts
The expected level15 pts
The long-run spread15 pts
Which rates stay positive15 pts
Match the vocabulary15 pts
Fitting an OU model to data10 pts
Why a spread suits an OU model15 pts

0/ 7 answered

1Numeric answer15 ptsNot answered

A spread follows an OU process with $\kappa=0.35$ per day. What is its half-life $\ln2/\kappa$ in days? Give two decimals.

h =

accepted: ± 0.01

2Numeric answer15 ptsNot answered

A rate starts at $X_0=0.08$ and follows an OU process with $\theta=0.04$ and $\kappa=0.5$ per year. What is $E[X_2]=\theta+(X_0-\theta)e^{-\kappa t}$ after $t=2$ years? Give four decimals.

m =

accepted: ± 0.0003

3Numeric answer15 ptsNot answered

An OU process has $\sigma=0.06$ and $\kappa=0.18$ . What is its long-run standard deviation $\sigma/\sqrt{2\kappa}$ ?

s =

accepted: ± 0.002

4Choice15 ptsNot answered

For a CIR rate, which parameter sets satisfy the Feller condition $2\kappa\theta\ge\sigma^{2}$ and so stay positive? Pick all that apply.

5Match15 ptsNot answered

Match each item with its meaning.

Pick the partner of each item on the left.

OU process

CIR process

\kappa

\theta

6Order the steps10 ptsNot answered

Put the steps in order.

Drag to reorder, or focus an item, press Space to lift it, the arrow keys to move it and Space to drop it.

1Compute the half-life
2Regress each change on the previous level
3Take the series and its changes
4Read the speed and the level from the fit

7Explain in words15 ptsNot answered

In one or two sentences, explain why the gap between two related stocks is modelled with an OU process and how a trader can use that.

Graded on:The gap is pulled back toward a long-run levelSo buy it when low and sell it when high0 / 400 characters

Math modeKeyboard or LaTeX inputTask 1 of 7 · Ctrl ⏎ to check

Mean-reverting processes OU and CIR

Prices that get pulled back

Lab · what's graded

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Mean-reverting processes OU and CIR

Prices that get pulled back

Lab · what's graded

Log in to run your code.