Gaussian emissions for returns

From size classes to real returns

Real returns are not Small or Large, they are numbers like $-3.0\%$ or $+0.4\%$ . So the emission of each state becomes a bell curve: in state $j$ the return is normal with its own mean $\mu_j$ and standard deviation $\sigma_j$ . The emission probability $B_j(r)$ is the height of that curve at the observed return $r$ . Everything else in the algorithms stays the same: forward, Viterbi, forward-backward and Baum-Welch only ask for the numbers $B_j(o_t)$ . In the M-step the new $\mu_j$ and $\sigma_j$ are the weighted mean and weighted standard deviation of the returns, with the smoothed probabilities $\gamma_t(j)$ as weights.

B_j(r)=\frac{1}{\sigma_j\sqrt{2\pi}}\exp\!\Big(-\frac{(r-\mu_j)^{2}}{2\sigma_j^{2}}\Big),\qquad \mu_j=\frac{\sum_t\gamma_t(j)r_t}{\sum_t\gamma_t(j)},\qquad \sigma_j^{2}=\frac{\sum_t\gamma_t(j)(r_t-\mu_j)^{2}}{\sum_t\gamma_t(j)}

(1)

A model with a calm and a stormy state is a mixture of two normal curves. Each curve alone has thin tails, but the mixture has fat tails: the stormy state, with its large $\sigma$ , produces big moves far more often than a single normal curve with the average volatility would. This is one reason HMMs fit real returns well. The states also tell a story. Typically one finds a calm state with a small positive mean and low volatility, and a stormy state with a negative mean and high volatility, and the stormy state is sticky, so crashes arrive in clusters.

A -3% dayLet Calm have mean

+0.05\%

and standard deviation

0.6\%

, and Stormy mean

-0.10\%

and standard deviation

2.0\%

a day. A return of

-3\%

(-3-0.05)/0.6=-5.1

standard deviations from Calm but only

(-3+0.10)/2=-1.45

from Stormy. The curve heights are

1.6\times10^{-6}

for Calm and

0.0697

for Stormy. With a prior of

0.9

for Calm and

0.1

for Stormy, the posterior chance of Stormy is

0.1\times0.0697\,/\,(0.1\times0.0697+0.9\times1.6\times10^{-6})=0.9998

. A daily volatility of

2\%

2\%\times\sqrt{252}=31.7\%

a year.

Gaussian emission: in each state the return is normal with its own mean and volatility.
M-step for returns: weighted mean and weighted standard deviation, with $\gamma_t(j)$ as weights.
Fat tails for free: a mixture of calm and stormy normals has heavier tails than one normal.

Lab · what's graded

A z-score15 pts
A curve height15 pts
An annual volatility15 pts
Which state was it?15 pts
Why a mixture has fat tails10 pts
Match the symbols15 pts
What the stormy state is15 pts

From size classes to real returns

B_j(r)=\frac{1}{\sigma_j\sqrt{2\pi}}\exp\!\Big(-\frac{(r-\mu_j)^{2}}{2\sigma_j^{2}}\Big),\qquad \mu_j=\frac{\sum_t\gamma_t(j)r_t}{\sum_t\gamma_t(j)},\qquad \sigma_j^{2}=\frac{\sum_t\gamma_t(j)(r_t-\mu_j)^{2}}{\sum_t\gamma_t(j)}

(1)

A -3% dayLet Calm have mean

+0.05\%

and standard deviation

0.6\%

, and Stormy mean

-0.10\%

and standard deviation

2.0\%

a day. A return of

-3\%

(-3-0.05)/0.6=-5.1

standard deviations from Calm but only

(-3+0.10)/2=-1.45

from Stormy. The curve heights are

1.6\times10^{-6}

for Calm and

0.0697

for Stormy. With a prior of

0.9

for Calm and

0.1

for Stormy, the posterior chance of Stormy is

0.1\times0.0697\,/\,(0.1\times0.0697+0.9\times1.6\times10^{-6})=0.9998

. A daily volatility of

2\%

2\%\times\sqrt{252}=31.7\%

a year.

Gaussian emission: in each state the return is normal with its own mean and volatility.
M-step for returns: weighted mean and weighted standard deviation, with $\gamma_t(j)$ as weights.
Fat tails for free: a mixture of calm and stormy normals has heavier tails than one normal.

Lab · what's graded

A z-score15 pts
A curve height15 pts
An annual volatility15 pts
Which state was it?15 pts
Why a mixture has fat tails10 pts
Match the symbols15 pts
What the stormy state is15 pts

From size classes to real returns

Lab · what's graded

Log in to run your code.

Gaussian emissions for returns

From size classes to real returns

Lab · what's graded

Log in to run your code.