User:Stevelihn/sandbox/Lihn's Lambda Distribution

Lihn's lambda distribution is a family of parametric continuous probability distributions on the real line. It is also called "version 1" of "generalized normal distribution", the exponential power distribution, or the generalized error distribution, with a different parametrization, but such nomenclature has not been standardized yet.

This distribution has been known as early as 1937. This page serves as a reference to the particular parametrization and notation used in Lihn's study (from 2015 to 2018). Lihn's contribution is to reveal the richness of this distribution family as following.


 * 1) Connected this distribution family with the stable law via lambda decomposition;
 * 2) Established it as a continuous leptokurtic distribution in which the normal distribution is a special case;
 * 3) Applied this distribution family in the Hidden Markov Model, where the lepkurtotic nature of financial data can be better captured;
 * 4) Established it as a stationary solution of leptokurtic extension of the Ornstein–Uhlenbeck process;
 * 5) Derived an analytic solution for S&P 500 option that contains the feature of volatility smile;
 * 6) As a special case of a larger "stable lambda distribution" family, which aims to describe the daily returns distribution of financial data more suitably.

Definition
Although Lihn studied both the symmetric and asymmetric distributions, only the symmetric formulation is mentioned here since the sequence of studies have showed that the symmetric formulation is far more important in applications.

The probability density function (PDF) is defined as


 * $$P(x)= \frac{1}{\sigma\lambda\Gamma(\frac{\lambda}{2})}

e^{-\left \vert \frac{x-\mu}{\sigma} \right \vert ^\frac{2}{\lambda}}, \qquad x \in \mathsf{R},$$

where $$\mu > 0$$is the location parameter, $$\sigma > 0$$ is the scale parameter, and $$\lambda > 0$$ is the shape parameter.

The cumulative distribution function (CDF) is


 * $$\Phi(x) = \frac{1}{2} + \frac{\sgn(x-\mu)}{2}

\left [ 1 - \frac{1}{\Gamma(\frac{\lambda}{2})} \Gamma(\frac{\lambda}{2}, {\left\vert \frac{x-\mu}{\sigma} \right\vert}^{\frac{2}{\lambda}}) \right ],$$

where $$\Gamma(s,x)=\int^\infty_x t^{s-1} e^{-t} \,dt$$ is the upper incomplete gamma function.

The $$\lambda $$ parameter is called the order of the distribution, which is structured such that its integer values bear important meaning. It is connected to the stability index in the stable distribution via $$\lambda = 2/\alpha$$. Using the inverse of stability index makes many formulas cleaner for this distribution, especially when gamma functions are involved.

It is easy to see that it becomes a normal distribution when $$\lambda = 1$$, and a Laplace distribution when $$\lambda = 2$$. When $$\lambda = 3$$, certain analytic solutions exist. Most importantly, $$\lambda = 4$$ is called "quartic lambda distribution", where there are many elegant analytic solutions.

This family allows for tails that are either heavier than normal (when $$\lambda>1$$) or lighter than normal (when $$\lambda<1$$). It is a useful way to parametrize a continuum of symmetric, platykurtic densities spanning from the normal ($$\lambda=1$$) to the uniform density ($$\lambda\rightarrow0$$), and a continuum of symmetric, leptokurtic densities spanning from the normal density to Laplace and beyond ($$\lambda>1$$).

The following table lists the variance and kurtosis of the most important integers $$\lambda = 1,2,3,4$$:
 * {| class="wikitable"

!$$\lambda$$ !var !kurtosis
 * +Variance and Kurtosis
 * 1
 * $$\frac{1}{2}\sigma^2$$
 * 3
 * 2
 * $$2\sigma^2$$
 * 6
 * 3
 * $$13.125\,\sigma^2$$
 * 12.257
 * 4
 * $$120\,\sigma^2$$
 * 25.2
 * } The kurtosis is approximately $$3\times\,2^{\lambda-1}$$ in this limited range.
 * 4
 * $$120\,\sigma^2$$
 * 25.2
 * } The kurtosis is approximately $$3\times\,2^{\lambda-1}$$ in this limited range.

Moments
The odd moments are zero. The n-th moment is



m_{(n)} = \begin{cases} 0 & \text{if }n\text{ is odd,} \\ \sigma^n \frac{\Gamma( \frac{\lambda}{2} (n+1))}{\Gamma( \frac{\lambda}{2})} & \text{if }k\text{ is even.} \end{cases} $$

Thus the variance is $$\sigma^2 \frac{\Gamma(\frac{3\lambda}{2})}{\Gamma(\frac{\lambda}{2})}$$ and the kurtosis is  $$\frac{\Gamma(\frac{\lambda}{2}) \Gamma(\frac{5\lambda}{2})}{\Gamma(\frac{3\lambda}{2})^2}$$.

Moment generating function
When all the moments are known, the moment generating function (MGF) can be calculated as



M(t) = E \left[ e^{tX} \right] = \begin{cases} \int_{-\infty}^{\infty} e^{tx} P(x)\,dx, & \text{the integral form;} \\ e^{t\mu} \left[ 1 + \sum_{n=2,4,...}^\infty \frac{m_{(n)}t^n}{n!} \right], & \text{the summation form.} \end{cases}

$$

However, both the integral and the sum diverge when $$\lambda>2$$. Lihn attributed this divergence as the root cause of "moment explosion", which has been a major obstacle in many stochastic volatility models (See Section 2.4 of ). This regime is called "the local regime", where MGF solution is available as long as $$\sigma$$is reasonably small. The "tail" of the integral and the sum must be truncated. The procedures are described as following.

Truncation of MGF integral
Assume $$\mu=0$$ in the following discussion. The truncation point in the right tail, that is, the upper limit of the integral, is where the integrand reaches its minimum: $$\frac{d}{dx}\left[ e^{tx} P(x) \right] = 0.$$This leads to the solution, (See Sections 3.1 of )


 * $$x_{\vec{\infty}} = \sigma \left( \frac{2}{\sigma t \lambda} \right)^{\frac{\lambda}{\lambda-2}}.$$

Truncation of MGF summation
The truncation point of the summation is where the summand reaches its minimum: $$\frac{d}{dn}\left[ \frac{m_{(n)}t^n}{n!} \right] = 0.$$This leads to the equation, (See Sections 2.5 of )


 * $$ \log(\sigma t) + \frac{\lambda}{2}

\psi\left( \frac{\lambda(n+1)}{2} \right) -\psi(n+1) = 0,$$

where $$\psi(x)$$is a digamma function. It has a large-n solution when $$\psi(x) \approx \log(x)+...$$,


 * $$n_{\vec{\infty}}+1 = \sigma t \left( \frac{2}{\sigma t \lambda} \right)^{\frac{\lambda}{\lambda-2}}.$$

We note that $$x_{\vec{\infty}} t \approx n_{\vec{\infty}}$$, in other words, they are equal in a dimensionless setting when $$t=1$$.

In Lihn's work of option pricing model, the risk-neutral drift $$\mu_D$$ is defined via MGF,


 * $$\mu_D = \log M(t=1; \mu=0).

$$ Therefore, it is essential that $$M(t=1)$$ can be calculated properly in order to price an option. The meaning of $$\mu_D$$ can be illustrated by the trivial case of a normal distribution, where $$M(t) = e^{t\mu + \sigma^2 t^2/4}$$leads to $$\mu_D(\lambda=1) = -\sigma^2/4$$. This is minus half of the variance, the well-known drift of a geometric Brownian motion.

Relation to symmetric alpha-stable distribution
TBD

Relation to one-sided stable distribution
TBD

Lambda Decomposition
TBD

Applications
This distribution can be used in modeling when the non-normality is prevailing. For instance, financial data is well known to violate normality almost all the time.

Leptokurtic extension of the Ornstein–Uhlenbeck process
TBD

Leptokurtic extension of Hidden Markov Model
TBD

Generalization - Stable Lambda Distribution
TBD.