Modeling Exit Likelihood with a Two Gaussian Mixture

Summary

The question seeks a probability density that is low at the mean and high for extreme moves in either direction, to represent the chance of exiting a position. The suggested construction is a mixture of two Gaussian distributions: draw from one normal component with probability p and from another with probability 1 − p. The resulting density is the weighted sum of the two component densities.

This provides a flexible way to model a bimodal distribution, with the component means and variances controlling the locations and spread of its peaks. The response gives the general mixture formula but does not specify parameters or show that a particular mixture has a minimum at the desired mean. It also does not develop an exit policy or test the model against market data, so traders would need to choose and validate parameters for their use case.

Key ideas

• A mixture of two Gaussian distributions can represent a bimodal probability density.
• The mixture density is the weighted sum of the component densities.
• Component means, variances, and mixture weights determine the shape of the distribution.
• A mixture model alone does not establish that its density is lowest at a chosen mean.

Tags

# Probability Distribution that fits my parameters?


# Probability Distribution that fits my parameters?












I'm trying to create a PDF that has the max values at its tails, and a P(x) of 0 at its mean.

Essentially it would be something like two normal distributions lined up side to side.

Is there any literature regarding such a distribution? I'm trying to model the probability of exiting a position. If the price of the position doesn't move, it has the lowest probability of being exited, but the probability of exit is highest with an extreme move in either direction.

## Answer by LocalVolatility (score 2)

https://quant.stackexchange.com/a/30013

> Essentially it would be something like two normal distributions lined up side to side.

This would e.g. be a mixture of two Gaussians. I.e. let $Y_+ \sim \mathcal{N} \left( \mu_+, \sigma_+^2 \right)$ and $Y_- \sim \mathcal{N} \left( \mu_-, \sigma_-^2 \right)$, then

\begin{equation} X \sim \begin{cases} Y_+ & \text{with probability } p\\ Y_- & \text{with probability } 1 - p \end{cases} \end{equation}

Then $X$ has the probability density function

\begin{equation} f_X(x) = p \phi_{Y_+}(x) + (1 - p) \phi_{Y_-}(x), \end{equation}

where $\phi_{Y_\pm}$ are the corresponding normal density functions.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.