Explain carefully why this Bernoulli-normal mixture is potentially a better mode

Question

Explain carefully why this Bernoulli-normal mixture is potentially a better model of

returns than the log-normal model.

Problem 1: Modeling heavy-tails with jumps The distribution of stock returns has fat tails (see lecture 1). As a result, we need models that deliver fat-tailed distributions (e.g., to price options on stocks). One way of doing this is to introduce jumps in the model. This problem describes the building blocks of how to do that. Consider a Bernoulli-distributed variable: o with probability 1 p 1 with probability p and two independent Standard Normal variables Et and ot Define the jump as

Explanation / Answer

A bernouli distribution is generally a discrete distribution and normal distribution is a continuous dsitribution, when these two distributions are combined, it helps represent the fat tails and the jump in an appropriate manner

The lognormal distribution on the other hand completely continuous distribution which fails to represent the jumps precisely.

Bernouli distribution thus provides a better fit to the model since the graphs represent the fat tails

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