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IAI Actuarial Core Principles · Economic Modelling · Stochastic models for security prices

A share price follows a model with jumps: between jumps it follows geometric Brownian motion, and jumps arrive as a Poisson process with rate 0.5 per year. Each jump multiplies the price by a factor Y independent of the past. Which feature distinguishes this model from pure geometric Brownian motion?

Sample paths show sudden discontinuities, and the return distribution has fatter tails than under pure geometric Brownian motion. Poisson jumps cause this, while independence of increments remains and the expected jump count is 0.5 a year.

  1. ASample paths are continuous everywhere
  2. BPrices can fall below zero
  3. CSample paths contain sudden discontinuities, giving fatter tails in return distributionsCorrect
  4. DLog-returns over disjoint intervals are dependent
  5. The expected number of jumps in a year is 2

Explanation

Poisson-driven jumps cause discontinuities in the path and produce heavier tails than the normal distribution of log-returns. Increments remain independent, and the expected number of jumps in a year is 0.5, not 2. Positive multipliers keep the price positive.

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