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

In a discrete-time model for a share price, the log-return over each year is assumed to be independent and normally distributed with the same mean and variance. Which statement about the share price process is correct?

The price is lognormally distributed. The log of the price is a sum of independent normal log-returns, so it is normal, and exponentiating gives a lognormal price that cannot be negative.

  1. AThe price at each future time is normally distributed
  2. BThe price at each future time is lognormally distributedCorrect
  3. CThe price at each future time is exponentially distributed
  4. DThe price can become negative with positive probability
  5. The price has a Poisson distribution at each future time

Explanation

If log-returns are independent normal, the log of the price is a sum of normal variables and so is normal. The price is the exponential of a normal variable, hence lognormal and always positive. Normality of the price itself would allow negative values, which this model does not.

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