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

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

The price is lognormally distributed. The log of the price is a sum of independent normal log-returns and so is normal, and exponentiating gives a lognormal variable, which is always positive. Negative prices are therefore impossible in this model.

  1. AS_t is normally distributed, so negative prices are possible
  2. BS_t is lognormally distributed, so the price cannot become negativeCorrect
  3. CS_t follows a Poisson distribution because time is discrete
  4. DS_t has a mean that falls as time increases, whatever the drift
  5. S_t is a martingale under the real-world measure

Explanation

If log(S_t/S_0) is a sum of independent identically distributed normal log-returns, it is normal. S_t is the exponential of that sum, so it is lognormal and strictly positive. Option 0 describes the price level of an arithmetic model, not this log-return model.

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