Lesson 2 of 6 · All lessons in this module
  1. The lognormal distribution
  2. From continuous returns to lognormal pricesVideo lesson
  3. Estimating and annualising volatility
  4. Monte Carlo simulation: what it is and what it is for
  5. Running a Monte Carlo simulation, step by step
  6. Bootstrap resampling

Lesson 2 of 6 · 14 min

From continuous returns to lognormal prices

Continuously compounded returns add up across periods, so the multi-period return is (at least approximately) normal and the price it produces is lognormal.

In short

  • Continuously compounded return: r=ln⁡(P1/P0)=ln⁡(1+R)r = \ln(P_1/P_0) = \ln(1 + R); the future price is PT=P0er0,TP_T = P_0 e^{r_{0,T}}.
  • Over several periods, continuous returns simply add: r0,Tr_{0,T} is the sum of the one-period returns.
  • i.i.d. returns are independent (the past does not predict the future) and identically distributed (stationary: same mean and variance every period).
  • With i.i.d. returns, the mean and the variance grow in proportion to T; the standard deviation grows with T\sqrt{T}.
  • A sum of normal returns is normal; a sum of non-normal i.i.d. returns is approximately normal by the central limit theorem. Either way, PTP_T is lognormal.

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