This post introduces discrete time price dynamics in a log-normal setup. This framework allows an easy translation of some of the continuous time results discussed on this site.

Reminder on log normality The random variable $$X$$ has a log-normal distribution if $$x=\log(X)$$ has a normal distribution. If the distribution of $$x$$ is $$\cal{N}(\mu,\sigma)$$, then we will note the distribution of $$X$$ $$\cal{L}(\mu,\sigma)$$. Its mean is: $\exp(\mu+\frac{1}{2}\sigma^{2}).$ This is consistent with the convexity of the exponential function and the Jensen effect1. Whereas $$x$$ has a distribution which is centered around its mean $$\mu$$, the distribution of $$X$$ is positively skewed. Its median is $$\exp(\mu)$$ which is lower than its mean $$\exp(\mu+\frac{1}{2}\sigma^{2})$$ by the factor $$\exp(-\frac{1}{2}\sigma^{2})$$. Log normal price dynamics Suppose now that we have a value process $$(P_{t})_{t \in \mathbb{N}}$$ such that as seen from date $$t$$, $$P_{t+1}/P_{t}$$ is log-normal $$\cal{L}(\mu_{t+1},\sigma_{t+1})$$. More explicitly, we assume that the variables $$\mu_{t+1}$$ and $$\sigma_{t+1}$$ are $$\cal{F}_{t}$$ measurable. Then: $E_{t}[\frac{P_{t+1}}{P_{t}}]=\exp(\mu_{t+1}+\frac{1}{2}\sigma_{t+1}^{2}).$ I will note $$r_{t+1}=\mu_{t+1}+\frac{1}{2}\sigma_{t+1}^{2}$$ the log of the sequential expected return as of date $$t$$. With this notation, we can now write: $\frac{P_{t+1}}{P_{t}}=\exp(r_{t+1}+\sigma_{t+1}\varepsilon_{t+1}-\frac{1}{2}\sigma_{t+1}^{2}),$ where by construction, $$\varepsilon_{t+1}$$ is $$\cal{N}(0,1)$$. We can also write: $\frac{P_{t+1}}{P_{t}}=\exp(r_{t+1})\exp(\sigma_{t+1}\varepsilon_{t+1}-\frac{1}{2}\sigma_{t+1}^{2}),$ where: $E_{t}[\exp(\sigma_{t+1}\varepsilon_{t+1}-\frac{1}{2}\sigma_{t+1}^{2})]=1,$ so that $$\exp(\sigma_{t+1}\varepsilon_{t+1}-\frac{1}{2}\sigma_{t+1}^{2})$$ can be interpreted as a multiplicative surprise. It is indeed the martingale ratio of a multiplicative martingale (see the notion of martingale difference in this post.). This multiplicative surprise has mean $$1$$ but median $$\exp(-\frac{1}{2}\sigma_{t+1}^{2})<1$$. We have thus decomposed the return into its expected component and a surprise. This is really a discrete time version of the usual geometric Brownian diffusion: $\frac{dP_{t}}{P_{t}}=r_{t}dt+\sigma_{t}dB_{t}.$ Intuition is sometimes easier to develop working on the discrete time model. As a final remark, we need to emphasize that although the shocks in the above discrete time model are conditionally log-normal, it does not imply that prices have log-normal marginal distributions. This is the case if $$(\mu_{t})_{t \in \mathbb{N}}$$ and $$(\sigma_{t})_{t \in \mathbb{N}}$$ are deterministic processes, but it fails to be the case in general.

1. If $$f(\cdot)$$ is a conxex function ($$\mathbb{R} \rightarrow \mathbb{R}$$) and $$z$$ is a random variable, $$E[f(z)]\geq f(E[z])$$.↩︎