A stock price \(S\) is governed by the model

where the period length is 1 month. Let \(u=\mathrm{E}[w(k)]\) and \(\sigma^{2}=\operatorname{var}[w(k)]\) for all \(k\). Now suppose the basic period length is changed to 1 year. Then the model is

where each movement in \(K\) corresponds to 1 year. What is the natural definition of \(W(K)\) ? Show that \(\mathrm{E}\left[W(K) \mid=12 u\right.\) and \(\operatorname{var}|W(K)|=12 \sigma^{2}\). Hence parameters scale in proportion to time.

Transcribed Image Text:

In S(k+1) In S(k)+w(k),