Transformations and change of variables

New variables from old ones

Finance constantly transforms random variables: prices into returns, returns into log returns, a position into its profit. For a linear change $Y=aX+b$ the rules are exact: $E[Y]=aE[X]+b$ and $\operatorname{Var}(Y)=a^2\operatorname{Var}(X)$, and a normal stays normal. Scaling a return by leverage $a$ scales its volatility by $|a|$ and its variance by $a^2$; adding a constant shifts the mean and leaves the spread alone.

For a nonlinear transformation $Y=g(X)$ you cannot just transform the mean. The density of $Y$ follows the change-of-variables formula: evaluate $f_X$ at the inverse and multiply by the stretching factor $|dx/dy|$. If $X$ is normal, $Y=e^{X}$ is lognormal with density $f_X(\ln y)/y$. Jensen's inequality says that for a convex $g$, $E[g(X)]\geq g(E[X])$: for example $E[e^{X}]>e^{E[X]}$ and $E[1/X]>1/E[X]$ for a positive non-constant $X$. Averages do not commute with curved functions.

• Linear: $E[aX+b]=aE[X]+b$, $\operatorname{Var}(aX+b)=a^2\operatorname{Var}(X)$.
• Change of variables: $f_Y(y)=f_X(g^{-1}(y))\,|dx/dy|$.
• Log returns add across time; Jensen: $E[g(X)]\geq g(E[X])$ for convex $g$.

Lab · what's graded

• A linear change: the mean10 pts
• A linear change: the variance10 pts
• Compounding log returns15 pts
• Jensen's gap15 pts
• Jensen with a reciprocal15 pts
• The lognormal density15 pts
• Why log returns20 pts