What is an accepted method to calculate percent PnL from a short position?

Question

Calculating the normalized (e.g., percent or logarithmic) return on investment on a long (equity, call option, etc...) position is fairly simple. The percent return on investment for any position which costs $\mathbb{P_t}$ is

$$r_{long} = \frac{\mathbb{P}_T-\mathbb{P}_t}{\mathbb{P}_t}$$

where, $\mathbb{P}$ is the Numeraire property of a position (and thereby discounts interest payments and/or dividends).

This property is "calcuable" simply because the upfront cost of the position is finite. However, the position cost of a short seems to be undefined. Rather, one who shorts an instrument, including options, receives $\mathbb{P_t}$, so the percent PnL seems to be undefined:

$$r_{short} = \frac{\mathbb{P}_t-\mathbb{P}_T}{0} = \text{undef.}$$

This intuition is backed by the possibility of infinite loss on a short. The PnL of a short put, on the other hand, is slightly better defined because the maximum risk is equivalent to long equity.

What are some ways to calculate a normalized and/or risk-adjusted PnL on a short position? Is there any way to calculate initial investment as capital at risk.

Answer 1

A short position is a liability on your books, as the borrowed asset has to be returned to the owner. The return is then the percentage return of that liability.

Assume that the shorted asset at initial time $t_0$ has price $p(t_0)$. The initial liability is then $p(t_0)$. At a future time $t$ the liability is $p(t)$. The return at time $t$ is hence $$ r(t,t_0) = \frac{p(t_0)-p(t)}{p(t_0)} $$ So if the asset decline in price the return is positive, and if the asset goes up in price the return is negative.

In addition to this would be any return you might generate on the capital received through the short sale. Without leverage, this capital is held as collateral against the liability.