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

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.

• It is safe to use the initial investment as capital at risk for a short equity position. This assumes you would not be holding onto a losing trade where the loss has become greater than 100%. For a short option position, assume the position went against you, was assigned and then treat it the same way you would treat an equity position. – amdopt May 22 '17 at 12:15

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.