Exercise: Trader's theta vs mathematical theta

Suppose a risk system reports, for a single long European call, the following three numbers under three different conventions:

• Convention A: $\Theta_A = -15.30$
• Convention B: $\Theta_B = +15.30$
• Convention C: $\Theta_C = -0.042$

You know the option value is approximately \2.36$ with one month to expiry, and the underlying doesn't pay dividends.

Tasks

1. Identify which convention each row uses, choosing from:

   • $\partial V/\partial t$ in units of "per year"
   • $-\partial V/\partial t$ in units of "per year" (trader's theta, per year)
   • $\partial V/\partial t$ in units of "per calendar day"
   • $-\partial V/\partial t$ in units of "per calendar day"

2. A trader says "this option decays fast — theta is fifteen." Is that plausible? Under which convention?

3. Your P&L calculator ingests a daily P&L estimate from theta and gamma. You pipe $\Theta_A$ into the formula $\text{PnL}_{\text{decay}} \approx \Theta \cdot \Delta t$ with $\Delta t = 1$ day. What does your calculator report, and what is the correct number?

Hint

Divide the per-year theta by $365$ to get per-calendar-day.