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Black-Scholes Theta Calculator

Enter your option parameters to find its theta.

How it is calculated

The tool prices a European option with no dividends. Time T is in years (0.5 is six months); the risk-free rate r and the volatility σ are annual percentages and are divided by 100 internally. It first computes d1 = [ln(S/K) + (r + σ²/2)·T] / (σ·√T) and d2 = d1 − σ·√T. The decay term −S·φ(d1)·σ / (2√T) is the same for calls and puts. The interest term r·K·e−rT·N(d2) is subtracted for a call, and r·K·e−rT·N(−d2) is added for a put. The yearly theta is divided by 365 for the per-day figure. The normal curve is approximated numerically, so the last digit can differ slightly from other software. Spot, strike, time and volatility must be above zero, otherwise no result is shown; the rate may be zero or negative.

Worked example

Take a call with spot 100, strike 100, 0.5 years to expiry, a 5% rate and 20% volatility. Then d1 = 0.035 ÷ 0.1414 = 0.2475 and d2 = 0.1061. The decay term is about −5.47 and the interest term about −2.64, so theta is -8.1160 /yr. Dividing by 365 gives -0.0222 /day, so the option loses roughly two cents of value per day with the other inputs unchanged. The same inputs as a put give -3.2394 /yr.

Try this example →

Common mistakes

  • Entering time in days or months. The field is in years, so 30 days must be typed as about 0.0822, otherwise the decay figure is far off.
  • Typing the rate or volatility as a decimal (0.2) instead of a percentage (20). Both are divided by 100 by the tool, so 0.2 would mean 0.2%.
  • Applying the result to an American option or to a stock that pays dividends. The formula is for European options and has no dividend input.
  • Reading the yearly figure as the daily loss. The per-year value is shown first; the per-day value is the one to compare with a quote that moves day by day.
  • Expecting a positive theta for a long position. A negative value is normal here and means the holder loses time value.

Sources

About this calculator

Theta measures how much an option's price is expected to decay per day (or per year) purely from the passage of time, holding everything else constant. Under the Black-Scholes model, theta for a call is Θ = −(S·N'(d1)·σ)/(2√T) − r·K·e^(−rT)·N(d2), and for a put it's Θ = −(S·N'(d1)·σ)/(2√T) + r·K·e^(−rT)·N(−d2), where S is spot price, K is strike, T is time to expiration, r is the risk-free rate, σ is volatility, and N(·)/N'(·) are the normal CDF and PDF.

Theta is almost always negative for long option positions — it's the daily “rent” an option buyer pays for holding time value, and it accelerates as expiration approaches, especially for at-the-money options. Options traders, market makers, and quantitative analysts monitor theta alongside the other Greeks (delta, gamma, vega) to manage time decay in a portfolio, and options sellers specifically structure trades to collect theta as their primary source of profit.

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