Charta is on Google Play: practise on the go. Get the app
Charta

Outline how delta, theta, gamma and vega are related to one another.

FRM® Part I · Valuation and Risk Models · Option Sensitivity Measures: The “Greeks” · FRM1.VR.16.g

The idea

The Greeks are tied together by the Black-Scholes-Merton differential equation, which can be written as Theta = r × V − r × S × Delta − ½ × σ² × S² × Gamma. Here V is the portfolio value. For a delta-neutral portfolio the delta term disappears, and if its value is near zero, the equation reduces approximately to Theta = −½ × σ² × S² × Gamma. Long gamma therefore goes with negative theta: the holder pays time decay in exchange for gains from large moves in the stock. Short gamma goes with positive theta: the writer collects decay but loses on large moves. Long options are also typically long vega, and short options short vega, but vega is not part of this equation. The usual mistake is to forget the one half or to use σ instead of σ². Exam questions solve for theta or gamma using it, with annual units.

Formulas

Theta from the Black-Scholes-Merton differential equation

Θ = r × Π − r × S × Δ − ½ × σ² × S² × Γ

r
the continuously compounded risk-free rate
Π
the value of the portfolio
S
the stock price
Δ
the portfolio delta
σ
the annualised volatility
Γ
the portfolio gamma

Theta is per year. For a delta-neutral portfolio worth about zero only the gamma term remains, so Θ ≈ −½ × σ² × S² × Γ. Dropping the ½ or using σ for σ² are the traps.

A practice question

From the question bank, with the reason each wrong option is wrong.

A delta-neutral portfolio with a value close to zero has a large positive gamma. What does the relationship among the Greeks imply about its theta?

  1. Theta is negative: the portfolio loses value as time passes. - correct
  2. Theta is positive: the portfolio gains value as time passes.Why not: Positive gamma is paid for with time decay; the signs of gamma and theta are opposite in this case.
  3. Theta is zero, because delta neutrality removes the effect of time.Why not: Delta neutrality removes first-order price risk, not time decay.
  4. Theta equals the portfolio's vega.Why not: The relationship links theta to gamma, not to vega; the two are different sensitivities.

For a delta-neutral portfolio, theta + (1/2) sigma² S² gamma = r x portfolio value. If value is near zero, theta is approximately -(1/2) sigma² S² gamma, so positive gamma goes with negative theta.

A harder one

A portfolio has delta 0.55 and gamma 0.04 per option, the stock is USD 50, r = 5%, volatility 30%, and the option is worth USD 3.00. Using theta + rS delta + (1/2) sigma² S² gamma = r x price, the option's theta per year is closest to:

  1. -5.725 - correct
  2. -5.875Why not: Omits the r x price term on the right-hand side of the equation.
  3. -4.350Why not: Omits the r S delta term.
  4. -2.975Why not: Adds the r S delta term instead of subtracting it when solving for theta.

Theta = r x 3.00 - r S delta - (1/2) sigma² S² gamma = 0.15 - (0.05 x 50 x 0.55 = 1.375) - (0.5 x 0.09 x 2,500 x 0.04 = 4.5) = -5.725 per year.

The hardest tier

A delta-neutral portfolio worth zero loses USD 2,000 per calendar day from time decay. The stock is at USD 40 and volatility is 25%. Using theta = -(1/2) sigma² S² gamma on an annual basis (365 days), the portfolio's gamma is closest to:

  1. 40Why not: Uses the daily theta without converting to an annual figure.
  2. 3,650Why not: Uses sigma (0.25) where the formula requires sigma² (0.0625).
  3. 14,600 - correct
  4. 7,300Why not: Omits the factor 2 when solving for gamma: gamma = -2 x theta / (sigma² S²).

Annual theta = -2,000 x 365 = -730,000. Gamma = -2 x theta / (sigma² S²) = 1,460,000 / (0.0625 x 1,600) = 14,600.

In the app: 6 practice questions on this outcome, spaced review that brings it back before you forget it.

Start free →

More in Option Sensitivity Measures: The “Greeks”