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

Describe vega, gamma, theta and rho as they apply to option positions, and compute an option's gamma and vega.

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

The idea

Gamma is the rate of change of delta with respect to the stock price. It is identical for a call and a put with the same strike and expiry. For a stock paying no dividends, Gamma = N'(d1) ÷ (S × σ × √T). Here N' is the standard normal density, not the cumulative probability. Vega is the sensitivity to volatility and is also the same for calls and puts, and for a stock paying no dividends Vega = S × √T × N'(d1). Vega is usually quoted per one percentage point, so divide by 100. Theta is time decay, normally negative for long options, and rho is sensitivity to the interest rate, positive for calls and negative for puts. Long options carry positive gamma and vega, and short options carry the reverse. Common errors are using N(d1) in place of the density N'(d1) and mis-scaling vega or rho.

Formulas

Gamma of a European option, no dividends

Γ = N′(d1) ÷ (S × σ × √T)

N′(d1)
the standard normal density at d1
S
the stock price
σ
the annualised volatility
T
the time to expiry in years

The same for a call and a put with the same strike and expiry. Using the cumulative N(d1) instead of the density is the classic slip.

Vega of a European option, no dividends

Vega = S × √T × N′(d1)

S
the stock price
T
the time to expiry in years
N′(d1)
the standard normal density at d1

Gives the change per 1.00 of volatility; divide by 100 for a percentage point. The same for calls and puts.

A practice question

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

Which statement correctly describes gamma?

  1. It is the rate of change of the option price with respect to volatility.Why not: This describes vega, the sensitivity to volatility, not gamma.
  2. It is the rate of change of the option price with respect to time.Why not: This describes theta, the sensitivity to time decay, not gamma.
  3. It is the rate of change of the option price with respect to the interest rate.Why not: This describes rho, the sensitivity to the interest rate, not gamma.
  4. It is the rate of change of delta with respect to the price of the underlying. - correct

Gamma = d(delta)/dS = d2C/dS2. It is the same for a call and a put with the same strike and expiry and is largest for at-the-money options near expiry.

A harder one

A European call on a non-dividend-paying stock has S = 50, K = 50, r = 5%, volatility 30% and T = 0.5 years. Its gamma, N'(d1) / (S x sigma x sqrt(T)), is closest to:

  1. 0.0519Why not: Divides by T instead of sqrt(T).
  2. 0.0367 - correct
  3. 1.8341Why not: Omits the stock price S from the denominator.
  4. 0.0555Why not: Uses the cumulative normal N(d1) where the standard normal density N'(d1) belongs.

d1 = [ln(1) + (0.05 + 0.045) x 0.5] / (0.30 x 0.7071) = 0.2239; N'(d1) = 0.3891. Gamma = 0.3891 / (50 x 0.30 x 0.7071) = 0.0367.

The hardest tier

A European call on a non-dividend-paying stock has S = 70, K = 65, r = 4%, volatility 20% and T = 1 year. Its rho, K x T x e^(-rT) x N(d2), for a one percentage point rise in the interest rate is closest to:

  1. 0.468Why not: Uses N(d1) instead of N(d2); rho involves the exercise probability N(d2).
  2. 42.5Why not: Gives rho per 1.00 change in the rate rather than per percentage point.
  3. 0.443Why not: Omits the discount factor e^(-rT).
  4. 0.425 - correct

d1 = 0.6705, d2 = 0.4705, N(d2) = 0.6810. Rho = 65 x 1 x e^(-0.04) x 0.6810 = 42.53 per 1.00, or 0.425 per percentage point.

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”