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

Work out a portfolio's delta, gamma and vega.

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

The idea

A portfolio's Greeks are the position-weighted sums of the individual Greeks. Portfolio delta = sum of (quantity × delta). Gamma and vega are built the same way. Long positions count positive and short positions negative, so a short put, whose own delta is negative, contributes positive delta. The underlying shares have a delta of one per share, and zero gamma and zero vega. Use consistent units: vega per percentage point is multiplied by the change in volatility in points. The Greeks give a quick profit estimate. For price moves, Change in value = delta × dS + ½ × gamma × dS². For volatility moves, Change in value = vega × change in volatility. The usual errors are sign mistakes on short positions, counting shares in gamma, and dropping the one half. Exam questions are often a multi-position table.

Formulas

Change in portfolio value from a price move, delta and gamma

ΔV = Δ × dS + ½ × Γ × dS²

Δ
the portfolio delta
dS
the change in the stock price
Γ
the portfolio gamma

Portfolio delta and gamma are position-weighted sums, shorts negative, shares counted in delta only. Dropping the ½ is the usual slip.

Change in portfolio value from a volatility move

ΔV = vega × Δσ

vega
the portfolio vega, position-weighted, shorts negative
Δσ
the change in volatility, in the units vega is quoted in

Vega per percentage point times the move in points. Multiplying by a decimal change instead mis-scales it by 100.

A practice question

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

A portfolio is long 4,000 call options (delta 0.60 each) and short 1,500 shares of the underlying. What is the portfolio delta, in shares?

  1. 900 - correct
  2. 3,900Why not: Adds the short share position as if it were long: 2,400 + 1,500.
  3. 2,400Why not: Ignores the share position and counts only the options.
  4. 2,500Why not: Subtracts the shares from the number of options (4,000 - 1,500) without multiplying by delta.

Delta of options = 4,000 x 0.60 = 2,400. Shares contribute -1,500. Portfolio delta = 2,400 - 1,500 = 900 shares.

A harder one

A portfolio is long 10,000 calls (gamma 0.035), short 4,000 puts (gamma 0.028) and long 3,000 shares. The portfolio gamma is closest to:

  1. 238 - correct
  2. 462Why not: Adds the short puts' gamma instead of subtracting it: 350 + 112.
  3. 3,238Why not: Counts the 3,000 shares as having gamma one; shares have zero gamma.
  4. 350Why not: Counts only the long calls and ignores the short put position.

Calls: 10,000 x 0.035 = 350. Short puts: -4,000 x 0.028 = -112. Shares: 0. Portfolio gamma = 350 - 112 = 238.

The hardest tier

A portfolio is long 4,000 calls (vega 0.12 per percentage point), short 6,000 puts (vega 0.10) and short 2,500 calls on another maturity (vega 0.08). Volatility rises by 2.5 percentage points across all options. The approximate change in portfolio value is:

  1. A gain of 3,200Why not: Treats all positions as long vega: (480 + 600 + 200) x 2.5.
  2. A loss of 8Why not: Multiplies the net vega (-320) by 0.025 instead of 2.5, mis-scaling the percentage points.
  3. A loss of 320Why not: Reports the net vega per point without multiplying by the 2.5 point move.
  4. A loss of 800 - correct

Net vega = 4,000 x 0.12 - 6,000 x 0.10 - 2,500 x 0.08 = 480 - 600 - 200 = -320 per point. For +2.5 points: -320 x 2.5 = -800. Short options carry negative vega, so the short positions outweigh the long calls.

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”