FINM6041 · Applied Derivatives
The Greek Letters & Dynamic Hedging
This chapter covers Lecture 09 (the Greek letters) and its Workshop 09 gamma extension, and it is one of the heaviest-weighted topics on the paper — delta-neutral hedging appears in both sample finals. The Greeks are the local sensitivities of an option's price: delta (Δ) to the underlying, gamma (Γ) to delta itself, vega (ν) to volatility, theta (Θ) to time, and rho (ρ) to the interest rate. In this course delta and gamma get the full quantitative treatment while vega, theta and rho stay conceptual (the formula sheet carries only delta and gamma). The workhorse result is that a call has Δ = e−qTN(d₁) and a put has Δ = e−qT[N(d₁) − 1], that a share has delta 1 and a forward on a yield asset has delta e−qT, and that the delta of a portfolio is simply the sum of its position deltas. That additivity is what makes a book delta-neutral: compute the net delta and trade the underlying to cancel it. Gamma tells you how quickly that hedge decays — it is largest for at-the-money options near expiry — and because the underlying has zero gamma, gamma-neutrality can only be reached by adding a second option.
What this chapter covers
- 01The five Greeks as local sensitivities: Δ (to S), Γ (to Δ), ν vega (to σ), Θ theta (to time), ρ rho (to r)
- 02Delta formulas: call Δ = e^{−qT}N(d₁) > 0; put Δ = e^{−qT}[N(d₁) − 1] < 0
- 03Reference deltas: one share = +1; a forward on a yield asset = e^{−qT}; a short position flips the sign
- 04Portfolio delta = Σ of position deltas — the additivity that turns hedging into arithmetic
- 05Building a delta-neutral book: net the deltas, then trade the underlying to cancel them
- 06Gamma: highest for at-the-money options near expiry; measures how fast delta moves
- 07Why gamma-neutrality needs a second option — the underlying has Γ = 0
- 08Practical dynamic hedging: rebalancing cost, and the 'buy high / sell low' bind of a short-option delta hedge
Making a mixed option-and-forward book delta-neutral
- 2 marksWrite the unit deltas: long call Δ = e^{−qT}N(d₁); short forward on a yield asset has delta −e^{−qT} (a long forward is +e^{−qT}); one share is +1. Positions carry their sign, and portfolio delta = Σ(position × unit delta).
- 2 marksCompute d₁ = [ln(31/30) + (r − q + σ²/2)T]/(σ√T). ln(31/30) = 0.032790; (0.04 − 0.03 + 0.0242) × 0.5 = 0.0171; numerator = 0.049890. Denominator = 0.22 × 0.70711 = 0.155564. d₁ = 0.3207, so N(d₁) = 0.6258.
- 1 markCall delta = e^{−qT}N(d₁) = e^{−0.015} × 0.6258 = 0.985112 × 0.6258 = 0.616483.
- 1 markForward delta (per contract, on a yield asset) = e^{−qT} = e^{−0.015} = 0.985112.
- 2 marksPortfolio delta = 80,000 × 0.616483 − 120,000 × 0.985112 = 49,318.6 − 118,213.4 = −68,894.8 ≈ −68,895.
- 2 marksThe book delta is negative, so buy the underlying to offset it: go long about 68,900 shares (each contributing +1 delta), which brings the total delta to approximately zero.
Key terms
- Delta (Δ)
- The sensitivity of an option's price to a small move in the underlying, ∂(price)/∂S — the slope of the price-versus-S curve. For a yield-paying asset a call has Δ = e^{−qT}N(d₁) (positive) and a put has Δ = e^{−qT}[N(d₁) − 1] (negative).
- Gamma (Γ)
- The rate of change of delta with the underlying, ∂Δ/∂S = ∂²(price)/∂S². It is the curvature of the price curve and is largest for at-the-money options near expiry, telling you how often a delta hedge must be rebalanced. Long options have Γ > 0; short options have Γ < 0.
- Delta-neutral
- A position whose net delta is zero, so it is (locally) insensitive to small moves in the underlying. Achieved by computing the portfolio delta and taking an offsetting position in the underlying; it must be re-established as the underlying moves.
- Portfolio delta (additivity)
- The delta of a book equals the sum of the deltas of its positions, each weighted by size and sign. This linearity is what makes hedging a mixed book of calls, puts, forwards and shares a single arithmetic step.
- Vega, theta, rho
- The remaining Greeks: vega (ν) is sensitivity to volatility σ, theta (Θ) to the passage of time, rho (ρ) to the interest rate r. In this course they are treated conceptually only — no formulas are provided for them on the exam formula sheet, which carries just delta and gamma.
- Dynamic (rebalancing) hedging
- Because delta changes as the underlying moves (that is gamma), a delta hedge must be adjusted repeatedly, which is costly. A hedger who is short options is forced to buy the underlying as it rises and sell as it falls — the 'buy high, sell low' cost of a short-option position.
The Greek Letters & Dynamic Hedging FAQ
What Greeks do I need for the FINM6041 exam?
Delta and gamma quantitatively; vega, theta and rho only conceptually. The exam formula sheet gives delta (all variants) and gamma but no vega/theta/rho formulas, which tells you exactly where the calculation questions can and cannot go. Know the delta and gamma formulas cold and be able to explain in words what vega, theta and rho measure and their signs.
How do you make a portfolio delta-neutral in FINM6041?
Compute the delta of every position (a call is e^{−qT}N(d₁), a put e^{−qT}[N(d₁) − 1], a share +1, a forward on a yield asset e^{−qT}, and short positions flip the sign), add them into a net portfolio delta, then trade the underlying against it: long the underlying if the net delta is negative, short it if positive. This is the single most repeated calculation in the two sample finals.
Why can't you gamma-hedge with the underlying?
Because the underlying asset is linear in itself — its price moves one-for-one with S — so its gamma is zero. Adding shares changes delta but never gamma. To reach gamma-neutrality you must trade a second option, size it by N₂ = −Γ_portfolio/Γ₂, and only then re-add shares to restore delta-neutrality.
Is delta hedging actually on the FINM6041 final?
Yes, reliably. A delta-neutral hedging question on a book of positions with a dividend yield appears in both the 2021 and 2022 sample papers and is flagged as part of the stable exam spine. Practise it until the sign conventions and the neutralising trade are automatic.
What is the delta of a call versus a put?
For an asset with continuous yield q, a call has Δ = e^{−qT}N(d₁), which is positive and between 0 and e^{−qT}; a put has Δ = e^{−qT}[N(d₁) − 1], which is negative. Without a dividend yield these reduce to N(d₁) and N(d₁) − 1. A call gains value as the stock rises, a put loses it — hence the opposite signs.
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Exam move
The exam move here is ruthless sign discipline plus additivity. Build a small reference card of unit deltas — long call +e^{−qT}N(d₁), long put e^{−qT}[N(d₁) − 1], share +1, long forward on a yield asset +e^{−qT} — and then a short of anything flips the sign. Every delta-neutral question is then the same three steps: get d₁ and N(d₁), turn each holding into a signed delta contribution, sum to the portfolio delta, and trade the underlying the opposite way. Because delta hedging recurs in both sample finals, rehearse it under the exam's decimals rule — keep N(d₁) and the products at full precision so your share count is right to the last hundred. Keep gamma conceptually clear too: it is the reason a delta hedge decays, it peaks for at-the-money options near expiry, and it is why any 'why does this hedge fail / what do I add?' prompt is answered with a second option, not more shares.
