QuantMax

The Greeks and hedging

4 lessons · 7 equations. Each lesson below gives its formulas and key rules; open the lesson for the full explanation.

Delta, gamma, vega and theta

The P&L equation

dΠ≈Δ dS+12Γ (dS)2+Θ dt+ν dσ\mathrm{d}\Pi \approx \Delta\,\mathrm{d}S + \tfrac{1}{2}\Gamma\,(\mathrm{d}S)^2 + \Theta\,\mathrm{d}t + \nu\,\mathrm{d}\sigma

A second-order Taylor expansion of the option price. Every day on an options desk is an argument about which of these four terms explained the day.

The Black–Scholes Greeks of a call

Δ=Φ(d1),Γ=φ(d1)SσT,V=ST φ(d1),ρ=KTe−rTΦ(d2)\Delta = \Phi(d_1),\quad \Gamma = \frac{\varphi(d_1)}{S\sigma\sqrt T},\quad \mathcal{V} = S\sqrt T\,\varphi(d_1),\quad \rho = KTe^{-rT}\Phi(d_2)

With no dividends. Put Greeks follow from parity: the put’s delta is Φ(d1)−1\Phi(d_1) - 1, its gamma and vega are the same as the call’s, and its rho is −KTe−rTΦ(−d2)-KTe^{-rT}\Phi(-d_2). Theta is −Sφ(d1)σ2T−rKe−rTΦ(d2)-\frac{S\varphi(d_1)\sigma}{2\sqrt T} - rKe^{-rT}\Phi(d_2) for the call.

Remember

  • Delta is the hedge ratio, the sensitivity and roughly the probability of finishing in the money.

Second-order Greeks: gamma, vanna, volga and the shapes they take

Gamma and vega at the money

Γ≈0.4SσT,ν≈0.4 ST\Gamma \approx \frac{0.4}{S\sigma\sqrt{T}}, \qquad \nu \approx 0.4\,S\sqrt{T}

Both from φ(0)≈0.4\varphi(0) \approx 0.4, the standard normal density at zero — the same constant as the at-the-money price approximation.

Vanna and volga in Black–Scholes

vanna=∂Δ∂σ=−φ(d1) d2σ,volga=∂V∂σ=V d1d2σ\text{vanna} = \frac{\partial\Delta}{\partial\sigma} = -\frac{\varphi(d_1)\,d_2}{\sigma}, \qquad \text{volga} = \frac{\partial\mathcal V}{\partial\sigma} = \mathcal V\,\frac{d_1 d_2}{\sigma}

Both vanish near the money forward and grow in the wings. Volga is positive whenever d1d_1 and d2d_2 share a sign — for every option except those very close to at-the-money — so long options are long volatility-of-volatility.

Remember

  • Gamma is ∂Δ/∂S\partial\Delta/\partial S; vanna is ∂Δ/∂σ\partial\Delta/\partial\sigma; volga is ∂ν/∂σ\partial\nu/\partial\sigma.

Delta hedging in practice: how often, and what it costs

What a hedged position earns

P&L≈12ΓS2(σrealised2−σimplied2)Δt\text{P\&L} \approx \tfrac{1}{2}\Gamma S^2\left(\sigma_{\text{realised}}^2 - \sigma_{\text{implied}}^2\right)\Delta t

Over a day, a delta-hedged long option earns the difference between the variance the stock delivered and the variance you paid for — weighted by dollar gamma.

How big is the hedging error?

σhedge≈π4  σ VN\sigma_{\text{hedge}} \approx \sqrt{\frac{\pi}{4}}\;\frac{\sigma\,\mathcal V}{\sqrt N}

For a delta-hedged at-the-money option rebalanced NN times over its life, the standard deviation of the final P&L. It falls with the square root of the number of rebalances and scales with vega — the error is, in effect, uncertainty about how much volatility was realised along the way.

Remember

  • A hedged long option earns 12ΓS2(σr2−σi2)Δt\tfrac12\Gamma S^2(\sigma_r^2 - \sigma_i^2)\Delta t.

Running a book: aggregating Greeks and neutralising them

Neutralise gamma, then delta

n=−ΓbookΓhedge,shares=−Δnewn=-\frac{\Gamma_{\mathrm{book}}}{\Gamma_{\mathrm{hedge}}},\qquad \text{shares}=-\Delta_{\mathrm{new}}

Use an option to remove gamma; stock has zero gamma and can then remove the remaining delta.

Remember

  • Greeks add within an underlying, weighted by size; they do not add across underlyings.

Detailed formula cards