Noesis
5.3.10The Greeks

Learn to manage a position's Greeks

1,686 words8 min readdifficulty · medium
🇮🇳 Yeh note Hinglish mein padho →

WHY manage Greeks at all?

WHY this is the master tool: because Greeks are derivatives and derivatives are linear. If P=∑iniViP = \sum_i n_i V_i then ∂P∂S=∑ini∂Vi∂S.\frac{\partial P}{\partial S} = \sum_i n_i \frac{\partial V_i}{\partial S}. That single fact — linearity of differentiation — is the reason you can "add up" and "cancel" risks at all. Everything below is just choosing nin_i to set some sum to zero.

HOW to neutralise each Greek

1. Delta-hedging (the first thing everyone does)

HOW: Add underlying (Delta = 1 per share/future) until the sum is zero. nstock=−Δoptionsn_{\text{stock}} = -\Delta_{\text{options}}

2. But Delta-neutral is NOT risk-free — enter Gamma

HOW to neutralise Gamma: stock has Γ=0\Gamma = 0 (its Delta is constant), so stock cannot fix Gamma. You need another option. Solve two equations at once.

Let option A (yours) and hedging option B, with nBn_B contracts and nSn_S shares: Γ:ΓA+nBΓB=0  ⇒  nB=−ΓAΓB\Gamma:\quad \Gamma_A + n_B \Gamma_B = 0 \;\Rightarrow\; n_B = -\frac{\Gamma_A}{\Gamma_B} Δ:ΔA+nBΔB+nS=0  ⇒  nS=−(ΔA+nBΔB)\Delta:\quad \Delta_A + n_B \Delta_B + n_S = 0 \;\Rightarrow\; n_S = -(\Delta_A + n_B\Delta_B)

Figure — Learn to manage a position's Greeks

3. Theta and Vega — the ones you can't hedge with stock either

HOW to neutralise Vega: again use another option — Vega is highest for at-the-money, longer-dated options, so pick a hedging option to solve the Vega equation, exactly like Gamma: nB=−νAνB.n_B = -\frac{\nu_A}{\nu_B}. With more Greeks to zero, you need more independent instruments (one option per Greek + stock for Delta).

Forecast-then-Verify drill

Common Mistakes

Flashcards

Why can't stock be used to hedge Gamma or Vega?
Stock's Delta is constant, so its Gamma and Vega are both 0; it only shifts Delta.
What is the portfolio Delta formula?
Δport=∑iniΔi\Delta_{port}=\sum_i n_i\Delta_i (linearity of derivatives).
To hedge Gamma with option B, how many contracts?
nB=−ΓA/ΓBn_B=-\Gamma_A/\Gamma_B.
After Gamma-neutralising, what fixes the leftover Delta?
Short/long nS=−(ΔA+nBΔB)n_S=-(\Delta_A+n_B\Delta_B) shares of the underlying.
Why hedge Gamma before Delta?
Stock only affects Delta, so the Delta step can't disturb the already-fixed Gamma (triangular structure).
Which two Greeks trade off with opposite signs for a long option?
Gamma (positive) vs Theta (negative).
A delta-neutral short straddle — what still hurts on a big move?
Negative Gamma (convex losses) plus you're short Vega if IV rises.
Which options have the largest Vega?
At-the-money, longer-dated options.
How many independent instruments to zero N Greeks?
Roughly N (one option per option-only Greek + stock for Delta).
Recall Feynman: explain to a 12-year-old

Imagine your bike can wobble in different ways: forward-back, side-to-side, up-down. Each "wobble" is a Greek. If you only care about going forward, you add little supports (extra options, some stock) so the other wobbles cancel out. Stock is a support that only fixes the side-to-side wobble; to stop the "up-down" (big-jump) wobble you need a different support — another option. Fix the tricky wobble first, then the easy one, so you don't undo your own work.

Connections

Concept Map

measured by

measured by

measured by

measured by

enables

choose quantities to zero

first, biggest risk

target of

makes Delta drift, needs re-hedge

stock has Gamma=0 so

fix Gamma first, then

triangular hedge

Option = bundle of risks

Delta price sensitivity

Gamma Delta change rate

Theta time decay

Vega vol sensitivity

Linearity of differentiation

Portfolio Greek = sum of scaled Greeks

Neutralise unwanted Greeks

Delta-hedge with stock

Gamma-neutral with 2nd option

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Dekho, ek option sirf "price upar/neeche" ka bet nahi hai — usme ek saath kai risks bundled hote hain, aur har Greek us risk ka ek dimension measure karta hai. Delta = price move ka effect, Gamma = Delta khud kitni tezi se badalta hai, Theta = time bitne se value ka decay, Vega = implied volatility ka effect. Position "manage" karne ka matlab hai — jo risk tum lena hi nahi chahte, use instruments combine karke zero pe le aao, taaki sirf tumhara actual view exposed rahe.

Sabse pehle log Delta-hedge karte hain, kyunki wahi sabse bada rupee risk hota hai — stock ya future add karke (nstock=−Δoptionsn_{stock} = -\Delta_{options}) Delta zero kar do. Lekin yaad rakho: Delta-neutral ka matlab risk-free NAHI hota. Gamma abhi bhi zinda hai — bade move pe Delta wapas aa jaata hai. Aur yahan crucial baat: stock se Gamma aur Vega hedge nahi hote, kyunki stock ka Gamma aur Vega dono zero hain. In risks ke liye ek doosra option chahiye.

Isliye order important hai: pehle option se Gamma (ya Vega) neutral karo (nB=−ΓA/ΓBn_B = -\Gamma_A/\Gamma_B), phir bache-khuche Delta ko stock se saaf karo. Kyun is order me? Kyunki stock sirf Delta ko chhuta hai, to Delta-step tumhare fix kiye Gamma ko disturb nahi karega — yeh ek "triangular" structure hai, bahut neat. Aur ek gehri baat samajh lo: long option me Gamma positive hai par Theta negative — yaani convexity ka fayda daily decay ke saath aata hai. Market ne yeh trade-off already price kiya hai, isliye "managing Greeks" ka matlab often hota hai choose karna ki kaunsa risk lena hai.

Test yourself — The Greeks

L1 RecognitionL2 RecallL3 ProductionL4 ApplicationL5 Mastery

Connections

Understand Theta and time decayStock-Market · 5.3.3Learn Vega and volatility sensitivityStock-Market · 5.3.4Learn implied vs historical volatilityStock-Market · 5.3.6