Understand Theta and time decay — Hinglish notes

5.3.3 · Stock-Market › The Greeks

Theta KYA hai?

Buyers ke liye negative KYU hota hai? Jab tum ek option khareedte ho, time tumhara dushman hai — tune optionality ke liye pay kiya jo erode hoti rehti hai. Jab tum option bechte (write karte) ho, time tumhara dost hai: tum woh decay collect karte ho, isliye ek short position effectively positive theta rakhti hai.

Time decay KYUN hoti hai? (First principles)

Ek option ka premium do parts mein banta hai:

$Premium = m a x (S - K, 0) for a call Intrinsic Value + extra paid for uncertainty Time Value$

Intrinsic value decay nahi hoti — yeh sirf aaj spot $S$ ka strike $K$ ke relative position par depend karti hai.
Time value isliye exist karti hai kyunki abhi bhi stock ke tumhare favour mein move karne ka waqt bacha hai. Zyada time = zyada possible acche outcomes = zyada pay karoge. Jaise expiry nazdik aati hai, fewer future paths remain, isliye time value zero ki taraf sidhti hai.

Kitni tezi se? Decay ki shape derive karna

Aao European call ke Black–Scholes formula se key qualitative result derive karte hain:

$C = S N (d_{1}) - K e^{- r τ} N (d_{2}), d_{1, 2} = \frac{l n ( S / K ) + ( r \pm \frac{1}{2} σ ^{2} ) τ}{σ τ}$

Yahaan $τ = T - t$ expiry tak bacha hua time hai. Theta $Θ = \partial C / \partial t = - \partial C / \partial τ$ hai. Differentiate karne par milta hai (non-dividend call ke liye):

$Θ = - \frac{S N ^{'} ( d _{1} ) σ}{2 τ} - r K e^{- r τ} N (d_{2})$

Formula ko scratch se padhna:

Denominator mein $τ$ kyun hai? — Kyunki uncertainty (ek diffusion) $τ$ ki tarah badhti hai: stock ke moves ka standard deviation $time$ ke saath scale hota hai. Time value $\propto τ$ near the money hoti hai, isliye uska derivative ek $\frac{1}{τ}$ lekar aata hai. Yahi crucial insight hai.

Figure — Understand Theta and time decay

Jaanney waale key behaviours

Situation	Theta ka magnitude	Kyun
At-the-money (ATM)	Sabse zyada	Khoone ke liye sabse zyada time value; max sensitivity
Deep in/out-of-money	Chhota	Decay karne ke liye thodi time value bachi
Expiry se bahut dur	Chhoti daily decay	$τ$ curve wahaan flat hoti hai
Expiry ke nazdik (ATM)	Bahut bada	$1/ τ$ blow up karta hai

Common mistakes (steel-manned)

Flashcards

Theta kya measure karta hai?

Ek option ki price ke time ke guzarne ke saath change ki rate (₹ lost per calendar day), baaki sab kuch constant rakhte hue.

Theta option buyer ke liye negative kyun hota hai?

Expiry nazdik aane par time value erode hoti hai; ek long option har din value khota hai, isliye

\partial V / \partial t < 0

Time decay linear hai ya non-linear, aur kyun?

Non-linear — near-ATM time value ∝

τ

hoti hai, isliye decay ∝

1/ τ

hoti hai, expiry ki taraf accelerate karti hai.

Theta sabse bada kis moneyness par hota hai?

At-the-money — iske paas khoने ke liye sabse zyada time value hoti hai.

Positive Theta kis position mein hota hai?

Ek short (written) option mein — seller decay collect karta hai.

Kya theta weekends par apply hota hai?

Haan — time value calendar time to expiry par depend karti hai, isliye options band dino mein bhi bleed karte hain.

Agar 16 din bache ho par ATM theta ₹0.50/day ho, toh 4 din par estimate karo.

16/4 = 2

→ lagbhag ₹1.00/day.

Theta hamesha kis Greek ke saath trade off hota hai?

Gamma — tum gamma risk lene ke liye theta paate ho.

Recall Feynman: 12-saal ke bacche ko samjhao

Sochо tum ek ticket khareedte ho jis par likha hai "agar Sunday se pehle baarish hoti hai, toh tum jeette ho." Har guzarta din bina baarish ke, tumhari ticket ki value kam hoti hai, kyunki baarish ke liye kam din bache hain. Sunday ki subah ticket almost worthless hai — jeetने ka koi waqt nahi bacha. Theta woh hai jitni value tumhari ticket sirf calendar flip hone se har din khoती hai. Aur yeh value tezi se aur tezi se khoती hai jaise Sunday nazdik aati hai.

Connections

The Greeks — theta paanch core sensitivities mein se ek hai
Understand Delta — directional sensitivity (spot), time ke saath contrast
Understand Gamma and its impact — high gamma ↔ high theta trade-off
Understand Vega and volatility — vega woh time value feed karta hai jo theta erode karta hai
Black-Scholes Model — $Θ$ ke liye source formula
Option Selling Strategies — "theta-gang," decay collect karna
Intrinsic vs Time Value — sirf time value decay hoti hai