Understand Black-Scholes intuition — Hinglish notes

5.3.9 · Stock-Market › The Greeks

BLACK-SCHOLES exist KYUN karta hai?

YEH kaunsi problem solve karta hai? 1973 se pehle, kisi ke paas bhi option ka non-arbitrage price nahi tha. Log andaza lagate the. Black, Scholes aur Merton ne dikhaya ki ek option ko perfectly replicate kiya ja sakta hai — continuously kuch stock aur kuch cash hold karke. Agar aap replicate kar sakte ho, toh option ka price forced hai — warna free money (arbitrage) milegi.

YEH kyun matter karta hai? Yeh ek gambling question ("kya stock upar jayegi?") ko ek engineering question mein convert karta hai ("kaunsa portfolio yeh payoff copy karta hai?"). Stock ka direction/drift nikal jaata hai — ek shocking result.

Formula ko scratch se banana

Step 1 — Stock ko random walk (GBM) ki tarah model karo

Maano ki stock Geometric Brownian Motion follow karta hai:

$d S = μ S d t + σ S d W$

Yeh step kyun? Returns (prices nahi) multiplicatively compound hote hain, isliye hum $\frac{d S}{S}$ model karte hain. $μ S d t$ term steady growth hai; $σ S d W$ price ke hisaab se scale kiye gaye random shocks hain. Yeh guarantee karta hai ki $S > 0$ hamesha.

Step 2 — Risk-free hedge banao (delta hedging)

Ek portfolio rakho: 1 option $V (S, t)$ long, $Δ$ shares short:

$Π = V - Δ S$

Itô's lemma (random-walk chain rule) use karke $d t$ mein change:

$d V = (\frac{\partial V}{\partial t} + \frac{1}{2} σ^{2} S^{2} \frac{\partial ^{2} V}{\partial S ^{2}}) d t + \frac{\partial V}{\partial S} d S$

Woh extra $\frac{1}{2} σ^{2} S^{2} V_{S S}$ term kyun? Kyunki random walk mein, $(d W)^{2} = d t$ negligible nahi hota — variance time mein linearly accumulate hoti hai. Yahi formula ki poori jaan hai.

$Δ = \frac{\partial V}{\partial S}$ choose karo. Tab random $d S$ terms cancel ho jaate hain:

$d Π = (\frac{\partial V}{\partial t} + \frac{1}{2} σ^{2} S^{2} \frac{\partial ^{2} V}{\partial S ^{2}}) d t$

Yeh step kyun? Humne randomness khatam kar di. Ek riskless portfolio ko risk-free rate $r$ kamaana hi chahiye, warna arbitrage hai:

$d Π = r Π d t = r (V - Δ S) d t$

Step 3 — Black-Scholes PDE

Dono expressions ko equate karo:

Itna beautiful kyun hai? Dekho $μ$ (real drift) chali gayi. Sirf $r$ aur $σ$ bacha.

Step 4 — European call ke liye solve karo

Boundary condition $V (S, T) = max (S - K, 0)$ ke saath, PDE solve karne par milta hai:

Figure — Understand Black-Scholes intuition

Worked examples

Common mistakes

Forecast-then-Verify

Flashcards

Black-Scholes fundamentally maanta hai ki stock kya follow karti hai?

Geometric Brownian Motion,

d S = μ S d t + σ S d W

(lognormal prices, normal log-returns).

Real drift

μ

option price se kyun gayab ho jaata hai?

Delta hedging stock exposure cancel karta hai; ek riskless portfolio ko

r

kamaana chahiye, toh pricing risk-neutral measure ke under hoti hai jahan drift

= r

hai.

Delta hedge ratio

Δ

kya hai?

Δ = \partial V / \partial S = N (d_{1})

ek call ke liye.

\frac{1}{2} σ^{2} S^{2} V_{S S}

term kahan se aata hai?

Itô's lemma se;

(d W)^{2} = d t

, toh variance

d t

mein negligible nahi hoti.

N (d_{2})

ko interpret karo.

Risk-neutral probability ki call in-the-money khatam ho (exercise hoga).

Black-Scholes call formula likho.

C = S N (d_{1}) - K e^{- r T} N (d_{2})

Quick ATM call approximation kya hai?

C \approx 0.4 S σ T

σ \to 0

aur

S > K

hone par, call kitni worth hogi?

max (S - K e^{- r T}, 0)

— sirf discounted intrinsic value.

Call price ki upper bound kya hai aur kyun?

C \leq S

; stock kharidne ka call, stock actually hold karne se better nahi ho sakta.

Black-Scholes PDE batao.

V_{t} + \frac{1}{2} σ^{2} S^{2} V_{S S} + r S V_{S} - r V = 0

Recall Feynman: 12-saal ke bachche ko samjhao

Socho ek coupon hai jo tumhe next month ek toy ₹100 mein kharidne deta hai. Agar toy ₹150 ki hogi, tumhara coupon ₹50 ka hai. Agar sirf ₹80 ka hoga, tum coupon phenk doge — uski value ₹0 hai, tumhara koi loss nahi. Toh coupon sirf help hi kar sakta hai. Black-Scholes ek clever recipe hai jo kehti hai: "Mujhe batao toy ki price kitni bouncy hai, coupon expire hone mein kitna time bacha hai, aur bank ka interest rate kya hai — main tumhe coupon ka fair price bataunga." Trick yeh hai: ek shopkeeper tumhara coupon copy kar sakta hai kuch toys aur kuch cash rakhke aur unhe daily adjust karke. Kyunki woh copy kar sakta hai, price fixed hai — koi cheating allowed nahi. Aur mazedaar baat, iska koi fark nahi padta ki tumhare hisaab se toy mahanga hoga; sirf yeh matter karta hai ki price kitni jumpy hai.

Connections

Geometric Brownian Motion — iske neeche ka stock model.
Itô's Lemma — jahan $\frac{1}{2} σ^{2} S^{2} V_{S S}$ term janm leti hai.
Delta Hedging — woh replication trick jo price fix karti hai.
Risk-Neutral Valuation — $μ \to r$ kyun hota hai.
The Greeks — Delta $= N (d_{1})$ , Gamma, Vega, Theta sab $C$ ko differentiate karke aate hain.
Put-Call Parity — call formula ko puts se jodta hai: $C - P = S - K e^{- r T}$ .
Implied Volatility — market prices se $σ$ nikalne ke liye Black-Scholes ko invert karo.