NISM VIII — Equity Derivatives ~5 min read

Option Pricing — Black-Scholes and the Greeks

The Black-Scholes-Merton model prices a European option on a non-dividend stock. Its inputs are Spot, Strike, Time, Volatility and Risk-free rate. The Greeks — ...

Definition

Pricing Model + Sensitivities

The Black-Scholes-Merton model prices a European option on a non-dividend stock. Its inputs are Spot, Strike, Time, Volatility and Risk-free rate. The Greeks — Delta, Gamma, Vega, Theta, Rho — measure how the option price changes when each input changes.

In Simple Words

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Black-Scholes (1973) revolutionised options by giving a closed-form fair price. The formula uses Spot (S), Strike (K), time to expiry (T), volatility (σ) and risk-free rate (r). The key insight: a portfolio of stock + short call can be made riskless over an instant, so it must earn the risk-free rate — from which the option's fair value pops out. The Greeks express sensitivity: Delta = ∂Price/∂Spot (0 to 1 for calls, 0 to −1 for puts), Gamma = ∂Delta/∂Spot (curvature), Vega = ∂Price/∂σ (volatility sensitivity), Theta = ∂Price/∂T (time decay), Rho = ∂Price/∂r (interest-rate sensitivity). ATM options have highest Gamma, Vega and Theta. Indian traders use Black-Scholes widely for Nifty and Bank Nifty options.

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Delta = Speed

How fast the option moves per ₹1 move in spot

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Gamma = Acceleration

How fast Delta itself changes — peaks at ATM

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Vega = Volatility Sensitivity

Option gains value when IV rises — pure vol play

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Theta = Clock Tax

Daily premium decay — buyer's enemy, writer's friend

Real-Life Scenario

A Practical Example

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The Five Greeks

What each Greek measures and whether the buyer wants it positive or negative

Greek	Measures	Long Call Sign	Long Put Sign	ATM / ITM / OTM Peak
Delta	Spot sensitivity	+0 to +1	−1 to 0	Deep ITM = ±1
Gamma	Delta sensitivity	Positive	Positive	Peaks at ATM
Vega	Volatility sensitivity	Positive	Positive	Peaks at ATM
Theta	Time decay	Negative	Negative	Largest at ATM, accelerates near expiry
Rho	Rate sensitivity	Positive	Negative	Larger for long-dated options

Key Points to Remember

What Makes This Important

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Black-Scholes: 5 inputs (S, K, T, σ, r); closed-form price for European options

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Delta — spot sensitivity; call +0 to +1, put −1 to 0

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Gamma — curvature; highest at ATM, near expiry

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Vega — vol sensitivity; ATM options benefit most from rising IV

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Theta — time decay; option buyers lose it daily

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Rho — rate sensitivity; small for short-dated options

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ATM options have highest Gamma, Vega AND Theta — all max uncertainty

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Writers are short-Gamma, short-Vega, long-Theta — opposite of buyers

The Formula

Black-Scholes European Option

Call = S·N(d1) − K·e^(−rT)·N(d2)
Put  = K·e^(−rT)·N(−d2) − S·N(−d1)
d1 = [ln(S/K) + (r + σ²/2)·T] / (σ·√T)
d2 = d1 − σ·√T

S — Spot price of underlying

K — Strike price

T — Time to expiry in years

σ — Annualised volatility

r — Risk-free rate (continuous)

N(·) — Standard-normal cumulative distribution

ℹ️

Exam note: NISM does not require you to compute Black-Scholes by hand, but you should know the 5 inputs and that volatility is the most important and the only unobservable one.

FAQs

Frequently Asked Questions

Theta. Every day the option loses value from time decay regardless of spot direction, and theta accelerates as expiry nears — especially for ATM options, where time value is largest.

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