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PART IX — MATHEMATICAL FINANCE🔴 Advanced

How Do You Calculate What a Stock Option Is Worth?

Black-Scholes Option Pricing Model

The Nobel Prize-winning formula that prices European options using five inputs: stock price, strike, volatility, time, and risk-free rate.

math financeoptionsBlack-ScholesderivativesvolatilityGreeksBSM

Live Result

⚠ Undefined function ln

ƒ(x)

Use variable names from the panel above (e.g. FV, r, n) — or type numbers directly: 10000 / (1 + 0.08)^10

Current stock price ($)

s = 100

Option strike price ($)

k = 105

r% as decimal

Continuously compounded risk-free rate (decimal)

r = 0.05

t% as decimal

Time to expiry in years

t = 0.5

v% as decimal

Annualised volatility σ (decimal)

v = 0.25

💡 You can also enter values directly in the formula: 10000 / (1 + 0.08)^10

⬇ Export Calculation

Exports a plain .txt file with your expression, formula, all variable values, result, and educational notes — ready to paste into any report, Word doc, Notion, or Google Docs.

The exported file includes the formula in standard mathematical notation — you can paste it directly into Excel, Google Sheets, or back into FinanceSheep.

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Learn: Black-Scholes Option Pricing Model

PART IX — MATHEMATICAL FINANCE · Educational Guide

The Core Idea

In 1973, Fischer Black and Myron Scholes published the most famous formula in finance. It answered the question traders had asked for decades: what is a stock option worth? The model earned Scholes and Robert Merton the 1997 Nobel Prize and transformed derivatives from a niche instrument into a $600+ trillion global market.

How It Works

Call = S × N(d₁) − K × e^(−rT) × N(d₂). Five inputs: S (stock price), K (strike), σ (volatility), T (time), r (risk-free rate). N(·) = cumulative normal CDF. d₁ = [ln(S/K) + (r + σ²/2)T] / (σ√T). d₂ = d₁ − σ√T. Put-Call Parity: Call + PV(K) = Put + S — if this holds, no arbitrage exists between puts, calls, and the stock. The Greeks: Delta (Δ) ≈ N(d₁) — hedge ratio; Vega — sensitivity to σ; Theta — time decay; Gamma — rate of delta change.

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Real-World Example: S=$100, K=$105, σ=25%, T=0.5yr, r=5%: d₁ = [ln(0.952) + (0.05+0.03125)×0.5] / (0.25×0.707) = [−0.0488+0.0406] / 0.1768 = −0.046. d₂ = −0.046 − 0.177 = −0.223. Call ≈ 100×N(−0.046) − 105×e^(−0.025)×N(−0.223) ≈ $6.04. Put via parity: $6.04 + $102.40 − $100 = $8.44. Delta ≈ N(−0.046) ≈ 0.48 — for every $1 the stock rises, the call gains ~$0.48.

How Do You Calculate What a Stock Option Is Worth?

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