How Are Options Priced? Black-Scholes and Binomial Models

Options give the right (not obligation) to buy or sell at a set price. Black-Scholes gives us the mathematically precise fair value of this right.

Black-ScholesOption GreeksCall OptionPut OptionBinomial

Live Result

12.61898

ƒ: S * N_d1 - K * exp(-r * T) * N_d2raw: 12.6190

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/asset price

S = 100

Strike price (exercise price)

K = 100

r% as decimal

Risk-free rate (decimal)

r = 0.05

T% as decimal

Time to expiration (years)

T = 1

N_d1% as decimal

Cumulative normal probability for d1

N_d1 = 0.637

N_d2% as decimal

Cumulative normal probability for d2

N_d2 = 0.537

sigma

Annual volatility (standard deviation) of asset

sigma = 0.20

💡 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: How Are Options Priced? Black-Scholes and Binomial Models

Contingent Claim Valuation · Educational Guide

The Core Idea

An option is the right to buy (call) or sell (put) something at a fixed price in the future. The question: how much should this right cost today? Black-Scholes gives a mathematical answer based on price, strike, time, interest rate, and volatility.

How It Works

Black-Scholes: Call value = S × N(d1) − K × e^(−rT) × N(d2). N(d1) is the probability of exercise (roughly). Higher stock price, lower strike, more time, and higher volatility all increase call value. The key insight: higher volatility is GOOD for option holders — more chance of big upside while downside is capped at $0.

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Real-World Example: Stock at $100, strike $100, 1 year, r=5%, σ=20%. d1 = 0.35, N(d1) = 0.637. d2 = 0.15, N(d2) = 0.560. Call = 100×0.637 − 100×exp(−0.05)×0.560 = 63.7 − 53.2 = $10.50. This call option is worth $10.50.

How Are Options Priced? Black-Scholes and Binomial Models

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