How this number is made
Black-Scholes-Merton prices a European option that cannot be exercised early. Volatility, the interest rate, and the dividend yield stay constant, and the share price is assumed to move as a lognormal random walk. Delta, gamma, vega, and theta are the model’s slopes, per share. One listed contract is 100 shares. An American put, a dividend that is a known cash amount, and a volatility smile are not in this price.
- Volatility is the annualized number the market is using, not last month’s move, unless you mean to use last month.
- Years can be a fraction. Three months is 0.25.
Formula
d1 = (ln(S/K) + (r − q + σ²/2)T) ÷ (σ√T). Call = Se^(−qT)N(d1) − Ke^(−rT)N(d2). Put = Ke^(−rT)N(−d2) − Se^(−qT)N(−d1).
Worked example
With the figures already in the form, call price per share is $10.45.
Questions
Why is my broker’s price different?
The broker is quoting a market with a bid, an ask, and a smile. This is one volatility and a European exercise.
Is vega in dollars?
Vega here is the change in the per-share price for a one-point move in volatility, from 20% to 21%, not from 20% to 120%.