Black-Scholes for Call Options
Black-Scholes prices a European call option from five inputs: stock price, strike, time to expiry, risk-free rate, and volatility. You calculate d1 and d2, look up their normal probabilities, and plug them into one formula. It looks scary, but it is really just arithmetic done in the right order.
What you will learn
- The five Black-Scholes inputs and the model's key assumptions
- How to calculate d1 and d2 without getting lost
- How to find N(d1) and N(d2) with a z-table or Excel NORM.S.DIST
- The call formula: C = S N(d1) − K e^(−rT) N(d2)
- Why higher volatility and more time make a call worth more
The formulas
- S
- current stock price
- K
- strike price
- r
- risk-free rate, continuously compounded, per year
- σ
- annual volatility of the stock's returns
- T
- time to expiration in years
- N(x)
- standard normal cumulative probability of x
- e^(−rT)
- continuous discount factor
Worked example
Price a European call with S = $50, K = $50, r = 5%, σ = 30%, T = 0.5 years (no dividends).
- ln(50 ÷ 50) = 0; (0.05 + 0.30² ÷ 2) × 0.5 = 0.0475; σ√T = 0.30 × √0.5 = 0.2121.
- d1 = 0.0475 ÷ 0.2121 = 0.2239; d2 = 0.2239 − 0.2121 = 0.0118.
- N(d1) = 0.5886 and N(d2) = 0.5047.
- PV of strike = 50 × e^(−0.025) = $48.7655.
- C = 50 × 0.5886 − 48.7655 × 0.5047 = 29.43 − 24.61 = $4.82.
Answer: The call is worth about $4.82 per share (normal probabilities computed exactly and shown to 4 decimals; price rounded to the cent).
Common questions
What do N(d1) and N(d2) mean in Black-Scholes?
N(d1) is the call's delta: roughly how many shares you would hold to copy the option, and how much the call price moves per $1 change in the stock. N(d2) is the risk-neutral probability that the call finishes in the money, meaning the stock ends above the strike.
What are the assumptions of the Black-Scholes model?
The basic model assumes European exercise, no dividends during the option's life, constant volatility and interest rates, lognormally distributed stock prices, no transaction costs or taxes, and the ability to trade continuously and borrow or lend at the risk-free rate.
How do you calculate N(d1) in Excel?
Use =NORM.S.DIST(d1, TRUE), which returns the standard normal cumulative probability. Do the same for d2. In older versions of Excel, =NORMSDIST(d1) gives the same result. Using Excel avoids rounding errors from reading a printed z-table.
Why does Black-Scholes use e^(−rT) to discount the strike?
The model is built in continuous time, so it uses continuous compounding. K × e^(−rT) is the present value of the strike paid at expiry. If you are given an annual rate, some courses convert it to a continuous rate first using ln(1 + r).