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.
Black-Scholes for call options, explained simply
Black-Scholes prices a European call from five inputs: stock price, strike, time, risk-free rate and volatility. Work out d1 and d2, find N(d1) and N(d2), and use C = S × N(d1) − K × e^(−rT) × N(d2). In the example (S = K = $50, r = 5%, σ = 30%, T = 0.5), the call is worth about $4.82.
What this Black-Scholes call lesson covers
- 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
Black-Scholes call formulas, explained
- 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
How to price a call with Black-Scholes step by step (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).
What happens to the call price when an input rises
| Input rises | Call price | Why |
|---|---|---|
| Stock price (S) | Rises | The right to buy at $50 is worth more |
| Strike (K) | Falls | You pay more to buy |
| Volatility (σ) | Rises | Bigger rises become more likely |
| Time to expiry (T) | Rises | More time for the stock to move |
| Interest rate (r) | Rises | The strike you pay later is worth less today |
How exact is a Black-Scholes price? David's take
The formula gives $4.82 to the cent. But volatility, the one input you can't observe, is a guess. At 20% volatility the call is worth $3.44, and at 40% it's worth $6.19. My concept 'High IQ, Wrong Variables' says smart people can reason perfectly with the variables they're given and still be poor at choosing them. The formula is precise; the variable isn't. Learn it, and always show the price for a range of volatilities.
From David's concept: High IQ, Wrong Variables. This is David's opinion, alongside the standard textbook answer above.
Harder Black-Scholes call questions
What if the option had 1 year or 3 months left instead of 6 months?
With everything else the same, about $7.12 for 1 year and $3.29 for 3 months. More time gives the stock more chance to rise.
What does N(d1) tell you?
N(d1), about 0.5886 here, is the call's delta: roughly how much the call price changes for a $1 move in the stock. It's also the number of shares needed to hedge one call.
Common Black-Scholes call 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).