Black-Scholes (Call Option)
Prices a European call option, where d₁ and d₂ depend on the stock price, strike, volatility, rate and time. N(·) is the standard normal CDF.
Formula
Variables
Example
Higher volatility or more time to expiry both raise the option’s value
How it works
The Black-Scholes call formula gives the fair price of a European call option by balancing two discounted quantities: the chance-weighted value of receiving the stock at expiry, and the present value of paying the strike price if exercise is worthwhile. The terms d1 and d2 convert the stock’s future uncertainty into standard-normal probabilities, so N(d1) and N(d2) act like risk-adjusted likelihoods of payoff in a no-arbitrage world. It works because, under the model’s assumptions, any mispriced option could be combined with the stock and a risk-free bond to create a profit-free arbitrage opportunity.
When to use it
Use this formula for pricing or checking a European call option on a non-dividend-paying stock when you know the stock price, strike, volatility, interest rate, and time to expiry. It is also useful in derivatives analysis, hedging, and comparing option values across different market conditions.
Worked example, step by step
Suppose a stock trades at S0 = 100, the strike is K = 95, the risk-free rate is r = 0.04, time to expiry is T = 0.5 years, and volatility is σ = 0.25. Step 1: Compute d1 = [ln(S0/K) + (r + σ^2/2)T] / (σ√T) = [ln(100/95) + (0.04 + 0.25^2/2)(0.5)] / (0.25√0.5) = [0.0513 + (0.04 + 0.03125)(0.5)] / 0.1768 = [0.0513 + 0.0356] / 0.1768 = 0.0869 / 0.1768 ≈ 0.492 Step 2: Compute d2 = d1 - σ√T = 0.492 - 0.25√0.5 = 0.492 - 0.1768 ≈ 0.315 Step 3: Look up standard normal CDF values N(d1) ≈ N(0.492) ≈ 0.689 N(d2) ≈ N(0.315) ≈ 0.624 Step 4: Discount the strike K e^{-rT} = 95 e^{-0.04(0.5)} = 95 e^{-0.02} ≈ 95(0.9802) ≈ 93.12 Step 5: Apply Black-Scholes C = S0 N(d1) - K e^{-rT} N(d2) C = 100(0.689) - 93.12(0.624) C = 68.90 - 58.11 C ≈ 10.79 Final answer: the call option is worth about 10.79 per share.
Common mistakes to avoid
- Using the spot stock price S0 in place of the discounted strike term K e^{-rT}
- Forgetting that d1 and d2 depend on volatility and time through σ√T
- Using this formula for American options or dividend-paying stocks without adjusting the model
Did You Know?
The Black-Scholes model earned Scholes and Merton the 1997 Nobel Prize in Economics (Black had died in 1995).
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