What is the Black-Scholes model, and how is it applied to price European call options? Provide a step-by-step example with sample inputs.

**Bottom line 🛠️** The Black–Scholes model is the industry-standard, closed-form formula for valuing European options. It prices a European call by calculating two probabilities— *d₁* and *d₂*—that a risk-neutral investor assigns to the option finishing in-the-money, and then discounting the strike price at the risk-free rate. Below is an intuitive walk-through and a numerical example you can replicate on a spreadsheet or calculator. 📈✨ --- ## 1. The intuition (why it works) 1. **Arbitrage logic** • Hold the stock *and* short a fraction of an option so that the combined position is risk-less over an instant. • Because the position is risk-less, it must earn the risk-free rate *r*. • Solving that no-arbitrage condition leads to a partial-differential equation whose solution for a European option is the Black–Scholes formula. 2. **Key assumptions** • Stock price follows a log-normal random walk with constant volatility σ. • No dividends (or they’re known and can be adjusted for). • Flat, known risk-free rate. • Continuous trading, frictionless markets, no arbitrage. --- ## 2. The formula for a European **call** ``` C = S·N(d₁) – K·e^(–rT)·N(d₂) d₁ = [ ln(S/K) + (r + ½σ²)T ] / [ σ√T ] d₂ = d₁ – σ√T ``` where • *C* = call price • *S* = current stock price • *K* = strike price • *T* = time to expiration (in years) • *r* = continuously-compounded risk-free rate • *σ* = annualised volatility of the stock • *N(·)* = cumulative normal-distribution function (the “risk-neutral probabilities”). --- ## 3. Step-by-step worked example Let’s price a **9-month (0.75 yr) European call** with these sample inputs: | Parameter | Symbol | Value | |-----------|--------|-------| | Stock price | S | 50.00 | | Strike price | K | 55.00 | | Time to expiration | T | 0.75 yr | | Risk-free rate (cont.) | r | 3 % (0.03) | | Volatility | σ | 30 % (0.30) | ### 3.1 Compute d₁ and d₂ 1. ln(S/K) = ln(50 / 55) = −0.09531 2. σ² / 2 = 0.30² ⁄ 2 = 0.045 3. (r + ½σ²)T = (0.03 + 0.045)·0.75 = 0.05625 4. Numerator = −0.09531 + 0.05625 = −0.03906 5. Denominator = σ√T = 0.30·√0.75 = 0.30·0.8660 = 0.2598 ``` d₁ = −0.03906 / 0.2598 = −0.1504 d₂ = d₁ – 0.2598 = −0.4102 ``` ### 3.2 Look up N(d₁) and N(d₂) Using a standard normal table or Excel’s `NORM.S.DIST`: ``` N(d₁) = N(−0.1504) ≈ 0.4404 N(d₂) = N(−0.4102) ≈ 0.3410 ``` ### 3.3 Plug into the pricing equation 1. Present-value the strike: K·e^(–rT) = 55·e^(−0.03·0.75) = 55·0.9777 = 53.77 2. Call value: ``` C = 50·0.4404 – 53.77·0.3410 = 22.02 – 18.37 ≈ $3.65 ``` > **Result:** the fair theoretical price of this European call is **≈ $3.65**. --- ## 4. How to use the model in practice 🧩 1. **Gather inputs** — market price of the underlying, strike, time to maturity, risk-free yield (use a Treasury bill/STRIP rate), and an estimate of σ (historical or option-implied). 2. **Calculate d₁ & d₂** — spreadsheet formulas make this trivial. 3. **Call/put parity** — once you have the call price, a European put price is `P = C + K·e^(−rT) − S`. 4. **Compare to market** — if the quoted option deviates materially from the model, ask why (dividends, discrete events, liquidity premia, changing volatility, early-exercise features, etc.). 5. **Scenario testing** — tweak σ or S to see how Vega/Delta affect value; that sensitivity analysis is essential for risk management. --- 🎯 **Take-away:** Black–Scholes gives you a quick, analytically clean price for European options under idealised conditions. While real markets add dividends, early exercise, or stochastic volatility, the model remains a foundational tool for gauging “fair value” and greeks—and an excellent starting point for more advanced option pricing frameworks. Happy modelling! 🔬💡