Black-Scholes Formula and European Option Pricing › Summary

The Black-Scholes formula for European option pricing is:

$$C = SN(d_1) - Ke^{-r\tau}N(d_2)$$

where:

• $C$: call option price
• $S$: stock price
• $N(\cdot)$: cumulative standard normal CDF
• $d_1, d_2$: standardized moneyness terms
• $K$: strike price
• $r$: risk-free interest rate
• $\tau$: time to expiration

$$P = Ke^{-r\tau}N(-d_2) - SN(-d_1)$$

where:

• $P$: put option price
• $K$: strike price
• $r$: risk-free interest rate
• $\tau$: time to expiration
• $N(\cdot)$: cumulative standard normal CDF
• $d_1, d_2$: standardized moneyness terms
• $S$: stock price

Key insights:

• Solution from the PDE: The Black-Scholes formula arises from solving the Black-Scholes PDE (derived in the previous chapter) subject to the option's terminal payoff condition. The solution involves transforming the PDE to the heat equation and applying the risk-neutral valuation principle.

• Probabilistic interpretation: For the European call, $N(d_2)$ is its risk-neutral exercise probability and $N(d_1)$ is its exercise probability under the share measure. The term $SN(d_1)$ is the discounted expected stock receipt restricted to exercise outcomes.

• Five inputs: The formula takes stock price, strike, remaining maturity, interest rate and volatility. Volatility must be forecast or inferred from a compatible market premium, and the pricing rate must be selected using suitable market conventions.

• Monotonic relationships: Call prices increase with stock price, volatility, and interest rate, and decrease with strike price. Put prices exhibit the opposite relationships for stock price and rate. Both option types increase with volatility; the remaining-time effect is not universal for European puts.

• The Greeks: Delta and Gamma describe stock-price sensitivity and curvature; Theta measures calendar-time sensitivity, Vega volatility sensitivity, and Rho interest-rate sensitivity. This list contains no strike derivative. The next chapter applies these measures to hedging and risk management.

• Model limitations: Constant volatility, normally distributed log-returns, continuous trading, and European exercise only are assumptions that don't perfectly match reality. Understanding these limitations guides when to use more sophisticated models.

The Black-Scholes formula provides a foundation for studying option valuation and hedging under a tractable model. Delta hedging, implied volatility and risk-neutral pricing are tools we can carry into the models and methods that follow.