Black-Scholes Formula and European Option Pricing › Put-Call Parity Revisited

As we learned when studying option basics, put-call parity is a fundamental relationship linking European call and put prices:

$$C - P = S - Ke^{-r\tau}$$

where:

• $C$: call option price
• $P$: put option price
• $S$: stock price
• $K$: strike price
• $r$: risk-free rate
• $\tau$: time to expiration

This equality does not require the Black-Scholes stock-price distribution. It follows from matching European calls and puts on a non-dividend stock under the stated frictionless trading and financing assumptions. The long-call, short-put portfolio pays $S_T-K$. Buying a share and borrowing $Ke^{-r\tau}$ today, with $K$ repaid at expiration, has the same payoff. No arbitrage therefore equates their current values.

Let's verify that the Black-Scholes formulas satisfy put-call parity. This verification serves two purposes: it confirms that our formulas are internally consistent, and it demonstrates the elegant symmetry properties of the normal distribution.

Starting with the Black-Scholes call and put formulas:

$$\begin{aligned} C &= S N(d_1) - K e^{-r\tau} N(d_2) \\ P &= K e^{-r\tau} N(-d_2) - S N(-d_1) \end{aligned}$$

where:

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

We compute the difference $C - P$ to verify the relationship. The key mathematical fact we will use is the symmetry of the standard normal distribution: $N(x) + N(-x) = 1$ for any value of $x$. This follows from the fact that the normal distribution is symmetric around zero, so the probability of being less than $x$ plus the probability of being greater than $x$ (which equals the probability of being less than $-x$) must sum to one.

$$\begin{aligned} C - P &= [S N(d_1) - K e^{-r\tau} N(d_2)] - [K e^{-r\tau} N(-d_2) - S N(-d_1)] && \text{(substitute formulas)} \\ &= S [N(d_1) + N(-d_1)] - K e^{-r\tau} [N(d_2) + N(-d_2)] && \text{(group terms)} \\ &= S \cdot 1 - K e^{-r\tau} \cdot 1 && \text{(symmetry: } N(x) + N(-x) = 1\text{)} \\ &= S - K e^{-r\tau} && \text{(simplify)} \end{aligned}$$

where:

• $C$: call price
• $P$: put price
• $S$: stock price
• $K$: strike price
• $r$: risk-free rate
• $\tau$: time to expiration

The Black-Scholes formulas are internally consistent and satisfy the no-arbitrage constraint of put-call parity. This verification is more than a mathematical exercise. It confirms that the formulas respect the fundamental arbitrage relationships that must hold in any well-functioning market. Any pricing model, no matter how sophisticated, must satisfy put-call parity. If a proposed option pricing formula violated this relationship, we would know immediately that it contains an error.