Black-Scholes Formula and Delta Hedging

In 1973, Black and Scholes published a formula that let any trader price a European option in seconds. It is the continuous-time limit of the binomial model (let periods go to infinity and the tree becomes a lognormal diffusion. For FAM, you need to apply the formula, compute delta, and understand delta hedging. No derivation required) just precise execution.

Use the Black-Scholes formula to calculate the price of a European option from the current asset price, strike, time to expiration, volatility, and risk-free interest rate. The delta hedge comes from how that price responds to a small change in the asset price.

HIGH-FREQUENCY: The Black-Scholes call and put formulas with $d_1$ and $d_2$ are tested repeatedly.

$$C = S \cdot N(d_1) - Ke^{-rT} \cdot N(d_2)$$

$$d_1 = \frac{\ln(S/K) + (r + \sigma^2/2)T}{\sigma\sqrt{T}}, \quad d_2 = d_1 - \sigma\sqrt{T}$$

$$P = Ke^{-rT} \cdot N(-d_2) - S \cdot N(-d_1)$$

• $\ln(S/K)$
• $(r + \sigma^2/2) \times T$

Common mistakes

• Confusing $d_1$ and $d_2$. Call uses $N(d_1)$ with $S$ and $N(d_2)$ with $Ke^{-rT}$. Swapping gives a materially wrong price. Trap: a price off by a few dollars.
• Using $\sigma^2$ instead of $\sigma^2/2$ in $d_1$. The numerator contains $r + \sigma^2/2$. Doubling the volatility term inflates $d_1$. Trap: a slightly too-large call price.
• Forgetting to discount the strike. Use $Ke^{-rT}$, not $K$. Trap: answer shifted by the discount factor.

Bottom line

• Call: $C = SN(d_1) - Ke^{-rT}N(d_2)$. Put: $P = Ke^{-rT}N(-d_2) - SN(-d_1)$.
• $d_1 = [\ln(S/K) + (r + \sigma^2/2)T]/(\sigma\sqrt{T})$, $d_2 = d_1 - \sigma\sqrt{T}$
• The call delta is $N(d_1)$; the put delta is $N(d_1) - 1$.
• Delta hedge a short call: buy $N(d_1)$ shares per call, then rebalance as $S$ moves.

Exam shortcut

Compute $d_1$ in pieces: numerator first, denominator second, then divide. Write every intermediate value. The most common exam error is an arithmetic mistake inside $d_1$ that cascades. Remember: "SND minus KND" (S times N(d1) minus K (discounted) times N(d2). "d1 is the big one") always larger than $d_2$. For delta hedging: shares = delta times number of options.

The full lesson (about 2,095 words, 14 min read) adds 3 worked examples, all 5 common mistakes, a self-check, free in the app.

Learning objectives

• 6c

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