Black-Scholes Formula and Delta Hedging | Free Exam FAM Lesson

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.

HIGH-FREQUENCY: The Black-Scholes call and put formulas with $d_{1}$ and $d_{2}$ are tested repeatedly.

$C = S \cdot N (d_{1}) - K e^{- r T} \cdot N (d_{2})$

$d_{1} = \frac{ln ( S / K ) + ( r + σ ^{2} /2 ) T}{σ T}, d_{2} = d_{1} - σ T$

$P = K e^{- r T} \cdot N (- d_{2}) - S \cdot N (- d_{1})$

$ln (S / K)$
$(r + σ^{2} /2) \times T$
Add results = numerator
$σ T$ = denominator
Divide = $d_{1}$
Subtract $σ T$ = $d_{2}$

HIGH-FREQUENCY: Delta hedging appears frequently.

$Δ_{call} = N (d_{1}), Δ_{put} = N (d_{1}) - 1$

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Common mistakes

Confusing $d_{1}$ and $d_{2}$ . Call uses $N (d_{1})$ with $S$ and $N (d_{2})$ with $K e^{- r T}$ . Swapping gives a materially wrong price. Trap: a price off by a few dollars.
Using $σ^{2}$ instead of $σ^{2} /2$ in $d_{1}$ . The numerator contains $r + σ^{2} /2$ . Doubling the volatility term inflates $d_{1}$ . Trap: a slightly too-large call price.
Forgetting to discount the strike. Use $K e^{- r T}$ , not $K$ . Trap: answer shifted by the discount factor.

Bottom line

Call: $C = S N (d_{1}) - K e^{- r T} N (d_{2})$ . Put: $P = K e^{- r T} N (- d_{2}) - S N (- d_{1})$ .
$d_{1} = [ln (S / K) + (r + σ^{2} /2) T] / (σ T)$ , $d_{2} = d_{1} - σ T$
Call delta = $N (d_{1})$ . Put delta = $N (d_{1}) - 1$ .
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 1,863 words, 12 min read) adds 3 worked examples, all 5 common mistakes, a self-check, free in the app.

Learning objectives

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