Lesson 119 — Synthesis: Black-Scholes revisited

In the Black-Scholes formula $C = SN(d_1) - Ke^{-rT}N(d_2)$, what does the variable $S$ represent?

In the Black-Scholes formula, what does $N(d)$ represent?

For $S = K = 50$, $r = 0{,}06$, $\sigma = 0{,}20$, $T = 1$, calculate $d_1$ and $d_2$.

With $d_1 = 0{,}40$ and $d_2 = 0{,}20$ from exercise 119.3, and knowing that $N(0{,}40) \approx 0{,}6554$, $N(0{,}20) \approx 0{,}5793$, calculate the call price ($S = K = 50$, $r = 0{,}06$, $T = 1$).

With the data from exercise 119.4 ($C = 5{,}50$, $S = K = 50$, $r = 0{,}06$, $T = 1$), use put-call parity to calculate the price of the European put.

With $\Delta = N(d_1) = 0{,}6554$ (exercise 119.3), how many shares should you short to delta-hedge a long position in 500 calls?

Which Greek measures the sensitivity of the option price to volatility $\sigma$?

Calculate $d_1$ and $d_2$ for PETR4: $S = 46{,}60$, $K = 47$, $r = 14{,}65\%$ p.a., $\sigma = 35\%$, $T = 30$ days.

With $d_1 \approx 0{,}085$ and $d_2 \approx -0{,}015$ (exercise 119.8), and $N(0{,}085) \approx 0{,}534$, $N(-0{,}015) \approx 0{,}494$, calculate the theoretical price of the PETR4 call.

With $C \approx 1{,}92$, $S = 46{,}60$, $K = 47$, $Ke^{-rT} \approx 46{,}43$, calculate the price of the PETR4 put by put-call parity.

The Black-Scholes PDE is mathematically analogous to which other classical equation in physics?

Repeat the calculation from Example 2 ($S = K = 100$, $r = 0{,}05$, $T = 1$) with $\sigma = 0{,}30$ (higher volatility). Compare with the result $C \approx 10{,}45$ for $\sigma = 0{,}20$. What happens to the call price?

With $S = K = 100$, $r = 0{,}05$, $\sigma = 0{,}20$, calculate $C$ for $T = 0{,}25$ (3 months). Compare with $C \approx 10{,}45$ for $T = 1$. Interpret the effect of time (Theta).

Read from the standard normal table the values of $N(d)$ for $d \in \{0; 0{,}50; 1{,}00; 1{,}96; 2{,}00\}$. Use the symmetry $N(-d) = 1 - N(d)$ to calculate $N(-1)$ and $N(-1{,}96)$.

Explain in one sentence the probabilistic interpretation of $N(d_2)$ in the Black-Scholes formula.

For $S = 100$, $K = 110$ (out-of-the-money), $r = 0{,}05$, $\sigma = 0{,}20$, $T = 1$, calculate $d_1$, $d_2$ and $C$. Compare with the at-the-money option from Example 2.

With $C = 6{,}0$ from exercise 119.16 ($S = 100$, $K = 110$, $r = 0{,}05$, $T = 1$), calculate the put price by parity. Compare put and call.

The market quoted the PETR4 call (exercise 119.9) at R$ 2.50 while the model with $\sigma = 35\%$ gave R$ 1.92. Explain the concept of implied volatility and how to calculate it.

Describe the 3 variable substitutions that transform the Black-Scholes PDE into the heat equation. Why is this useful?

Describe a delta-neutral dynamic hedge for 1,000 PETR4 calls ($\Delta = 0{,}534$) over 2 days. Why is this hedge "dynamic" and what is its real cost that BS ignores?

Explain how the Central Limit Theorem (Lesson 77) justifies the assumption of log-normality of $S_T$ in the Black-Scholes model.

Why is the Black-Scholes PDE classified as parabolic? What does this mean physically?

In the special case $S = K$ and $r = 0$, show that $C = S[2N(\sigma\sqrt{T}/2) - 1]$. Use the symmetry of the standard normal distribution.

The function $N(d)$ has no closed-form formula. Describe a polynomial approximation (Abramowitz-Stegun) used in implementations without a statistical library. Calculate $N(0{,}35)$ by the approximation and compare with the table value ($\approx 0{,}6368$).

Sketch the derivation of the Black-Scholes PDE: construct the replicating portfolio $\Pi = \Delta S - C$, apply Itô's Lemma to $dC$, eliminate risk by choosing $\Delta$ appropriately, and use no-arbitrage to obtain the PDE.

List 3 Black-Scholes assumptions that clearly failed during the 2008 financial crisis. For each one, cite an alternative model that relaxes it.

Verify numerically that the BS formula satisfies the BS PDE, using the values from Example 2 ($S = K = 100$, $r = 0{,}05$, $\sigma = 0{,}20$, $T = 1$). Calculate each term of the PDE.

Prove put-call parity $C - P = S - Ke^{-r(T-t)}$ via a no-arbitrage argument with two portfolios of the same payoff.

Calculate the limit $\lim_{\sigma\to 0^+} C(S, t)$ by the Black-Scholes formula. Use the property $N(-\infty) = 0$ and $N(+\infty) = 1$. Interpret financially.

How does the Selic rate affect the price of a European call according to BS? Calculate the Rho Greek ($\rho = \partial C / \partial r$) for the PETR4 data (exercise 119.9) and interpret.

Calculate the Vega of the call from Example 1 ($S = K = 100$, $d_1 = 0{,}35$, $T = 1$, $\sigma = 0{,}20$). Use $\phi(0{,}35) \approx 0{,}375$. Interpret: how much does the call change if $\sigma$ rises from 20% to 21%?

Calculate the Gamma of the call from Example 1 ($S = K = 100$, $d_1 = 0{,}35$, $\sigma = 0{,}20$, $T = 1$). Interpret: if $S$ rises R$ 1, how much does Delta change?

In a portfolio with 2 equally weighted assets, with volatilities $\sigma_1 = \sigma_2 = 25\%$ and correlation $\rho = 0{,}5$, calculate the portfolio volatility. How does this connect to the covariance matrix used in multi-asset BS?

Sketch the payoff diagram of a European call at expiration with $K = 100$ and premium paid $C = 10{,}45$. Identify the break-even point and the profit/loss zones.

Write $N(d)$ as an integral and explain why this integral has no closed-form solution in elementary functions. How does this connect to the Fundamental Theorem of Calculus (Lesson 83)?

Compare geometric Brownian motion (GBM) with a simple (discrete) random walk. Why is GBM more appropriate for modeling stock prices?

PETR4 pays continuous dividends at a rate $q = 5\%$ per year. How to adjust the BS formula? Calculate the new $S_{\text{adj}} = Se^{-qT}$ for the data from exercise 119.8 and describe the effect on the call price.

In Brazilian practice, the implied volatility curve of PETR4 is not flat — it is a "smirk" (asymmetric smile). Explain what this means and why it contradicts the assumptions of Black-Scholes.

Describe the Newton-Raphson algorithm to calculate implied volatility, given that the PETR4 call is quoted at R$ 2.50 (vs. R$ 1.92 theoretical with $\sigma = 35\%$). Use Vega as the derivative. Calculate the first iteration.

Describe the Cox-Ross-Rubinstein (CRR) binomial model as an alternative to BS for pricing an American call option ($S = K = 100$, $r = 0{,}05$, $\sigma = 0{,}20$, $T = 1$ year, $n = 3$ steps). Why does BS not work for American options?

Explain the concept of "real option" and how the BS formula can be used to evaluate the right to expand a factory. Identify $S$, $K$, $\sigma$ in this context.

From the Black-Scholes PDE, prove the relationship $\Theta + \frac{1}{2}\sigma^2 S^2 \Gamma + rS\Delta = rC$ between the Greeks. Interpret financially: why does a long call position delta-hedged lose money over time but gain from abrupt moves in the underlying?