Quiz: Black-Scholes: Derivation, Greeks, Limitations

1. In the Black-Scholes delta-hedging argument, which term in Itô's lemma applied to $dC(t, S_t)$ is eliminated by choosing $\Delta_t = \partial C / \partial S$?

2. Why does the physical drift $\mu$ of the stock not appear in the Black-Scholes formula?

3. The Gamma of a European call and a European put with the same strike $K$ and maturity $T$ are:

4. The P&L of a delta-hedged long call position over a small time interval $dt$, when realised volatility $\sigma_R$ differs from implied volatility $\hat{\sigma}$, is approximately:

5. The Breeden-Litzenberger result states that $\partial^2 C / \partial K^2 = e^{-rT} p^{\mathbb{Q}}_{S_T}(K)$, where $p^{\mathbb{Q}}$ is the risk-neutral density. This means that a full call price surface uniquely determines the risk-neutral marginal distribution of $S_T$ for each $T$.

6. Which of the following is the most direct empirical failure of the Black-Scholes model?