← Morgan Stanley Interview Insights
Start by recalling the Black-Scholes formulas for delta, gamma, and vega, then use simple approximations and rules of thumb to estimate them mentally. Focus on the key drivers: moneyness, time to expiration, and volatility, and relate them to intuitive ranges.
Pro tip: Mention that for at-the-money options, delta is approximately 0.5, gamma is highest and roughly 0.4/(Sσ√T), and vega is around 0.4S√T. This shows you know the practical shortcuts used on trading floors.
State the exact Black-Scholes formulas for delta (N(d1) for call, N(d1)-1 for put), gamma (N'(d1)/(Sσ√T)), and vega (S√T N'(d1)).
Note that d1 depends on S, K, r, σ, and T. For mental math, assume r=0 and focus on moneyness (S/K) and total volatility σ√T.
For at-the-money (ATM), d1≈0, so N(d1)≈0.5 and N'(d1)≈0.4. For away-from-the-money, use linear approximations or known values (e.g., delta≈1 for deep ITM call).
Gamma is highest ATM and decreases as you move away. Vega is also highest ATM and increases with √T. Use the ATM approximations: gamma≈0.4/(Sσ√T), vega≈0.4S√T.
Ensure estimates are within plausible bounds: delta between 0 and 1 (or -1 and 0 for put), gamma positive and typically <0.1 for equities, vega positive and often in the tens or hundreds depending on S and T.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
Start by acknowledging that exact Gaussian CDF values are rarely needed for quick estimates; instead, use approximations like the 68-95-99.7 rule and the logistic approximation. Then explain how these approximations translate to estimating Greeks (delta, gamma, vega, theta) by leveraging the relationships between the CDF, PDF, and option sensitivities. Finally, emphasize the importance of understanding the qualitative behavior of Greeks and how they change with moneyness, time to expiration, and volatility.
Pro tip: Mention that in practice, traders often use the '1-2-3' rule for delta: ATM delta ≈ 0.5, 1 standard deviation ITM/OTM delta ≈ 0.84/0.16, and 2 standard deviations ≈ 0.98/0.02. This shows practical intuition beyond textbook formulas.
Use the empirical rule for normal distributions: about 68% of values lie within 1 standard deviation, 95% within 2, and 99.7% within 3. This gives quick CDF estimates at key points: N(1) ≈ 0.84, N(2) ≈ 0.975, N(3) ≈ 0.9985.
Approximate the normal CDF with the logistic function: N(x) ≈ 1/(1 + exp(-1.702x)). This is easy to compute mentally for any x and is accurate to within 0.01 for most x.
For a European call, delta = N(d1), so the CDF approximation directly gives delta. Gamma and vega involve the PDF, which can be approximated by the derivative of the logistic or by using the peak value at ATM (≈0.4) and decaying exponentially.
Recall that d1 = [ln(S/K) + (r + σ²/2)T] / (σ√T) and d2 = d1 - σ√T. For quick estimates, ignore small r and use ln(S/K) ≈ (S-K)/K for near-the-money options.
Delta is roughly 0.5 ATM, approaches 1 deep ITM and 0 deep OTM. Gamma peaks ATM and decreases as you move away. Vega peaks ATM and increases with time to expiration. Theta is most negative ATM and decreases with time.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
Start by explaining that the PnL of a delta-hedged option position under constant implied volatility but stochastic realized volatility is driven by the difference between implied and realized variance. Then describe how the PnL can be decomposed into a gamma/theta trade-off and a vega-like exposure to realized volatility, and specify the conditions under which the position makes or loses money.
Pro tip: Emphasize that the sign of the PnL depends on whether realized volatility exceeds implied volatility, and that the magnitude scales with the option's gamma and the square of the volatility difference. Mention that in practice, transaction costs and discrete hedging can erode profits, so the theoretical edge may not be fully realized.
Explain that the PnL of a delta-hedged option position can be approximated by the difference between the option's theta and the gamma-weighted realized variance. Under constant implied volatility, the expected PnL is zero if realized volatility equals implied volatility.
Define realized volatility as the actual volatility of the underlying over the hedging period, and implied volatility as the constant volatility used to price and hedge. The PnL depends on the difference between these two volatilities.
Present the standard result: PnL ≈ 0.5 * Gamma * S^2 * (σ_realized^2 - σ_implied^2) * Δt, summed over the hedging period. This shows that the position profits when realized volatility exceeds implied volatility and loses when it is lower.
Explain that the magnitude of PnL is proportional to the option's gamma, which is largest for at-the-money options near expiration. Also note that discrete hedging introduces hedging errors that can add noise to the PnL.
Summarize that the hedged position makes money if realized volatility is higher than implied volatility (long gamma) and loses if realized volatility is lower (short gamma). The breakeven point is when realized volatility equals implied volatility.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.