Options & Volatility

Systematic Vol QR 面试备考指南

📚 核心资源链接

🎯 Systematic Vol 组专项面试题

模块一：Vol 基础与定价

1. RV vs IV 的本质区别是什么？它们如何驱动期权策略的盈亏？

2. 解释 delta-hedged straddle 的 P&L 公式： $$dP&L = \frac{1}{2}\Gamma S^2(\sigma_{RV}^2 - \sigma_{IV}^2)dt$$ 为什么 IV−RV 是 carry 策略的核心？

3. Long vol book 在市场剧烈波动时仍亏钱，可能的原因是什么？（答：RV 没有超过买入时的 IV；Theta 拖累）

4. Forward vol 和 implied vol 有什么区别？如何从 term structure 中提取 forward vol？

5. 如何用两种不同方法计算 earnings event vol，并 strip 掉 vol surface 中的 event vol？（这直接对应你在 Millennium 的工作）

6. 为什么 1-month close-close RV 不是预测未来 vol 的好指标？还有哪些更好的 RV estimator？（答：Parkinson、Garman-Klass、Yang-Zhang 等高低价估计量）

模块二：Vol Surface 与 Skew

1. 股票市场为什么有负 skew（OTM put IV 更高）？请从需求/供给和 leverage effect 两个角度解释。

2. Sticky-strike 和 sticky-delta 的区别是什么？各自在什么市场环境下更合适？对 delta/vanna/vomma 对冲有什么影响？

3. Local vol model 和 Heston（stochastic vol）model 各自的优缺点是什么？
• Local vol：精确拟合当前 surface，但 skew 动态（smile dynamics）错误
• Heston：smile dynamics 更真实，但拟合精度不如 local vol

4. Vol surface 实际上告诉我们什么？它与风险中性概率分布的关系是什么？

5. 如何判断某个 strike 的 vol 是 rich 还是cheap？你会用哪些信号？

模块三：Systematic 信号研究

1. 你会如何构建一个预测 3M ATM IV 变动的系统性信号？从数据源、特征工程、模型选择到回测验证，完整描述你的流程。（这是对你 Millennium 工作的深挖）

2. 描述你设计的 Sharpe-based model combining 方案。为什么选 Sharpe 而不是 IC 来权重？

3. 如何量化并控制信号之间的相关性（signal correlation）？信号冗余对组合 Sharpe 的影响是什么？

4. IV 的定价 building blocks 有哪些？除了 RV 预测，还有哪些因子影响 IV 的定价？（答：流动性溢价、事件 vol、周末效应、季节性等）

5. Variance Risk Premium（VRP）是什么？如何系统性地捕捉它？有哪些常见的 regime 下 VRP 会反转？

6. 你如何回测一个 vol 策略同时避免 look-ahead bias？事件 vol cleaning 的 point-in-time 要求如何实现？

模块四：Greeks 与对冲实践

1. Vanna 和 Volga 是什么？为什么做市商在对冲 vol surface 时需要关注它们？

2. 如何构建一个 trade 来表达"方向性看涨 + 财报后 vol crush"的复合观点？（答：买垂直价差，降低 vega 敞口）

3. Gamma scalping 在什么条件下盈利？如何确定最优 delta 再对冲频率？

4. 在大型到期日前后，dealer 的 gamma 对冲行为如何影响市场微观结构？（答：Gamma pinning 效应）

模块五：概率/统计/ML（通用）

1. OLS 的假设有哪些？异方差（Heteroscedasticity）如何检验和处理？

2. Ridge 和 Lasso 的区别？在信号选择场景下各自适合什么情况？

3. 给定两个 portfolio 的 Sharpe 和它们之间的相关系数，如何求最优权重组合？

4. 解释 Sharpe ratio 在时间序列上的统计显著性检验（t-stat of Sharpe）。多少年的数据可以在 95% 置信度下证明 Sharpe > 0？

5. 信息系数（IC）和 ICIR 是什么？如何用它们来评估和组合信号？

🔑 Millennium Vol 组面试的特殊注意事项

• 你在 Millennium 实习中构建的每一个信号的经济直觉

• Event vol stripping 的两种方法的数学细节和各自 trade-off

• 为什么选择 Sharpe-based combining 而不是 IC-based 的具体论据

• 你的 vol surface 特征的截面和时序属性

Questions

🧩 1. Option Pricing Fundamentals

1. Derive the Black–Scholes PDE starting from a self-financing, delta-hedged portfolio.

2. Why is the discounted price process a martingale under the risk-neutral measure?

3. Show that under risk neutrality, follows

4. Derive the closed-form Black–Scholes call option formula.

5. What assumptions does Black–Scholes rely on? Which of these fail in reality?

6. What is the relationship between the forward price and the spot price under continuous compounding?

7. Explain put–call parity and when it may break down.

8. What is the difference between European and American options, and when is early exercise optimal?

9. How does dividend yield enter into option pricing?

10. If interest rates increase, what happens to call and put prices, all else equal?

📈 2. Greeks & Sensitivities

1. Define and interpret Delta, Gamma, Vega, Theta, Rho.

2. Derive analytical formulas for these Greeks under Black–Scholes.

3. What is the sign of Gamma for calls and puts? Why?

4. How does Vega behave as maturity approaches zero?

5. When is Theta positive, when is it negative?

6. What does Gamma exposure mean for a delta-hedged book?

7. If volatility doubles, how approximately does the call price change (use Vega linear approximation)?

8. Show mathematically that Gamma is always positive for convex payoffs.

9. Why do option market makers like to be long Gamma and short Vega?

10. What is the Vega–Gamma trade-off?

🔍 3. Implied Volatility & Smiles

1. Define implied volatility.

2. Why does implied volatility typically vary with strike and maturity?

3. What are the main shapes of the volatility smile/skew in equity vs. FX vs. rates markets?

4. What information about the underlying distribution can be inferred from the smile?

5. How do leverage effects cause negative skew in equities?

6. What is the volatility surface, and how is it constructed?

7. How do you interpolate/extrapolate a volatility surface consistently (no static arbitrage)?

8. What is calendar arbitrage, and how can it appear on a vol surface?

9. What is butterfly arbitrage, and how can you detect it numerically?

10. How can you convert a vol surface to an implied probability density? (Breeden–Litzenberger)

11. Explain how local volatility models attempt to fit the smile.

12. What is the Dupire formula for local volatility?

13. Why does local volatility fail to capture forward smile dynamics?

14. Describe the stochastic volatility approach (e.g., Heston).

15. Compare the implied smile shape under local vol vs stochastic vol models.

🔧 4. Stochastic Volatility Models

1. Write down the Heston model SDEs and describe each parameter’s role.

2. Derive or sketch the characteristic function of log-price under Heston.

3. How is Heston calibrated to market data?

4. Why can the variance process in Heston be negative in simulation, and how is it fixed?

5. Compare Heston, SABR, and Hull–White volatility models.

6. What is the intuition behind the SABR model and its β parameter?

7. Derive the SABR implied vol approximation (Hagan formula).

8. How does correlation between spot and volatility drive the skew?

9. What is the Volga–Vanna approximation used for Vega-convexity corrections?

10. Why is volatility-of-volatility important for exotic options?

11. What is the forward volatility and how is it related to the variance swap curve?

12. Explain the concept of volatility clustering in empirical data.

13. What’s the intuition behind using GARCH to model volatility dynamics?

14. Compare GARCH volatility forecasts vs. implied volatility forecasts empirically.

💹 5. Volatility Derivatives & Variance Swaps

1. What is a variance swap, and how is its payoff defined?

2. Derive the replication formula for a variance swap using a continuum of options.

3. What is the difference between realized variance and implied variance?

4. Why is the variance risk premium usually positive in equities?

5. How can you delta-hedge a variance swap?

6. Explain how a volatility swap differs from a variance swap.

7. What is log-contract replication, and why is it important?

8. How can you infer the risk-neutral variance from listed option prices?

9. How does VVIX relate to the volatility of volatility?

10. Why might realized volatility exceed implied volatility before major events?

11. Derive an expression for the fair variance swap rate in discrete strikes approximation.

⚙️ 6. Advanced Volatility Concepts

1. Explain the concept of stochastic volatility with jumps (SVJ, Bates model).

2. Derive the price of an option when the underlying follows a jump-diffusion process.

3. How do jumps affect the implied volatility smile?

4. What’s the intuition behind volatility convexity adjustments?

5. What is the volatility beta between two assets, and how could you trade it?

6. Explain the volatility term structure — why does short-term vol usually spike more?

7. How would you arbitrage the volatility surface if mispricing is detected?

8. What are sticky-delta and sticky-strike rules in quoting conventions?

9. Explain how to perform Vega hedging for a portfolio of options.

10. How do cross-asset volatility correlations affect multi-asset option pricing?

11. What is the smile dynamics problem (Derman–Kani, Bergomi)?

12. How can forward variance curves be extracted from market data?

13. Derive the relationship between implied volatility term structure and forward variance.

14. What is the Volatility Cube, and why do we need it?

15. What does the Skew Stickiness Ratio (SSR) measure?

🧠 7. Research-Level / Hard Quant Questions

1. Show that under Black–Scholes, implied vol is monotonic in strike only if volatility is constant.

2. Prove the Dupire local volatility formula starting from call price derivatives.

3. Derive the Fokker–Planck equation for transition density of GBM.

4. Derive the moment generating function for log returns under Heston.

5. Prove that the instantaneous variance of log-returns under Heston is νt\nu_tνt.

6. Compute E[σt2]E[\sigma_t^2]E[σt2] for an OU process volatility model.

7. Show that the stochastic volatility model leads to non-log-normal return distribution with skew and kurtosis.

8. Show that the implied vol skew under stochastic vol models behaves linearly in small time.

9. Prove the small-time expansion of implied volatility under jumps.

10. Derive the Dupire forward PDE for implied vol dynamics.

11. Discuss the difference between forward smile and spot smile.

12. How do you calibrate a local–stochastic volatility (LSV) model?

13. Explain why LSV models can preserve both spot smile fit and realistic dynamics.

14. Given empirical returns, design a volatility forecasting model and describe how to backtest it.

15. Explain the impact of volatility risk premia on delta-hedged P&L distributions.

16. Derive the hedging error when actual vol ≠ implied vol.

17. Show that the expected P&L of a delta-hedged option equals the variance risk premium.

18. Explain implied volatility decomposition into instantaneous forward variance components.

19. Describe how to use principal component analysis on the volatility surface for factor modeling.

20. Explain how VIX is constructed from S&P 500 options.

21. Derive the VIX formula in continuous strike space.

22. Why is the VIX not exactly the expectation of future realized volatility?

23. Derive the term structure of variance swaps from the forward variance curve.

24. What is volatility spillover and how would you measure it?

25. Suppose volatility follows a rough process (e.g., rough Bergomi). Explain what “rough volatility” means and why it fits empirical data better.