Markets: A Rigorous Course
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Problem sets

Problems

Every problem in the course. Solutions sit behind a deliberate step on each problem page.

to difficulty Check numeric answers checked on the page

M00 Market on-ramp: accounts, screens, instruments, money and tax 15

  1. P1 SIP future value ★ Check
  2. P2 The cost of waiting ★ Check
  3. P3 XIRR with a partial withdrawal ★★ Check
  4. P4 Step-up SIP ★★ Check
  5. P5 A delivery round trip and its tax ★ Check
  6. P6 Intraday costs and the brokerage cap ★ Check
  7. P7 An option round trip that loses ★ Check
  8. P8 Reading a depth snapshot ★★ Check
  9. P9 Choosing product, order type and validity ★
  10. P10 Stops and gaps ★ Check
  11. P11 Retail allotment odds ★★ Check
  12. P12 Sampling without replacement helps, slightly ★★ Check
  13. P13 Tax on a mixed year ★★ Check
  14. P14 F&O turnover ★★ Check
  15. P15 A satellite risk budget ★ Check

M01 Market structure, microstructure and costs 18

  1. P1 Crossing of smooth schedules ★
  2. P2 An auction with ties ★★ Check
  3. P3 The residual book is not crossed ★★
  4. P4 Cost arithmetic ★ Check
  5. P5 Short-side break-even ★★ Check
  6. P6 Two brokerage regimes ★★ Check
  7. P7 Option costs ★★ Check
  8. P8 The 2026 STT increase and a futures scalper ★★ Check
  9. P9 Capital gains ★ Check
  10. P10 Set-off ★★
  11. P11 Index return identity over a basket change ★★
  12. P12 Rights issue and the divisor ★★ Check
  13. P13 Adjusting a series for rights ★★ Check
  14. P14 Dividend adjustment conventions ★★★
  15. P15 Total return index ★★ Check
  16. P16 Reading the SEBI FY26 study ★★ Check
  17. P17 DvP and the 1992 scam ★★★
  18. P18 Circuit breakers ★ Check

M02 Statistics of returns 16

  1. P1 Return arithmetic on election week ★ Check
  2. P2 The portfolio log-return gap ★★ Check
  3. P3 Leveraged index funds and the drag ★★ Check
  4. P4 Mean versus median ★ Check
  5. P5 AM–GM on a fund's track record ★★ Check
  6. P6 (Variance ratio of an MA(1)) ★★★ Check
  7. P7 How precisely can you estimate volatility? ★★ Check
  8. P8 Judging a track record ★★ Check
  9. P9 Hermite polynomials and Jarque–Bera ★★ Check
  10. P10 The Hill estimator by hand ★★★ Check
  11. P11 Regimes make fat tails ★★ Check
  12. P12 VaR for a ₹1 crore Nifty portfolio ★★ Check
  13. P13 Is correlation really rising? ★★ Check
  14. P14 Case analysis: pricing 4 June 2024 ★★★ Check
  15. P15 Black Monday under fatter models ★★ Check
  16. P16 Variance ratio from autocorrelations ★★ Check

M03 The mathematics of betting: expectancy, Kelly, ruin 16

  1. P1 A trade journal in R ★ Check
  2. P2 Binary Kelly and its growth rate ★★ Check
  3. P3 Kelly versus expected wealth ★★ Check
  4. P4 Where does overbetting turn negative? ★★★ Check
  5. P5 Continuous-time Kelly from a discrete step ★★ Check
  6. P6 Choosing a Kelly fraction from a drawdown limit ★★ Check
  7. P7 Estimation risk ★★ Check
  8. P8 Two-asset Kelly ★★ Check
  9. P9 Gaps change everything ★★★ Check
  10. P10 Gambler's ruin with pushes ★★ Check
  11. P11 Expected duration by martingales ★★ Check
  12. P12 The Lundberg exponent at half Kelly ★★★ Check
  13. P13 Drawdowns of a Sharpe-0.8 strategy ★★ Check
  14. P14 Position sizing across instruments ★ Check
  15. P15 Case analysis: LTCM's leverage arithmetic ★★★ Check
  16. P16 Case analysis: the arithmetic of retail option buying ★★ Check

M04 Systematic trading research and swing trading 18

  1. P1 The golden cross as a return filter ★ Check
  2. P2 EMA and MACD weights ★★ Check
  3. P3 Sign rules under Gaussianity ★★
  4. P4 RSI by hand ★ Check
  5. P5 Overlapping returns ★★ Check
  6. P6 Lo–MacKinlay with two stocks ★★ Check
  7. P7 Momentum's beta after a bear market ★★ Check
  8. P8 Breakeven win rate with Indian costs and gaps ★ Check
  9. P9 PSR and minimum track record ★★ Check
  10. P10 Survivorship ★ Check
  11. P11 Cost drag and rebalancing frequency ★★ Check
  12. P12 Annualising with autocorrelation ★★ Check
  13. P13 Exact expected maxima ★★★ Check
  14. P14 Deflating a backtest ★★ Check
  15. P15 Indian momentum: a replication plan ★★
  16. P16 Case analysis: pre-registering a data-snooping test ★★
  17. P17 Multiple testing by hand ★★ Check
  18. P18 More on Benjamini–Hochberg ★★★

M04A Reading markets: charts, volume, breadth and trade management 16

  1. P1 Aggregating bars and reading candles ★ Check
  2. P2 Bracket with drift ★★ Check
  3. P3 Stop-out probabilities ★ Check
  4. P4 Expected time in a bracket ★★ Check
  5. P5 What does a win rate imply? ★★ Check
  6. P6 ATR, Parkinson and volatility-scaled stops ★★ Check
  7. P7 Trailing stops ★★★ Check
  8. P8 Closing-basis stops ★★ Check
  9. P9 VWAP with U-shaped volume ★★ Check
  10. P10 Gap fills ★★ Check
  11. P11 Scaling out and pyramiding ★★ Check
  12. P12 Widening a stop ★★ Check
  13. P13 Reading participant-wise OI ★★ Check
  14. P14 Breadth in a factor model ★★ Check
  15. P15 Sizing for an event day ★★ Check
  16. P16 Testing support and resistance ★★★ Check

M05 Intraday trading and execution 18

  1. P1 Roll arithmetic ★ Check
  2. P2 Roll with unbalanced and persistent order flow ★★ Check
  3. P3 Roll with adverse selection ★★
  4. P4 Glosten–Milgrom dynamics ★★ Check
  5. P5 Kyle: commitment does not help ★★★
  6. P6 Kyle with Indian numbers ★★ Check
  7. P7 Estimating Kyle's lambda ★★ Check
  8. P8 Square-root impact, capacity and splitting ★★ Check
  9. P9 When TWAP and when VWAP is optimal ★★
  10. P10 Continuous Almgren–Chriss in closed form ★★★ Check
  11. P11 Almgren–Chriss with drift ★★★ Check
  12. P12 VWAP and implementation shortfall ★ Check
  13. P13 What R2R^2 does an intraday signal need? ★★ Check
  14. P14 Cost drag and the optimal holding period ★★★ Check
  15. P15 Intraday versus swing in a mid-cap ★★ Check
  16. P16 How long a record proves skill? ★★★ Check
  17. P17 A stylised model of expiry-day index pushing ★★★
  18. P18 ORB and multiple testing ★★ Check

M06 Fundamental analysis and valuation 17

  1. P1 Reconciling free cash flow ★★ Check
  2. P2 Annuity variants ★ Check
  3. P3 A two-stage DCF ★ Check
  4. P4 Terminal value must pay for its growth ★★ Check
  5. P5 Economic profit ★★
  6. P6 Competitive-advantage period ★★★ Check
  7. P7 Banks on book ★★ Check
  8. P8 CAPM from the tangency condition ★★
  9. P9 An INR cost of capital ★★ Check
  10. P10 Reverse DCF ★★ Check
  11. P11 Correlated inputs ★★★ Check
  12. P12 Beneish by hand ★★ Check
  13. P13 Piotroski by hand ★ Check
  14. P14 Coffee Can arithmetic ★★ Check
  15. P15 Satyam forensics ★★ Check
  16. P16 Divergence and the P/B of a bank ★★ Check
  17. P17 What a shareholding pattern can and cannot tell you ★★ Check

M06A Corporate events, earnings, sectors and factors 21

  1. P1 Computing SUE ★★ Check
  2. P2 Drift from naive expectations ★★★ Check
  3. P3 Extreme-group spreads ★ Check
  4. P4 Implied earnings move ★★ Check
  5. P5 Buyback arithmetic ★★ Check
  6. P6 Tendering in three tax regimes ★★ Check
  7. P7 Demerger cost and value ★ Check
  8. P8 Open-offer pricing and tendering ★★ Check
  9. P9 Reverse book building ★★ Check
  10. P10 HDFC and HDFC Bank ★★ Check
  11. P11 Merger arbitrage as a short put ★★★ Check
  12. P12 Rock's model ★★★ Check
  13. P13 Pledge invocation ★ Check
  14. P14 Residual income for a bank ★★ Check
  15. P15 ROE tree and asset quality ★★ Check
  16. P16 A steel producer and an IT company ★★ Check
  17. P17 Factors ★★★ Check
  18. P18 Event study of a short report ★★ Check
  19. P19 Cleaning a quarter: TCS, Q2 FY27 ★★ Check
  20. P20 Order-book arithmetic: BEL ★★ Check
  21. P21 APE, VNB margin and mix ★★ Check

M06B Funds: mutual funds, ETFs and the asset-management business 18

  1. P1 Computing a NAV ★ Check
  2. P2 Stale-NAV arbitrage ★★ Check
  3. P3 Swing pricing ★★ Check
  4. P4 Fee drag on a lump sum ★ Check
  5. P5 Direct and regular SIPs ★★ Check
  6. P6 An AMC's operating leverage ★★ Check
  7. P7 The value of a SIP customer ★★ Check
  8. P8 Harmonic mean ★ Check
  9. P9 Lump sum or averaging? ★★ Check
  10. P10 Luck versus skill ★★★ Check
  11. P11 Berk–Green ★★★ Check
  12. P12 Tracking difference ★★ Check
  13. P13 An ETF's premium ★★ Check
  14. P14 ETF or index fund? ★ Check
  15. P15 A debt fund's NAV ★★ Check
  16. P16 A run on a debt fund ★★★ Check
  17. P17 Tax ★★ Check
  18. P18 Overlap and the active expense ratio ★★ Check

M07 Forwards and futures 17

  1. P1 Discrete dividends, law of one price ★
  2. P2 The Nifty term structure on 5 Nov 2026 ★ Check
  3. P3 No-arbitrage band ★★ Check
  4. P4 Value of an existing forward ★ Check
  5. P5 MTM and margin calls on Bank Nifty ★★ Check
  6. P6 Futures = forward, general deterministic rates ★★
  7. P7 The futures–forward gap under Vasicek rates ★★★ Check
  8. P8 Minimum-variance hedging, properties ★★ Check
  9. P9 Beta hedging and what is left ★★ Check
  10. P10 Basis risk ★★ Check
  11. P11 Open interest accounting ★★ Check
  12. P12 Leverage and tails ★★ Check
  13. P13 Futures as financed stock ★★ Check
  14. P14 Storage and convenience yield ★★ Check
  15. P15 Metallgesellschaft: stack-and-roll hedge ratio ★★★ Check
  16. P16 March 2020: was there an arbitrage? ★★ Check
  17. P17 Negative prices and storage ★★ Check

M08 Option pricing: from replication to Black–Scholes 17

  1. P1 Bull call spread on Nifty ★ Check
  2. P2 A parity violation ★ Check
  3. P3 Strike monotonicity and convexity ★★
  4. P4 Homogeneity and the risk-neutral density ★★
  5. P5 Early exercise of puts ★★
  6. P6 American calls and a discrete dividend ★★ Check
  7. P7 Incompleteness in a trinomial model ★★★
  8. P8 A two-step tree by hand ★★ Check
  9. P9 Itô's lemma in use ★★
  10. P10 Solutions of the Black–Scholes PDE ★★
  11. P11 Digital options ★★ Check
  12. P12 Greeks in numbers ★ Check
  13. P13 Hedged P&L of a short straddle ★★★ Check
  14. P14 Implied volatility ★★ Check
  15. P15 Black-76 on MCX crude oil ★★ Check
  16. P16 Case analysis: moving the reference price ★★★
  17. P17 CRR details ★★ Check

M09 Volatility and option trading 17

  1. P1 Realised volatility from OHLC ★ Check
  2. P2 The fourth moment of the range and Parkinson's efficiency ★★ Check
  3. P3 Unbiasedness of Garman–Klass and Rogers–Satchell ★★★
  4. P4 A day in the life of a hedged option seller ★ Check
  5. P5 Expected P&L and drift ★★★
  6. P6 Robustness and its limits ★★
  7. P7 The variance risk premium in a two-state economy ★★ Check
  8. P8 Variance-swap replication and discretisation ★★ Check
  9. P9 Variance notional and convexity ★ Check
  10. P10 Iron condor arithmetic ★★ Check
  11. P11 Expiry-day gamma ★★ Check
  12. P12 Kelly with a fat left tail ★★ Check
  13. P13 Sticky delta ★★ Check
  14. P14 Case analysis: leveraged volatility products ★★★ Check
  15. P15 Case analysis: expiry-day leverage in the Jane Street interim order ★★ Check
  16. P16 Costs and margin for an Indian short strangle ★★ Check
  17. P17 Reading India VIX ★ Check

M09A The Indian options desk 17

  1. P1 Reading one row of the chain ★ Check
  2. P2 Max pain on a small chain ★★ Check
  3. P3 Max pain as a median ★★
  4. P4 What moves the PCR ★★ Check
  5. P5 The ATM straddle formula ★★ Check
  6. P6 Straddle to implied volatility ★ Check
  7. P7 Event variance from two expiries ★★★ Check
  8. P8 One expiry before, one after ★★ Check
  9. P9 IV rank versus IV percentile ★ Check
  10. P10 Calendar monotonicity and forward volatility ★★ Check
  11. P11 Reading participant-wise OI ★★ Check
  12. P12 Scenario margin of hedged positions ★★ Check
  13. P13 Adjusting a tested strangle ★★ Check
  14. P14 Physical settlement ★ Check
  15. P15 Hedging feedback and pinning ★★★ Check
  16. P16 Testing max pain ★★ Check
  17. P17 A results-day gap on stock futures ★ Check

M10 Commodities: gold, silver, crude 27

  1. P1 Storage as a rupee cost ★ Check
  2. P2 No lower bound, and the inventory holder's bound ★★
  3. P3 A two-period storage equilibrium ★★ Check
  4. P4 Implied convenience yield ★ Check
  5. P5 Roll yield with a moving curve ★★ Check
  6. P6 The index investor's year ★★ Check
  7. P7 Hotelling with linear demand ★★ Check
  8. P8 Gold's duration ★★ Check
  9. P9 The May 2026 duty hike ★ Check
  10. P10 Refining the parity ★★ Check
  11. P11 Decomposing the 2026 rupee gold move ★★ Check
  12. P12 Half-life: estimate, bias, interval ★★★ Check
  13. P13 Reading ADF output ★★
  14. P14 Crack spread and a refiner hedge ★ Check
  15. P15 Black-76 on an MCX silver option ★★ Check
  16. P16 Case analysis: negative WTI on MCX ★★★ Check
  17. P17 Case analysis: gold in March 2026 ★★
  18. P18 Case analysis: the Hormuz convenience yield ★★ Check
  19. P19 Reading a COT line ★ Check
  20. P20 Who is on the other side? ★★ Check
  21. P21 The seasonal FF-test ★★ Check
  22. P22 Three commodities, 36 months ★★ Check
  23. P23 Roll gaps in a continuous series ★★ Check
  24. P24 Festival demand: where should it show up? ★★★
  25. P25 Sizing silver for a ₹10 lakh account ★ Check
  26. P26 A margin hike on an existing position ★ Check
  27. P27 CTT against STT, and the cost of exercise ★★ Check

M11 Macro: rates, FX and flows 23

  1. P1 Premium and discount ★
  2. P2 Pricing a G-sec ★ Check
  3. P3 Perpetuity and zero ★★
  4. P4 Barbell versus bullet ★★ Check
  5. P5 Equity duration ★★ Check
  6. P6 Forward rates and the term premium ★★ Check
  7. P7 Fisher and breakevens ★ Check
  8. P8 Taylor rule sensitivity ★★ Check
  9. P9 Taylor principle with lagged inflation ★★★
  10. P10 CIP arbitrage ★★ Check
  11. P11 Oil and the current account ★★ Check
  12. P12 Fama decomposition ★★★ Check
  13. P13 The FPI's dollar return ★ Check
  14. P14 Debt dynamics ★★ Check
  15. P15 Inflation and the P/E ★★ Check
  16. P16 DXY arithmetic ★ Check
  17. P17 Case analysis: 2013 and 2026 ★★
  18. P18 Contract arithmetic ★ Check
  19. P19 Fair value and the arbitrage band ★★ Check
  20. P20 Importer with a date mismatch ★★ Check
  21. P21 Exporter and the minimum-variance ratio ★★ Check
  22. P22 Garman–Kohlhagen ★★★ Check
  23. P23 The rules and what they cost ★★ Check

M12 Geopolitics and event studies 16

  1. P1 Single-day AR variance ★ Check
  2. P2 Constant-mean model ★★ Check
  3. P3 Power and window length ★★ Check
  4. P4 Exactness of the tt-test ★★★
  5. P5 Event-induced variance ★★ Check
  6. P6 Clustering ★★ Check
  7. P7 Event dates ★
  8. P8 Building GPR ★ Check
  9. P9 Bayes by hand ★★ Check
  10. P10 The value of waiting ★★ Check
  11. P11 Breakeven tail probability ★★ Check
  12. P12 Correlated news ★★★ Check
  13. P13 Supply-shock arithmetic ★★ Check
  14. P14 GPR regression design ★★
  15. P15 Case analysis: 1990 and 2026 ★★★
  16. P16 Convexity ★★

M13 Forecasting and market prediction 17

  1. P1 Noise predictors have negative OOS R2R^2 ★★
  2. P2 (AR(1) for a futures basis) ★ Check
  3. P3 (AR(2): stationarity and Yule–Walker) ★★ Check
  4. P4 (MA(1) and the bid-ask bounce) ★★ Check
  5. P5 Sanity-checking a claimed R2R^2 ★★ Check
  6. P6 Exact Sharpe of a linear timer with drift ★★★ Check
  7. P7 Stambaugh bias on an Indian valuation predictor ★★ Check
  8. P8 GARCH term structure after a shock ★★ Check
  9. P9 GARCH fourth moment ★★★ Check
  10. P10 Two steps of the Hamilton filter by hand ★★ Check
  11. P11 Ergodic probabilities and durations ★ Check
  12. P12 Purged CV arithmetic ★★ Check
  13. P13 Multi-category Brier ★★ Check
  14. P14 Reading a reliability table ★★ Check
  15. P15 Case analysis: a 100% recession call ★★★ Check
  16. P16 Odds-form updating for an RBI decision ★★ Check
  17. P17 Meta-labelling arithmetic ★★ Check

M13A Cointegration, pairs and statistical arbitrage 18

  1. P1 The DF distribution and the probability of "mean reversion" ★★ Check
  2. P2 ADF reparametrisation ★ Check
  3. P3 How spurious tt-statistics grow ★★ Check
  4. P4 (a cointegrated VAR(1) by hand) ★★ Check
  5. P5 Engle–Granger in practice ★★
  6. P6 Johansen arithmetic ★★ Check
  7. P7 (OU parameters from an AR(1)) ★ Check
  8. P8 Passage times and a band ★★ Check
  9. P9 Bertram's optimal symmetric band ★★ Check
  10. P10 Three hedge ratios ★ Check
  11. P11 Kalman steady state ★★ Check
  12. P12 One Kalman step by hand ★ Check
  13. P13 A Chow test ★★ Check
  14. P14 Data snooping in break tests ★★★ Check
  15. P15 Lots and hedge ratios ★ Check
  16. P16 Is the gold/silver ratio tradeable? ★★ Check
  17. P17 Case analysis: testing whether a merger broke a pair ★★★
  18. P18 A merger as a known cointegrating vector ★ Check

M14 Portfolio construction, risk, psychology and going live 29

  1. P1 Every portfolio covaries equally with the GMV ★
  2. P2 Zero-beta portfolios ★★★
  3. P3 Tangency by hand ★ Check
  4. P4 Error amplification ★★ Check
  5. P5 Shrinkage intensity in closed form ★★ Check
  6. P6 Risk contributions for two assets ★ Check
  7. P7 When inverse vol is not ERC ★★ Check
  8. P8 Volatility targeting ★★ Check
  9. P9 Conditional correlation ★ Check
  10. P10 Two books ★ Check
  11. P11 Drawdown control with gaps ★★★ Check
  12. P12 Prospect theory numbers ★★ Check
  13. P13 Measure your own disposition effect ★★ Check
  14. P14 Graduation arithmetic ★★ Check
  15. P15 Calibrating a daily loss limit ★★ Check
  16. P16 An LTCM-style stress ★★ Check
  17. P17 Jhunjhunwala's two books ★★ Check
  18. P18 Two short-option desks ★★ Check
  19. P19 Student-tt Expected Shortfall ★★ Check
  20. P20 Why the definition of ES matters ★★★ Check
  21. P21 Delta–gamma on a short straddle ★★ Check
  22. P22 Aggregating rupee Greeks ★★ Check
  23. P23 An MTF margin call ★★ Check
  24. P24 Wrong-way collateral and endogenous margin ★★ Check
  25. P25 Probability of a call ★★ Check
  26. P26 Short via SLB or via the future ★★ Check
  27. P27 Static versus constant leverage ★ Check
  28. P28 A beta-matched put hedge ★★ Check
  29. P29 The tax mismatch ★★ Check

M15 Trade playbooks: real Indian trades from start to finish 22

  1. P1 Sizing and break-even for the BEL breakout ★ Check
  2. P2 What seven trades can say ★★ Check
  3. P3 The Coal India plan and the chance of being stopped on noise ★★ Check
  4. P4 A setup against its null ★★ Check
  5. P5 The TCS opening-range trade ★ Check
  6. P6 How much evidence is in 56 intraday trades? ★★★ Check
  7. P7 The market's expected move for TCS's results ★★ Check
  8. P8 The defined-risk alternative ★★ Check
  9. P9 Building the weekly condor ★★ Check
  10. P10 Daily attribution and expiry-day margin ★★★ Check
  11. P11 Sizing, costing and breaking even on the RBI put spread ★★ Check
  12. P12 Greek attribution, the vanna residual, and what nine trades can prove ★★★ Check
  13. P13 Reverse DCF in years of advantage ★★ Check
  14. P14 Decomposing a Beneish flag ★★★ Check
  15. P15 Decomposing the 13 May 2026 duty jump ★★ Check
  16. P16 Implied grams, a contaminated baseline, and the cost of proving an edge ★★★ Check
  17. P17 Gap-adjusted sizing for HPCL on the same signal ★★ Check
  18. P18 IOC event study and the trades needed to validate the rule ★★★ Check
  19. P19 Lot granularity and the 1% rule for a cash-plus-futures pair ★★ Check
  20. P20 How long is the half-life, and how long to learn the edge? ★★★ Check
  21. P21 Sizing the crude-mini trade, and the contracts that do not fit ★★ Check
  22. P22 What a continuous-series backtest measures ★★★ Check

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