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Industry problems from pricing, risk, portfolio and trading desks, across the whole curriculum. Each one is graded by tests, ranked by difficulty and tagged with the modules it draws on.
- Easy · 10 points
- Medium · 25 points
- Hard · 50 points
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26 of the problems below are open now.
Showing 104 problems
Turn deposit offers quoted with different compounding conventions into one comparable rate and rank them.
Foundations 1.4Fixed income+10
15 min
Produce the monthly payment, the interest and principal split each month, and the effect of an overpayment.
Foundations 1.5Banking and corporate finance+10
20 min
Find the yield to maturity of each bond in a ladder from its price, and the yield of the whole ladder.
+10
25 min
Rank five projects by net present value and internal rate of return, and find the rate at which two of them swap places.
+10
25 min
Combine calls, puts and stock into one payoff function, then find the break-even prices and the worst case.
Foundations 1.1Options+10
20 min
Compute simple and log returns, annualise them correctly and explain why the two figures differ.
Foundations 1.4Risk+10
20 min
Find the weights that minimise the variance of a two-fund portfolio and show how the answer moves with correlation.
Foundations 1.2Portfolios+10
20 min
Value a growing income paid continuously, then compare it with the same income paid monthly.
Foundations 2.5Banking and corporate finance+10
20 min
Strip zero rates from par yields, interpolate between them, and price an off-the-run bond on the curve.
+25
40 min
Size a hedge in two bonds that protects the book against small parallel moves, then test it on a large shock.
+25
45 min
Back out implied volatilities from a day's call quotes with a safeguarded Newton solver, and flag the bad quotes.
+25
45 min
Calibrate a constant hazard rate to a bond spread and value a risky cash-flow schedule with it.
Foundations 2.5Credit+25
35 min
Approximate bond price changes with first- and second-order terms and measure where each approximation breaks down.
Foundations 2.7Fixed income+25
35 min
Choose step sizes for bumped Greeks that balance truncation and rounding error, and check them against exact values.
+25
40 min
Solve a linear system for the bond holdings whose cash flows exactly meet a schedule of pension payments.
Foundations 3.3Fixed income+25
35 min
Diagnose why a hand-edited covariance matrix is invalid, find the nearest valid one, and repair it again keeping the variances.
+25
40 min
Use a Cholesky factor to simulate joint moves in rates, equities and FX that match a target correlation matrix.
+25
35 min
Derive the weights with a Lagrange multiplier, compute them for ten funds, and check them with a numerical optimiser, with and without short sales.
Foundations 4.5Portfolios+25
40 min
Find the growth-optimal fraction for a repeated bet and show by simulation what over-betting does to wealth.
+25
35 min
Compute a one-day 99% VaR from a normal model and from history, bootstrap the historical figure and backtest the normal one on fat-tailed returns.
+25
35 min
Estimate a default probability from a loan book's history, with a confidence interval the credit committee can use.
Foundations 5.9Credit+25
30 min
Regress a stock's excess returns on the market's, report beta with its standard error, and test whether it differs from one.
Foundations 5.10Portfolios+25
30 min
Simulate a strategy's equity curve as a random walk and estimate how deep and how long its drawdowns get.
Foundations 5.11Trading+25
35 min
Price a cash-or-nothing call by integrating against the lognormal density, and check it with simulation.
+25
35 min
Solve the mean-reversion equation for a policy rate, fit its speed to a central bank's projected path, and read off the yield curve it implies.
Foundations 6.1Fixed income+25
30 min
Solve the heat equation with an explicit scheme, find its stability limit, and use it to price a European option.
+50
60 min
Align, de-duplicate and fill daily prices from three vendors with pandas, and report every change you made.
+10
30 min
Rewrite a loop-based daily P&L over hundreds of positions with NumPy and prove the results are unchanged.
+10
25 min
Model fixed, floating and amortising legs as classes with a shared interface, with tests for each.
+25
45 min
Fit a Nelson–Siegel curve to government bond yields and judge whether its parameters are stable from day to day.
+25
45 min
Price European calls by Fourier inversion and match the Black–Scholes price to many decimal places.
+50
70 min
Price an American put with early exercise, find the exercise boundary, and measure the early-exercise premium.
+25
40 min
Solve the pricing PDE for a down-and-out call and handle the barrier and the payoff kink without spurious oscillation.
+50
75 min
Price an Asian option with antithetic paths and a control variate, and report the error saved for the same run time.
+25
40 min
Value a portfolio of vanilla and path-dependent options with one simulation, and report its Greeks.
+50
80 min
Split a notebook into a package with type hints, tests and a command-line entry point.
+25
45 min
Build the information available at each step of a three-step tree and compute conditional expectations at every node.
+10
30 min
Show by simulation and by argument that a doubling strategy on a fair game is a martingale, and find its hidden risk.
+25
35 min
Price the same claim with real-world probabilities and a deflator, and with risk-neutral probabilities, and reconcile them.
+25
40 min
Simulate Brownian paths and confirm how their spread scales with time and how their quadratic variation behaves.
+10
25 min
- Simulate an Itô integralMedium
Approximate an Itô integral with left-point sums, confirm its mean and variance, and show what goes wrong at the midpoint.
+25
35 min
Apply Itô's lemma to the log of a stock price and confirm the minus one-half sigma squared term by simulation.
+25
35 min
Simulate a mean-reverting short rate, price a zero-coupon bond by Monte Carlo, and compare with the closed form.
+25
45 min
Use Feynman–Kac to price one claim by solving a PDE and by simulating paths, and explain any gap.
+50
60 min
Price an option from real-world paths by weighting each with the Girsanov density, and compare with direct pricing.
+50
60 min
Derive the optimal risky share for a CRRA investor and check by simulation that small changes to it lower expected utility.
+50
70 min
Test daily index returns for fat tails, volatility clustering and the leverage effect, with a chart for each.
+10
25 min
Regress a fund's returns on market, size and value factors, and decide whether its alpha survives robust errors.
+25
40 min
Identify, fit and check an ARMA model for a credit spread, and compare its forecasts with a random walk.
+25
40 min
Run Engle–Granger tests on candidate pairs, estimate the hedge ratio and the half-life of the spread.
+25
45 min
Run a principal component analysis on daily curve changes and hedge a bond portfolio against the first three factors.
+25
40 min
Estimate daily volatility with an exponentially weighted average and flag the days when it jumps.
+10
25 min
Fit a GARCH(1,1) by maximum likelihood, forecast ten days ahead and explain what the persistence means for risk.
+50
70 min
Estimate time-varying correlations between asset classes and show how diversification behaved in a sell-off.
+50
70 min
Simulate correlated defaults in a basket of names and price first- and second-to-default protection.
+50
70 min
Fit a two-state Markov-switching model to returns and use the regime probabilities to scale risk limits.
+50
70 min
Count exceptions for a year of daily VaR forecasts, run coverage tests and write the verdict for the risk committee.
+25
45 min
Rank EWMA, GARCH and historical forecasts with loss functions that are robust to noisy volatility proxies.
+25
45 min
Find the ways future information leaks into a feature set, fix them, and measure the honest score.
+10
30 min
Label each trade by which barrier it hits first, profit, loss or time, and check the balance of the labels.
+25
40 min
Fit regularised regressions to many weak predictors and show how the penalty trades bias for out-of-sample stability.
+25
40 min
- Build a credit scorecardMedium
Fit a logistic regression to loan applications, turn it into a points scorecard and choose a cut-off by expected loss.
+25
45 min
Set up purged walk-forward splits for a direction classifier and show how naive validation flatters it.
+50
80 min
Compare boosted trees with the scorecard, then explain individual decisions to a credit officer.
+25
50 min
Cluster a universe of stocks on return correlations and compare the clusters with official sectors.
+10
30 min
Build a sentiment score for financial headlines and test whether it carries information about next-day returns.
+25
45 min
Train a small neural network on Black–Scholes prices and measure where its errors are largest.
+25
50 min
Train a recurrent network on realised volatility and test whether it beats a GARCH benchmark out of sample.
+50
80 min
Train an agent to hedge an option with transaction costs and compare its P&L with delta hedging.
+50
90 min
Turn a model's predictions into positions, add costs and risk limits, and decide whether the strategy is worth running.
+50
80 min
Price forwards on a stock, a commodity and a currency, and find the quote that allows a cash-and-carry arbitrage.
+10
25 min
Value a seasoned swap from a discount curve, find its par rate, and compute its sensitivity to each curve point.
+25
40 min
Check a sheet of call and put quotes against put–call parity and the no-arbitrage bounds, and build the trade.
+10
25 min
Price a call on ever finer trees, measure how fast the price approaches Black–Scholes, and explain the oscillation.
+10
30 min
Compute delta, gamma, vega and theta for a book of options, aggregate them, and suggest the hedge.
+25
45 min
Rebalance a hedge daily along simulated paths, then split the P&L into gamma, theta and hedging error.
+50
75 min
Fit a parametric smile to quotes and check the fitted prices for butterfly and calendar arbitrage.
+50
70 min
Price a knock-out call with a formula, a tree and simulation, and explain which method to trust near the barrier.
+25
50 min
Calibrate Heston parameters to a volatility smile and judge how well the fitted model prices options it was not fitted to.
+50
80 min
Turn at-the-money, risk-reversal and butterfly quotes into strikes and volatilities, following FX market conventions.
+25
50 min
Bootstrap caplet volatilities from quoted cap prices and smooth them without creating arbitrage.
+50
70 min
Bootstrap survival probabilities from a term structure of CDS spreads and value an off-market contract.
+25
45 min
Price an autocallable with a knock-in put, and show how its value depends on volatility and correlation.
+50
75 min
Revalue an options book under historical and hypothetical scenarios and report the worst losses and their drivers.
+25
45 min
Compare sample and shrinkage covariance estimates by the out-of-sample risk of minimum-variance portfolios.
+25
50 min
Trace the frontier with and without a no-short-selling rule, and explain where the two diverge.
+10
30 min
Move an existing portfolio towards a target while respecting turnover, position and sector limits.
+25
45 min
Build a minimum expected-shortfall portfolio as a linear programme and compare its tails with minimum variance.
+25
45 min
Find the weights at which every asset contributes the same risk, and compare the result with equal weights.
+25
40 min
Build both portfolios on the same universe and compare their out-of-sample risk and stability.
+50
60 min
Combine market-implied returns with two analyst views, and show how each view's confidence moves the weights.
+50
60 min
Compare calendar and threshold rebalancing for a multi-asset fund once trading costs are counted.
+25
45 min
Choose a small basket of stocks and weights that tracks an index, and report the tracking error out of sample.
+50
70 min
Backtest an allocation walk-forward, adjust for the number of variants tried, and report it on one page.
+25
45 min
Replay add, cancel and trade messages into a limit order book, and report the spread and depth over the day.
+25
45 min
Compute quoted, effective and realised spreads for a day of trades, and explain what each one measures.
Microstructure 1.3Microstructure 1.4Trading· Opens with Market Microstructure and Algorithmic Trading+10
30 min
Estimate how price impact grows with order size from a set of executions, and test the square-root law.
+25
45 min
Compare a broker's executions with arrival price and VWAP benchmarks, and write the findings for a portfolio manager.
+25
45 min
Trade a large order over a day, trading off impact against risk, and compare the schedule with TWAP.
+50
70 min
Simulate an Avellaneda–Stoikov market maker and tune its risk aversion for profit against inventory risk.
Microstructure 3.1Microstructure 3.2Trading· Opens with Market Microstructure and Algorithmic Trading+50
80 min
Build a cointegrated basket, set entry and exit bands, and backtest the strategy with costs.
+25
50 min
Combine time-series momentum and carry signals across futures markets with volatility-scaled positions.
+25
50 min
Measure order-flow imbalance from book updates and test how well it predicts the next price move.
+50
70 min
Scale a strategy to a volatility target, add a drawdown brake, and show the effect on returns and risk.
+25
40 min