Chapter 15: Exercises

Overround and Probability Extraction (Exercises 1--8)

Exercise 1: Basic Overround Calculation

A three-candidate election market has the following prices: - Candidate A: 0.52 - Candidate B: 0.35 - Candidate C: 0.20

(a) Calculate the overround. (b) Calculate the overround percentage. (c) If you bought one contract of each candidate, what would be your guaranteed loss?

Exercise 2: Multiplicative Overround Removal

Using the prices from Exercise 1: (a) Apply multiplicative (proportional) overround removal to obtain estimated true probabilities. (b) Verify that your probabilities sum to 1. (c) Which candidate's probability changed the most in absolute terms? In relative terms?

Exercise 3: Additive Overround Removal

A four-outcome market has prices: 0.40, 0.30, 0.25, 0.15. (a) Apply additive overround removal. (b) Are all resulting probabilities valid (non-negative)? (c) Now consider prices: 0.40, 0.30, 0.25, 0.03. What happens with additive removal? Why?

Exercise 4: Power Method

For the market with prices [0.45, 0.35, 0.28]: (a) Write the equation that the power method exponent $k$ must satisfy. (b) Show that $k = 1$ gives a sum greater than 1. (c) Show that $k = 2$ gives a sum less than 1 (compute $0.45^2 + 0.35^2 + 0.28^2$). (d) Conclude that the solution $k$ lies between 1 and 2. Use a calculator or Python to find $k$ to 3 decimal places.

Exercise 5: Method Comparison

A horse racing market has 6 runners with prices: [0.35, 0.22, 0.18, 0.15, 0.10, 0.08]. The overround is 8%. (a) Apply multiplicative removal. (b) Apply the power method. (c) Compare the results for the favorite (0.35) and the longest shot (0.08). Which method adjusts the longshot more aggressively? (d) Which method would you trust more if you believed there was informed trading in the market? Why?

Exercise 6: Shin's Method Intuition

(a) Explain in your own words why Shin's method distributes overround unevenly. (b) In Shin's model, what does the parameter $z$ represent? Why would a higher $z$ lead to larger probability adjustments for longshots? (c) If a market has $z = 0.03$, what does this suggest about the level of informed trading?

Exercise 7: Overround Across Platforms

The same 5-candidate election is traded on two platforms:

Platform A: [0.30, 0.25, 0.22, 0.15, 0.12] Platform B: [0.32, 0.26, 0.20, 0.14, 0.15]

(a) Calculate the overround on each platform. (b) After multiplicative removal, which platform gives higher implied probability for Candidate 1? (c) Can you construct any cross-platform arbitrage? What would you need to check?

Exercise 8: Dynamic Overround

A market starts with 4% overround and increases to 12% overround after a period of low liquidity. The true probabilities remain [0.40, 0.35, 0.25]. (a) What are the market prices at 4% overround (assuming multiplicative overround)? (b) What are the prices at 12% overround? (c) How does higher overround affect the profitability of informed trading?

Dutch Books and Arbitrage (Exercises 9--13)

Exercise 9: Simple Dutch Book Detection

A market has three outcomes with the following bid and ask prices:

(a) Does a buyer's Dutch book exist (can you buy all outcomes for less than 1)? (b) Does a seller's Dutch book exist (can you sell all outcomes for more than 1)? (c) What is the "tightest" the market could be while still having no Dutch book?

Exercise 10: Constructing a Dutch Book

A poorly calibrated market has three outcomes with ask prices summing to 0.95: - A: 0.40 - B: 0.30 - C: 0.25

(a) Construct the buyer's Dutch book. How much do you invest? What do you receive? (b) What is the guaranteed profit per dollar invested? (c) How many contracts of each should you buy to guarantee a profit of exactly $100?

Exercise 11: Cross-Market Arbitrage

Two platforms trade the same event: - Platform X: "Will Team A win?" Yes: 0.65, No: 0.40 - Platform Y: "Will Team A win?" Yes: 0.55, No: 0.50

(a) Does a Dutch book exist on Platform X alone? On Platform Y alone? (b) Can you construct a cross-platform arbitrage? Describe the trades. (c) What practical obstacles might prevent you from executing this arbitrage?

Exercise 12: Dutch Book with Transaction Costs

Using the market from Exercise 10, suppose each trade incurs a 2% fee on the transaction amount. (a) Recalculate the Dutch book profit after fees. (b) At what fee level does the Dutch book become unprofitable? (c) How does this relate to the concept of "no-arbitrage bounds"?

Exercise 13: LP Formulation

Write out the full linear programming formulation for detecting a Dutch book in a 4-outcome market with bid prices $b_1, \ldots, b_4$ and ask prices $a_1, \ldots, a_4$. Define all variables, the objective function, and all constraints.

Portfolio Optimization (Exercises 14--19)

Exercise 14: Multi-Outcome Kelly --- Simple Case

A 3-outcome market has prices [0.35, 0.35, 0.35] (overround = 5%) and your true probability estimates are [0.50, 0.30, 0.20]. (a) Which outcome has the highest edge? Calculate the edge for each. (b) Should you bet on outcomes where your estimated probability is lower than the market price? Why or why not? (c) Using Kelly criterion, if you could only bet on one outcome, which would it be and why?

Exercise 15: Return Calculation

You invest 10% of your bankroll on Outcome A (priced at 0.30) and 5% on Outcome B (priced at 0.20) in a 4-outcome market. (a) What is your return if A wins? (b) What is your return if B wins? (c) What is your return if C or D wins? (d) What is your expected return if true probabilities are [0.35, 0.25, 0.25, 0.15]?

Exercise 16: Fractional Kelly Comparison

Your full Kelly solution for a 5-outcome market recommends betting: [12%, 8%, 0%, 3%, 0%] of bankroll. (a) What is the half-Kelly allocation? (b) What is the quarter-Kelly allocation? (c) If the expected growth rate at full Kelly is 2.5% per bet, estimate the growth rate at half-Kelly. (Hint: fractional Kelly growth rate approximation is $\alpha(2-\alpha)$ times full Kelly growth.) (d) Why might quarter-Kelly be more appropriate for this 5-outcome market than for a binary market?

Exercise 17: Budget Constraint Impact

A 4-outcome market has true probabilities [0.40, 0.30, 0.20, 0.10] and prices [0.30, 0.28, 0.25, 0.22]. (a) Which outcomes have positive edge? (b) If your budget constraint limits total wagering to 20% of bankroll, how does this affect the optimization compared to unlimited budget? (c) Intuitively, which outcomes should get priority in a budget-constrained setting?

Exercise 18: Comparing Kelly and Mean-Variance

Explain in your own words: (a) What is the key difference in objective function between Kelly and mean-variance optimization? (b) In what situation would mean-variance and Kelly give similar recommendations? (c) In what situation would they give very different recommendations? (d) Which approach is more appropriate for a prediction market trader who makes 50 bets per year vs. 5,000 bets per year?

Exercise 19: Sensitivity Analysis

Your Kelly-optimal allocation depends on your probability estimates. For the market [0.30, 0.25, 0.25, 0.20] with your estimates [0.35, 0.28, 0.22, 0.15]: (a) Recalculate the edge on each outcome if your probability estimates are each off by +/- 3 percentage points. (b) How does this uncertainty affect your confidence in the Kelly allocation? (c) Propose a method to account for probability estimation uncertainty in the Kelly framework.

Relative Value and Pair Trades (Exercises 20--23)

Exercise 20: Odds Ratio Calculation

A 6-candidate market has: - Market prices: [0.28, 0.22, 0.18, 0.15, 0.10, 0.07] - Your estimates: [0.25, 0.25, 0.20, 0.15, 0.10, 0.05]

(a) Calculate the odds ratio for every adjacent pair (1 vs 2, 2 vs 3, etc.). (b) Which pair has the most attractive relative value opportunity? (c) Describe the pair trade you would make.

Exercise 21: Pair Trade Payoff Analysis

You go long Candidate B (price 0.22) and short Candidate A (price 0.28) with 100 contracts each. (a) What is your net cost/credit to enter the trade? (b) Calculate your payoff in each of the 6 possible outcomes (using the prices from Exercise 20). (c) In how many outcomes do you profit? In how many do you lose? (d) What is your expected profit using your probability estimates from Exercise 20?

Exercise 22: Rank Order Disagreement

Your model ranks 5 candidates as: C > A > B > D > E. The market prices imply: A > B > C > D > E. (a) Identify all rank-order disagreements. (b) Which disagreements represent the strongest trading opportunities? Why? (c) Construct a portfolio of pair trades that profits if your ranking is correct.

Exercise 23: Correlation in Pair Trades

In a primary election market, if Candidate A drops out, Candidate B (from the same ideological wing) would likely benefit. (a) How does this "succession" effect influence the relative value between A and B? (b) If you are long A, is being long B a hedge or a correlated bet? Explain. (c) How might you hedge an A position given this correlation structure?

Scalar Markets (Exercises 24--27)

Exercise 24: Implied Distribution

A GDP growth bracket market has:

(a) Remove the overround using the multiplicative method. (b) Compute the implied mean using bracket midpoints (use -1% for the first bracket and 5% for the last). (c) Compute the implied standard deviation. (d) Does the implied distribution look roughly normal? How would you check?

Exercise 25: Distribution Fitting

Using the data from Exercise 24: (a) Fit a normal distribution by matching the mean and variance to the implied moments. (b) Compare the bracket probabilities implied by this normal fit to the market probabilities. (c) Which brackets are most mispriced relative to the normal fit? (d) If you trusted the normal distribution assumption, what trades would you make?

Exercise 26: Tail Trading

A market for annual S&P 500 returns has brackets:

(a) Calculate the overround. (b) Your analysis suggests the market underprices tail risk (both positive and negative extremes). Design a tail-trading strategy. (c) What is the cost of your strategy (what do you need to sell to fund the tail purchases)? (d) Under what conditions does your strategy profit? Under what conditions does it lose?

Exercise 27: Moment Trading

Two traders disagree about GDP growth: - Trader X: Mean 2.0%, StdDev 0.8% - Trader Y: Mean 2.5%, StdDev 1.5%

(a) Which brackets would Trader X buy relative to Trader Y? (b) Describe the "mean shift" trade between X and Y. (c) Describe the "variance expansion" trade between X and Y. (d) Can you construct a trade that profits from the variance disagreement without being exposed to the mean disagreement?

Hedging and Market Making (Exercises 28--30)

Exercise 28: Hedging Calculation

You hold the following positions in a 4-outcome market: - Outcome A: +500 contracts (long) - Outcome B: +200 contracts (long) - Outcome C: 0 contracts - Outcome D: -100 contracts (short)

Current prices: [0.35, 0.28, 0.22, 0.15].

(a) Calculate your payoff in each of the 4 possible outcomes. (b) What is your worst-case outcome? Best-case outcome? (c) Design a hedge that reduces your worst-case loss by 50% while minimizing the cost.

Exercise 29: Market Maker Inventory

A market maker in a 3-outcome market has accumulated inventory of [+300, -150, +50] contracts. Current fair values are [0.45, 0.35, 0.20]. (a) What is the market maker's PnL if each outcome wins? (b) How should the market maker adjust bid/ask quotes to reduce inventory imbalance? (c) If the inventory adjustment parameter is $\delta = 0.0001$ per contract, what are the adjusted midpoints?

Exercise 30: LMSR Market Making

An LMSR market maker has liquidity parameter $b = 100$ and 3 outcomes with current quantities purchased $q = [50, 30, 20]$. (a) Calculate the current price of each outcome using the LMSR formula. (b) If someone buys 10 contracts of Outcome 1, what are the new prices? (c) What is the cost to the buyer for those 10 contracts? (Hint: integrate the price function.) (d) What is the maximum possible loss for this LMSR market maker?