Chapter 10 Exercises

Part A: Spread Fundamentals (Exercises 1-6)

Exercise 1: Calculating Basic Spread Measures

A prediction market for "Will the Federal Reserve raise rates in March?" shows the following order book:

(a) Calculate the quoted spread, the midpoint, and the relative spread.

(b) If you submit a market order to buy 100 contracts, what is your average execution price? Calculate the effective spread.

(c) If you submit a market order to buy 300 contracts, what is your average execution price? How much additional cost do you pay compared to buying at the best ask?

(d) If, 5 minutes after your buy of 100 contracts at the ask, the midpoint has moved to $0.66, what is the realized spread from the market maker's perspective?

Exercise 2: Spread Decomposition

Over the course of a trading day, the following trades occur in a prediction market (midpoint at time of trade shown):

(a) Calculate the effective spread for each trade.

(b) Calculate the realized spread for each trade (using the 5-minute-later midpoint).

(c) Calculate the adverse selection component for each trade.

(d) What is the volume-weighted average effective spread? Volume-weighted average adverse selection cost?

Exercise 3: Spread Comparisons Across Markets

You are monitoring three prediction markets on the same platform:

• Market A (Presidential Election Winner): Bid 0.48, Ask 0.52, Daily Volume 50,000
• Market B (Will GDP growth exceed 3%?): Bid 0.30, Ask 0.37, Daily Volume 2,000
• Market C (Super Bowl Winner): Bid 0.71, Ask 0.73, Daily Volume 25,000

(a) Calculate the quoted spread and relative spread for each market.

(b) Rank the markets from cheapest to most expensive to trade, using the relative spread.

(c) Explain why Market B might have a wider spread despite being a simpler question than Market A.

(d) If you had to buy 500 contracts in each market, which would likely have the most slippage? Why?

Exercise 4: Spread and Probability Extremes

Consider two binary prediction markets, both with a 2-cent quoted spread:

• Market X: Bid 0.49, Ask 0.51
• Market Y: Bid 0.94, Ask 0.96

(a) Calculate the relative spread for each market.

(b) For Market Y, calculate the maximum profit per contract for a buyer and a seller. What is the asymmetry?

(c) Explain why a 2-cent spread is "cheaper" in absolute terms but "more expensive" in relative terms for Market Y.

(d) If both markets have the same amount of informed trading (same $\pi$ in the Glosten-Milgrom model), which market should theoretically have a wider absolute spread? Why?

Exercise 5: Time-Varying Spreads

A prediction market for an election has the following average spreads at different times:

(a) What is the relationship between volume and spreads in the pre-election period?

(b) Why do spreads widen on election day despite volume being much higher?

(c) Why are spreads widest during vote counting, when the answer is about to be revealed?

(d) If you wanted to establish a position as cheaply as possible, when would you trade? What trade-off do you face?

Exercise 6: Cross-Market Spread Arbitrage

Two platforms are offering contracts on the same event ("Will it snow in New York on December 25?"):

• Platform A: Bid 0.18, Ask 0.22
• Platform B: Bid 0.24, Ask 0.28

(a) Is there an arbitrage opportunity? If so, describe the trades.

(b) What is the guaranteed profit per contract, ignoring fees?

(c) If Platform A charges a 1-cent taker fee and Platform B charges a 2% fee on notional, is the arbitrage still profitable?

(d) What risks remain even with the "guaranteed" profit?

Part B: Transaction Costs and Fees (Exercises 7-12)

Exercise 7: Total Cost Calculation

You want to buy 200 "Yes" contracts on Kalshi for an event currently priced at Bid 0.55, Ask 0.58. The order book shows:

• 80 contracts available at 0.58
• 60 contracts available at 0.59
• 100 contracts available at 0.60

Kalshi's taker fee is min($0.01, price/15) per contract. There are no maker fees.

(a) Calculate the average execution price for a 200-contract market buy order.

(b) Calculate the total explicit fee.

(c) Calculate the total implicit cost (spread + slippage).

(d) What is the total cost of this trade as a percentage of notional value?

(e) If the event resolves "Yes," what is your total profit net of all costs? Net of fees on exit?

Exercise 8: PredictIt Fee Analysis

You buy 100 shares of "Yes" at $0.60 on PredictIt (historical analysis). PredictIt charges 10% of profits and 5% on withdrawals.

(a) If the event occurs (you win), calculate your gross profit, profit fee, and net profit before withdrawal.

(b) Calculate the withdrawal fee if you withdraw your entire balance (original investment + net profit).

(c) What is your total return after all fees?

(d) What is the minimum true probability for this trade to have positive expected value after all fees?

(e) Compare this to the breakeven probability on Polymarket (zero fees, 2-cent spread) for the same $0.60 entry price.

Exercise 9: Gas Fee Impact Analysis

You are trading on a blockchain-based prediction market where gas fees are $0.03 per transaction.

(a) If you buy 1 contract at $0.50, what percentage of your investment is consumed by the gas fee?

(b) If you buy 10 contracts at $0.50 each, what percentage?

(c) If you buy 100 contracts at $0.50 each, what percentage?

(d) Derive a formula for the minimum trade size $N$ such that gas fees are less than $x$% of the notional value, given gas fee $G$ and contract price $P$.

(e) During a period of high network congestion, gas fees increase to $2.50 per transaction. What is the minimum trade size (at $0.50 per contract) to keep gas fees below 1% of notional?

Exercise 10: Maker vs. Taker Strategy

On a platform with the following fee schedule: - Maker fee: -$0.002 per contract (rebate) - Taker fee: $0.008 per contract

A market has Bid 0.50, Ask 0.54.

(a) If you buy using a market order (taker), what is your effective purchase price including fees?

(b) If you place a limit buy at $0.51 (maker) and it fills, what is your effective purchase price including the rebate?

(c) What is the cost savings per contract from using a maker order?

(d) If your limit order at $0.51 has only a 60% chance of filling within the next hour, and the midpoint is expected to drift up by $0.005 per hour, what is the expected cost of the maker strategy including the opportunity cost of non-fills?

(e) At what fill probability does the maker strategy become worse than the taker strategy?

Exercise 11: Multi-Leg Trade Costs

You want to execute the following combined trade on Kalshi: - Buy 100 "Yes" contracts on Market A at $0.40 (taker) - Sell 100 "Yes" contracts on Market B at $0.65 (taker)

Both markets have taker fees of min($0.01, price/15).

(a) Calculate the total fees for both legs of the trade.

(b) If Markets A and B are correlated (both are about the same underlying event but framed differently), and you expect both to resolve the same way 90% of the time, calculate your expected P&L before and after fees.

(c) What correlation between the outcomes would make this trade break even after fees?

(d) How would the analysis change if you could execute both legs as maker orders?

Exercise 12: Cost Comparison Across Platforms

You have $1,000 to invest and want to buy "Yes" contracts on an event priced at approximately $0.50 across platforms:

(a) For each platform, calculate how many contracts you can buy with $1,000 (round down to whole numbers).

(b) For each platform, calculate the total cost of entry (including all fees).

(c) If the true probability is 0.58, calculate the expected profit after ALL costs for each platform.

(d) Rank the platforms from most to least profitable for this specific trade.

Part C: Breakeven Edge and Overround (Exercises 13-18)

Exercise 13: Breakeven Edge Derivation

(a) Starting from the expected profit equation $E[\pi] = p \cdot (1 - P_{\text{ask}}) - (1-p) \cdot P_{\text{ask}} - C$, derive the breakeven probability formula.

(b) Show that the breakeven edge (relative to the midpoint) equals half the spread plus other costs.

(c) For a market with a 4-cent spread and $0.02 in fees per contract, what is the breakeven edge?

(d) If your model estimates the true probability is 3 cents above the midpoint, should you trade? Why or why not?

Exercise 14: Breakeven with Profit Fees

A platform charges a 15% fee on profits (no other fees). The market asks 0.45 for a "Yes" contract.

(a) Derive the breakeven probability formula for this fee structure.

(b) Calculate the breakeven probability.

(c) How does the breakeven change if the ask price increases to 0.70?

(d) At what ask price is the breakeven probability exactly 1.0 (meaning the trade can never be profitable)?

(e) Plot (or calculate a table of) breakeven edge as a function of ask price for profit fee rates of 5%, 10%, 15%, and 20%.

Exercise 15: Overround Calculation

An election market has four candidates with the following ask prices:

(a) Calculate the overround.

(b) Using proportional normalization, calculate the implied true probability for each candidate.

(c) Using midpoint normalization (bids are: A=0.38, B=0.27, C=0.14, D=0.10), calculate the implied true probabilities.

(d) Which normalization method gives probabilities closer to what you would expect? Discuss.

(e) If you believed Candidate C's true probability was 0.20, calculate your edge using ask prices, midpoints, and normalized probabilities. Which should you use for trading decisions?

Exercise 16: Multi-Outcome Overround Decomposition

A market for "Which party wins the Senate?" has three outcomes:

(a) Calculate the overround using ask prices.

(b) Calculate the overround using bid prices (note: sum should be less than 1).

(c) Calculate the overround using midpoint prices.

(d) Why is the spread on "Other/Tie" so much wider than the other outcomes?

(e) If you want to buy all three outcomes to guarantee a $1 payout, what do you pay and what is your guaranteed loss?

Exercise 17: Sensitivity Analysis Table

Create a sensitivity analysis table showing the breakeven edge (in cents) for the following combinations:

• Ask prices: 0.20, 0.40, 0.60, 0.80
• Spread sizes: 0.02, 0.04, 0.06
• Fee per contract: 0.00, 0.01, 0.02

(a) Fill in the complete table (24 cells).

(b) Which factor has the largest impact on breakeven edge: ask price, spread size, or fee?

(c) For each ask price, what is the maximum spread at which a trader with 5 cents of edge can profitably trade (with $0.01 fee)?

Exercise 18: The "Hidden" Cost of the Overround

You want to buy a "basket" of all outcomes in a 5-candidate election market as a hedge. Ask prices are:

(a) What is the total cost of the basket? What is the overround?

(b) If one contract of each candidate costs $1.05 total and pays $1.00, what is the percentage loss?

(c) If instead of buying all five at the ask, you place limit orders at the midpoint (assume spreads are all 4 cents, so midpoints are 2 cents below asks), what would the basket cost?

(d) An index fund manager wants to hold all outcomes proportionally for 30 days and then sell. If they rebalance weekly and each rebalance costs the overround, what is the cumulative cost over 30 days?

Part D: Market Impact and Execution (Exercises 19-24)

Exercise 19: Square-Root Impact Model

A prediction market has daily volume of 5,000 contracts and price volatility of $0.025 per day. The impact parameter $\alpha = 0.6$.

(a) Using the square-root impact model $\Delta P = \alpha \cdot \sigma \cdot \sqrt{Q/V}$, calculate the expected impact for orders of 100, 500, 1000, and 2500 contracts.

(b) For each order size, calculate the total impact cost ($\Delta P \times Q$).

(c) Show that total impact cost scales as $Q^{3/2}$, not $Q^2$.

(d) If you could split a 1,000-contract order into two 500-contract pieces (executed on different days), what would be the total impact cost? How much do you save compared to executing all at once?

Exercise 20: Order Book Walking

An order book has the following ask-side depth:

(a) Calculate the average execution price and total cost for orders of 100, 300, 500, and 800 contracts.

(b) Plot (or tabulate) the marginal cost of the next contract as a function of cumulative quantity.

(c) Where is the "liquidity cliff" — the point at which marginal costs increase sharply?

(d) If you are buying 500 contracts, how much do you save by placing a limit order at $0.56 (and waiting for fills) versus a market order?

Exercise 21: TWAP Execution Simulation

You need to buy 1,000 contracts over 5 days. Daily volume is 2,000 contracts. You plan to use a TWAP strategy (200 contracts per day).

(a) If temporary market impact is $\Delta P_{\text{temp}} = 0.01 \times \sqrt{Q_{\text{day}}/V_{\text{day}}}$ and permanent impact is 40% of temporary impact, what is the impact on day 1?

(b) What is the starting price on day 2 (accounting for permanent impact from day 1)?

(c) Complete the calculation for all 5 days, tracking the cumulative average execution price.

(d) Compare the total cost to executing all 1,000 contracts on day 1.

Exercise 22: AMM vs. Order Book Impact

A market exists on both an order book platform and an AMM platform.

Order book: Best ask 0.50, depth as in Exercise 20 (scaled to this price).

AMM: Constant-product formula with reserves of 10,000 "Yes" tokens and 10,000 "No" tokens. Price = No/(Yes+No) = 0.50.

(a) Calculate the cost of buying 100 "Yes" tokens on each platform.

(b) Calculate the cost of buying 500 "Yes" tokens on each platform.

(c) Calculate the cost of buying 2,000 "Yes" tokens on each platform.

(d) At what order size does the AMM become cheaper than the order book (approximately)?

(e) Discuss the advantages and disadvantages of each mechanism for large traders.

Exercise 23: Optimal Order Splitting

You have a 600-contract order to execute in a market with daily volume of 3,000 contracts, volatility of $0.02, and impact parameter $\alpha = 0.5$. The price is expected to drift up by $0.003 per day (unfavorable drift).

(a) Calculate the total execution cost if you execute all at once (market impact only, no drift).

(b) Calculate the total execution cost if you split into 3 equal pieces over 3 days (include both impact and drift costs).

(c) Calculate the total execution cost if you split into 6 equal pieces over 6 days.

(d) Find the approximate optimal number of slices that minimizes total cost (impact + drift).

(e) How does the optimal number change if the drift doubles to $0.006 per day?

Exercise 24: Real-World Execution Analysis

You executed the following trades over a week on Polymarket:

(a) Calculate the effective spread for each completed trade.

(b) Calculate the realized spread for each completed trade.

(c) Calculate the adverse selection component for each trade.

(d) What was the average execution price across all filled contracts?

(e) The Friday limit order did not fill, and by the end of Friday the midpoint was $0.57. Calculate the opportunity cost of the missed fill. Was the limit order strategy optimal in hindsight?

Part E: Synthesis and Advanced Problems (Exercises 25-30)

Exercise 25: Platform Selection Optimizer

You have identified an opportunity where your model estimates a 65% probability, and the market midpoint is $0.58. You want to buy 200 contracts. Compare your expected net profit across three platforms:

(a) For each platform, calculate the expected profit per contract before costs.

(b) For each platform, calculate the expected cost per contract (entry + exit + conditional fees).

(c) For each platform, calculate the expected net profit for 200 contracts.

(d) Which platform should you choose? Does the answer change if your edge is only 2 cents (true probability = 0.60)?

Exercise 26: Market Making Profitability

You are considering market making in a prediction market. You plan to quote Bid 0.48, Ask 0.52 (4-cent spread). You expect: - 100 contracts traded per day on each side (buy and sell) - 20% of counterparties are informed traders - When informed traders trade, the price moves 5 cents in their direction - Taker fee rebate (you earn): $0.002 per contract - Your technology costs: $5 per day

(a) Calculate your gross spread revenue per day.

(b) Calculate your expected adverse selection loss per day. (Hint: 20% of flow is from informed traders who move the price 5 cents against you.)

(c) Calculate your net daily profit/loss.

(d) What spread would you need to charge to break even?

(e) How does your answer change if informed trading increases to 40%?

Exercise 27: Dynamic Spread Management

As a market maker, you start the day with zero inventory and a 3-cent spread centered at 0.50 (Bid 0.485, Ask 0.515).

Over the day, the following sequence of trades occurs against you: 1. Buy 50 from you at 0.515 (you sell, inventory = -50) 2. Buy 30 from you at 0.515 (inventory = -80) 3. Sell 20 to you at 0.485 (inventory = -60) 4. Buy 100 from you at 0.515 (inventory = -160)

(a) After trade 4, you are short 160 contracts. If you keep your spread centered at 0.50, what is your maximum loss if the event resolves "Yes"?

(b) To manage inventory risk, you want to shift your quotes up to encourage sells and discourage buys. If you shift to Bid 0.50, Ask 0.53, how does this affect your expected fill rates?

(c) Suppose the order flow pattern (mostly buys) is because informed traders know the true probability is 0.60. Calculate your expected loss on the 160 short contracts.

(d) At what inventory level should you widen your spread from 3 cents to 5 cents? Propose a rule of thumb.

Exercise 28: Fee-Adjusted Kelly Criterion

The Kelly Criterion (without costs) for a binary prediction market bet is:

$$f^* = p - \frac{1 - p}{b}$$

where $p$ is the true probability, $b$ is the net payout odds ($(1-P_{\text{ask}})/P_{\text{ask}}$), and $f^*$ is the fraction of bankroll to bet.

(a) For a market at Ask = $0.55 with true probability 0.65, calculate the Kelly fraction (no costs).

(b) Now include a round-trip cost of $0.04 per contract. Adjust the Kelly fraction by reducing $p$ by the cost. What is the new Kelly fraction?

(c) With a bankroll of $10,000, how many contracts should you buy in each case?

(d) Calculate the expected growth rate of your bankroll (Kelly criterion log growth) for both cases.

(e) If costs reduce the Kelly fraction by more than 50%, what does this suggest about the quality of the trading opportunity?

Exercise 29: End-to-End Strategy Evaluation

You develop a model that identifies 50 trading opportunities per month. On average: - Your model estimates 3 cents of edge per trade (above the midpoint) - Average trade size: 100 contracts - Average midpoint price: 0.50 - Average spread: 0.03 - Platform: Kalshi (taker fee $0.01/contract) - Your model accuracy: probability estimates are correct on average (unbiased) with standard deviation of 0.05

(a) Calculate the expected gross profit per trade (before costs).

(b) Calculate the expected cost per trade (spread + fees).

(c) Calculate the expected net profit per trade and per month.

(d) What percentage of your trades are actually profitable after costs? (Hint: your edge is 3 cents, your cost is some amount, and your estimation error has standard deviation 0.05.)

(e) If you could switch to maker orders (zero fee, but only 50% of orders fill), how does your monthly expected profit change?

Exercise 30: Comprehensive Case — Election Market Trading

You are trading election markets across multiple platforms. The presidential election is 45 days away. You believe Candidate X has a 55% chance of winning. Current market prices:

Your total capital is $50,000.

(a) Calculate the breakeven probability for buying "Yes" on each platform.

(b) Calculate the optimal position size using the cost-adjusted Kelly Criterion on each platform.

(c) If you split your capital across both platforms, what is the optimal allocation?

(d) Over the next 45 days, you expect to trade 10 times (rebalancing as your model updates). Calculate the cumulative transaction costs.

(e) What is the minimum edge (in probability points) you need to make money over the entire 45-day period, accounting for all cumulative costs?

(f) Write pseudocode for an automated system that monitors both platforms and executes trades when the edge exceeds the cost threshold.