Derivatives Foundation puzzles, solved step by step

Puzzles
100
Traced to a firm
66
Topics
12
Hard
29
Topic
All topicsMental maths and estimation9Random walks and Markov chains7Conditional probability and Bayes7Volatility and correlation7Option pricing intuition7Expected value and optimal stopping10Market making11Option payoffs and no-arbitrage10Probability and counting11Distributions and statistics8Games and logic8Betting and sizing5
Level
AnyWarm upCoreHard
Source
AnyReported at a firmStandard
Showing 1–1 of 1 · filtered from 100Clear filters
  1. 030A coin falls your way 60% of the time and pays even money. What fraction of your capital should you bet each time, and what growth rate does that give?Betting and sizingCoreOptiverAustin · 2025

    Try it first

    What fraction of your capital goes on each flip?

    Show the worked solution

    Bet 20% of your capital each time, for a growth rate of about 2.0% per bet. The Kelly fraction at even money is the edge, p minus q, which is 0.6 minus 0.4. The expected log growth is 0.6 x ln(1.2) + 0.4 x ln(0.8) = 0.0201, so capital typically grows by a factor of about 7.5 over 100 bets. Bet twice that, 40%, and the growth turns slightly negative, -0.24% a bet: anything beyond about 39% loses money despite the edge.

    Why not bet everything on a coin that favours you?

    A street vendor with a stall that makes money six days in seven does not spend the whole float on stock each morning, because the seventh day would end the business. Expected value is the right guide for a single bet with money you can replace, but when you must survive to play again, what matters is the growth of your capital over many bets, and that is governed by the average of the logarithm, not the average of the rupees. Betting all of it has the highest expected value and a certainty of ruin: one loss and there is nothing left to compound. The logarithm punishes that loss infinitely, which is the mathematical way of saying you cannot come back from zero.

    The relationship
    g(f)=pln⁡(1+f)+qln⁡(1−f),f∗=p−q=0.2,g(0.2)=0.6ln⁡1.2+0.4ln⁡0.8≈0.0201g(f) = p\ln(1+f) + q\ln(1-f), \qquad f^* = p - q = 0.2, \qquad g(0.2) = 0.6\ln 1.2 + 0.4\ln 0.8 \approx 0.0201
    fthe fraction of capital staked on each bet
    p, qthe chance of winning and of losing, 0.6 and 0.4
    f starthe fraction that maximises long-run growth; at even money it is the edge
    gthe expected log growth per bet, about 2% here
    What it says in wordsMaximise the average log of your wealth after one bet, and at even money the best stake is simply the edge.
    Growth per bet against the fraction staked: the peak is at edge over odds, 20%-3%-2%-1%0+1%+2%0%10%20%30%40%50%60%fraction of capital staked each betexpected log growth per betKelly 20%: 2.01% a bethalf Kelly 10%: 1.50%double Kelly 40%: -0.24%turns negative at about 39%ruin in slow motionOver 100 bets the typical path multiplies capital by 7.5x at 20%, 4.5x at 10%, and 0.78x at 40%The flat top means under-betting costs little; the steep right side means over-betting costs everything
    Expected log growth per bet rises from zero to a peak of 2.01% at a 20% stake and falls back through zero at about 39%, so half Kelly keeps three quarters of the growth while double Kelly, just past the crossing, loses a little on every bet.

    Where does the 20% come from, and what if the odds are not even?

    Differentiate g with respect to f and set it to zero: p over (1 + f) equals q over (1 - f), which gives f = p - q. At even money that is the edge, 0.2. For a bet that pays b to 1 the formula becomes (bp - q) over b, edge over odds. Say the general form, because the interviewer will often move the payout: at 2 to 1 on the same coin the stake becomes (1.2 - 0.4) / 2 = 40%, and at odds where bp is below q the right stake is zero. The growth at 20% is 0.0201 per bet, which compounds to a factor of e to the 2.01, about 7.5 times, over 100 bets on the typical path. That is the median outcome, not the average: a few lucky paths do far better and pull the mean up.

    Why do desks bet less than Kelly?

    The curve is flat on the left and steep on the right. Half Kelly, 10%, still delivers 1.50% a bet, three quarters of the peak, with far smaller swings. Double Kelly, 40%, delivers -0.24%, already below zero, with wild swings: the crossing point is about 39%, and anything beyond it shrinks capital in the long run even though every single bet has positive expected value. Over-betting is punished much harder than under-betting, and in real trading the 60% is an estimate rather than a fact, so sizing at a fraction of Kelly is the standard discipline. The limitation to state: Kelly assumes you know p, that bets are independent and repeated, and that you can resize freely; on a desk, position limits and the uncertainty in your edge usually bind before the formula does.

    Where candidates lose it

    The common loss is answering with the expected value: 60% of the time you win, so bet big, or even bet everything. The interviewer wants to hear that growth is about the log, that ruin ends the game, and that the stake is the edge, 20%.

    The second is giving 20% but no growth rate, or quoting the 20% expected return per rupee as the growth rate. The growth per bet is about 2%, far below the 20% edge, because the losses bite on a shrinking base.

    What the interviewer asks next

    • The coin pays 2 to 1 instead of even money. What is the stake now?
    • Why does half Kelly give three quarters of the growth? What is the general relationship?
    • You are only 60% sure the coin is 60% biased. How does that change the stake?
    • What is the probability that a full-Kelly bettor halves their capital at some point before doubling it?

    Asked at Optiver, Quantitative Research, Austin, 2025 (Wall Street Oasis): Use Kelly Criterion