Derivatives Foundation puzzles, solved step by step

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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
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  1. 007A stock goes up 10% one day and down 10% the next, and keeps alternating for 250 trading days. Where does it end relative to its start? And where does a fund that delivers three times the stock's daily move end up?Volatility and correlationCoreVolatility tradingWealth management

    Try it first

    Before multiplying: after one up day and one down day, is the stock back where it started?

    Show the worked solution

    The stock ends at about 28% of its start; the three-times fund at roughly 8 millionths of its start, effectively zero. Each up-and-down pair multiplies the stock by 1.1 x 0.9 = 0.99, and 125 pairs give 0.99^125 = 0.285. The leveraged fund moves 30% each way, so each pair is 1.3 x 0.7 = 0.91, and 0.91^125 is about 7.6e-06. The arithmetic average return is zero in both cases; the compounded return is not.

    Why does a zero average return lose money?

    Take a 100 rupee note to a shop that marks everything up 10% in the morning and discounts 10% in the afternoon. The afternoon discount is taken off a bigger number, so the price ends at 99, not 100. A gain and a loss of the same percentage do not cancel, because the loss acts on the larger base; the pair costs the square of the move, 1% for a 10% swing. Repeat that 125 times and the 1% losses compound to a 72% fall. This is volatility drag: the gap between the average return and the compounded return.

    Alternating moves on a log scale: both drift down, and leverage multiplies the drift10.10.010.00110^-410^-510^-6050100150200250trading daystart = 1.0stock: 0.99^125 = 28% of start3x fund: 0.91^125 = 8 millionthseach up-down pair: 1.1 x 0.9 = 0.99, a loss of 1%with 3x: 1.3 x 0.7 = 0.91, a loss of 9%, nine times worse
    On a log scale both paths step down in straight lines: the stock loses 1% per up-and-down pair and ends at 28% of its start after 250 days, while the three-times fund loses 9% per pair and ends at about 8 millionths of its start, so leverage multiplies the drag by far more than three.
    The relationship
    (1+m)(1−m)=1−m20.99125=0.2850.91125≈7.8×10−6(1+m)(1-m) = 1 - m^2 \qquad 0.99^{125} = 0.285 \qquad 0.91^{125} \approx 7.8 \times 10^{-6}
    mthe daily move, 0.10 for the stock and 0.30 for the three-times fund
    1 - m^2what one up-and-down pair leaves of the value
    125the number of pairs in 250 days
    What it says in wordsEach pair loses the square of the move, and the leveraged fund's loss per pair is nine times the stock's because 0.3 squared is nine times 0.1 squared.

    Why is three times the move so much worse than three times the loss?

    The drag per pair is the square of the move. Tripling the move multiplies the drag by nine, not three: the stock loses 1% per pair, the fund loses 9%. That is why a daily-rebalanced leveraged fund in a choppy, sideways market bleeds even when the underlying ends flat. The general rule you can quote: over many periods the compounded growth rate is roughly the average return minus half the variance, and leverage multiplies the variance by the square of the leverage.

    What would you say to a client who holds the three-times fund?

    That the product tracks three times the daily move, exactly as promised, and that this is not the same as three times the return over a year. A leveraged fund is a tool for a view on the next day or week; held through a sideways year it loses to its own rebalancing. The limitation to state: the alternating path is the worst case for drag, and a strongly trending market can make a leveraged fund return more than three times the underlying. The drag is about path, not just direction.

    Where candidates lose it

    The common answer is that the stock ends flat, because plus 10 and minus 10 seem to cancel. They cancel in arithmetic and not in compounding; the second move acts on a different base.

    The second loss is saying the leveraged fund ends at three times the stock's loss, or at 28% cubed. The right route is per pair: 1.3 x 0.7 = 0.91, then raise to the 125th power. The drag scales with the square of the leverage.

    What the interviewer asks next

    • Make the daily move 1% instead of 10%. Where does the stock end after 250 days?
    • Over a year with 16% annual volatility and zero average daily return, roughly what is the compounded return?
    • Why do leveraged funds rebalance daily, and what would change if they rebalanced monthly?
  2. 032One-month implied volatility is 20% and three-month implied volatility is 25%. What volatility is implied for the period from month one to month three?Volatility and correlationCoreVolatility tradingEquity derivatives

    Try it first

    What is the forward volatility for months two and three?

    Show the worked solution

    About 27.2%. Variance is volatility squared times time, and variances add across periods. The three months carry 0.25 squared x 3 = 0.1875 of variance; the first month carries 0.20 squared x 1 = 0.04; so months two and three carry 0.1475 between them, 0.0737 per month on an annualised basis. The square root is 0.272. The forward volatility must sit above 25% because the quarter's average has to be pulled up from the 20% start.

    Why do you add variances and not volatilities?

    If you walk a random distance each hour, your spread after three hours is not three times the hourly spread, because some hours cancel others. What grows in a straight line is the variance, the spread squared. Implied volatility is quoted per year, so the variance an option carries is volatility squared times its life, and a longer option's variance is the sum of the variances of the pieces of time inside it. That is the whole mechanism. Twenty per cent for one month is 0.04 of variance; twenty-five per cent for three months is 0.1875; the difference belongs to the two months in between, and dividing by two months and taking the square root turns it back into a volatility.

    Variance is area: height is volatility squared, width is time, and areas add0.020.040.060.08variance per yeartodaymonth 1month 2month 3month 120% squared = 0.04area 0.04 x 1months 2 and 3: the unknownheight 0.0737, so volatility sqrt = 27.2%area 0.0737 x 2 = 0.147525% squared= 0.0625x 3 months= 0.1875Dashed area 0.1875 = 0.04 + 0.1475, so the forward variance is 0.1475 / 2 = 0.0737 and the forward volatility is 27.2%Averaging volatilities instead of variances would give 27.5%, which is the wrong quantity to add
    Drawn as area, the first month's variance of 0.04 plus the two forward months' variance of 0.0737 each must fill the quarter's 0.1875, so the forward box has height 0.0737 and its square root is a forward volatility of 27.2%.
    The relationship
    σ1,32=σ32T3−σ12T1T3−T1=0.0625×3−0.04×12=0.07375σ1,3=0.07375≈27.2%\sigma_{1,3}^2 = \frac{\sigma_3^2 T_3 - \sigma_1^2 T_1}{T_3 - T_1} = \frac{0.0625 \times 3 - 0.04 \times 1}{2} = 0.07375 \qquad \sigma_{1,3} = \sqrt{0.07375} \approx 27.2\%
    sigma 3, sigma 1the three-month and one-month implied volatilities, 25% and 20%
    Tthe option life in months; the units cancel as long as both use the same one
    sigma 1,3the forward volatility for the period between the two expiries
    What it says in wordsSubtract the short period's variance from the long period's and spread what is left over the time in between.

    What does the number tell a trader?

    The 27.2% is the volatility you would lock in for months two and three by selling the one-month option and buying the three-month one in variance-weighted sizes, a calendar spread. If you believe realised volatility in those two months will be well below 27.2%, the three-month option is rich relative to the one-month, and the calendar is the trade that expresses it. Say the rounding: an upward sloping term structure, 20% then 25%, hides a forward that is steeper than either quote, 27.2%, and the trap of reading 25% as the forward is what the question is built to catch. The calculation uses calendar time in months; a desk would use trading days or variance-weighted business days, which shifts the number by a few tenths.

    What is the limit of the calculation?

    The subtraction has to leave something positive. If the three-month volatility were below 11.5% with the one-month still at 20%, the quarter would carry less variance than its first month alone, which is impossible without arbitrage: you could sell the one-month, buy the three-month and hold a position with negative forward variance. Forward variance can be small but never negative, so a term structure that inverts too sharply is a mispricing, not a forecast. The second limit is that both quotes must refer to the same strike in forward terms; mixing an at-the-money one-month with a three-month that has rolled away from the money adds skew to the comparison and the forward you compute is no longer clean.

    Where candidates lose it

    The common loss is averaging volatilities: 25% over three months with 20% in the first month gives 27.5% for the rest if you treat volatility as additive. It is close, which is why it survives, but it is the wrong quantity and on a steeper curve the gap is large.

    The second is forgetting to weight by time: subtracting 0.04 from 0.0625 and taking the root. Write variance times time for each leg, every time.

    What the interviewer asks next

    • Six-month volatility is also 25%. What is the forward volatility for months four to six?
    • The one-month is 30% and the three-month is 25%. What is the forward, and what does it say about the market?
    • How would you actually lock in that forward volatility with listed options, and what would break the hedge?
    • Why do desks compute this in trading days rather than calendar months?