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 1–2 of 2 · filtered from 100Clear filters
  1. 018A stock trades at Rs 1,000 and its options are priced at 16% implied volatility. You buy an at-the-money option and delta-hedge it every day. Roughly how large a daily move does the stock need to make for you to break even?Option pricing intuitionWarm upVolatility tradingMarket making

    Try it first

    Answer in your head before reading on: the breakeven daily move is about

    Show the worked solution

    About 1% a day, Rs 10. With roughly 256 trading days in a year and volatility growing with the square root of time, daily volatility is annual volatility divided by 16, so 16% a year is 1% a day. A delta-hedged long option earns half its gamma times the square of each day's move and pays theta every day; the two cancel when the move equals the implied daily move. Using 252 days gives 1.008%, still Rs 10.

    Where does dividing by 16 come from?

    Walk randomly on a straight road, one step forward or back each second, and after 100 seconds you are typically about 10 steps from where you began, not 100, because most steps undo each other. Price moves add up the same way. Volatility scales with the square root of time, and the square root of 256 trading days is 16, so an annual volatility divided by 16 is the standard deviation of one day's move. Traders call this the rule of 16. It makes 16% implied volatility the cleanest number on the screen: 1% a day, Rs 10 on this stock.

    The relationship
    σday=σyear256=16%16=1%,1%×1,000=Rs 10\sigma_{\text{day}} = \frac{\sigma_{\text{year}}}{\sqrt{256}} = \frac{16\%}{16} = 1\%,\qquad 1\% \times 1{,}000 = \text{Rs } 10
    sigma yearthe implied volatility, quoted per year
    256trading days in a year, rounded so the square root is a whole number
    sigma daythe standard deviation of one day's percentage move
    What it says in wordsOne day's typical move is the annual volatility divided by the square root of the number of trading days.

    Why does that daily move decide whether the hedged option makes money?

    Once the delta is hedged, the option's daily P&L is two pieces. Gamma pays you half gamma times the square of the move, in either direction; theta charges you a fixed amount for the day passing. For a one-month at-the-money option here, gamma is 0.0087 per rupee and theta is 0.435 a day. A Rs 10 move earns 0.5 x 0.0087 x 100 = 0.435, exactly the theta, because the option's price was built so that theta pays for a move of one implied standard deviation. A flat day loses 0.435; a Rs 20 day makes 1.31.

    Delta-hedged long option: one day's P&L against the day's move-20-100+10+20-0.5012Stock's move today, Rsloses thetamoves under Rs 10flat day -0.44+1.31 at Rs 20break evenRs per optionThe rule of 16256 trading days a yearsquare root of 256 = 1616% a year / 16 = 1% a day1% of Rs 1,000 = Rs 10gamma 0.0087, theta 0.435 a daygain = half x gamma x move squaredequal to theta at a move of Rs 10with 252 days: 1.008%, still Rs 10
    A delta-hedged long at-the-money option loses its theta of 0.435 on a flat day, breaks even when the stock moves Rs 10 either way, and makes 1.31 on a Rs 20 move, because the gain grows with the square of the move while theta is fixed, and Rs 10 is 16% divided by 16.

    What does the quick answer leave out?

    Three things, and naming one earns the follow-up. First, the breakeven is on the average squared move, not the average move. If the stock's moves are normal with a standard deviation of Rs 10, its average absolute move is only about Rs 7.98, so a stock that typically moves Rs 8 a day is already moving enough to break even. Second, gamma changes as the stock drifts away from the strike and as expiry nears, so the Rs 10 holds for an at-the-money option on the day you measure it. Third, hedging once a day adds noise to the P&L even when realised volatility exactly matches implied. The rule of 16 is a desk shortcut, not a pricing model.

    Where candidates lose it

    The common slip is dividing 16% by the number of trading days, or by 365, and quoting a breakeven of a few paise. Volatility adds in squares, so time enters under a square root; dividing by days is the mistake the question is built to catch.

    The second loss is quoting Rs 10 as an average move to expect every day. It is a standard deviation: plenty of days will move Rs 2 and a few will move Rs 25, and the hedged option breaks even only if the average of the squared moves matches 100.

    What the interviewer asks next

    • The same stock's options are priced at 32% volatility. What is the breakeven move, and what is the theta in terms of gamma?
    • Over a week the stock moves 5, minus 12, 3, 15 and minus 9. Did a delta-hedged long option make or lose money, roughly?
    • Why might a trader quote 252 days rather than 256, and when does the difference matter?
  2. 031A stock is at 1,000, volatility is 20% and rates are near zero. Estimate the three-month at-the-money call in your head, and the straddle.Option pricing intuitionWarm upMarket makingVolatility trading

    Try it first

    Say the call price before you reach for a formula.

    Show the worked solution

    Call about 40, straddle about 80. The rule is 0.4 x S x sigma x sqrt T. Three months is a quarter of a year, so sqrt T is 0.5 and the volatility over the period is 10%; 0.4 x 1,000 x 0.1 = 40. With rates at zero the at-the-money put is worth the same, so the straddle is 80. The full model gives 39.88 for the call, because the exact constant is 1 over sqrt(2 pi), 0.3989, not 0.4.

    Where does the 0.4 come from?

    A tailor who knows a customer's height is normally distributed around 170 cm with a spread of 10 cm can say how far above 170 the average tall customer stands: about 0.4 of the spread, which is 4 cm, because the mean of the positive half of a normal is sigma over sqrt(2 pi). An at-the-money call pays the positive half of the stock's move, and the average of the positive half of a normal is 0.4 of its standard deviation, so the call is worth 0.4 times the standard deviation of the move over its life. The standard deviation of the move is S x sigma x sqrt T, which is 1,000 x 0.2 x 0.5 = 100 here, and 0.4 of 100 is 40. The exact constant is 1 / sqrt(2 pi) = 0.3989, so the rule gives 40 where the precise version gives 39.89, and the model 39.88.

    The relationship
    CATM≈0.4 S σT=0.4×1000×0.20×0.5=40Cexact=S(2N ⁣(σT2)−1)C_{ATM} \approx 0.4\, S\, \sigma \sqrt{T} = 0.4 \times 1000 \times 0.20 \times 0.5 = 40 \qquad C_{exact} = S\left(2N\!\left(\tfrac{\sigma\sqrt{T}}{2}\right) - 1\right)
    Sthe stock price, 1,000
    sigma sqrt Tthe volatility scaled to the option's life: 20% x 0.5 = 10%
    0.4the approximation to 1 over root 2 pi, which is 0.3989
    Nthe standard normal distribution function
    What it says in wordsAn at-the-money call is about four tenths of one standard deviation of the stock's move over its life.
    The 0.4 rule against the full model, three-month at-the-money call on a stock at 1,0000501001502000%20%40%60%80%100%implied volatilitycall pricerule 40, model 39.9rule 120, model 119.2rule 200, model 197.420% vol: call 40, straddle 80rule: 0.4 x S x sigma x sqrt Tfull model, rates zeroExact constant 1 / sqrt(2 pi) = 0.3989; the model price S(2N(sigma sqrt T / 2) - 1) bends below the straight ruleAt 20% for three months the bend costs 0.12 on 40; at 100% it costs 2.6 on 200
    Across volatilities from 0 to 100% the rule 0.4 x S x sigma x sqrt T sits almost on top of the full model price for a three-month at-the-money call, giving 40 against 39.88 at 20% volatility, and bends below it only at high volatility where the model's price curves.

    Why is the straddle just double, and when is it not?

    With rates at zero and no dividends, the forward equals the spot, so an at-the-money call and put have the same value by put-call parity, and the straddle is simply two calls, about 80. The straddle costs 8% of the stock for a three-month bet, which is the whole quarter's one-standard-deviation move of 10% times 0.8: that is the number a volatility trader carries in their head. With rates or dividends the forward moves away from spot, at-the-money means at-the-forward, and the call and put split the straddle unevenly, though their sum barely changes. The rule also assumes the volatility is the right one for this strike, which on a real surface with skew it may not be.

    How far can you push the rule?

    The rule is linear in volatility and time, the model is not. For three months the gap is 0.12 on 40 at 20% volatility, and even at 60% volatility the rule gives 120 against 119.2. Over one year at 20% the rule gives 80 against the model's 79.66. Up to a total move of about 50% the rule is within a couple of percent, which is every interview case and most of the real book; beyond that the model price flattens because a call can never be worth more than the stock. Say the limitation and then use the rule anyway: on a desk the question is never whether 40 is exactly right, it is whether 44 on the screen is rich or cheap.

    Where candidates lose it

    The common loss is forgetting to scale the volatility to the horizon and quoting 0.4 x 1,000 x 0.2 = 80 for the call, which is the one-year number. Say sqrt T out loud: three months is a half.

    The second is giving the call and then stalling on the straddle, or doubling the call without saying why. The put equals the call only because rates are zero and there is no dividend; name parity and the interviewer knows you understand what at the money means.

    What the interviewer asks next

    • Rates are now 8%. Which is worth more at the money, the call or the put, and by roughly how much?
    • The stock pays a 2% dividend before expiry. What changes?
    • Quote me the one-month straddle on the same stock, then the one-year.
    • The market is paying 44 for the call. What volatility is it implying, roughly?