Hedge Funds puzzles, solved step by step

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  1. 009How would you price a digital option that pays Rs 100 if the index is above 11,000 at expiry, using only the prices of ordinary call options?Options and payoffsHardVolatility and relative value fundsProp and quant trading firms

    Try it first

    A digital paying Rs 100 above 11,000 is closest to which position?

    Show the worked solution

    Replicate it with a tight call spread: buy 5 calls at 10,990 and sell 5 at 11,010. The position pays 0 below 10,990 and 100 above 11,010, a steep ramp standing in for the step. So the digital costs about 5 x (C at 10,990 minus C at 11,010). With calls at 212.40 and 203.60 that is 5 x 8.80 = Rs 44. In the limit, the price is minus 100 times the slope of call prices against strike.

    Why does a call spread look like a step?

    A steep enough ramp can stand in for a stair. A call spread's payoff is a ramp: nothing below the lower strike, rising point for point between the strikes, flat above the upper strike. Narrow the strikes and scale up the size, and the ramp tightens into the step a digital pays. Here the strikes are 20 points apart, so each spread pays at most 20, and five spreads pay at most 100, the digital's payout.

    Five tight call spreads are a steep ramp standing in for the step010010,97010,99011,00011,01011,030Index at expirydigital: pays 100 above 11,0005 x call spread10,990 / 11,010spread pays morespread pays lessPrice = 5 x (212.40 - 203.60)= 5 x 8.80 = Rs 44
    Five 10,990 / 11,010 call spreads pay 0 below 10,990 and 100 above 11,010, overpaying the digital just below 11,000 and underpaying just above it, and with illustrative calls at 212.40 and 203.60 the position costs Rs 44.

    What does the price of the spread tell you?

    The spread costs the difference in call prices, so the digital costs five times that. As the strikes close in, the price becomes minus 100 times the slope of the call price against strike, and that slope is the discounted market-implied chance of finishing above the strike. With the illustrative quotes, 8.80 across 20 points is a slope of 0.44, a digital worth Rs 44 and an implied chance of about 44% before discounting.

    The relationship
    D≈100×C(K−h)−C(K+h)2h  ⟶  −100 ∂C∂KD \approx 100 \times \frac{C(K-h) - C(K+h)}{2h} \;\longrightarrow\; -100\,\frac{\partial C}{\partial K}
    Dthe digital's price
    C(K)the price of a call struck at K
    hhalf the gap between the strikes, here 10
    What it says in wordsA digital is a call spread scaled up as it narrows, so its price is the slope of call prices with strike.

    Which spread does a desk that sold the digital actually buy?

    A desk that has sold the digital wants a hedge that pays at least 100 wherever the digital does. The centred spread overpays just below the strike and underpays just above it, so a seller hedges with five 10,980 / 11,000 spreads, which pay the full 100 by the strike and cost a little more. That difference is what the desk charges for an index that settles right at the strike. One more point marks a strong answer: the slope of call prices includes the change in implied volatility across strikes, so with the usual equity skew, where lower strikes carry higher volatility, the digital is worth more than a flat-volatility model says.

    Where candidates lose it

    Candidates reach for a pricing formula straight away. The question said using only call prices, and the interviewer wants the replication argument; the formula comes after, if at all.

    The second loss is the size. A call spread 20 points wide pays at most 20, so it takes five of them to pay 100; a candidate who buys one spread prices the digital at a fifth of its value.

    What the interviewer asks next

    • How would you replicate a digital that pays 100 below 11,000?
    • What do the digital call and the digital put at the same strike cost together?
    • Why is a digital close to expiry, with the index at the strike, so hard to hedge?
  2. 094You buy an at-the-money option at 20% implied volatility and delta-hedge it. Over its life the stock realises 30% volatility. Where does your P&L come from, and does the direction of the moves matter?Options and payoffsHardVolatility and relative value fundsProp and quant trading firms

    Try it first

    What drives the hedged position's P&L?

    Show the worked solution

    From gamma: the hedged option earns on the size of moves, not their direction, and pays theta for the privilege. Each day the P&L is about half gamma times the squared move, minus theta. Theta is priced for 20% volatility, so a daily move of about 1.26% breaks even. At 30%, typical moves are 1.89%, and the squared gain is 2.25 times the decay. Up or down makes no difference; the path still does.

    What is left after the delta hedge?

    Think of a street vendor who sells umbrellas and sunglasses from the same cart: she no longer cares whether it rains or shines, only whether the weather changes enough to bring people out. A delta hedge sells enough stock to cancel the option's first-order bet on direction, so what remains is the curvature: the option gains a little more on the way up than the hedge loses, and loses a little less on the way down than the hedge gains. That curvature is gamma, and it pays on moves either way.

    The hedge line is straight, the option is curved: any move lands above the linestart: stock Rs 1,000, option Rs 39.9option valuedelta hedge lineshaded gap: what a moveearns, up or down9501,0001,0501,100Stock price, Rs050100One day at 30% realised, per optionGamma gain on a Rs 18.9 move+0.71Time decay priced at 20%-0.32Net, up or down+0.40
    The option's value curves above the straight delta hedge line on both sides of Rs 1,000, so a move either way earns the shaded gap; a day at 30% realised volatility earns about Rs 0.71 of it per option against Rs 0.32 of time decay priced at 20%.

    How big is the daily P&L in numbers?

    Take a stock at Rs 1,000 and a three-month at-the-money call priced at 20% volatility, with zero interest rates. Its gamma is about 0.0040 per rupee and its time decay about Rs 0.32 a day over 252 trading days. A day's move of 1.89%, typical of 30% volatility, is Rs 18.9, and half of 0.0040 times 18.9 squared is a gain of Rs 0.71, against Rs 0.32 of decay: about Rs 0.40 a day per option. The decay is exactly what a move of 1.26%, typical of 20% volatility, would pay back.

    The relationship
    P&Lday≈12 Γ (ΔS)2−Θ=12 ΓS2(σr2−σi2)Δt\text{P\&L}_{\text{day}} \approx \tfrac{1}{2}\,\Gamma\,(\Delta S)^2 - \Theta = \tfrac{1}{2}\,\Gamma S^2\left(\sigma_r^2 - \sigma_i^2\right)\Delta t
    Gammahow fast the option's delta changes with the stock price
    Delta Sthe day's move in the stock price
    Thetathe option's daily time decay
    sigma_r, sigma_irealised and implied volatility, 30% and 20%
    What it says in wordsEach day a hedged long option earns half its gamma times the squared move and pays its time decay; on average that is the gap between realised and implied variance.

    Why does direction not matter, but the path does?

    The move enters squared, so plus Rs 18.9 and minus Rs 18.9 earn the same. The P&L depends on realised volatility against implied, not on where the stock ends up. But gamma is largest near the strike and shrinks as the stock drifts away, so 30% realised in a trend that carries the stock far from the strike early earns less than 30% realised while the stock chops around the strike. Hedging frequency matters too: hedge rarely and the P&L becomes noisy, even if its average barely changes.

    Where candidates lose it

    The common loss is saying the option made money because the stock went up. With a delta hedge in place, direction has been sold away; a candidate who talks about direction has not understood what the hedge does.

    The second loss is saying a hedged position has no P&L. It has exactly one exposure left, realised against implied volatility, and naming it is the whole answer.

    What the interviewer asks next

    • Realised volatility comes in at 15% instead. What happens to the P&L, and why?
    • Why is the same realised volatility worth more while the stock stays near the strike?
    • You think implied volatility is too low but have no view on direction. What position expresses that?