On a Rs 1,000 stock, compare a covered call, a protective put and a long straddle over three equally likely outcomes of 850, 1,000 and 1,150. Which makes more money on average, which has the more volatile P&L, and where does each come from?

The protective put averages Rs +15.33 a share and the straddle Rs +15, while the covered call averages Rs -11.33. The covered call and the protective put have the same spread, a standard deviation of Rs 85, as mirror images; the straddle's is lower at Rs 70.7. The averages come from one source: these outcomes are wider than the option prices assume, so whoever buys options gains and whoever sells them pays.

Write each payoff as its pieces and add. The covered call is the stock's move plus the Rs 22 collected, less anything above 1,050: minus 128 at 850, plus 22 at 1,000 and plus 72 at 1,150. The protective put is the stock's move less the Rs 18 paid, plus anything below 950: minus 68, minus 18 and plus 132. The straddle is the distance from 1,000 less the Rs 85 paid: plus 65, minus 85 and plus 65. Think of three ways to hold a house: rent out the top floor and cap what you can sell it for, insure it against a crash, or bet only that its price moves a lot either way. Each of these is the stock with a different slice of the range sold or bought.

Average each row with weights of one third. The protective put averages Rs +15.33, the straddle Rs +15.00, the stock alone zero, and the covered call Rs -11.33. For spread, the covered call and the protective put have exactly the same standard deviationA measure of how far outcomes sit from their average: the square root of the average squared distance. Here it is computed over the three equally likely outcomes., Rs 85.0, because their payoffs are mirror images: one has a big loss and two modest gains, the other a big gain and two modest losses. The straddle's is lower, Rs 70.7, and all three are far below the stock's Rs 122.5. So the standard deviation alone does not separate them; the shape does. The covered call's risk is a large loss in the crash, the protective put's a small steady bleed, and the straddle's a loss only if nothing happens.

Every strategy here is the stock, which averages zero, plus options. So the average P&L of each strategy is exactly the average P&L of the options in it, and that depends on one comparison: how wide are these outcomes compared with the width the option prices assume? The three outcomes, 15% either way or no move, have a standard deviation of about 12.2% in three months, about 24.5% a year. The straddle at Rs 85 implies roughly 21.2% a year, using the rule of thumb that an at-the-money straddle is worth about 0.8 times spot times volatility times the square root of time. When the world is wider than the price, option buyers win on average: the put buyer and the straddle buyer collect the surplus, and the covered call writer, who sold the 1,050 call for Rs 22 when it pays an average of Rs 33.33, pays it. In a world narrower than the price, every sign flips.

Say the limits. Three equally likely outcomes is a toy distribution with fat tails and no skew, chosen by the interviewer; a real stock has many outcomes and usually more weight on large falls than on large rises, which is why puts trade at higher implied volatility than calls. The strategies also differ in what they cost to hold and in margin: the covered call needs no extra capital beyond the stock, the straddle pays its whole premium up front. The judgement the interviewer wants is the sentence above: an option strategy is a view on the width of the distribution, and its average P&L is the gap between the width you believe and the width you pay for.