Hedge Funds puzzles, solved step by step

Yes: the direct quote is rich by 60 paise, so buy euros through dollars and sell them for rupees at 93.00, making Rs 6 lakh on EUR 1 million before costs. The implied rate is 1.10 x 84.00 = 92.40 rupees per euro. Spend Rs 9.24 crore on USD 1.1 million, turn that into EUR 1 million, and sell the euros for Rs 9.30 crore. You finish in rupees, where you started, with Rs 6 lakh more.

How do you spot the mispricing in one line?

If one stall sells mangoes at Rs 100 a dozen and the stall next to it buys them back at Rs 10 each, you would buy dozens and sell singles all day. Every pair of currencies can be priced two ways, directly or through a third currency, and when the two prices differ you buy on the cheap route and sell on the rich one. Through dollars a euro costs 1.10 dollars at Rs 84.00 each: Rs 92.40. The direct dealer pays Rs 93.00. That 60 paise gap is the whole trade.

What stops this from being free money in practice?

Three things. Real quotes have a bid and an offer, and the gap survives only if it is wider than the three spreads you cross plus the cost of dealing. A 60 paise gap on a 92 rupee price is about 0.65%, far wider than dealer spreads in major currencies, which is why a gap that size would be traded away almost at once. And the legs must be done together: if one price moves before you finish, you are left holding an open currency position instead of a locked-in profit.

The spread must be Rs 0.60: a bid Rs 0.30 below the mid and an offer Rs 0.30 above it. Call the half-spread s. Each uninformed order earns her s on average. Each informed order trades on the side where the value really is Rs 1 away, so she loses 1 minus s. Breaking even needs 0.7s = 0.3(1 - s), which gives s = 0.30.

Why do some of the people who trade with you cost you money?

A shop that buys second-hand phones pays the same price to every seller, but the sellers who know their phone is faulty are the keenest to sell. A quote is an offer made to everyone, and the people who know more take it only when it hurts you. The spread is the fee charged to the whole crowd to pay for the few who know more, which is called adverse selectionThe tendency for the counterparties who choose to trade with you to be the ones who know the trade is good for them..

How do you set up the break-even?

Charge a half-spread s on each side. An uninformed order is as likely to be a buy as a sell and carries no information, so on average it earns you s; an informed order buys at your offer only when the value is Rs 1 above the mid, so it costs you 1 minus s. Weight each by how often it arrives and set the total to zero: 0.7s - 0.3(1 - s) = 0, so s = 0.30 and the spread is Rs 0.60.

What moves the spread?

Two things: how much of the flow is informed and how much the informed know. In general the half-spread is s = aV, where a is the informed share and V is how far the true value sits from the mid, so doubling either doubles the spread. Here 0.3 x 1 = 0.3. That is why quotes widen ahead of a results announcement, when more of the flow may know something, and in small stocks, where one piece of news moves the value a long way. The model leaves out inventory risk and competing quoters, both of which move a real spread; say so.

Sell at the 55 bid; the edge is Rs 5 a contract, five points of probability. A contract paying Rs 100 on an up close is worth Rs 100 times the probability, so your model values it at Rs 50. Selling at 55 collects Rs 55 for a liability worth Rs 50 on average. Buying at the 60 offer would give away Rs 10 of expected value. The edge is only as good as the model behind it.

Why is a binary price just a probability?

Think of a bet with a friend: you pay a fixed sum now and get Rs 100 back if it rains tomorrow. If you think rain is a 30% chance, the most you would pay is Rs 30. A contract paying Rs 100 or nothing is worth 100 times the probability of the payout, so a price of 55 is the market saying 55%. Read as probabilities, the quote says between 55% and 60%, and your model says 50%.

Which side of the quote can you trade?

You buy at the offer and sell at the bid, never the other way round. Buying costs 60 for something your model values at 50, a loss of 10 in expectation, while selling receives 55 for the same thing, a gain of 5. So the trade is to sell at 55. Sell 100 contracts and you receive Rs 5,500; half the time you pay out Rs 10,000 and half the time nothing, so the expected payout is Rs 5,000 and the expected profit Rs 500.

When would you do nothing?

If your model said 57%, the fair value of 57 would sit inside the quote: selling at 55 loses 2 and buying at 60 loses 3. An edge exists only when your value falls outside the bid and offer, not whenever you disagree with the middle of the market. Then add the honest caveat. A market pricing an up close near 57.5% may know something your model does not, and each contract swings between plus 55 and minus 45, so a 5-point edge needs many independent trades, and a model you trust, before it shows up in the P&L.