Case 046Statistical arbitrage and event tradesHard

The near-month index future is at 22,150 and the next month at 22,280, 28 days apart, with funding at 6.8% and a dividend yield of 1.2%. Is the calendar roll rich or cheap, and what does the roll trader do?

The situation

Sampravah Quant runs a roll desk on futures of the Avantara 50, an invented broad index. Three days before the near-month expiry, the near future trades at 22,150 and the next-month future at 22,280, so the calendar spread, the roll, is 130 points. The two contracts expire 28 days apart. One lot is 50 units of the index.

The desk can fund a long position in the index basket at 6.8% a year, and the expected dividend yield on the index over the window is 1.2% a year. Trading a futures leg costs about 1 point a lot; buying or selling the whole cash basket costs about 5 basis points a side. The desk's limit for this trade is 400 lots. All levels are illustrative.

Your task

Work out the fair value of the roll, say whether 130 is rich or cheap and by how much, describe the trade and how its edge is actually captured, and name what could make it lose money.

Quick check

What should the 28-day roll be worth on these inputs?

Worked solution

Try it on paper, then open one step at a time.

30-second answerThe answer to give first

The roll is about 35 points rich: fair carry is about 95 points and it trades at 130, which implies funding at 8.85% instead of 6.8%. The trader sells the roll: sells the next month and buys the near month. If the spread falls back towards carry, it is closed in futures for about 31 points; if not, owning the basket between expiries locks about 11 points after costs.

Step 1What should the gap between two futures be?

Think of buying a flat today against agreeing to buy it in a month. If you buy today you pay interest on the money for a month but collect the month's rent; if you agree to buy later you do neither. The fair price for later is today's price plus the interest less the rent. Two index futures a month apart are the same contract on the same index at two dates, so the gap between them should equal the cost of carrying the index across the gap: funding minus dividends, 95.2 points here. Funding at 6.8% for 28 days on 22,150 is 115.5 points; dividends at 1.2% are 20.4. That cost of carryWhat it costs to hold an asset from one date to another: the financing on the money tied up, less any income the asset pays in the meantime. is the roll's fair value.

The relationship

F_2 - F_1 \approx F_1\,(r - q)\,\frac{28}{365} = 22{,}150 \times 0.056 \times 0.0767 \approx 95.2

F_1, F_2	near-month and next-month futures prices
r	funding rate, 6.8% a year
q	dividend yield over the window, 1.2% a year
28/365	the time between the two expiries, in years

What it says in wordsThe next-month future should sit above the near month by the funding cost of holding the index for 28 days, less the dividends received while holding it.

Funding the index for 28 days adds 115.5 points and dividends take off 20.4, so the roll is worth about 95.2 points; at 130 it is 34.8 points rich, as if the market were charging 8.85% to fund the index instead of 6.8%.

Step 2Why would a roll trade rich at all?

Because of who is trading it in the last days before expiry. Investors who hold long index exposure through futures must roll every month: sell the expiring contract, buy the next. When long rollers dominate, they all want the same side of the spread, buy next month and sell near month, and they push the roll above carry until someone with a balance sheet takes the other side. The roll desk is paid for supplying that balance sheet. Expressing the richness as a rate makes it plain: 130 points means the market is paying 8.85% a year to have someone hold the index for 28 days, against the desk's own cost of 6.8%.

Step 3What exactly is the trade, and how is the edge captured?

Sell the roll: sell the next-month future at 22,280 and buy the near month at 22,150, in equal lots. There are two ways out. If the roll falls back towards carry before expiry, the desk buys it back in futures and keeps the difference, about 31 points a unit after four futures legs at 1 point each. If it does not, the desk carries the position through: at near expiry the long near future settles at the index level, and the desk buys the basket at that same level, so it now owns the index at an effective 22,150. It funds the basket for 28 days, collects the dividends, and sells the basket at next-month expiry, where the short future settles. The futures and the basket offset each other at both dates; what remains is the 130-point roll, less carry, less costs.

Selling the roll and then owning the basket between the two expiries turns the 130-point spread into the index bought at an effective 22,150 and sold at 22,280; after 115.5 points of funding and 20.4 of dividends the edge is 34.8 points, of which about 10.7 survive futures and basket costs.

On 400 lots, a notional of about Rs 44.3 crore, the convergence exit is worth about Rs 6.2 lakh and the carry-through exit about Rs 2.1 lakh. The carry-through route is the floor, and it is thin: each point on the funding rate is worth 17 index points over 28 days, so the 10.7-point floor disappears if funding rises to about 7.4%. The convergence exit is the reason to do the trade; the carry-through is what makes it possible to wait for it.

Scenario	Fair roll	Rich by	Carry-through, after costs
Base case	95.2	34.8	10.7
Funding 1 point higher	112.1	17.9	-6.3
Dividends in the window halve	105.3	24.7	0.5
Funding 1 point lower	78.2	51.8	27.7

Index points per unit. The base case leaves 10.7 points after costs if carried to expiry; a funding rate one point higher turns that into -6.3, and dividends in the window coming in at half the assumed yield cut it to 0.5.

Step 4What can go wrong?

Three things, in the order a risk manager asks about them. Funding: the desk's 6.8% is a forecast for the next month, and a rate rise or a tight quarter-end raises it. Dividends: the yield assumed for the window is a guess about which companies declare and when; a large dividend postponed out of the window lowers the cash the basket earns and raises the fair roll. Mark to market: a rich roll can get richer before it converges, so the position can show a loss and draw margin on both legs even when the expiry arithmetic is sound. A roll that is rich because balance sheets are scarce is rich precisely when the desk's own balance sheet is under strain.

Close with the view the desk head wants. The roll is about 35 points rich, far beyond the 4 points of futures costs and enough to survive the basket route; sell it within the limit, plan to exit on convergence, and keep the funding line committed for the 28 days so the carry-through stays available. The limit of the analysis is that the dividend estimate and the funding rate are the trade, so both are written down before it is put on.

Where candidates lose it

The usual loss is pricing the roll on funding alone, about 116 points, and calling 130 only slightly rich, or forgetting that dividends go to whoever holds the basket. Carry is funding minus dividends, and missing the dividend leg understates the richness by about 20 points.

The second is calling the trade riskless. It is locked only if the desk owns the basket between expiries, pays the basket costs and gets the funding it assumed; and before expiry the spread can widen, which costs margin even when the end point is safe.

What the interviewer asks next

The roll trades at 80 instead. What is the trade and what do you need to be able to do it?
A large index constituent announces a special dividend payable inside the window. Does the roll move, and which way?
How would you size this against a limit stated in Rs crore of notional rather than lots?
Why do rolls tend to trade richest in the last two or three days before expiry?

Company names and figures are illustrative.