Case 086Position sizing and bankrollCore

Tamrisk's market-making book has a one-day 99% VaR budget of Rs 30 lakh, and the contract moves with a daily standard deviation of Rs 1,500 per lot. What is the maximum inventory, and how should the quotes skew as inventory approaches it?

The situation

Tamrisk Markets quotes a two-way price all day in an invented commodity future. Customers hit its bid or lift its offer, and Tamrisk earns the spread, Rs 200 a lot either side of its fair value, but holds whatever inventory the flow leaves it with. The contract's value moves with a daily standard deviation of Rs 1,500 per lot.

Risk has given the book a one-day 99% value at risk budget of Rs 30 lakh, measured on net inventory. Over the morning, sellers have dominated and Tamrisk is accumulating a long position. The head of desk asks two questions: how long can the book get, and what should the quotes do on the way there?

Your task

Turn the VaR budget into an inventory limit, then design a quote skew that keeps the book from reaching it, and say what the budget does not cover.

Quick check

What is the largest net inventory the budget allows?

Worked solution

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

30-second answerThe answer to give first

The limit is about 858 lots: each lot carries 2.33 x Rs 1,500 = Rs 3,495 of one-day 99% VaR. Skew both quotes down as long inventory builds, here Rs 0.5 a lot, so the offer reaches fair value at about half the limit and the bid is withdrawn at the limit. Skewing early sells inventory back to the flow cheaply, long before a forced exit, which would cost far more than the spread given up.

Step 1How does a VaR budget become an inventory limit?

A shopkeeper with room for 50 sacks of rice does not need a theory of storage: the shelf sets the limit. For a market maker the shelf is the risk budget, and because P&L on inventory scales with the number of lots, the budget converts directly into a maximum position. One lot's one-day P&L has a standard deviation of Rs 1,500. Assuming normal moves, the 99% one-day loss is 2.33 standard deviations, Rs 3,495 a lot. Dividing the Rs 30 lakh budget by that gives 858.4, so the limit is 858 lots, long or short.

The relationship

q_{\max} = \frac{\text{VaR budget}}{z_{0.99}\,\sigma_{\text{lot}}} = \frac{30{,}00{,}000}{2.33\times 1{,}500} \approx 858 \text{ lots}

VaR budget	one-day 99% value at risk allowed, Rs 30 lakh
z	2.33, the 99% point of a normal distribution
sigma lot	daily standard deviation of one lot's value, Rs 1,500

What it says in wordsThe most inventory the book can hold is the budget divided by the loss one lot can produce on a one-in-a-hundred day.

Tamrisk's one-day 99% VaR grows by Rs 3.5 lakh for every 100 lots of inventory and reaches the Rs 30 lakh budget at 858 lots, so the budget is an inventory limit.

Step 2How should the quotes move as inventory builds?

Waiting until the limit and then dumping the position is the expensive way to stay inside it: a forced sale of hundreds of lots crosses the spread and moves the price. The cheap way is to make the flow do the work: as the book gets longer, shade both quotes down, so the offer becomes more attractive to buyers and the bid less attractive to sellers. A simple linear rule shifts both quotes by Rs 0.5 for every lot held. At 429 lots, half the limit, the shift is Rs 214, so the offer sits just below fair value: Tamrisk is happy to sell at no edge to shed risk. At the limit the bid is withdrawn altogether and the offer sits Rs 229 below fair.

Share of limit	Inventory, lots	Skew, Rs	Bid vs fair	Offer vs fair	VaR, Rs lakh
0%	0	0	-200	+200	0.0
25%	214	107	-307	+93	7.5
50%	429	214	-414	-14	15.0
75%	644	322	-522	-122	22.5
100%	858	429	withdrawn	-229	30.0

Tamrisk's quote ladder under a skew of Rs 0.5 a lot. The offer reaches fair value at about half the limit, and at the full 858 lots the bid is pulled and VaR equals the Rs 30 lakh budget.

As Tamrisk's long inventory grows from zero to the 858-lot limit, both quotes slide down by Rs 0.5 a lot, the offer reaching fair value at about half the limit and the bid disappearing at the limit, so buyers are invited in well before the book is full.

Step 3What does the budget not capture?

Three things the desk head should hear. First, the one-day horizon is a choice: if the book could always be flattened within an hour, the same budget would allow about 2,146 lots, but in a fast market an hour's exit is exactly what is not available, so the one-day figure is the honest one. Second, a normal 2.33 understates losses on the days that matter; commodity futures have fat tails, so a stress limit sits beside the VaR limit. Third, the skew size is a judgement: too small and the book drifts to its limit, too large and Tamrisk gives away edge to every buyer. The rule here costs at most Rs 229 a lot on the last lots sold, small against a 2.33 standard deviation day of Rs 3,495.

Where candidates lose it

The common loss is dividing the budget by one standard deviation and quoting a limit of 2,000 lots. A 99% budget is sized for a 2.33 standard deviation day, and the first bad day would breach the budget by more than double.

The second is treating the limit as the only control: quoting symmetrically until the limit is hit, then stopping. Skew is what keeps a market maker away from the wall; the limit is the wall.

What the interviewer asks next

How would you change the skew if the contract's volatility doubled during the day?
The flow is one-way because a large client is selling. How should that change the skew?
Why might the desk prefer a limit in lots to a limit in VaR during the day?
How would you set the skew if the book also held a related contract with correlation 0.8?

Company names and figures are illustrative.

1The situation

2Your task