Case 009Volatility tradingCore

A fund is short one-month and one-year index straddles, each with vega of Rs 1 crore per point. A 5% one-day fall lifts one-month volatility 12 points and one-year 4. Which loses more, and what else hits the one-month?

The situation

Kalu Macro Fund has sold at-the-money Satpura 50 straddles in two tenors, one month and one year, sized so that each position has vega of Rs 1 crore per volatility point. Before the shock, one-month implied volatility was 15% and one-year 16%. Both positions were delta-hedged at the close.

Overnight news takes the index down 5% in a single session. One-month implied volatility jumps 12 points to 27% and one-year rises 4 points to 20%. The desk wants the loss on each position before any rehedging.

Your task

Which position loses more and by how much from the volatility move, what additional loss does the one-month position suffer that the one-year position largely escapes, and what does this say about sizing by vega?

Quick check

Before the Greeks: both positions have the same vega. Which one loses more on this day?

Worked solution

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

30-second answerThe answer to give first

The one-month straddle loses about Rs 22 crore and the one-year about Rs 4.8 crore. On vega alone it is Rs 12 crore against Rs 4 crore, because short-dated volatility moved three times as far. The one-month position also loses about Rs 10 crore from gamma, the cost of being short a 5% move before rehedging, while the one-year's gamma loss is under Rs 1 crore. Equal vega is not equal risk: the short end carries the volatility of volatility and the gamma.

Step 1What does vega alone say, and why is that only the first line?

Vega of Rs 1 crore per point means each point of implied volatility is worth Rs 1 crore to the position. A 12 point rise on the one-month costs Rs 12 crore and a 4 point rise on the one-year costs Rs 4 crore, so the positions that looked identical on the risk report lose in a ratio of three to one. Two umbrellas with the same price tag are not the same bet if one is for Mumbai in July and the other for Jaipur. Short-dated implied volatilityThe volatility that, put into the pricing model, reproduces the quoted option price; what the market charges for movement over the life of the option. reprices violently on news because it covers only the next few weeks, which the shock dominates; a year's volatility averages the shock with eleven calmer months.

Step 2What else does the one-month position lose?

Gamma. A short straddle that was delta-flat at the close is short the move itself: as the index falls, the put gains delta faster than the hedge can follow, and the loss grows with the square of the move. For an at-the-money option, the gamma loss on a move x relates to the vega per point by the factor x squared over two, divided by 0.01 times volatility times time: for the one-month at 15% that factor is about 10, so a 5% gap costs about Rs 10 crore; for the one-year at 16% it is about 0.8, under Rs 1 crore. The one-month position is therefore down about Rs 22 crore before anyone has touched it, against Rs 4.8 crore on the one-year.

The relationship

\frac{\text{gamma loss}}{\text{vega per point}} = \frac{x^2 / 2}{0.01\,\sigma\,T} \quad\Rightarrow\quad \frac{0.05^2/2}{0.01 \times 0.15 \times 1/12} \approx 10, \qquad \frac{0.05^2/2}{0.01 \times 0.16 \times 1} \approx 0.78

x	the one-day move in the index, 5%
\sigma	implied volatility before the shock, 15% and 16%
T	time to expiry in years, one twelfth and one

What it says in wordsFor an at-the-money option the gamma loss on a move is the vega per point scaled by the move squared over volatility times time, so the short-dated option's gamma loss is about ten times its vega per point while the one-year's is under one times.

On a 5% one-day fall the one-month short straddle loses Rs 12 crore from a 12 point rise in volatility plus about Rs 10 crore from gamma, about Rs 22 crore in all, while the one-year loses Rs 4 crore from vega plus under Rs 1 crore from gamma, about Rs 4.8 crore, from positions that showed the same vega.

Step 3What does this say about how the book should be sized and reported?

A risk report that adds vega across tenors treats one point of one-month volatility as equal to one point of one-year volatility, and the market does not. Desks weight vega by tenor, typically scaling by the square root of time so that a one-month point counts for more than a one-year point, and they show gamma and a scenario loss alongside, because the shock loss is the sum of both. Say the limits of the numbers here: vega is not constant, and a 12 point rise in a 15 vol option roughly doubles its vega on the way, so the true one-month loss is convex and larger than Rs 12 crore; the gamma figure is an at-the-money approximation before rehedging; and a day of theta, about Rs 25 lakh on the one-month, is nowhere near enough to matter on a day like this.

Where candidates lose it

Candidates answer that the two positions lose the same because the vegas are equal, which is the reading the question is designed to punish. Vega prices one point; the shock delivers twelve points to one tenor and four to the other.

The second loss is forgetting gamma entirely and reporting Rs 12 crore against Rs 4 crore. The one-month straddle is short a 5% move, and that costs about as much again as the volatility move did. An options interviewer will not let a short-dated loss pass without the gamma line.

What the interviewer asks next

The index recovers the full 5% the next day and volatility stays elevated. What is the one-month position's P&L over the two days?
How would you set a vega limit so that the two tenors carry the same risk?
Why might one-year volatility rise more than one-month after a different kind of shock, say a change in the rules on index derivatives?
What is vega of vega, and which of the two positions has more of it?

Company names and figures are illustrative.