Case 088Volatility, options and convertiblesCore

Talwin Vol Fund is long a variance swap struck at 20% volatility with a vega notional of Rs 1 crore. Realised volatility comes in at 30%. What does it make, and why is the payoff larger than on a volatility swap?

The situation

Talwin Vol Fund thinks index volatility is underpriced ahead of a crowded election calendar. It buys a one-year variance swap on the index struck at 20 volatility points, with a vega notional of Rs 1 crore: about Rs 1 crore of profit for each point realised volatility beats the strike, for small moves.

Over the year, realised volatility, computed from daily log returns and annualised, comes in at 30. A colleague holds a volatility swap with the same strike and the same Rs 1 crore per point.

Your task

What does the variance swap pay, why is it more than the volatility swap, and what does that difference cost?

Quick check

With realised volatility at 30 against a strike of 20, which pays more?

Worked solution

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

30-second answerThe answer to give first

The variance swap makes about Rs 12.5 crore against Rs 10 crore on a volatility swap. Its variance notional is Rs 1 crore over 2 x 20, Rs 2.5 lakh per variance point, and 30 squared minus 20 squared is 500 points. Because it pays on the square, gains grow faster than the move and losses shrink: at 10 realised it loses Rs 7.5 crore, not Rs 10 crore. That convexity is why variance strikes sit above volatility strikes.

Step 1How do you turn a vega notional into a payoff?

A variance swap settles on variance, volatility squared, but traders size it in vega so it compares with options. The link is that near the strike, one point of volatility is worth about 2 x 20, or 40, points of variance, so variance notional is vega notional divided by twice the strike: Rs 1 crore over 40, Rs 2.5 lakh per variance point. Realised 30 against 20 is 900 less 400, 500 variance points, and 500 times Rs 2.5 lakh is Rs 12.5 crore.

The relationship
Payoff=Nvega2K (σR2−K2)=1 crore40 (302−202)=12.5 crore\text{Payoff} = \frac{N_{vega}}{2K}\,(\sigma_R^2 - K^2) = \frac{1 \text{ crore}}{40}\,(30^2 - 20^2) = 12.5 \text{ crore}
N vegavega notional, Rs 1 crore per volatility point near the strike
Kstrike in volatility points, 20
sigma Rrealised volatility over the life of the swap, 30
What it says in wordsThe swap pays the variance notional times the gap between realised variance and the strike's variance.
Step 2Why does the variance swap beat the volatility swap on both sides?

Think of the area of a square garden. Adding a metre to a 20-metre side adds 41 square metres; adding ten metres adds 500, not 400. Variance grows with the square of volatility, so the payoff curve bends upward: a big rise in volatility pays more than the straight line and a big fall costs less. The gap has a clean formula: the variance payoff equals the volatility payoff plus Rs 1 crore over 40 times the squared miss, 10 squared times Rs 2.5 lakh, Rs 2.5 crore here.

Variance pays on the square: the curve sits above the line on both sides-20-10+10+20+300variance +12.5vol +10.0variance -7.5vol -10.0strike 20010203040Rs crore
With a Rs 1 crore vega notional struck at 20, the variance swap pays Rs 12.5 crore at 30 realised against Rs 10 crore on a volatility swap, and loses Rs 7.5 crore at 10 against Rs 10 crore, because the variance curve lies above the line on both sides.
Realised volatilityVariance swap, Rs croreVolatility swap, Rs croreDifference
10-7.50-10.00+2.50
15-4.38-5.00+0.62
20+0.00+0.00+0.00
25+5.62+5.00+0.62
30+12.50+10.00+2.50
40+30.00+20.00+10.00
The variance swap pays more than the volatility swap at every realised level except the strike, and the difference grows with the square of the miss, from Rs 0.63 crore at five points away to Rs 10 crore at twenty.
Step 3If convexity always helps, what does it cost?

Nothing is free: the dealer who sells the variance swap knows the curve bends, so variance swaps are struck above volatility swaps on the same index and date, and that premium is the price of the convexity. A variance swap can be replicated with a strip of options across strikes, weighted more heavily at low strikes, which is why its strike reflects the whole skew, including expensive downside puts. Single-name variance swaps usually carry a cap on realised variance, because a crash can make the uncapped payoff unbounded for the seller.

Close with what Talwin actually bet on. The fund was long realised volatility against a fixed price, so it made money from the path of daily moves, not from where the index ended. A strong answer adds the risk on the other side of the trade: a quiet year at 12 realised would have cost about Rs 6.4 crore, less than a volatility swap would have lost but a loss all the same.

Where candidates lose it

The frequent slip is applying the Rs 1 crore vega notional to variance points directly, 500 times Rs 1 crore, which gives Rs 500 crore, or to the volatility gap, which gives Rs 10 crore and misses the convexity entirely.

The second is saying the variance swap is simply better. Its extra payoff on large moves is paid for up front in a higher strike, and the interviewer wants to hear that the edge is priced.

What the interviewer asks next

  • What realised volatility makes the variance swap and the volatility swap pay the same, other than 20?
  • How would you replicate a variance swap with listed options?
  • Why do sellers insist on a cap for single-stock variance swaps?

Company names and figures are illustrative.