Relative-Value Volatility Trades Across Different Underlyings

# How to perform volatility arbitrage between two instruments with different prices but the same realized volatility


# How to perform volatility arbitrage between two instruments with different prices but the same realized volatility












Suppose We have two assets $S_1$ and $S_2$. They have different price, but share the same realized vol. They have corresponding options $O_1$ and $O_2$. When the ATM IV of $O_1$ and $O_2$ differ too much, I think it is possible to make arbitrage by buying low IV and selling high IV.

I know how to do this arbitrage between different strike for the same product. The basic concept is balance cash gamma and hedge delta every day. Black Scholes tell us $pl=50*Cash Gamma*(iv_1^2-iv_2^2)*T$. But how to do this arbitrage between $S_1$ and $S_2$? How to calculate risk of this portfolio and how to hedge this portfolio to realize this profit?

## Answer by liz pang (score 1)

https://quant.stackexchange.com/a/85384

There is no true arbitrage here unless (S_1) and (S_2) have identical payoffs in all states.

Buying low IV on (S_1) and selling high IV on (S_2) is a relative-value volatility trade, not arbitrage. Different ATM IVs are allowed because the assets are different and can realize different future variance.

What you can do:

```
•   Buy ATM option on low-IV asset
•   Sell ATM option on high-IV asset
•   Size positions to be delta-neutral (and often vega-neutral)
•   Delta-hedge each underlying separately
```

P&L after hedging comes from the difference in realized variance, not directly from the IV gap. Your Black–Scholes cash-gamma formula only applies to options on the same underlying.

Main risks (cannot be hedged away):

```
•   Relative realized volatility
•   Correlation and jump risk
•   Skew/smile differences
```

## Answer by Newquant (score 0)

https://quant.stackexchange.com/a/80061

Each leg of the trade will have a positive expected return (buying cheap vol and selling expensive vol), which you can treat as orthogonal. Assuming that asset volatilities move in tandem, you have already neutralised some of the risk of the overall movement of volatility.

If you think the dynamics of each asset's realised volatility are related, with some beta to the realised volatility of the market, then you are able to create a beta to that market realised volatility for each leg. You now scale your individual leg's cash gamma positions by this beta so that the scaled beta positions offset. Now you have neutralised your risk to the market's volatility, all that remains is the idiosyncratic risk of each asset's volatility. By trading a large number of long/short pairs you can minimise that idiosyncratic risk.

That's a simplified example I think, and I'm sure there's other ways to do it, but that's my first thought. It's the same concept as long/short stock trading, except your alpha of the stock's direction is your forecast for the rv-iv spread. There's just a good deal more estimation difficulty in creating that volatility beta as rv is an instantaneous process and hard to get a good estimate for. You could try generating your beta with some estimate for the continuous N day period ATM IV for each asset, and calculating betas using that.

## Answer by Soumirai (score 0)

https://quant.stackexchange.com/a/80329

If your two assets have a non-arbitrage type of relation that links their realized volatilities (e.g. one is a leveraged version of the other), then it's obvious that you have some risk-free P&L if you sell the high vol, buy the low vol.

Now if the two assets just happen to have the same realized vol at a given date, but their realized can deviate in the future, then you can't arbitrage that. It's just a view on their respective realized vols. You probably get a higher Sharpe ratio by trading the spread (sell expensive buy cheap) rather than just one of the two options outright, because you hedge some overall market vol risk by trading a spread.