Keep in mind that 4 billion (likely Softbank invested more) means 100x in stocks. i.e. a 10 billion investment in call options 'generates' 1 trillion in extra market cap (because of dealers hedging). That is 1/3 of AMZN or 2x TSLA and so on at current (not March) valuations. Such concentrated volume CAN move big stocks and in turn indexes.

Even if the market let go a bit now - Softbank has literally cornered the market for so long - they still make a killing in the process.

This is totally incorrect. Individual options contracts are indeed written against 100x of the underlying asset but this has absolutely nothing to do with the relationship between the price (premium) paid per contract and that of the underlying share. To given an example, you could find a put option whose premium is actually equal to the current market value of the underlying (i.e. a merely 1 to 1 relationship) or one that's so far out of the money that it's a 1 to 1000 relationship. Without knowing more details about the precise options SoftBank bought, we cannot infer how much market cap in underlying shares the $4bil corresponds to.

The concept of notional value is more complicated than that. Dealers don’t hedge the full notional value they hedge only enough to cover the delta and gamma on the option.

True but at the current high implied volatility of the affected symbols the delta/ gamma has been high —> lots of hedging needed. Big loop.

Correct, but not 100:1 hedging.

In addition to that, they match orders in real-time, so they might not even need to hedge partially because they have offsetting positions coming in.

how are retail options different from softbanks options. if there are 34 b of retail calls purchased that would dwarf softbank if done in the same time period

I don't think the retail purchases were as concentrated.

If that’s true, that’s 34B in the total options market, this 4B is targeted at like 5 companies.

Retail buys that much total, not net. And retail mostly buys puts as a hedge to their existing position. SoftBank introduced $4b of premium directed in the same direction for new deltas.

source on retail investors? I also believe retail is having a larger effect, I just haven't seen this number before.