Useful equation to remember when thinking about leverage, forced trading and impact on prices is X = L * (L - 1) * R.

Here R is the return on the underlying stock/futures/whatever, L is leverage and X is the trade required to maintain constant leverage after a price move.
To get there, imagine a portfolio L:1 leverage i.e. you have $L of positions for every $1 of equity.

If the stock has a return of R, you now have equity of (1 + LR) and stock worth L(1+R)
To maintain leverage of L you need the stock you hold to be worth L(1+LR) so you need to buy/sell

X = L(1+LR) - L(1+R) = L + L^2R - L - LR = L(L-1)R

Remember we are doing this per $1 of equity, so multiply by your initial account size to get a $ trade size.
Couple of interesting things:

1. If L = 0 (you hold cash) or L = 1 (unlevered) then you never need to trade, your portfolio automatically tracks.

2. If 0 < L < 1 then L(L-1) is negative so you sell on price increases and buy on decreases, which is stabilising
3. If L < 0 or L > 1 then you need to buy when the stock is up and sell when it is down, which is trend-following/destabilising behaviour. A recent L < 0 case is the XIV blowup and a recent L > 1 case is the Archegos blowup
4. L(L-1) is quadratic, so when L gets big, X gets really big. For Archegos if we assume L=5 then L(L-1)=20 so whenever the stock went up 5% they would need to buy 100% of their initial position size to maintain leverage -- in retrospect not surprising that VIAC was up so much.
Either big leverage (high L) or moderate leverage and volatile stocks (high R) is enough to create interesting feedback effects that can lead to blowups in less liquid markets like VIX futures or mid cap stocks.

Big leverage *and* volatile stocks is spectacular :)
This should say they need to buy 100% of their initial *account* size, not position size.

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More from @macrocephalopod

6 Apr
This did some numbers over the weekend and was rightly mocked. Source of the claim was this chart in the FT showing an estimate of the size of Archegos' portfolio, which shows a drop of over $100 billion in March. Source of the chart is Bespoke Investment Group.
Obviously Archegos did not lose anywhere near $110 billion - they lost something like $10-15 billion plus maybe another $7-10 billion lost by prime brokers (CS, Nomura etc).

So did the FT or Bespoke fuck up?
No! Pink line on the chart shows an estimate of Archegos' portfolio size (reconstructed from bank 13-F filings) assuming *constant 5:1 leverage*

The constant leverage is important as it means that portfolio size grows/shrinks very quickly even when gains/losses are modest.
Read 10 tweets
3 Apr
This is a nice question -- roughly "What does the futures curve look like for quarterly futures that settle into quarterly GDP % growth estimates"?
Quarterly GDP numbers are not tradable which means there is no cash and carry arbitrage. If you can't trade spot then futures will be priced on three things --

1. Expectations
2. Seasonality
3. Risk premium
1. and 2. are kind of the same thing, basically "what is your best guess of GDP prints for the next three years?" This depends on long term expectations, seasonal fluctuations, and other factors.
Read 16 tweets
31 Mar
Article seems to be wrong on several major points?

• "The fund was not overly levered"
• "Its risk was not hidden"
• "Hwang typically ran dollar-neutral portfolios"
• "Analysts had full insight into Archegos’ ... position sizes"
bloomberg.com/opinion/articl…
Like given the fund went from $10-15bn to ~$0 in like a week, without a major event or liquidity crisis, I would probably say it was overly levered?
There is a *massive* difference between a quant equity book (carefully hedged to sector, country and factor exposures) which can run 10-15x leverage and still be pretty low volatility vs. levered 5x long in concentrated shitcos "hedged" with index futures
Read 7 tweets
29 Mar
We’re going to hear a lot about equity total return swaps over the next couple of days but let me take a second to point out how extremely normal they are and how unsurprising it should be to find out that Archegos was using them.
Some people have a mental model for leveraged longs which is like, you negotiate with your broker to lend you cash, when the cash hits your account you go out an buy stocks with it so you now have eg a 3x long stock position and a -2x loan.
If you are more sophisticated you may imagine this happens in a margin account where the whole thing is streamlined and the loan settlement and stock purchase happen at the same time, but it’s still basically a long stock position plus a loan.
Read 11 tweets
25 Mar
This is amazing. The strategy here is to immediately sell half of the BTC received for anyone who pays in Bitcoin.

If BTC goes up, refund them in USD, if Bitcoin goes down, refund in Bitcoin. Tesla makes money regardless of the outcome.
e.g. you receive 1 BTC worth $50k. Sell half so you have 0.5 BTC + $25k.

If BTC goes to $100k you refund the $50k, left with 0.5 BTC and -$25k cash, profit $25k

If BTC goes to $25k you refund 1 BTC, left with $25k cash and -0.5 BTC, profit $12.5k
This is an application of put-call parity btw. Immediately selling half of the BTC you received converts half of your implicit BTC call into a BTC put, so you now have a BTC straddle, which is guaranteed to profit in every outcome - optionality without price exposure.
Read 4 tweets
21 Mar
This is a good question! Some possible answers -

1. Infrastructure that is too expensive or too hard to build yourself (eg colocation, low latency data feeds, compute clusters, fpgas)

2. Data that is too expensive or simply impossible to buy (eg historical tick data)
3. Scale. Say you have $100k in your account. If you are willing to have 30% annual vol and Sharpe 2 you can make $60k in your first year, $96k in your second year, $150k in your third year (minus data and infrastructure costs of course).
If you trade for a hedge fund you maybe get a 15% cut and a $50m allocation with 5% annual vol so you make $750k ($50m x 5% x 2 x 15%) in your first year (minus data and infrastructure costs).
Read 7 tweets

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