Synthetic Straddle via Leveraged ETF's

[TheBigShort]

I am exploring leveraged/inverse ETF's and came across a paper on how the leverage + daily rebalancing effect works. https://www.math.nyu.edu/faculty/avellane/LeveragedETF20090515.pdf.

I have a math question I need some help on. It's how to distinguish the break evens for a long L(everaged)ETF vs a short LETF.

For a portfolio where I am long a 3x LETF and short 3*regular ETF. My break-even over a 21 day(one month) period would be:

St/S0 = exp(sigmaS^2*(21/252))*(1+/- sqrt(1-exp(-sigmaS^2*(21/252))

Where S is the non-levered ETF.

However, I am really confused about the portfolio where we are long a single 2x inverse ETF and long 2x regular etf. The author of the above paper states it as:

Where Vt is

I am not too sure why we are finding the roots of the cubic equation.
I would like to understand the math here rather than plug it into a calculator (starting from bottom-up).

Thanks for your time

 

[betcashrun]

You got it: the idea is to compare two portfolios.
First one: one dollar invested in a 2x inverse ETF
Second one: two dollars invested in the regular ETF
Over a period of time which one is the better?
The values for which the first portfolio equals the second portfolio are the solutions of equation (15). X+ and X- are the positive roots.
If X > X+ or X < X- it is better to have invested in the first portfolio.
If X is inside ]X-, X+[ it is better to have invested in the second portfolio.

 

[qazwsxedc]

Check your inbox. (Top right, next to your name.)