This model rebalances between cash and the sp500 in accordance with the Kelly criterion.
We assume:
- returns from SPY are generated via geometric brownian motion (GBM) https://en.wikipedia.org/wiki/Geometric_Brownian_motion
- maximizing the expected geometric growth rate is the optimal strategy https://en.wikipedia.org/wiki/Kelly_criterion
- the GBM has a drift parameter of 7% (maximum likelihood estimate from the past 30 years of data)
- the expected variance of the sp500 is given by the square of the VIX https://en.wikipedia.org/wiki/VIX
The expected geometric growth rate is given by lambda * mu - lambda ^2 * sigma^2 / 2 , where mu is the drift parameter, sigma^2 the (expected) variance, lambda is the sp500 leverage.
Maximizing this gives the optimal leverage of lambda = mu / sigma^2.
i.e. lambda = 0.07 / VIX^2
Our strategy seeks to keep leverage equal to this value.
