How to apply Kelly criterion to a portfolio made by a stock plus a option?

Question

First of all, assuming a Gaussian, Markowitz, well behaved world. Extensions for non-well behaved world will be welcomed.

I know that by a portfolio made by only by one stock (and a risk free bond) I can use the formula:

f* = (R-Rf)/d^2

f* - Wealth fraction that maximize the log return 
R - Asset Return 
Rf - Risk-free return d - Standard Deviation

But in a portfolio made by a Stock and a option, considering I am only interested in the return at 'the end of the month', so I'm not gonna hedge continually, only buy a stock and a option and wait until expiration. Is there any closed form solution? I currently doing it by Monte Carlo in Excel, but it is a little bit unpractical since I have to simulate all the wealth fraction to see which one maximizes de log return on average (times all the options available).

Any idea?

Answer 1

Kelly is mostly based upon assets with zero correlation made independent of each other. The way I approximate Kelly for multiple bets with correlation is:

1. Assume after your first bet the capital is gone.
2. Place a second bet based upon the Kelly of the remaining capital.
3. Factor in correlation..

Part 3 is the challenging part. I assume that with multiple bets at zero correlation placed simultaneously that I would bet the full Kelly per bet made. I assume that with multiple bets at a correlation of 1 I would divide the Kelly by the number of bets. So if for example I were to make 5 bets with a Kelly of 20%... a correlation of 1 would be 20% divided by 5 or 4% per bet. A correlation of zero would be 1-(0.80^5) to determine total capital at risk and then divide by 5 which is ~13.45% per bet. A correlation of 50% is the average of the two or ~8.7% Anything else is a weighted average but you have to be careful not to get the weightings backwards. For example a correlation of 20% you take 80% of the Kelly amount 13.45 and 20% of 4% and sum them together.

I don't believe this is totally accurate because I'm not sure the relationship of correlation is linear and also the assumption that the capital is gone after a single Kelly bet in reality would instead follow more of a decision tree of possible futures and grow increasingly complex with each additional bet... but it serves its purpose.

Major warning: the markets are totally uncertain. Given uncertain edges and conditions that change there is FAR greater problems with overbetting than under betting. Even an extremely small probability of overbetting by more than 2x Kelly or brief period of substantially overbetting by more than 2x the Kelly introduces the risk of ruin and may turn a positive edge negative. However... the costs of fees going up as a percentage of the trade goes up as a trade gets smaller which is problematic for smaller accounts. Smaller accounts are more likely to recover losses through additional deposits so this may suggest a slightly more aggressive approach initially and a plan to scale back risk for awhile.

However, there are utility functions of wealth and psychological ones which suggests that a dollar lost costs more than a dollar gain, particularly as it pertains to producing an annual (or monthly) gain vs an annual loss.

As such, one possible modification is to bet a fractional Kelly initially and then as you start to produce gains increase the fraction of the Kelly.

You may wish to run Monte Carlo simulations to determine the probability of bad results and seek to minimize those results... unfortunately I don't really know how correlation relates to probability of a similar outcome so a I haven't been able to effectively model a multi variable portfolio with an average correlation or correlation matrix or matrices.

I personally look at risk of stock a little different because I am very unlikely to lose more than 10% in a stock so instead I factor in the likely capital at risk on a stock trade to be 1% for a 10% position or 2% if I'm being more conservative about the possibility of an overnight gap down or making poor decisions due to psychological factors. If I calculate the Kelly to be something like just over 2% per out of the money option position for 20 option positions, (totalling about 45%), I'd prefer to replace some of the options with stocks and reduce the risk by estimating a higher correlation and even modifying my initial approach and only considering that the upper limit.

I basically am aiming for 1% per option for up to 20 positions plus 5 optional "double position size" and up to 2 of those 5 optionally "triple". This gets me to a maximum of 27% at risk in options which allows me to put 180% of stock at 1% risk or 90% at risk. I prefer to have capital available at all times and a much lower profile so I instead might aim for 1/3rd of that and up to 60% stock which leaves a minimum of 13% cash. Since I'm unlikely to use the upper end limit of capital and unlikely to require the margin I'll put a 10% position into an income basket.

I also like to use a portion of my portfolio based upon a more game theoretic solution of positioning based upon the probability that a bullish trade will produce a win vs a bearish trade. So I'll establish a minimum of 20% bearish (or bullish) option positions. I'll also diversify over multiple timelines and since it is likely I'll have unused capital at specific times I will buy VXX puts and/or XIV position to use up capital when the implied volatility (measured by the VIX index) of the market rises too high.

I sort of want to have a portfolio that isn't just playing the individual stocks and cash, and there are several other money management problems that come up as one or more position that is active increases by a large amount, but selling before a target price is undesirable and exiting early can turn a profitable system unprofitable... So I am considering owning a set portion of that 20 positions (or additional position) in SPY calls when the market is low or oversold and reducing when the market is high (or using VXX puts or XIV as a proxy following vix spikes) as that will allow me to solve overallocation into the market by reducing SPY calls with a more mean reversion philosophy. There are also other low correlation assets, some of which may be contained in the income basket (such as bonds) and others of which have not been mentioned like commodities and currency trades that are generally strong in game theoretic equilibrium solutions that may be worth integrating. Ultimately, reducing position sizes of individual trades may provide some trade off. Also, Options not only have potential correlation of underlying instruments but of expiry cycles which can be mitigated through diversification of expiry and possibly mixing of strategies from long put/call and short/long bear put spreads and bull call spreads. There are mathematical solutions in scholarly papers if you are really advanced that better answer this many of which are covered in the book Kelly Capital Growth investment criterion.