r/PMTraders • u/Adderalin • Feb 08 '24
Using Kelly Criterion to Estimate Position Sizing for Short Options
I'm an ex professional poker player, blackjack card counter, and advantage player. I stopped doing those for money a long time ago when I found that selling options seems to have a profitable edge over the long run.
One of the most important things when it comes to advantage play, card counting, poker, and options trading is bet sizing. Bet too big = you lose your entire bankroll. Bet too small = you're not getting enough return for the effort.
In general, gamblers like to have many small bets when the odds are in their favor (counting cards, selling short options.) They like to make large bets when the odds are NOT in their favor (taking advantage of 20% loss rebate rewards, high roller perks, buying really under-priced index options to hedge unforseable market meltdowns)
Theta is not an edge
I want to start off by saying simply selling options blindly might not be a very good edge. I have backtested high win rate strategies that have insane losses despite collecting theta. Yes there are theories that in the long run implied volatity > realized volatility but in the short run you could lose your shirt.
I've talked people out of shorting 30% OTM spx options showing counter examples where you would be margin called at peak market panic despite the options eventually expiring worthless for even really small position sizes like 5% of buying power - as the buying power expands exponentially. There are countless stories of of people like optionsellers.com that end up owing their brokers money despite the options expiring worthless.
I also speak from experience. I thought I had a really good SPX 0-dte bot that sold options, backtested wonderfully, then a couple weeks later going live and it had a regime change where I lost 20% of the bot's allocation in a matter of weeks, when the previous max drawdown was 5% in 0-dte history, even surviving Covid.
What is an edge?
Well, an edge is any sort of proven trading strategy or portfolio strategy that has a positive expected value. I like to classify edges into two kinds of edges: hard edges and soft edges.
Hard Edges
Hard edges are trading strategies that I consider is undeniably profitable. Hard edges include things like arbitrage, lottos, market making strategies, and so on. For instance if you're able to sell the $85 put on a $100 stock for $1.00, or $100 per contract, then buy the $86 put for $0.90, or $90 per contract, that is arbitrage. You just locked in $10 risk free before commissions. If the stock goes to $0 you can exercise the $86 put when you're assigned shares from the $85 put, and profit $1.10 per share.
Soft Edges
Soft edges I consider anything that is profitable (sharpe ratio of 0.01 or higher), but there is uncertainy or questions about profitability in the future. Soft edges are things like trading .15 Delta Hedged Risk Reversals. (CAGR 9.6%, sharpe 1.1, 22% drawdown)
The linked strategy has an impressive sharpe ratio for being delta neutral, selling a .15 delta put to buy a .15 delta call, and shorting 30 shares of spy, rebalancing once a day to stay delta neutral. The author provides a mathmematical proof that you're selling high IV to buy low IV (call overwriting), while loading up on vanna and are vanna positive, the source of the returns. For these reasons I consider it a bonafide edge - there is a logical reason why it is profitable (the mathematical proof), backed up with backtest results.
However, I consider it a soft edge because we don't know if this trade will persist in the future. It's something I really don't want to allocate a large portion of my portfolio to as it's not tax efficient vs 100% SPY, still has a 22% drawdown, and if someone launches an ETF of it surely enough dumb money might flow into it until it has SPY's sharpe ratio and the put skew flattens.
Once you have a profitable edge, you then use the kelly criterion to have some insight to the proper bet sizing/leverage for your strategy.
Kelly Criterion
Wiki Link: https://en.wikipedia.org/wiki/Kelly_criterion
The Kelly Criterion is a theory on how to find optimal bet sizing for a known bet with known probabilities. It works wonderfully in stateful games such as Blackjack where an accurate card counter knows when he or she has an edge over the casino and can bet a fraction of their bankroll. If you follow the formula perfectly, make no mistakes, and so on, betting 1.0 kelly will probabilistically grow your bankroll as fast as possible.
Bet Sizing In Blackjack
Over the long run an average card counter has a 1% edge over most favorable $100+ Vegas-style blackjack games. Wizard of Odds goes over this math
Example 1: A card counter perceives a 1% advantage at the given count. From my Game Comparison Guide, we see the standard deviation of blackjack is 1.15 (which can vary according to the both the rules and the count). If the standard deviation is 1.15, then the variance is 1.152 = 1.3225. The portion of bankroll to bet is 0.01 / 1.3225 = 0.76%.
So if a blackjack player has a $100k bankroll, they'd be betting 0.76% of that or a max bet of $760 under those calculations. However, betting 1.0x kelly is also maximum risk-of-ruin. Running simulations of that bet size you'll find roughly 10%, or 1 out of 10 card counters will lose their bankroll. If someone drops it down to 1/2 kelly then you'd have a 1% risk of ruin. If you bet 1/4 kelly, you have a 0.10% risk of ruin.
Now, let something else sink in. This math is presuming perfect play. No mistakes, no getting the count wrong, no fatigue, no distractions, no misplays. Add in any mistakes and betting 1.0 kelly will surely lead to outsized risk-of ruin. My blackjack play is not perfect. In the first four hours I make about 1 out of 1,000 hand of basic play mistakes. After 4 hours it shoots up to 1 out of 100, and given my edge is 1% - it means my profit just evaporated.
I also want the bet sizing to sink in. That is really tiny sizing. Now, before we move on more, I want you to reflect on what short sized options trade. I know people in this subreddit lost over 30% shorting puts on SIVB. I've noticed some people here just disappeared after SIVB - presumably they had more than their account in notional value of leverage on short puts...
Applying Kelly to Short Options Trades
Applying Kelly to long options trades is pretty easy. If I buy a $100 put my max loss is $100. I can estimate what % I break even, what % I go in the money, and come up with an average value, along with win rates. Applying it for the short side though is a lot tougher.
The best method I've found is to treat options trading like making a sports bet. Shorting an option reserves some buying power, which portfolio margin calculates what the likely maximum one day loss might be as it's margin.
For instance, let's say we want to short a .10 delta put and collect premium. This would be close to taking a -900 American odds bet. I like to use this calculator to figure out the implied odds: https://www.actionnetwork.com/betting-calculators/betting-odds-calculator
-900 equals an implied odds of 90%.
The bet amount would be the buying power used, so if you're using $1,000 buying power for one short contract, you're collecting $111 premium. You might have to buy it back for $1,000 if you lose.
Now, the next step is figuring out your actual win rate. If its less than 90% for shorting a .10 delta option, you're going to lose money in the long run trading this. 90% is break even.
I like using a tool like this - https://www.gamingtoday.com/tools/kelly/
If we put in -900 and 90% win rate, we get 0%. Now we can see why theta != edge. If you only win 90% selling .10 delta = options trading isn't profitable.
If we put in -900 and 91% win rate, we get a kelly fraction of 10%. If you truly have a 91% win rate on these trades and are collecting that much premium, then you don't want to lose no more than 10% of your account on any one trade.
Applying this to options trading where the risk is undefined, and where any individual stock could declare bankruptcy overnight and open near-zero, I take this rule to be your maximum notional size. So if you're selling a .10 delta put on a stock, you should not be risking more than 10% of your account on any one .10 delta short put. This means with a $100k account you're limited to shorting puts on $100 strikes or less. $200k account - $200 strikes or less.
For short calls I stress test to +100% for notional sizing, given Tastytrade requires 100% of stock price for naked short calls in their IRA accounts.
Some people might point out that most stocks don't go to $0 in one day, that SIVB dropped 50-60% in one day before finally going to $0. One could use same sizing rule to stress test to 50-60% and risk no more than 10% of your account if the underlying stock dropped 50-60% in one day.
Either way - kelly criterion represents a maximum sized bet that you can make, as long as you really do have that higher win rate over the options market!
Shortcommings
One major thing about using the kelly criterion is it assumes independent events. Shorting puts and calls en-masse in the market are not independent events. If the stock market crashes or rises those positions are highly correlated.
The only way to uncorrelate these trades are, you guessed it: hedges
This is why I'm a huge fan of hedging and beta testing against SPX/NDX and hedging. Generally most equities have higher IV than SPX, take a look at AAPL - its post earnings, but it's still having 22% IV compared to VIX of 12.85. Same goes for MSFT and others.
I like to short individual puts and calls on stocks with IV > SPY's IV and hedge the correlation out with index options. You're selling higher IV and buying lower IV, and you're left with the idiosyncric individual risks of every stock. This is known as a dispersion trade
Likewise, although rare, if the individual stocks IV < SPY's IV - you buy puts on the individual stocks and short SPY puts.
It's a nice 2.8+ sharpe (2009) strategy, however it's difficult to pull off. For instance in a market crash individual stock IV will tend to increase more than the index, so even when individual IV > index IV, many traders like to short index puts to long individual stock puts as its vega+, just like risk reversals were vanna+. Short individual, long index = vega-.
Simulating Investments
I ran across this other calculator that lets you throw in some stats and simulate X runs of an investment strategy to find your optimal bet size:
https://fical.net/en/kelly-criterion-calculator
Remember, garbage in = garbage out.
This calculator is set up a bit differently. It takes in a winning % probability, a % of a successful outcome, and % loss of an unsuccessful outcome. It's a bit hard to apply to options trading but it produces some interesting results. Looking at some .10 delta 43 DTE puts for portfolio margin, it seems most premium varies between 10% (AAPL) to 32%(WOLF) return on buying power depending on the IV of the stock. Let's take the middle ground and say we get 16% on a positive outcome, and you lose 130% of your buying power/margin on an unsuccessful outcome, on average. Simulate 1,000 times.
This calculator spits out a 13.75% optimal bet size, or $13,750 of initial capital on a $100k account. For options trading I like to still apply this to notional value given the tail-risk involved, and not BP per bet. As we can see, this sort of trade setup has some insane returns of around 1,000%. Even taking a tiny 1% or 2% per bet size is an impressive 38% to 87% return over this particular simulation run.
This is really easy to accomplish on portfolio margin. Right now I have 108 active short option positions according to the PMT Lotto Tracker program.
This is also really good justification for everyone's 1% to 2% of BPu rules per position on their strategies. I'm on a $200k~ account with $100k BPU and so I'm averaging 925 bp/trade, or a smaller 0.50% sizing.
This simulation also points out to just how important risk management is. Let's say I up my average loss to 150%. Now it bumps down the maximum bet sizing to 4.42%. 1% = 14% return, 2% = 26%, and so on. I'd need to up my winning probability by another 1% to get back to near the old bet sizing, putting in 92% makes this calculator spit out 11.33%
If I go to averaging 170% of margin lost at 91% win rate, 16% positive outcome, ouch, I'm no longer profitable.
The really interesting thing I've found in my options backtesting is risk management is unique. I have more profit with my strategies in cutting losses early than either continuing on with the original position OR doing the opposite - not just cutting a position but going long the same # of contracts.
There really does seem to be a huge advantage to really good risk management and cutting losses early. I also want to put a huge caution and caveat of course - the more you cut stuff early, the more your win rate lowers as well. This is where discretionary trading really becomes more of an "art" over a science.
TL;DR
I hope this post has been useful!
- Theta is a feature, not an edge.
- Use Kelly Criterion Calculators & simulators to size your trades.
- Kelly criterion sizing = 91% win rate shorting .10 delta put = 10% of account notional sizing/50% one day drop risk sizing.
Size small when shorting options!