Insights From The Warrant Puzzle via Financial Hacking

Financial Hacking by Philip Maymin is one of the best books for developing intuition for quant finance. These are my notes on the book.

It’s hard to pick one of these to explore here, but I’m going to go with the warrant one for 3 reasons:

  1. If you are unfamiliar with warrants (I never traded them myself) you’ll acquire basic background knowledge.

  2. The “devious” part of the puzzle is a great test of your arbitrage-goggles.(Totally self-serving comment but I did get the right answer to this puzzle as well as most of the questions in the book — when you don’t spend your days in the mines anymore you worry about how rusty you are getting so the check-up is nice.)

  3. I weaponized the logic of the puzzle’s solution — it’s implicitly a proof that “selling calls is not income”!

I’ll get you started here.

The setup:

Warrants are call options issued by the company itself (as opposed to a call option written in a listed market by an arbitrary counterparty).

Imagine 2 identical stocks. One with a call option outstanding and the other with a warrant outstanding. These respective derivative contracts have the same exact terms (expiration date, expiration style, etc) and the stocks themselves have the same attributes but are distinct entities. This is not a trick — you can accept the assumptions — the terms of the contracts and the behavior of the 2 different stocks is identical.

Question 1: What is worth more — the call option on Stock A or the warrant on Stock B?

Then the more “devious” version”

Question 2: What if a single company has both a warrant and call option (again identical terms) outstanding…which is worth more?

You can find the answers to the questions as well as my case for how this proves that “selling calls for income” is a nonsense statement. [In that explanation, you will also have more mental foundations shaken when you start to consider the ramifications of “implied delta”]


I’ll mention one more thing that doesn’t have to do with the book. Again, this concept of “selling calls for income”. If you overwrite calls each month on a stock you own you are doing something very similar to just selling a portion of your position every month. If you own 100 shares of X, and you sell a .20 delta call, you theoretically liquidate 20% of your position. It may not feel like that because usually the calls expire worthless but look at 2 approaches:

Strategy #1: You sell 20% of your holdings a month instead of selling calls

Your position shrinks by 20% each month. In 10 months your position is .80¹⁰ or 10% of what you started with. Zeno’s paradox aside, in about a year you are out of your position.

Strategy #2: You overwrite .20 delta calls every month

Most months you keep the call premium, approximately 1 in 5 months, your entire position gets called away.

The relative performance of the 2 strategies is going to depend on the volatility and path of the stock!

[This is a significant insight to noodle on by the way]

It’s tempting to think “well if I get assigned on those .20d calls less than 1-in-5 times then I’m selling the calls for more than they are worth”.

Sorry. That’s not the full test.

Just think of the scenario where you sell 20% of the holding instead of selling the call option — then the stock drops to zero. You will never have been assigned on your call but that call-overwriting strategy will have much worse results than the “sell a portion of your holdings” strategy. And what did the performance disparity depend on? The volatility.

I know it’s hard to believe — but calls are puts and puts are calls and this demonstration didn’t rely on the put-call parity formula to make the point. Mayim gives an intuitive proof of p-c parity as well.


Stocks that pay dividends are the equivalent of strategy #1 (although on a much smaller scale since dividends are closer to 2% than 20%). If the stock didn’t pay a dividend you can create your own by just selling a portion of your holdings. The same logic holds for selling calls.

That there is a whole asset management sales-machine that revolves around call-selling and dividend-paying stocks obscures the reality that the economics are pretty similar (taxes are a central difference). That a company making $10 in earnings might retain/reinvest it instead of paying it is an overrated distinction.¹

[After all, once a company pays a dividend, the stocks drops by the amount of the dividend since its assets fall by the cash amount. This is literally why you exercise in-the-money call options early — the stock is going to drop and you need to own the shares to receive the dividend that compensates the call holder for share decline.

Mayim even uses this point in a separate context — to show that even spot prices are forwards! This is a more important point for arbitrage traders than investors — a handful of readers might recall a nefarious strategy of picking off stock specialists by requesting a different settlement date than the standard T+3 ahead of a special dividend. This strategy ended up in court. I’ve always thought those extreme couponers who read fine print as a form of offense would enjoy high-finance shenanigans. I see such cleverness as Moloch embodied — when you actively hunt for the limit of where the spirit of a law gives way to the letter you take the free-rider problem and make it a feature for personal gain. Congratulations on your yacht, I guess.]


  1. It might even pose an anomaly that can be traded against depending on how overrated it is —which depends on effective the marketers are at convincing large pools of money and the amount of risk capital that may find fading the anomaly worthwhile. I’m not saying there is an anomaly, I haven’t and normally don’t do the kind of work that would reveal that, but framings like this can be the type of place to hunt for an anomaly.

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