Drawbacks of the Black–Scholes Model

From time to time, I observe the emergence of characters trying to build a “holy grail” options trading strategy based on selling far out-of-the-money options — selling strikes beyond the third sigma.

And although real-life examples rarely teach anyone anything, I’ll still try to speak on this topic. Perhaps my words will reach someone after all.

I understand where the “far-tail sellers’” faith in the third sigma comes from. In the option pricing framework accepted by the options community, it is assumed that the probability distribution of logarithmic daily returns of the underlying asset follows a normal distribution.

One of its properties is that daily returns exceeding ±3 sigmas from the mean occur extremely rarely. I even had a suitable illustration of this pop up somewhere this month.

It is precisely this theoretical fact that ignites hope among those who want to earn systematically and “without risk” (or with minimal risk). I don’t feel like talking now about fat tails, the probabilistic nature of the world in general and speculative trading in particular, or the imperfections of most models used by humanity.

Let’s move straight to practice.

Let’s Look at an Example As part of an options position, there is a sold DOGE January put option: 77 contracts, strike 0.34. The put was sold at a price of 0.0051 USDT.

Effectively, for this small premium, if the option expires out of the money, we would receive around 40 USDT (and perhaps still will).

Suppose the sold put, taking into account time value and increased IV, generates a loss roughly three times larger — about 120 USDT.

If DOGE continues to decline, our losses will grow much faster.

At the same time, if DOGE starts to rise, our profits will grow far less enthusiastically.

There are only three possible actions in such a situation:

1. rolling the position;
2. delta hedging;
3. taking ownership of the underlying asset and holding or trading it.

Let’s go through them.

Margin Requirements
First, we need to estimate the margin requirements at the moment the position was opened. When I sold the put, I didn’t know I would later write this article and didn’t record the GM value, but we can estimate it.

Let’s go to the Deribit options calculator and find a put option priced close to 0.0051.
There is a DOGE January put with a 0.28 strike whose theoretical price is currently just under 0.005. We’ll attribute the difference to time decay (those who wish can also calculate the sigma).

We construct a position of 77 sold contracts.

Approximately 450 USDT would have been locked by the exchange as margin upon entry. This is a rough estimate, but it adequately reflects how the situation develops qualitatively.
In real life, if this truly were a third-sigma option, the difference between GM at entry and GM when price moves into the sold tail would be far more dramatic.

And sellers of third sigmas tend to load up almost “all-in” on margin. Otherwise, they won’t even see 20–30% annual returns.

If we also consider exchanges that like to raise margin requirements during periods of sharp price movements, or platforms offering leverage of 10× or more, then trouble for far-tail sellers is inevitable — and it arrives much faster.

Rolling the position

Now let’s consider rolling to a farther strike — effectively pushing the risk lower in price. In our case, everything is bad. Here’s why.

To roll, we must first close the existing position, i.e., buy back the 77 sold options. This locks in a loss of 120 USDT (in real options trading, likely more).

Now, when rolling, we must not only recover this loss but also ideally earn something on top — say the same 40 USDT. That makes a total of 160 USDT.

To collect 160 USDT, we would need to sell about 220 put option contracts with a 0.3 strike. This would require roughly 1050 USDT in margin.

At this point, two questions arise:

1. Does the third-sigma seller even have that kind of money in their trading account?
2. What happens if the price continues to fall?

Put yourself in the place of this naïve dreamer and answer honestly.

Let’s end the discussion of rolling here — along with the idea that a third-sigma seller is likely to roll successfully at all, let alone twice. The price of DOGE can go anywhere; it knows nothing about the three-sigma rule.

Delta Hedging

Next comes delta hedging. I’ll say it upfront: things are no better here.

Theoretically, over a certain number of delta hedging operations, the cost of the hedge will approximately equal the option’s time value fixed at the moment the hedger is launched.

If, when selling volatility, you configure your delta hedger with an IV slightly lower than the exchange-implied IV, then theoretically, on average, you will incur a slightly lower hedging cost.
In practice, things will be worse.

How much worse is a multifactor question, and I won’t attempt a quantitative estimate. I fully allow that in a single, random case the delta hedge cost may turn out better than what I described.

But over a long series of trials (delta hedge activations), the result will be exactly as outlined above.

Owning the Underlying

That leaves the third approach — buying DOGE and holding or trading it.
Buying DOGE without leverage and waiting until the price “does its thing” works reasonably well in the U.S. stock market, which has been growing for about 150 years.

In our reality and in crypto, this approach can be deeply disappointing and confusing — especially if you like leverage. As for “actively trading” DOGE, I’ve seen plenty of storytellers claiming special knowledge.

By the way, let’s look at the Gross Margin required for a position of 77 long DOGE index futures on Deribit.

It would be roughly twice the size of our original deposit.

Conclusion

So if you still want to build a systematic options business by selling the “third sigma” — go ahead. Take the flag, strap on the drum, and march toward millions.
My advice, however, is simple: put your money in a bank deposit. It will be safer there, and you’ll have more free time to think.