### Blowing No-Arbitrage Pricing Theory to Hell. Is There Anything Fintech Can’t Do?

Standard no-arbitrage derivatives pricing theory rules out certain trading strategies because they are seemingly unrealistic. For example, if the set of trading strategies is unrestricted, the following strategy is guaranteed to make any arbitrarily large profit (pick a number, any number! e.g., $10 quadrillion) prior to some finite time *T* almost surely–and don’t call me Shirley!: specifically, at time *s*, hold 1/(*T*–*s*) shares of stock in your trading portfolio, and cash out when you’ve made $10 quadrillion.

Obviously, this requires you to hold an arbitrarily large–and arbitrarily expensive–position in the stock. No problem: just borrow! Further, this obviously exposes you to risk of infinite loss: the variance of your portfolio value goes to infinity as you approach *T*. Again, no problem!: just borrow! The self-financing constraint is satisfied, so dream of how you’ll spend your immense riches.

Well, borrowing infinite sums seems a tad unrealistic, so it is standard to rule out such doubling strategies. One good explanation is in the very good book by J. Michael Steele, *Stochastic Calculus and Financial Applications*:

Nothing is wrong with the model or with the verification of the self-financing property, yet something has gone wrong. The problem is that our model diverges from the real world in a way that any decent banker would spot in an instant. Just consider what happens as the time gets nearer to T if the investor in charge of the portfolio V(t) has not yet reached his goal. In that case, the investor borrows more and more heavily, and he pours the borrowed funds into the risky asset. Any banker worth his salt who observes such investment behavior will pull the investor’s credit line immediately. The simplest rules of lending practice are enough to prohibit the management of a portfolio by any strategy [a(t), b(t)] like that defined by equations (14.36) and (14.38).

But maybe not, thanks to the miracles of fintech! Apparently a glitch in the Robinhood app has given users in the know “infinite leverage”:

The cheat code was being shared on social media site Reddit, with one trader claiming he took a $1,000,000 position in stock using only a $4,000 deposit. Through Robinhood Gold, the start-up’s subscription service, users can borrow money from the company to make trades. The backdoor was essentially free money and was being called “infinite leverage” and the “infinite money cheat code” by Reddit users who discovered it.

Looks like we can’t rule out “doubling” strategies, and impose a credit constraint restrict trading strategies to what Steele denotes the set *SF*+ (“self-financing plus”). I guess Robinhood isn’t “a decent banker.” LOL.

This is what happens when you make coders bankers. They blow no-arbitrage pricing theory all to hell.

Looks like imma gonna haveta rework my lecture notes for my PhD derivatives pricing course.