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title: 1-2-3 Model author: Keith A. Lewis institution: KALX, LLC email: [email protected] classoption: fleqn abstract: Not as easy as 1, 2, 3. ...

Back when I was interviewing quants I gave them a simple puzzle to find out if they understood the difference between "real-world" measure and "risk-neutral" measure. In the mid 90's most of them gave the "wrong" answer. By the late 90's most candidates had been through a Mathematical Finance program and gave the "right" answer. By that time I learned from the traders I worked with that the right answer was the wrong answer.

At a recent NYU faculty meeting my colleagues were concerned about the effect of ChatGPT and other Large Language Models. I use Copilot in the course I teach there. It is great for generating homework problems. It spits out incorrect code and the student's assignment is to correct that. It is a skill they will need to be successful in their future career.

My colleagues were worried the tests they have been developing over the years would become generally available to students. None of them were amused by my quip "Keep the same tests, just change the answers."

This is a singular example of that.

Suppose a one-period market has a bond with price 1 at the beginning of the period that goes to price 2 at the end of the period, and a stock with price 1 at the beginning of the period that goes to price 1 with probability $0.1$ or price 3 with probability $0.9$ at the end of the period. What is the value of a call with strike 2?

If you are vaguely familiar with the Black-Scholes/Merton theory then you know the value is the expected value of the discounted payoff. The call has payoff 0 if the stock ends at 1 and payoff 1 if the stock ends at 3. Since the discount is $1/2$ one might think the value is ${(0(0.1) + 1(0.9))/2 = 0.45}$. This fails if used on the stock, ${(1(0.1) + 3(0.9))/2 = 1.4 \not= 1}$. To reprice the stock we need ${(1(0.5) + 3(0.5))/2 = 1}$. This gives the "correct" call price ${(0(0.5) + 1(0.5))/2 - 0.25}$.

As B-S/M showed us, the value of an option is the cost of setting up the hedge. The call can be perfectly hedged for $0.25$: borrow another $0.25$ using the bond and buy $0.5$ in the stock. No matter what, you have to pay back $0.5$ on the bond. If the stock stays at 1 sell the stock for $0.5$ to pay off the bond and owe nothing on the call. If the stock goes to $3$ you get $1.50$ from selling the stock, use $0.5$ to pay off the bond, and have $1$ left over to cover the call obligation.

When I was proudly showing off this mathematically correct analysis to a trader he looked at me as if I had lost my mind. "Wait, wat? I can give you $0.25$ to get back a dollar 90% of the time? If I have to borrow at 100% interest that is still a quarter to get half a buck. I'll take that trade all day long!"

John Illuzzi pointed out, when we were at Banc of America Securities back in the day when we had to spell bank "banc", he might give a different answer if losing money on a single trade meant he would be taken out back and shot in the head.

That is when I realized "risk-neutral" meant risk blind. The mathematical theory did not provide tools traders found useful for managing risk. Scholes and Merton won Nobel prizes for showing how to replicate options without knowing the "real-world" growth rate of a stock. Their assumptions stock price can be modeled by geometric Brownian motion and it is possible to hedge in continuous time were not realistic. That had a deleterious effect on subsequent research. Instead of coming up with a theory that accurately modeled what traders do, academics used more math to publish papers trying patch up a fundamentally flawed theory.

There have been many theoretical advances to address these impractical assumptions over the past half century, but there is still no generally accepted answer to an even simpler question.

What is the value of an instrument based on a fair coin flip that pays $1MM if it comes up heads and -$1MM if it comes up tails?

One thing missing from the classical theory is who is offering to back the (bit?)coin flip and who they can convince, or are willing to allow, to accept it.

Legal entities do trades. There has to be a knob for that in any realistic theory of trading.