1-2-3 Model

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. It still does not have an answer that is generally accepted in the Mathematical Finance world.

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 familiar with the Black-Scholes/Merton theory then you know the price 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 “price” of an option is the cost of setting up a “perfect” hedge. The call can be perfectly hedged for 0.25: borrow another .25 using the bond and buy 0.5 in the stock. No matter what, we have to pay back .5 on the bond. If the stock stays at 1 we can sell the stock for 0.5 to pay off the bond and owe nothing on the call. If the stock goes to 3 we 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 all day long!”

John Illuzzi pointed out when we were at Banc of America Securities that 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 a good first approbation.

There have been many theoretical advances to address these unrealistic 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?