We sat down with Ed Thorp, a pioneer in the mathematical analysis of casino games and investing, to get his insights on an array of topics from casino gambling to quantitative investing.

Ed Thorp, author of the famed bestseller Beat the Dealer (1966) and A Man for All Markets (2017), is regarded by many as the father of quantitative investing. He recently sat down with Aaron Brown, Antti Ilmanen and Rodney N. Sullivan to discuss contemporary challenges and best practices in investing. This is the seventh in a series of Words From the Wise interviews to be published on AQR.com. Following is an executive summary and the full interview with Mr. Thorp.

Although not well known among investment managers, Ed Thorp is nonetheless recognized by many as the father of quantitative investing. He initially applied his training in mathematics — he earned a PhD in mathematics from University of California, Los Angeles — to solving casino games such as blackjack and roulette, and then moved to tackling the challenges of option pricing and statistical arbitrage. His hedge fund strategies, which ran principally at Princeton Newport Partners and later at Ridgeline Partners, were profitable every year for 37 years, from 1966-2002. His recently published autobiography, A Man for All Markets, chronicles a man who thinks independently about how markets and games operate, and then tests his ideas against economic rationale and real world data. He currently resides in Newport Beach, California where he oversees his family office.

### Executive Summary

Ed Thorp, a pioneer in the mathematical analysis of casino games and investing, is part of an elite group of early thought leaders responsible for creating tools and techniques for use in both gambling and modern investment management. He famously authored Beat the Dealer (1966), a book in which he describes in detail a mathematical system he devised for beating the house in blackjack. He then turned his attention to solving challenges in investment management, specifically option pricing — this is where we begin our conversation. We discuss how he came to investments and his thought process behind creating and implementing his option pricing model, independent of, and a few years ahead of, Black and Scholes. We then turn to get his thoughts on more contemporary topics, including factor modeling, the challenges of consistently generating unique alpha, and the current state of retirement plans. We conclude with hearing about his heroes.

### Early Career

**Sullivan**: You came into the practice of finance not from a finance background, but as an academic from the hard sciences. In retrospect, what were some of the cornerstones of finance, if any, that would have been helpful for you to have known early in your career?

**Thorp**: Well, I had no contact with the academic finance and economics community at all. I was unaware of their existence. I was focused on math, physics and other things in science. In 1962 I started teaching myself about investing. I had money from my gambling experience and from writing books.

At first, I made some investment mistakes, foolish ones that many people make. As a result, I decided to really buckle down and try to understand investments. After a couple of summers of reading, I felt that I had a good idea to pursue. I wanted to work out a mathematical analysis of common stock purchase warrants, which were similar to call options today. The idea was appealing because I could get rid of most of the variables that people commonly use to evaluate companies. Also, I could hedge the risk in the warrant by using common stock because the price of a stock and its corresponding warrant move up and down together. So, instead of having to do a lot of fundamental analysis (like go out and talk to CEOs), I just needed data for a few readily available variables like stock price volatility and the riskless interest rate. That finding was a great revelation to me and I thought I could use it to make steady profits.

**Sullivan**: Were you aware of others working on the option pricing problem at that time?

**Thorp**: I began to discover that there were people in this vast world of finance and economics who’d been working away at this, and lots of other interesting problems. So, my good fortune, as well as my misfortune, was that I didn’t know anything about finance. I didn’t have any academic finance background or connections. That also meant that I had fewer set ideas going in. So, I thought things through ab initio for myself.

**Sullivan**: Did you realize at the time how revolutionary your options formula would be to finance?

**Thorp**: No, not at all. Since I had no connection with the academic world of finance or economics, I had no idea this was that important a problem, and that it would have such widespread application.

**Sullivan**: At what point did you become aware that others were working on this same formula?

**Thorp**: Around 1967 I was thinking about developing a warrant pricing formula, and as a first step I integrated the lognormal distribution to see what I would get. And then I realized that the folks at MIT, Samuelson, among others, had already done some work on it.1 The relevant formula contained two unknowns. One was the rate of growth of the common stock, and the other was the discount rate applied to get a risky terminal distribution for the payoff on the warrant. That’s where everybody had gotten to, and they were all stuck there.

This idea didn’t come right away, but after thinking about it for a while, I thought that in this situation, I could just plug in a sensible estimate of the discount rate instead of attempting to calculate a solution for every possible situation. Then, I thought, what if you were in a world that was risk-neutral? Well, then, all the various rates would become the riskless rate. So, let’s try that out and see what happens — and bingo, you got a really simple, beautiful formula.

So then, by extension, I thought that although the various stock hedges are not exactly riskless, because it’s too costly to adjust them continuously, I can adjust them fairly frequently whenever there’s a moderate deviation from balance. As I have a collection of them, the deviations between my discrete adjustments and continuous adjustment are like a diversified pool of random noise. So the more positions I have, the more the deviations wash away and the appropriate discount rate becomes the riskless rate. In short, what I’ve got is a pot of hedges which pay out, if fairly priced, at what ought to be the riskless rate. So, it made sense to use this formula using the riskless rate for the growth rate and the discount rate.

In thinking about this just qualitatively, I realized that if a stock is more risky, then the expected growth rate will be greater but the discount rate for that payoff will also be greater. So, there’s going to be substantial cancellation between the two. Maybe it’s incomplete. Who knows? After some thought, I decided that this formula holds all the properties that you want in such a formula, and it’s the answer in the risk-neutral universe. So, if there is a single answer, this formula has to be it. It’s plausible that it’s not the single answer for everything. But I decided to use it, and I did.

**Sullivan**: Ben Graham used to say successful investing requires you to have the courage of your convictions. How did you come to be so confident that you had solved this challenge?

**Thorp**: Well, I thought I had a very good formula that at least qualitatively did everything right. If it was off, it wasn’t off by very much. So, now I had this tool that nobody else apparently had. But that wasn’t important to me; it was just a tool for managing my warrant hedges. This was prior to the launch of the CBOE.

In talking to Fischer Black long afterwards, I found out that he and Myron Scholes had also figured out this formula two years after I did. Their related research article was not well received by the journals at that time, so, the formula wasn’t published until 1972, and then another paper in 1973.2

*Article by Aaron C. Brown, Antti Ilmanen, Rodney N. Sullivan, Ed Thorp – AQR*

See the full PDF below.