After thinking more about the article that I posted last week, it became clear to me that the most important part of the story had less to do with math or finance than with the limits we quietly accept about ourselves and the effort needed to push beyond them.
This is the personal side of that journey.
In several of my previous posts I have mentioned (more accurately confessed and pleaded) that I am not a mathematician. In fact, as a young student I never went beyond advanced algebra in high school. Though I got a final grade of "A" in all my math classes (I did get a midterm grade of "C" in basic algebra), I was told by someone in authority that I did not have the ability for more advanced classes. Perhaps they were right. In any event, I never took a math class in college and always viewed mathematics as outside of my core competencies and strengths.
Much later in life, after I had finished careers in law and banking, I found myself teaching a class which I had developed entitled "Corporate Finance for Non-Finance Majors." At the very end were the materials for a class I sometimes chose to include depending on how the semester had gone. It was basically a very non-technical view of options. I explained what options were and how they worked — the inputs being strike price, interest rates, volatility and time to expiration. I showed my students the Black-Scholes option pricing formula, which is certainly one of the most extraordinary achievements in finance. I did some hand waving, told my students not to freak out and showed where each of the inputs could be found in the formula. That was certainly enough for non-finance majors. And to be perfectly honest, it was pretty much all I was capable of doing.
But that bothered me — a lot. I wanted a better understanding, and I suppose I also wanted to fight back against the narrative that I just wasn't smart enough to figure it out. So, several years ago, with the encouragement of my sons and their wives — one of whom is a PhD mathematician — I started taking some online courses. First, statistics and then calculus on the Coursera platform. I spent innumerable hours (actually weeks) studying and going over materials. And then last summer and into the fall I repeated the whole process over again — new courses on MathandScience.com that were more comprehensive and challenging.
And every once in a while, I would open up my computer and look at the Black-Scholes model for pricing options. Slowly, ever so slowly I came to believe that I could do this — understand this Noble Prize-winning achievement at a deep and intuitive level.
Looking back on all of the time, effort and struggle, on the moments of discouragement as well as the feelings of achievement, the long journey has been an intellectually and emotionally satisfying one. In one respect it began relatively recently, just months or years ago when I embarked on an exploration of the mathematical underpinnings of finance. But if I am honest with myself, this journey began back in what was then called junior high school. I was told that I didn't have the intellectual capacity for any type of advanced mathematics or that the amount of effort I would need to put into it would be too much. I believed what I was told. Why wouldn't I? And so, I steered away from anything mathematical, never even pursuing mathematics in college and also abandoned my love of physics.
However, it is hard to predict where life will take you and an early professional career in the law led to a longer career in banking and finance and ultimately teaching. I started wondering about all sort of things mathematical — taking online courses in statistics and calculus. The more I learned the more I wanted to know. There was Euler's number; the normal (Gaussian) curve; and always in the background that incredibly intimidating Black-Scholes option pricing formula.
I knew that at some point I would have to face all the symbols in that model. When I finally felt ready, I began very tentatively. I spent what was probably a ridiculous amount of time thinking about it. Sometimes I even woke up in the middle of the night still turning it over in my mind. None of it made much sense until one key insight finally clicked into place. Once that happened, what had seemed like an impenetrable wall of symbols began to reveal an underlying logic and elegance that I could finally grasp.
In the end, this journey was about far more than an option pricing formula. It was about discovering that understanding does not belong exclusively to specialists, and that what once appears impenetrable can give way, slowly and sometimes painfully, to persistence, curiosity, and the willingness to keep asking better questions. This last point was crucial. What I lacked in mathematical experience and formal training I made up for with the most rigorous thinking I could manage. I worked hard to make sure every single dot was connected. I fought against hand waving even from experts.
This journey through Black-Scholes gave me something I never expected: not just intuition and understanding but a renewed sense that the boundaries of our understanding are often far wider than the stories we inherit about ourselves. In my personal case the journey gave me a very much needed and desired new perspective on my own capabilities.