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This chapter is from the book

When I Realized It Might Be Easier

Starbucks, late. I clearly remember looking at the number: $11.93. It was only 1 cent higher than the number I had gotten using the Black-Scholes formula, $11.92. But I wasn’t using the Black-Scholes formula. I was using a spreadsheet—a simple one.

After a few years as a hedge fund manager, I had finally settled into a strategy I felt comfortable with. What I didn’t know how to do was describe it. I didn’t even know what to call it. For lack of anything better, I called it a synthetic annuity. I used synthetic because of the risk-management features that I guessed would qualify as a synthetic hedge, and annuity because it involved selling options to generate monthly income.

I knew I needed to devote time to communicating the strategy in a way that the average investor could understand. At a minimum, I needed to put it in context of the various traditional and hedge fund strategies. I struggled with this. Because it involved trading options, I was concerned that it would get the typical bad rap of being too risky or too complex, neither of which I think is true. But it was a form of managed structured security, so I would have to explain the basics of structured securities and how they worked.

The previous day, I had been flipping through one of my go-to texts, McDonald’s Derivative Markets, looking for something that might give me a starting point. I saw this:

  • The Black-Scholes formula arises from a straightforward lognormal probability calculation using risk-neutral probabilities. The contribution of Black and Scholes was not the particular formula but rather the appearance of the risk-free rate in the formula. (p. 613)8

I had already been thinking about Black-Scholes, having just reread Peter Bernstein’s beautifully written books on the history and evolution in investment thinking, Capital Ideas and Capital Ideas Evolving. Bernstein referred to options and the pricing model as “the most powerful financial invention in history.” And I remembered the emphasis Paul Samuelson put on option pricing when he gave his advice to anyone entering the investment field: “Learn the Black-Scholes option-pricing model.”9

My immediate interest was more in tailoring risk and reward profiles, but I had reached a point where I needed to construct a reasonable basis for comparing alternative structures. I was skeptical about using the Black-Scholes framework because of its well-publicized limitations, such as not handling fat tails and assuming constant volatility.

Then I changed my mind. I wasn’t trying to weigh something precisely, so I didn’t need a very accurate scale. I was measuring the difference in two things, which even an inaccurate scale can do. And using Black-Scholes had the advantage of making the structure approximately hedgeable, which is more important in my work than being precise.

So I decided to try what McDonald had suggested: to derive the Black-Scholes formula. I could either start with a differential equation or start with a spreadsheet approximation. I chose the spreadsheet. I was hoping to build something that would fit on a page or two of Excel—and I’m not crazy about differential equations.

I began with one of the assumptions used to derive the Black-Scholes formula:

  • Continuously compounded returns on the stock are normally distributed.

Excel has a built-in function for the standard version of the normal distribution. That was the first step. Then I went through the process of converting it into a stock return distribution and then a stock price distribution. The option payoff was straightforward, as was weighting the payoffs by probabilities.

The entire calculation fit in six columns, and there were no complicated formulas. It was symmetric and simple.

Too bad it was wrong.

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