Derivatives Models on Models

10

A Look in the Antimatter Mirror^*

Take a look at yourself in the mirror. You will hopefully see a reasonably symmetric image of the real you, though your left and right sides have been reversed. In this chapter I will explore an amazing antimatter-mirror that I stumbled across recently in the files of quantitative finance.

1 Garbage in, Garbage Out?

One day when the market was very quiet, I became bored. As an option trader (and formula collector) what else could I do but play around with my Black-Scholes-Merton calculator (a computer implementation of the Black and Scholes (1973)/Merton (1973) formula¹)? Just for fun, I wondered what would happen if I dared to input negative volatility in the formula. My state-of-the-art self-made options system returned a negative number; at least I didn't get a “catastrophic error: restart immediately” message. Options cannot take negative values, so I guess garbage in, garbage out.

But wait, could there be something to it? By playing around with it a bit more I found, quite interestingly, that inputting negative volatility for a call option, then multiplying the result by negative one actually gave me the value of a put, and vice versa. Some time ago I had listened to Nobel prize winner Richard Feynman lecture on “Einstein's Relativity, Symmetry, and Space-Time”, see Feynman (1997). The talk on symmetry, matter and antimatter had been particularly fascinating. Had I just looked at a financial parallel to matter and antimatter? Before we move ...