Stock prices don’t behave in a “random walk.” The Black-Scholes-Merton model’s fundamental assumption, that stock prices exhibit Brownian motion and the normal (Gaussian) probability distribution function (PDF) describes their behavior, fails every practical test.
The Gaussian probability PDF has skinny tails. Our PDF does not.
The Gaussian PDF deems big market swings, the so called “black swans,” nearly impossible: our PDF does not.
We derive the equity’s implied volatility from the prices of its options.
Because our PDF has fat tails, the implied volatility remains constant over a wide range of strike prices – no “smile.”
Our
derivation results in closed-form value equations. See Detailed
Presentation
Because the first and second derivatives of our option value equations also have closed forms, we need no “Greeks” to describe options’ price behavior.
Closed-form value equations make for easy profitability projections of spreads, collars, strangles, etc. A constant volatility identifies opportunities for hedging and arbitrage of over- or under-bought options.
Attending California Institute of Technology, he changed his major from mathematics to engineering. He writes a column for PokerPlayerNewspaper wherein he solves common and rare poker problems using computational mathematics, keeping his math skills sharp.
Soon after newspapers published the prices of exchange-traded options, Mr. Burke noticed the square-root-of-time behavior of at-the-money option prices. Recently he succeeded in developing the complete equations for describing American-style option values.