Determination of Parameters of Moving Up and Down of the Price of a Share in the Binomial Option Pricing Model Using the Ito Lemma. A Short Research Memo
Moscow State Institute (University) for International Relations,
The article discusses the approach to determining the parameters of the binomial model for estimating the premium of the American stock option, which is presented in the article "Option Pricing: A Simplified Approach" by Cox et al. (1979), and an alternative solution to the problem using Ito's lemma. They determined the parameters of the model based on the return on the underlying stock in the real world, and not on the risk-free rate, as required by theory. However, when passing to the limiting case, their solution to the problem turns out to be correct. The use of Ito's lemma made it possible to obtain a solution similar to the solution of Cox et al. (1979) and to answer the question why, when passing to the limiting case, the technically incorrect solution of Cox et al. turned out to be correct. From the solution of the stochastic differential equation, it can be seen that, at infinitely small time intervals, the variance is the determining parameter of the dynamics of the share price, and the trend becomes virtually indistinguishable. Accordingly, the fact that when finding the model parameters they used the actual stock return as a trend, and not the risk-free rate, did not affect the model parameters in the limiting case.
Keywords: lemma Ito, risk-neutral valuation, geometric Brownian motion, binomial model, Girsanov theorem.
JEL codes: G12, G13, G17