Joachim Kuczynski, 24 October 2023
In this article I want to derive the explicit relationship between an option value and the probability of occurrence of its event states in the binomial model of Cox, Ross and Rubinstein. In many cases I read that the risk neutral probabilities and therefore the option value do not depend on the probabilities of the real state values. But the options values depend on them implicitly. That is what I will derive in this post.
Binomial Model by Cox, Ross and Rubinstein
Options can be valued with the binomial model from Ross, Cox and Rubinstein. The value of an option at time is given by:
and are the option values of the up and down development at time . is the risk free rate and T is the time between and , . is the risk neutral probability of the up movement in , is the risk neutral probability of the down movement in . The binomial model provides the following relationship:
Including provides this expression for
Hence we obtain:
and are defined as ratio of up and down movement in relation to the expected value in , :
Up to now the probabilities of up state and down state have not occured. Many times that leads to the argument that these probabilities do not influence the option value. But that is not true. The expected value of the state and therefore depends on the probabilities. The expected value of the event state in is the discounted value of event state in . With as yearly constant discount rate we get:
For and we get the following:
As final result we obtain:
This is the basic relationship between the value of an option at a time and explicit problem specific variables.
We realize that the option value expicitely depends on the probability of the up state , and of the down state respectively. That is what we wanted to prove. The argument that this dependency does not exist, does not take into account that the value of state depends on the state probabilities in . Hence there is no disappearance mystery of real life or real states probabilities in options valuation. q.e.d.