Replication Portfolio in Binomial Model

In this post I a want to give a short derivation of the replication portfolio and the risk neutral probabilities in the binomial model from Cox-Ross-Rubinstein. Let $V$ be the value of the underlying asset in $t_0$ . In a project or investment this might be the present value of the project’s contribution (market related) cash flows. The positive development of $V$ at time $t_1$ , $V^+$ , occurs with probability $p$ , the negative development with value $V^-$ in $t_1$ with probability $1-p$ . The twin security of the underlying in the open market takes a similar notation $S$ , $S^+$ , $S^-$ , $p$ . We consider an option with option value $E$ in $t_0$ that leads to an option value of $E^+$ in the upper state $V^+$ and to an option value of $E^-$ in the lower state $V^-$ . As result we are searching the option value $E$ at time $t=0$ .

Next we replicate the option value in $t_1$ by a portfolio of $n$ shares of twin security $S$ partly financed by borrowings of amount $B$ at the risk-free rate $r$ . The values of the upper and lower state in $t_1$ are $E^+=nS^+-\left(1+r\right)B$ and $E^-=nS^--\left(1+r\right)B$ .

In efficient markets there exist no profitable arbitarge opportunities. Therefore the outcome of the option value $E$ in $t=1$ must be the same in the upper and in the lower state. Setting $E^+-nS^+=E^--nS^-$ we get:

$n=\frac{E^+-E^-}{S^+-S^-}$

Replacing $n$ in the previous equations we obtain the value borrowed at the risk-free rate r:

$B=\frac{1}{1+r}\frac{{E^+S^--E}^-S^+}{S^+-S^-}$

The law of one price tells us that the value of assets that lead to the same cash flows must be the same. That means that the value of the option at time $t_0$ must be the same as the value of the portfolio at time $t_0$ . Therefore we can set $E=nS-B$ . With that we calculate the value of the option in $t_0$ :

$E=\frac{\left(\frac{S\left(1+r\right)-S^-}{S^+-S^-}\right)E^++\left(1-\frac{S\left(1+r\right)-S^-}{S^+-S^-}\right)E^-}{1+r}$

We create a new variable $p^\prime$ to simplify the previous expression.

$p^\prime=\frac{S\left(1+r\right)-S^-}{S^+-S^-}$

Hence we obtain:

$E=\frac{p\prime E^++\left(1-p\prime\right)E^-}{1+r}$

$p^\prime$ can be interpreted as probability for $E^+$ , $1-p^\prime$ for $E^-$ . $p^\prime$ and $1-p^\prime$ are known as risk-neutral probabilities. Note that the value of the option does not explicitly involve the actual probabilities $p$ and $1-p$ of the underlying. Instead, it is expressed in terms of risk-neutral probabilities. They allow to discount the expected future values at the risk-free rate.

Discounting at the risk-free rate is the main difference between decision tree analysis (DTA) and contingent claim analysis (CCA) or real options analysis (ROA). DTA does not take into account that the risk of the cash flow streams changes when you consider options and opportunities. ROA implements this issue correctly.