Discrete-time binomial market

At each period a risky asset in a binomial market has one of two returns, while the risk-free asset grows by a fixed factor. When the risk-free return lies strictly between the two stock returns, the market is arbitrage-free and complete.

If the stock factors are $1+a$ and $1+b$ and the risk-free factor is $1+r$, the discounted stock is a martingale under the probabilityof an up move. The probability of a down move is $(b-r)/(b-a)$.

For one-period successor claim values $V_u,V_d$ and stock prices $S_u,S_d$, the replicating stock holding isThe remaining value is placed in the risk-free asset. In a complete binomial market, backward replication determines the unique no-arbitrage claim price.

At each node of a binomial tree, a claim with next-period values $V_u,V_d$ has valuewhere $q$ is the local risk-neutral probability in a binomial market. For an American claim, replace this continuation value by the maximum of continuation and immediate exercise value.

Let $Q$ be the risk-neutral measure in an $N$-period binomial market. The stock-numeraire measure is defined on terminal events byIf the $Q$-probability of an up move is $q$, its up probability under $\hat{Q}$ isThis change of measure converts discounted expectations containing a factor $S_N$ into probabilities under $\hat{Q}$.