The Binomial Tree

The Recombining Tree

A binomial tree lets the stock move up or down each step.

highlighted = computed this step

Start and moves

The stock starts at $100.00. Each up move multiplies by 6/5, and each down move multiplies by 9/10.

S0=$100.00,u=6/5,d=9/10S_0=\$100.00,\quad u=6/5,\quad d=9/10
Two-period binomial treeThe recombining stock tree and call values are recomputed.updowntodayS $100.00t1 u0S $90.00t1 u1S $120.00t2 u0S $81.00t2 u1S $108.00t2 u2S $144.00

Terminal stock prices

After two moves, the all-up stock price is $144.00, the recombined middle price is $108.00, and the all-down price is $81.00.

Suu=$144.00,Sud=Sdu=$108.00,Sdd=$81.00S_{uu}=\$144.00,\quad S_{ud}=S_{du}=\$108.00,\quad S_{dd}=\$81.00

Model note

Recombining means an up-then-down path equals a down-then-up path, so the tree stays small. The constant move factors and single stated rate are model inputs, not forecasts.

S0⋅u⋅d=S0⋅d⋅uS_0\cdot u\cdot d=S_0\cdot d\cdot u