Risk-Neutral Valuation

The Whole Tree at Once

Risk-neutral valuation can price all terminal payoffs in one expectation.

highlighted = computed this step

Terminal probabilities

Under the risk-neutral probability 1/3, the all-up terminal probability is 1/9, the recombined middle probability is 4/9, and the all-down probability is 4/9.

Puu=1/9,Pud+Pdu=4/9,Pdd=4/9P_{uu}=1/9,\quad P_{ud}+P_{du}=4/9,\quad P_{dd}=4/9
Risk-neutral binomial valueThe full two-step tree is recomputed from the public inputs.updowntodayS $100.00V $5.67t1 u0S $90.00V $1.00t1 u1S $120.00V $15.00t2 u0S $81.00V $0.00t2 u1S $108.00V $3.00t2 u2S $144.00V $39.00

Expected terminal payoff

The whole tree can be priced at once as the discounted risk-neutral expected terminal payoff.

V0=1/9⋅$39.00+4/9⋅$3.00+4/9⋅$0.00(1+r)2V_0=\frac{1/9\cdot \$39.00+4/9\cdot \$3.00+4/9\cdot \$0.00}{(1+r)^2}

Same answer

The result is exact 1700/3 cents, displaying as $5.67. This matches backward induction.

V0=1700/3 cents≈$5.67V_0=1700/3\text{ cents}\approx \$5.67