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 1/2, the all-up terminal probability is 1/4, the recombined middle probability is 1/2, and the all-down probability is 1/4.

Puu=1/4,Pud+Pdu=1/2,Pdd=1/4P_{uu}=1/4,\quad P_{ud}+P_{du}=1/2,\quad P_{dd}=1/4
Risk-neutral binomial valueThe full two-step tree is recomputed from the public inputs.todaydownupS $100.00V $10.20S $90.00V $1.43S $120.00V $20.00S $81.00V $0.00S $108.00V $3.00S $144.00V $39.00

Expected terminal payoff

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

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

Same answer

The result is exact 50000/49 cents, displaying as $10.20. This matches backward induction.

V0=50000/49 cents≈$10.20V_0=50000/49\text{ cents}\approx \$10.20

Model note

The whole-tree result is a model price under the stated risk-neutral pricing assumptions, not a market price. The model is frictionless and uses one stated rate with no transaction costs, fees, taxes, credit risk, or liquidity limits. This is descriptive, not investment advice.

risk-neutral value is a model price\text{risk-neutral value is a model price}