Risk-Neutral Valuation

What the Model Assumes

A binomial price is only as honest as the model assumptions behind it.

highlighted = computed this step

Model assumptions

The model uses constant up and down factors, one stated risk-free rate, European exercise at expiry, frictionless trading, and no transaction costs, fees, taxes, credit risk, or liquidity limits.

d<1+r<ud<1+r<u
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

Pricing probability

The risk-neutral probability is a pricing device, not a real-world or forecast probability. It ignores actual odds and risk preferences.

q=1/2prices payoffs, not forecast oddsq=1/2\quad \text{prices payoffs, not forecast odds}

Scope note

The binomial price is a model price under the stated assumptions, not a market price. More steps can refine a binomial tree, and real models calibrate the move factors, for example to volatility, with estimation uncertainty. Convergence is only a qualitative idea here. This is descriptive, not investment advice.

more steps refine the approximation\text{more steps refine the approximation}