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, and frictionless trading.

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

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=1/3prices payoffs, not forecast oddsq=1/3\quad \text{prices payoffs, not forecast odds}

Scope note

More steps can refine a binomial tree, and real models calibrate the move factors, for example to volatility. Convergence is only a qualitative idea here. This is descriptive, not investment advice.

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