State Prices & Risk-Neutral

Risk-Neutral Probabilities

Risk-neutral probabilities re-express state prices as discounted expectations.

highlighted = computed this step

Risk-neutral probabilities

Risk-neutral probabilities multiply state prices by one plus the risk-free rate.

qu=πu(1+r),qd=πd(1+r)q_u=\pi_u(1+r),\quad q_d=\pi_d(1+r)
State-price call valueThe lattice recomputes the call value from state payoffs.updownTodayS0 $100.00V0 $5.00Up stateSu $120.00Vu $15.00Down stateSd $90.00Vd $0.00

Risk-neutral pricing

Under the risk-neutral probabilities, the price is the expected payoff discounted at the risk-free rate.

V0=quVu+qdVd1+rV_0=\frac{q_uV_u+q_dV_d}{1+r}

Zero-rate case

At 0% interest, the up risk-neutral probability is 1/3, equal to the up state price. The discounted expected payoff is $5.00, matching replication and state prices.

V0=1/3⋅$15.00+2/3⋅$0.001+0=$5.00V_0=\frac{1/3\cdot \$15.00+2/3\cdot \$0.00}{1+0}=\$5.00

Why q differs from state prices

State prices sum to the discount factor. At 5% interest, the state prices sum to 20/21, while risk-neutral probabilities sum to 1. Multiplying state prices by one plus the rate renormalizes them into probabilities, so they coincide only at 0% interest.

πu+πd=20/21=11+r,qu+qd=1\pi_u+\pi_d=20/21=\frac{1}{1+r},\quad q_u+q_d=1

Positive-rate aside

At 5% interest, the up risk-neutral probability is 1/2 but the up state price is 10/21. The risk-neutral formula gives exact price 5000/7 cents, which displays as $7.14.

V0=1/2⋅$15.00+1/2⋅$0.001.05=5000/7 cents≈$7.14V_0=\frac{1/2\cdot \$15.00+1/2\cdot \$0.00}{1.05}=5000/7\text{ cents}\approx \$7.14

Model note

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

risk-neutral is pricing, not forecasting\text{risk-neutral is pricing, not forecasting}