State Prices & Risk-Neutral

State Prices

State prices value payoffs one state at a time.

highlighted = computed this step

State-price definition

A state price is the price today of $1.00 paid only in one future state.

πu=1/3,πd=2/3\pi_u=1/3,\quad \pi_d=2/3
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

Positive state prices

Here the up state price is 1/3 and the down state price is 2/3. Both are positive, which is the no-arbitrage signal.

1/3>0,2/3>01/3>0,\quad 2/3>0

State-price value

Any payoff is priced by multiplying each state payoff by its state price. The call price is $5.00, matching replication.

V0=1/3⋅$15.00+2/3⋅$0.00=$5.00V_0=1/3\cdot \$15.00+2/3\cdot \$0.00=\$5.00

The same idea as linear-programming duality

The state prices are dual variables. A set of positive state prices that reprices the traded assets is exactly a linear-programming duality certificate for no-arbitrage. The optimization track develops that duality directly, so one certificate underlies both an arbitrage-free price here and a duality proof there.

state prices=dual variables\text{state prices}=\text{dual variables}

Model note

The state-price value is a model price under the stated one-period, two-state assumptions, not a market price. It assumes frictionless trading, one stated rate, no transaction costs, fees, taxes, credit risk, or liquidity limits. Positive state prices are the no-arbitrage signal. This is descriptive, not investment advice.

state prices are no-arbitrage model inputs\text{state prices are no-arbitrage model inputs}