The One-Period Model

When No-Arbitrage Fails

A violated no-arbitrage bound creates a visible money pump.

highlighted = computed this step

Guard violation

The no-arbitrage guard requires the bond growth factor to sit strictly between the down and up stock factors. Here the bond grows by 5/4, above the up factor 6/5, so the guard rejects the market.

9/10<1+r<6/5fails because1+r=5/4>6/59/10<1+r<6/5\quad\text{fails because}\quad 1+r=5/4>6/5

Zero-cost money pump

Short 1 share for $100.00 and lend that $100.00 at 25% interest. The cost today is $0.00, but next period the portfolio pays $5.00 in the up state and $35.00 in the down state.

cost today=$100.00−$100.00=$0.00,Vu=$5.00,Vd=$35.00\text{cost today}=\$100.00-\$100.00=\$0.00,\quad V_u=\$5.00,\quad V_d=\$35.00
No-arbitrage violationThe table recomputes a zero-cost high-rate arbitrage.Money-pump payoffStateLend legShort-cover legNet payoffUp$125.00$-120.00$+5.00Down$125.00$-90.00$+35.00

Valid-rate contrast

At 10% interest, the bond growth factor is 11/10, inside the band. The state prices are 20/33 and 10/33; both are positive. That is the model signal that no zero-cost portfolio has nonnegative payoff in every state with a positive payoff somewhere.

9/10<11/10<6/5,πu=20/33,πd=10/339/10<11/10<6/5,\quad \pi_u=20/33,\quad \pi_d=10/33

Model note

This is a model illustration under frictionless assumptions: one stated rate, no transaction costs, fees, taxes, credit risk, or liquidity limits, and short-selling allowed. Real markets do not offer persistent riskless profits; prices adjust to remove them. This is descriptive, not investment advice.

the guard excludes money-pump markets\text{the guard excludes money-pump markets}