= Two-stock binomial market without a supplied bond
Suppose two positive stock prices have joint one-period multipliers $(u_1,u_2)$ or $(d_1,d_2)$, with $u_1>u_2$. If $d_1<d_2$, the <Arrow state prices> are $(d_2-d_1)/(u_1d_2-u_2d_1)$ and $(u_1-u_2)/(u_1d_2-u_2d_1)$, both strictly positive. The two stocks therefore span both state payoffs, including a <synthetic zero-coupon bond>, and backward induction excludes <arbitrage> in the corresponding finite multiperiod market. If $d_1\geq d_2$, equal initial currency exposures long the first stock and short the second give an arbitrage when both states have positive probability.
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