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_{1} d_{2} - u_{2} d_{1})

and

(u_{1} - u_{2}) / (u_{1} d_{2} - u_{2} d_{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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Past exam of the martingales and financial mathematics course of the University of Oxford / 2009 / B / 6 / a / Solution

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