= Terminal proportionality identifies bounded stock volatility
For two strictly positive zero-rate stock <martingales> with bounded volatilities and the same initial value, proportional terminal prices must be equal because their <expectations> coincide. <Conditional expectation> then makes the whole paths equal. Comparing their stochastic integrals by the <Itô isometry> gives equality of the volatility coefficients almost everywhere in time and probability. <Perfect positive correlation> alone does not provide the required proportionality.
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