The Arbitrage Pricing Theory (APT) concerns a central problem of modern finance theory - the trade - off between risk and expected rate of asset return. It was discovered by Ross (1976). His proof of the approximate arbitrage pricing formula was not based on a formal definition of arbitrage. The first precise proof without the estimation of pricing errors was given by Huberman (1982) and with that estimation by Chamberlein and Rothschild (1983). A necessary and sufficient condition for an analogue of approximate arbitrage pricing, called `approximate factor pricing', is given. The proof is very simple and is based on the Riesz representation theorem of a continuous linear functional on Hilbert space. It gives the upper and lower bounds on the pricing errors. The lower bound is attained. As corollaries one obtains the Chamberlain - Rothschild theorem on approximate arbitrage pricing and the Reisman theorem on approximate factor pricing.

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