We use minimum price variation rules (tick size rules) in the French stock market to discriminate between stocks with high and low transaction costs. If the stock price exceeds a threshold of 500 francs, the minimal tick size for quotes increases from 10 centimes to 1 franc. The increased tick size induces a considerable transaction cost increase and generates a natural experiment. The average cost of a roundtrip approximately doubles for stocks in the price range of 500 to 600 francs relative to the roundtrip costs for stocks with prices between 400 and 500 francs. We explore if this important increase in transaction costs is accompanied by a reduction in return volatility for the stocks in the high cost regime.Our data show a statistically significant, but economically insignificant reduction in the return volatility for daily, weekly and monthly measures of return volatility. This leads to a pessimistic assessment of the suitability of a security transaction tax as a suitable policy measure to reduce asset return volatility.