Volatility can be detrimental to option values!

The value of digital options (both European and American types) can have an inverse-U shape relationship with the volatility of the underlying process! This seemingly counterintuitive proposition is driven by a particular feature of Martingale processes bounded from below (including the geometric Brownian motion (GBM) process). We show that in such processes a higher variance parameter may reduce the probability mass of realizations above the expected value. When the volatility approaches infinity, the probability of hitting a barrier above the mean goes to zero. Our finding is in contrast to the common belief that a higher volatility always increases all option values. Digital options are observed in a variety of economic applications, including mortgage tax, emission fines, venture capital, and credit risk models.