  We solve the first-passage problem for the Heston random diffusion model. We
obtain exact analytical expressions for the survival and hitting probabilities
to a given level of return. We study several asymptotic behaviors and obtain
approximate forms of these probabilities which prove, among other interesting
properties, the non-existence of a mean first-passage time. One significant
result is the evidence of extreme deviations --which implies a high risk of
default-- when certain dimensionless parameter, related to the strength of the
volatility fluctuations, increases. We believe that this may provide an
effective tool for risk control which can be readily applicable to real
markets.
