  We study and solve the worst-case optimal portfolio problem as pioneered by
Korn and Wilmott (2002) of an investor with logarithmic preferences facing the
possibility of a market crash with stochastic market coefficients by enhancing
the martingale approach developed by Seifried in 2010. With the help of
backward stochastic differential equations (BSDEs), we are able to characterize
the resulting indifference optimal strategies in a fairly general setting. We
also deal with the question of existence of those indifference strategies for
market models with an unbounded market price of risk. We therefore solve the
corresponding BSDEs via solving their associated PDEs using a utility
crash-exposure transformation. Our approach is subsequently demonstrated for
Heston's stochastic volatility model, Bates' stochastic volatility model
including jumps, and Kim-Omberg's model for a stochastic excess return.
