Higher or Lower on the Next Card

A dealer shuffles a standard $52$-card deck and turns one card face up. Ranks run from ace low ($1$) to king ($13$); suits are irrelevant. Looking at that card, you call whether the *next* card off the deck will be higher or lower in rank. A tie -- the next card matching the rank of the face-up card -- counts as a loss.

Playing optimally, what is your probability of winning, and what is the fair price to enter a game that pays $\$1$ on a win and nothing on a loss?