MILAN—Troubled lender Banca Monte dei Paschi di Siena SpA on Monday launched a €4.3 billion ($4.6 billion) debt-to-equity swap offer, the first step of an ambitious recapitalization plan it needs to carry out by the end of the year.

Over the summer the bank pledged to shed around €28 billion worth of bad loans and raise in the market €5 billion of fresh capital to cover losses ensuing from the sale, as it will sell the loans at a lower price than it is has written in its books. The plan is aimed at putting an end to its financial...