Before the financial crisis started in 2007, the forward rate agreement contracts could be perfectly replicated by overnight indexed swap zero coupon bonds. After the crisis, the simply compounded risk-free overnight indexed swap forward rate became less than the forward rate agreement rate. Using an approach proposed by Cuchiero, Klein, and Teichmann, we construct an arbitrage-free market model, where the forward spread curves for a given finite tenor structure are described as a mild solution to a boundary value problem for a system of infinite-dimensional stochastic differential equations. The constructed financial market is large: it contains infinitely many overnight indexed swap zero coupon bonds and forward rate agreement contracts with all possible maturities. We also investigate the necessary assumptions and conditions which guarantee existence, uniqueness and non-negativity of solutions to the obtained boundary value problem.