Limited data availability poses a major obstacle in training deep learning models for financial applications. Synthesizing financial time series to augment real-world data is challenging due to the irregular and scale-invariant patterns uniquely associated with financial time series - temporal dynamics that repeat with varying duration and magnitude. Such dynamics cannot be captured by existing approaches, which often assume regularity and uniformity in the underlying data. We develop a novel generative framework called FTS-Diffusion to model irregular and scale-invariant patterns that consists of three modules. First, we develop a scale-invariant pattern recognition algorithm to extract recurring patterns that vary in duration and magnitude. Second, we construct a diffusion-based generative network to synthesize segments of patterns. Third, we model the temporal transition of patterns in order to aggregate the generated segments. Extensive experiments show that FTS-Diffusion generates synthetic financial time series highly resembling observed data, outperforming state-of-the-art alternatives. Two downstream experiments demonstrate that augmenting real-world data with synthetic data generated by FTS-Diffusion reduces the error of stock market prediction by up to 17.9%. To the best of our knowledge, this is the first work on generating intricate time series with irregular and scale-invariant patterns, addressing data limitation issues in finance.