A 65-year-old chronic smoker presents with cough, weight loss, and progressive dyspnea. Chest imaging reveals a centrally located hilar lung mass. Histological examination shows sheets of small round blue cells with scant cytoplasm, hyperchromatic nuclei, a high nuclear-to-cytoplasmic ratio, nuclear molding, frequent mitotic figures, and areas of necrosis. Immunohistochemistry shows tumor cells that are positive for synaptophysin. What is the most likely diagnosis?