Cook the spaghetti al dente, according to the package directions. Reserve 1/2 cup of the pasta water and drain.
Meanwhile, in a food processor, combine the garlic, green olives, parsley, and basil. With the motor running, slowly add the olive oil and process until the oil is fully incorporated. Reserve 1/4 cup of the Green Olive Pesto for another use. Roughly chop the black olives and reserve half for another use. In a large skillet, over medium heat, cook the remaining pesto for 2 minutes. Add the spaghetti, reserved pasta water, remaining black olives, and salt and cook until the water is absorbed and the spaghetti is heated through.