Revenue from Contracts with Clients - Revenue from Contracts with Clients (Details) - USD ($) $ in Millions	12 Months Ended
	Sep. 30, 2023	Sep. 30, 2022	Sep. 30, 2021
Disaggregation of Revenue [Line Items]
Revenue from contract with customer	$ 3,493.4	$ 3,607.5	$ 2,119.5
Principal gains, net	1,079.9	1,145.2	892.0
Interest income	987.6	219.0	102.4
Total revenues	60,856.1	66,036.0	42,534.2
Point-in-time
Disaggregation of Revenue [Line Items]
Revenue from contract with customer	3,338.1	3,489.5	2,021.8
Time elapsed
Disaggregation of Revenue [Line Items]
Revenue from contract with customer	155.3	118.0	97.7
Exchange-traded futures and options
Disaggregation of Revenue [Line Items]
Revenue from contract with customer	214.1	210.7	190.6
OTC derivative brokerage
Disaggregation of Revenue [Line Items]
Revenue from contract with customer	14.5	16.8	15.9
Equities and fixed income
Disaggregation of Revenue [Line Items]
Revenue from contract with customer	57.8	62.9	60.5
Mutual funds
Disaggregation of Revenue [Line Items]
Revenue from contract with customer	3.0	4.1	5.5
Insurance and annuity products
Disaggregation of Revenue [Line Items]
Revenue from contract with customer	9.2	9.3	9.7
Other
Disaggregation of Revenue [Line Items]
Revenue from contract with customer	3.4	3.1	2.3
Total sales-based commission
Disaggregation of Revenue [Line Items]
Revenue from contract with customer	302.0	306.9	284.5
Mutual funds
Disaggregation of Revenue [Line Items]
Revenue from contract with customer	12.4	14.1	14.5
Insurance and annuity products
Disaggregation of Revenue [Line Items]
Revenue from contract with customer	14.2	16.0	17.0
Total trailing commission
Disaggregation of Revenue [Line Items]
Revenue from contract with customer	26.6	30.1	31.5
Clearing fees
Disaggregation of Revenue [Line Items]
Revenue from contract with customer	153.3	153.2	150.9
Trade conversion fees
Disaggregation of Revenue [Line Items]
Revenue from contract with customer	8.5	11.5	11.2
Other
Disaggregation of Revenue [Line Items]
Revenue from contract with customer	8.0	6.2	9.1
Total commission and clearing fees
Disaggregation of Revenue [Line Items]
Revenue from contract with customer	498.4	507.9	487.2
Total revenues	498.4	507.9	487.2
Underwriting fees
Disaggregation of Revenue [Line Items]
Revenue from contract with customer	0.7	0.5	0.6
Asset management fees
Disaggregation of Revenue [Line Items]
Revenue from contract with customer	45.1	43.9	38.3
Advisory and consulting fees
Disaggregation of Revenue [Line Items]
Revenue from contract with customer	35.0	30.9	24.9
Sweep program fees
Disaggregation of Revenue [Line Items]
Revenue from contract with customer	48.6	13.1	3.0
Client account fees
Disaggregation of Revenue [Line Items]
Revenue from contract with customer	15.9	16.0	15.8
Other
Disaggregation of Revenue [Line Items]
Revenue from contract with customer	13.7	6.9	8.4
Total consulting, management, and account fees
Disaggregation of Revenue [Line Items]
Revenue from contract with customer	159.0	111.3	91.0
Total revenues	159.0	111.3	91.0
Precious metals sales under ASC Topic 606
Disaggregation of Revenue [Line Items]
Revenue from contract with customer	2,836.0	2,988.3	1,541.3
Physical precious metals under ASC Topic 815
Disaggregation of Revenue [Line Items]
Sales of physical commodities	50,979.5	57,404.3	37,250.4
Physical agricultural and energy products
Disaggregation of Revenue [Line Items]
Sales of physical commodities	$ 4,315.7	$ 3,660.0	$ 2,169.9