Revenue from Contracts with Clients - Revenue from Contracts with Clients (Details) - USD ($) $ in Millions	12 Months Ended
	Sep. 30, 2022	Sep. 30, 2021	Sep. 30, 2020
Disaggregation of Revenue [Line Items]
Revenue from contract with customer	$ 3,607.5	$ 2,119.5	$ 487.3
Principal gains, net	1,145.2	892.0	622.2
Interest income	219.0	102.4	130.9
Total revenues	66,036.0	42,534.2	54,139.6
Point-in-time
Disaggregation of Revenue [Line Items]
Revenue from contract with customer	3,489.5	2,021.8	396.1
Time elapsed
Disaggregation of Revenue [Line Items]
Revenue from contract with customer	118.0	97.7	91.2
Exchange-traded futures and options
Disaggregation of Revenue [Line Items]
Revenue from contract with customer	210.7	190.6	143.7
OTC derivative brokerage
Disaggregation of Revenue [Line Items]
Revenue from contract with customer	16.8	15.9	19.7
Equities and fixed income
Disaggregation of Revenue [Line Items]
Revenue from contract with customer	62.9	60.5	39.9
Mutual funds
Disaggregation of Revenue [Line Items]
Revenue from contract with customer	4.1	5.5	5.2
Insurance and annuity products
Disaggregation of Revenue [Line Items]
Revenue from contract with customer	9.3	9.7	8.4
Other
Disaggregation of Revenue [Line Items]
Revenue from contract with customer	3.1	2.3	1.4
Total sales-based commission
Disaggregation of Revenue [Line Items]
Revenue from contract with customer	306.9	284.5	218.3
Mutual funds
Disaggregation of Revenue [Line Items]
Revenue from contract with customer	14.1	14.5	12.9
Insurance and annuity products
Disaggregation of Revenue [Line Items]
Revenue from contract with customer	16.0	17.0	15.3
Total trailing commission
Disaggregation of Revenue [Line Items]
Revenue from contract with customer	30.1	31.5	28.2
Clearing fees
Disaggregation of Revenue [Line Items]
Revenue from contract with customer	153.2	150.9	139.0
Trade conversion fees
Disaggregation of Revenue [Line Items]
Revenue from contract with customer	11.5	11.2	8.9
Other
Disaggregation of Revenue [Line Items]
Revenue from contract with customer	6.2	9.1	9.2
Total commission and clearing fees
Disaggregation of Revenue [Line Items]
Revenue from contract with customer	507.9	487.2	403.6
Underwriting fees
Disaggregation of Revenue [Line Items]
Revenue from contract with customer	0.5	0.6	0.6
Asset management fees
Disaggregation of Revenue [Line Items]
Revenue from contract with customer	43.9	38.3	31.3
Advisory and consulting fees
Disaggregation of Revenue [Line Items]
Revenue from contract with customer	30.9	24.9	22.2
Sweep program fees
Disaggregation of Revenue [Line Items]
Revenue from contract with customer	13.1	3.0	9.5
Client account fees
Disaggregation of Revenue [Line Items]
Revenue from contract with customer	16.0	15.8	12.3
Other
Disaggregation of Revenue [Line Items]
Revenue from contract with customer	6.9	8.4	7.8
Total consulting, management, and account fees
Disaggregation of Revenue [Line Items]
Revenue from contract with customer	111.3	91.0	83.7
Precious metals retail sales
Disaggregation of Revenue [Line Items]
Revenue from contract with customer	2,988.3	1,541.3	0.0
Physical precious metals trading
Disaggregation of Revenue [Line Items]
Sales of physical commodities	57,404.3	37,250.4	51,598.5
Physical agricultural and energy product trading
Disaggregation of Revenue [Line Items]
Sales of physical commodities	$ 3,660.0	$ 2,169.9	$ 1,300.7