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Revenue from Contracts with Clients - Schedule of Disaggregation of Revenue (Details) - USD ($)
$ in Millions
 12 Months Ended
Sep. 30, 2024
Sep. 30, 2023
Sep. 30, 2022
Disaggregation of Revenue [Line Items]      
Revenue from contract with customer $ 2,215.9 $ 3,493.4 $ 3,607.5
Principal gains, net 1,189.6 1,079.9 1,145.2
Interest income 1,396.8 987.6 219.0
Total revenues 99,887.8 60,856.1 66,036.0
Point-in-time      
Disaggregation of Revenue [Line Items]      
Revenue from contract with customer 2,054.5 3,338.1 3,489.5
Time elapsed      
Disaggregation of Revenue [Line Items]      
Revenue from contract with customer 161.4 155.3 118.0
Total commission and clearing fees      
Disaggregation of Revenue [Line Items]      
Revenue from contract with customer 548.0 498.4 507.9
Total revenues 548.0 498.4 507.9
Total sales-based commission      
Disaggregation of Revenue [Line Items]      
Revenue from contract with customer 317.4 302.0 306.9
Exchange-traded futures and options      
Disaggregation of Revenue [Line Items]      
Revenue from contract with customer 226.1 214.1 210.7
OTC derivative brokerage      
Disaggregation of Revenue [Line Items]      
Revenue from contract with customer 11.3 14.5 16.8
Equities and fixed income      
Disaggregation of Revenue [Line Items]      
Revenue from contract with customer 65.9 57.8 62.9
Mutual funds      
Disaggregation of Revenue [Line Items]      
Revenue from contract with customer 3.1 3.0 4.1
Insurance and annuity products      
Disaggregation of Revenue [Line Items]      
Revenue from contract with customer 11.0 9.2 9.3
Other      
Disaggregation of Revenue [Line Items]      
Revenue from contract with customer 0.0 3.4 3.1
Total trailing commission      
Disaggregation of Revenue [Line Items]      
Revenue from contract with customer 27.9 26.6 30.1
Mutual funds      
Disaggregation of Revenue [Line Items]      
Revenue from contract with customer 12.7 12.4 14.1
Insurance and annuity products      
Disaggregation of Revenue [Line Items]      
Revenue from contract with customer 15.2 14.2 16.0
Clearing fees      
Disaggregation of Revenue [Line Items]      
Revenue from contract with customer 180.2 153.3 153.2
Trade conversion fees      
Disaggregation of Revenue [Line Items]      
Revenue from contract with customer 13.1 8.5 11.5
Other      
Disaggregation of Revenue [Line Items]      
Revenue from contract with customer 9.4 8.0 6.2
Total consulting, management, and account fees      
Disaggregation of Revenue [Line Items]      
Revenue from contract with customer 167.2 159.0 111.3
Total revenues 167.2 159.0 111.3
Underwriting fees      
Disaggregation of Revenue [Line Items]      
Revenue from contract with customer 0.4 0.7 0.5
Asset management fees      
Disaggregation of Revenue [Line Items]      
Revenue from contract with customer 50.9 45.1 43.9
Advisory and consulting fees      
Disaggregation of Revenue [Line Items]      
Revenue from contract with customer 38.2 35.0 30.9
Sweep program fees      
Disaggregation of Revenue [Line Items]      
Revenue from contract with customer 44.4 48.6 13.1
Client account fees      
Disaggregation of Revenue [Line Items]      
Revenue from contract with customer 19.8 15.9 16.0
Other      
Disaggregation of Revenue [Line Items]      
Revenue from contract with customer 13.5 13.7 6.9
Precious metals sales under ASC Topic 606      
Disaggregation of Revenue [Line Items]      
Revenue from contract with customer 1,500.7 2,836.0 2,988.3
Physical precious metals under ASC Topic 815      
Disaggregation of Revenue [Line Items]      
Sales of physical commodities 91,241.1 50,979.5 57,404.3
Physical agricultural and energy products      
Disaggregation of Revenue [Line Items]      
Sales of physical commodities $ 3,844.4 $ 4,315.7 $ 3,660.0