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Revenue from Contracts with Clients - Schedule of Revenue from Contracts with Clients (Details) - USD ($)
$ in Millions
 3 Months Ended
Dec. 31, 2024
Dec. 31, 2023
Disaggregation of Revenue [Line Items]    
Revenue from contract with customer $ 663.1 $ 627.4
Principal gains, net 308.9 293.8
Interest income 378.2 290.1
Total revenues 27,935.3 19,573.0
Total operating revenues 944.3 784.2
United States    
Disaggregation of Revenue [Line Items]    
Total revenues 1,695.6 1,603.5
Total operating revenues 659.1 568.0
Europe    
Disaggregation of Revenue [Line Items]    
Total revenues 782.5 658.4
Total operating revenues 189.6 135.6
South America    
Disaggregation of Revenue [Line Items]    
Total revenues 168.0 128.0
Total operating revenues 35.9 41.2
Middle East and Asia    
Disaggregation of Revenue [Line Items]    
Total revenues 25,285.4 17,175.6
Total operating revenues 55.9 32.0
Other    
Disaggregation of Revenue [Line Items]    
Total revenues 3.8 7.5
Total operating revenues 3.8 7.4
Point-in-time    
Disaggregation of Revenue [Line Items]    
Revenue from contract with customer 620.7 589.7
Time elapsed    
Disaggregation of Revenue [Line Items]    
Revenue from contract with customer 42.4 37.7
Exchange-traded futures and options    
Disaggregation of Revenue [Line Items]    
Revenue from contract with customer 63.2 51.4
OTC derivative brokerage    
Disaggregation of Revenue [Line Items]    
Revenue from contract with customer 2.8 2.8
Equities and fixed income    
Disaggregation of Revenue [Line Items]    
Revenue from contract with customer 15.1 16.5
Mutual funds    
Disaggregation of Revenue [Line Items]    
Revenue from contract with customer 0.8 0.7
Insurance and annuity products    
Disaggregation of Revenue [Line Items]    
Revenue from contract with customer 3.2 2.0
Other    
Disaggregation of Revenue [Line Items]    
Revenue from contract with customer 0.4 (0.1)
Total sales-based commission    
Disaggregation of Revenue [Line Items]    
Revenue from contract with customer 85.5 73.3
Mutual funds    
Disaggregation of Revenue [Line Items]    
Revenue from contract with customer 3.4 3.0
Insurance and annuity products    
Disaggregation of Revenue [Line Items]    
Revenue from contract with customer 4.0 3.7
Total trailing commission    
Disaggregation of Revenue [Line Items]    
Revenue from contract with customer 7.4 6.7
Clearing fees    
Disaggregation of Revenue [Line Items]    
Revenue from contract with customer 44.1 43.3
Trade conversion fees    
Disaggregation of Revenue [Line Items]    
Revenue from contract with customer 2.2 4.0
Other    
Disaggregation of Revenue [Line Items]    
Revenue from contract with customer 10.1 2.4
Total commission and clearing fees    
Disaggregation of Revenue [Line Items]    
Revenue from contract with customer 149.3 129.7
Total revenues 149.3 129.7
Underwriting fees    
Disaggregation of Revenue [Line Items]    
Revenue from contract with customer 0.3 0.0
Asset management fees    
Disaggregation of Revenue [Line Items]    
Revenue from contract with customer 14.8 11.4
Advisory and consulting fees    
Disaggregation of Revenue [Line Items]    
Revenue from contract with customer 8.6 8.2
Sweep program fees    
Disaggregation of Revenue [Line Items]    
Revenue from contract with customer 11.6 11.4
Client account fees    
Disaggregation of Revenue [Line Items]    
Revenue from contract with customer 6.2 4.2
Other    
Disaggregation of Revenue [Line Items]    
Revenue from contract with customer 6.3 3.3
Total consulting, management, and account fees    
Disaggregation of Revenue [Line Items]    
Revenue from contract with customer 47.8 38.5
Total revenues 47.8 38.5
Precious metals sales under ASC Topic 606    
Disaggregation of Revenue [Line Items]    
Revenue from contract with customer 466.0 459.2
Physical precious metals under ASC Topic 815    
Disaggregation of Revenue [Line Items]    
Sales of physical commodities 25,522.4 17,162.5
Physical agricultural and energy products    
Disaggregation of Revenue [Line Items]    
Sales of physical commodities $ 1,062.7 $ 1,199.2
Revenue from Contract with Customer Benchmark | Customer Concentration Risk | Clients    
Disaggregation of Revenue [Line Items]    
Revenues from contracts with clients as a percentage of total revenues 2.40% 3.20%