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# Applications of the PMP. Margin Calculations

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AACIMP 2010 Summer School lecture by Dmitry Krushinsky. "Applied Mathematics" stream. "The p-Median Problem and Its Applications" course. Part 4.

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### Applications of the PMP. Margin Calculations

1. 1. Applications of the PMP Margin Calculations
2. 2. Outline • Introduction • Possible models • Possible impacts
3. 3. Background: how the stocks are traded Buy on cash client cash securities full price broker cash full price Stock Market
4. 4. Background: how the stocks are traded Buy on margin client cash securities margin broker cash full price Stock Market
5. 5. Background: definitions Margin – the amount of cash Margin – the money that that must be paid at the are lying without use moment of a buy-sell transaction Margin rate – percentage of the price that must be paid at the moment of a buy-sell transaction Types of Securities Simple: Derivatives: -shares -options -bonds -[futures] -etc.
6. 6. Background: options Option – a contract that gives its owner a right to buy/sell a specified amount of specified stock within a specified time period for a specified price (exercise price) call option put option good for buyer right to buy right to sell raising market seller obligation to sell obligation to buy good for falling market
7. 7. Background: options • A profit-loss diagram (acquired call) critical PROFIT point market price of amount paid the underlying LOSS for the option stock exercise price
8. 8. Background: options acquired call sold call acquired put sold put
9. 9. Margining and risk management Simple securities Spreads (combinations) profit profit profit price price price loss loss loss Options profit profit profit underlying underlying underlying loss price loss price loss price
10. 10. Motivation: Euronext Amsterdam stock market  average daily turnover of options in January 2009 was 3,356,541 Euro  average margin rate is 40%  1,342,616 Euro were kept as margin daily more that 1 million Euro are kept away from investments every day
11. 11. Motivation: World Crisis World search of economic lack of additional crisis funds sources … possible sources of monetary funds
12. 12. Motivation: World Crisis World search of economic lack of additional crisis funds sources margins … possible sources of monetary funds
13. 13. Zero Margin Rate: Cure or Disaster? For clients: • unlimited investments For brokers: • attractive for clients • more money is lent −−> higher income But: Margining is the only mechanism that protects brokers from clients’ default!
14. 14. Optimal pairing and the AP AP = Assignment Problem bullish side bearish side Securities that Securities that bring profit bring profit when the when the (underlying) (underlying) price goes up price goes down
15. 15. Optimal pairing and the AP AP = Assignment Problem bullish side bearish side Securities that Securities that bring profit bring profit when the when the (underlying) (underlying) price goes up price goes down
16. 16. Optimal pairing and the AP bullish side bearish side I J s.t. - dummy stocks
17. 17. N-tuples and the Multidimensional AP structure of the cost objective function matrix pairs 2D AP triples 3D AP N-tuples … MAP (ND AP)
18. 18. MAP and PMP MAP = Multidimensional AP PMP = p-Median Problem MAP clustering PMP bounded number of bounded number of components in a cluster clusters =N =p no dummies !
19. 19. Conclusions: a note on importance • Existing tools use heuristic procedures and AP model • Tools approved by stock exchanges can catch only two- component spreads • Tools capable of catching spreads with more than four components do not exist • If the number of components in a spread is doubled, the margin is halved
20. 20. Conclusions: impact Euronext Amsterdam:  average daily turnover of options in January 2009 was 3,356,541 Euro  average margin rate is 40%  1,342,616 Euro were kept as margin daily This amount exceeds Dutch Government gross If 4-component spreads are considered: external debt in  margin rate is halved 2008  671,308 Euro are set free daily  additional ~200,000,000 Euro are available yearly
21. 21. Conclusions • A possible model based on the PMP – fast – flexible: • no limit on spread size • any margining rules can be “inserted”