AACIMP 2010 Summer School lecture by Dmitry Krushinsky. "Applied Mathematics" stream. "The p-Median Problem and Its Applications" course. Part 4. More info at http://summerschool.ssa.org.ua

1. Applications of the PMP
Margin Calculations

2. Outline
• Introduction
• Possible models
• Possible impacts

3. Background: how the stocks are traded
Buy on cash
client cash securities
full price
broker cash
full price
Stock Market

4. Background: how the stocks are traded
Buy on margin
client cash securities
margin
broker cash
full price
Stock Market

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. 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. 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. Background: options
acquired call sold call
acquired put sold put

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. 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. Motivation: World Crisis
World search of
economic lack of
additional
crisis funds sources
…
possible sources of monetary funds

12. Motivation: World Crisis
World search of
economic lack of
additional
crisis funds sources
margins …
possible sources of monetary funds

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. 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. 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. Optimal pairing and the AP
bullish side bearish side
I J
s.t.
- dummy stocks

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. 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. 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. 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. Conclusions
• A possible model based on the PMP
– fast
– flexible:
• no limit on spread size
• any margining rules can be “inserted”