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ESTIMATION OF REQUIRED LIQUIDITY FOR INVESTMENT POSITION

IN FUTURES OF THE ATHENS DERIVATIVES EXCHANGE MARKET.

 

By Dr COSTAS KYRITSIS

University of Portsmouth UK

Department of Mathematics

and Computer Science

Software Laboratory

National Technical University

of Athens

 

 

                                                            Abstract

In this paper we discuss the risk of mark-to-market  loss of positions with leverage, in futures. We make the usual assumptions of Lognormal distribution and geometric Brownian motion for the underlying  as in the Black-Scholes options pricing model. With these assumptions  we estimate tables of  required liquidity for futures on FTSE-20 and FTSE-40 in the Athens Derivative Exchange Market, during the year 2000.

Key words

Derivatives, Geometric Brownian Motion, Stochastic Differential Equations, Simulation, Investment, Liquidity

 

§1 Introduction  Since August 1999 the Athens Derivative Exchange Market (ADEX) introduced for  the first time futures on the Index FTSE-20, and soon afterwards on the Index FTSE-40. At present the average amount of money that is traded in ADEX is 10-15 billion drachmas that is almost 3% of the traded money in the Athens Stock Exchange Market (ASE, October 2000). The interest is increasing with the introduction of options and stock lending. Nevertheless the peculiarities and risks of investing to futures are not quite clear to the present average investor. For example it is highly risky and not advisable to leave investment positions of futures that have loss till the market has reverse trend and they became profitable, as is done with securities. There is the risk of running out of the budget and eventually closing the position before it becomes profitable. In this paper we analyze the source of this risk and we estimate the required amount of money that should be kept in cash, for probable losses.

§2 Leverage in positions on futures

When investing in positions on futures we do not pay all the money of the investment.

Instead it is calculated daily the profit or loss of the investment position (called mark-to-market) and is paid by the investors and Broker Companies to appropriate clearance bank (Alpha Credit Bank). In addition it is paid a percentage only of the height of the position, as much as it is considered it is risked for 1-2 days for ADEX to close the position if anything goes wrong (default position). The percentage is estimated according to the volatility (standard deviation) of the daily percentage changes of the underlying Index and is called Margin . This percentage at present is 12% for futures on the Index FTSE-20 and 16% for futures on the Index FTSE-40. This makes an advantage for the investor as he must only pay 12%-16% of the height of a position when he opens it. This is called the leverage of the position and is a multiplier of  1/12%=8.33 times and 1/16%=6.25 times respectively. Of course not only the rate of return is multiplied with this numbers but also the Beta (or Elasticity) of the position. The advantage of leverage has also its risks , as the profit or loss is paid daily on 100% of the height of the position and in a reverse trend of the market can easily lead to bankruptcy, something not really possible with investment positions in securities.

§3 Volatility of the price of the underlying .

In order to estimate the required Liquidity we need a forecasting of the underlying Index for the investment horizon, and  the correlation and coupling of the future with the underlying Index .

The most difficult parameter is that of the Market, in other words the forecasting of the Index. This obviously depends on the season and the particular trend of the market at the Investment time.

We shall go around this difficulty in a way that is standard in the pricing of Derivatives and is also used by ADEX in the estimation of the percentages of 12% and 16% for the margin. We shall assume a neutral market, neither growing neither decaying, but with a trend equal to the risk free rate (5.59% in a year base, at present October 2000). So the model of the underlying index is, as in the Black-Scholes option pricing model ,a Geometric Brownian Motion (continuous time random compound interest) of normally distributed rate r and volatility ó. .For the definition of the stochastic differential equations and the geometric Brownian motion see Oksental p121 Chpt. V p 60 ,exerc.7.9 ,p 121,example 5.1 p 60

The stochastic differential equation of Brownian motion (Ito interpretation)  is:

 

                                                                                (1)

The exact interpretation of the symbols requires the concepts and definitions of stochastic Integrals and is outside the scope of this paper. For the definitions see Oksental 1995.

            The distribution of the prices Xt is Lognormal.

The solution of this stochastic differential equation is given by the formula:

                                                               (2)

where Bt is a Brownian Motion

The logarithm of this process Xt/ X0  is an ordinary Brownian motion with drift:

log(Xt/ X0)=(r-(1/2) s2)t + s Bt .                                                                      (3)

The average time T that it reaches X for the first time starting from x0 

Is T=log(X/ x0)/(r-(1/2) s2)                                                                               (4)

 

If  â is the beta (elasticity)  of the future Yt over the index Xt we  assume that the futures also, follows the equation

                                                                                        (5)

 

If l is the leverage of the investment the value of the investment position on the future follwos actually the equation

 

                                                                                      (6)

 

The conditional variance of the Index price Xt  is calculated to have the formula

(see Karlin S-Taylor H.M.(1975)  p  357)

 

Var(X(t)/X(0)=X0 )=X2Exp(2t(r+ó2/2))*(Exp(tó2)-1)                           (7)       

§4 Required Liquidity tables .

For the required liquidity of position of horizon t we may take the conditional volatility (standard deviation) of the price of the future .We should not forget that these formulas assume a neutral market and the liquidity is the average and not a worse scenario.

From the above  formulae and assuming â=1 ,r=5.59% ,ó=40% (historic volatility of 1 year) for the FTSE-20 and ó=47% for the FTSE-40 (values that are used by ADEX at present and are estimated by the daily data) we compute the next tables.

Days-FTSE-20

Liquidity Percentage

Days-FTSE-20

Liquidity Percentage

Days-FTSE-20

Liquidity Percentage

Days-FTSE-20

Liquidity Percentage

1

2.53%

21

11.77%

41

16.68%

61

20.65%

2

3.58%

22

12.06%

42

16.90%

62

20.83%

3

4.39%

23

12.34%

43

17.11%

63

21.01%

4

5.07%

24

12.61%

44

17.32%

64

21.19%

5

5.68%

25

12.88%

45

17.53%

65

21.37%

6

6.22%

26

13.14%

46

17.74%

66

21.55%

7

6.73%

27

13.40%

47

17.94%

67

21.73%

8

7.20%

28

13.66%

48

18.14%

68

21.91%

9

7.64%

29

13.91%

49

18.35%

69

22.09%

10

8.06%

30

14.16%

50

18.54%

70

22.26%

11

8.46%

31

14.40%

51

18.74%

71

22.44%

12

8.84%

32

14.64%

52

18.94%

72

22.61%

13

9.21%

33

14.88%

53

19.13%

73

22.78%

14

9.56%

34

15.12%

54

19.33%

74

22.95%

15

9.90%

35

15.35%

55

19.52%

75

23.13%

16

10.24%

36

15.58%

56

19.71%

76

23.30%

17

10.56%

37

15.80%

57

19.90%

77

23.47%

18

10.87%

38

16.03%

58

20.09%

78

23.63%

19

11.18%

39

16.25%

59

20.28%

 

 

20

11.48%

40

16.47%

60

20.46%

 

 

 

Days/FTSE-40

Liquidity Percentage

Days/FTSE-40

Liquidity Percentage

Days/FTSE-40

Liquidity Percentage

Days/FTSE-40

Liquidity Percentage

1

2.98%

21

13.88%

41

19.75%

61

24.53%

2

4.21%

22

14.22%

42

20.01%

62

24.75%

3

5.16%

23

14.56%

43

20.26%

63

24.98%

4

5.97%

24

14.88%

44

20.52%

64

25.20%

5

6.68%

25

15.20%

45

20.77%

65

25.42%

6

7.32%

26

15.52%

46

21.02%

66

25.63%

7

7.91%

27

15.83%

47

21.26%

67

25.85%

8

8.47%

28

16.13%

48

21.51%

68

26.07%

9

8.99%

29

16.43%

49

21.75%

69

26.28%

10

9.49%

30

16.73%

50

21.99%

70

26.49%

11

9.96%

31

17.02%

51

22.23%

71

26.71%

12

10.41%

32

17.31%

52

22.47%

72

26.92%

13

10.84%

33

17.59%

53

22.70%

73

27.13%

14

11.26%

34

17.87%

54

22.94%

74

27.34%

15

11.67%

35

18.15%

55

23.17%

75

27.55%

16

12.06%

36

18.42%

56

23.40%

76

27.76%

17

12.45%

37

18.70%

57

23.63%

77

27.96%

18

12.82%

38

18.96%

58

23.86%

78

28.17%

19

13.18%

39

19.23%

59

24.08%

 

 

20

13.54%

40

19.49%

60

24.31%

 

 


 

 

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