1. University of Economics, Faculty of Informatics Dolnozemská cesta 1, 852 35 Bratislava Slovak Republic Financial Mathematics in Derivative Securities and Risk Reduction Fundamental Role of Derivative Securities and Portfolio Insurance Ass. Prof. Ľudovít Pinda, CSc. Department of Mathematics, Tel.:++421 2 67295 813, ++421 2 67295 711 Fax:++421 2 62412195 e-mail: pinda@dec.euba.sk

2. Sylabus of the lecture

3. Statistic concepts and measures Fig. 1

4. .

5. Securitties and market portfolio

7. Fig. 2

9. The risk of a portfolio Tab. 1 Variance-covariance matrix with weights for M assets

10. Fig. 3 Tab. 2

11. Fig. 4

12. The fundamental role of derivative securities.

13. Fig. 5

14. Fig. 6.

16. Example 1 Tab. 3

17. Fig. 7

18. Fig. 8

19. Tab. 4 Example 2

21. Contract specifications of stock index futures SP 500 Tab. 5

22. Stock index arbitrage on SP 500, the futures contracts are underpriced

25. Fig. 9

27. Fig. 10

28. The interest rate futures Tab. 6

31. Determining the future price

32. Duration-based hedging strategies

33. Example 3 Tab. 7 The calculation of duration

35. Maximum profit Maximum loss Profit Loss Writter Investor Maximum profit Maximum loss Profit Loss Writer Investor INSURANCE PORTFOLIO Call option ( long and short position ) Put option ( long and short position )

36. The Generalized Black – Scholes Option Pricing Formula Black-Scholes (1973) stock option model, Merton (1973) stock option model with continuous dividend yield, Black (1976) futures option model, Garman and Kohlhagen (1983) currency option model.

37. The cumulative normal distribution function

38. Portfolio insurance Fig. 11 Value of no insure portfolio M KL Value of insure portfolio

39. F % Normal distribution Expected return Proba bility Preferred distribution Fig. 12

40. SITUATION British fund - £ 10 mil. – diversified portfolio, FTSE Index – 2 000 index points, 1 index point - £ 10 Short – term interest rate – 10 %, Problem: To protect the fund from a fall in FTSE below 1 800 index points. The simple solution: To buy put options on the index at an exercise price of 1 800 index points Investor need: £ 10 000 000 / (£ 10 · 2 000) = 500 put contracts

41. If index fall to 1 700 index poins: The total fund value = (1 700/2 000) · £10 mil. + 500 · £ 10 · 100 = = £ 8.5 mil. +£ 0.5 mil. Insurance is not free: The price of an 1 800 put option = 40 index poins Total insurance cost: 500 · 40 · £ 10 = £ 0.2 mil. Manager need: £ 10.2 mil. to implement the strategy To rescale by a factor 10/10.2 = 0.980 then a £ 10 mil. fund need 490 put options, The fund = insurance + invest in shars = 490 · 40 · £ 10 + £ 9.804 mil. = = £ 0.196 mil. + £ 9.804 mil. The fund would garantee a terminal value: (1 800/2000) · £ 9.804 = = £ 8.823 mil.

42. Level of index All-share portfolio value Insured portfolio Value of shares Value of puts at index 1 800 Total value of fund 140076.8631,968.823 15007.57.3531,478.823 160087.8430,988.823 17008.58.3330,498.823 180098.8230 19009.59.3140 2000109.8040 210010.510.2940 22001110.7840 230011.511.2750 24001211.7650 Tab. 8

43. 140016001800200022002400 8 9 10 11 12 13 Insured portfolio Shares only Index level Port- folio value Fig. 13

44. Put-call parita p– Price of Europen call option, c– Price of Europen put option, S– Share, PV(E)– Present value of lending equity The garanteed minimum - £ 8.823 The amount at 10 % : £ 8.823/1.1 = £ 8.021 and £ 10 - £ 8.021 = £ 1.979 mil.

45. index points The number of calls Level of index Value of T-bills Value of 490 calls at E = 1800 Total value of fund 16008,8230 17008,8230 18008,8230 19008,8230,499,313 20008,8230,989,803 21008,8231,4710,293 Tab. 9

46. Assume: Investor was not willing to accept any loss a one – year horizont He invested at risk – free rate £ 10 mil / 1.1 = £ 9.091 mil. £ 10 mil. – £ 9.091 = £ 0.909 mil. in calls. Let calls cost an exercise price 2 000 index points 250 points for one year Buy £ 909 000 / (£ 10 ·250 ) = 363.3 calls The manager participate in £ 3 633 / £ 5 000 = 72.72 % of any rise in market ( £ 5 000 = £ 10 000 000 / 2 000 ). The higher the guaranteed value of the fund => the smaller participation in any rise

47. 5 % gain => £ 10.5 mil. => equivalent index level 2 100 => £ 10 500 000 / 1.1 = £ 9. 545 000 invested in the risk – free assets and £ 0.455 mil. in calls with price 160 points, £ 455 000 / ( 10 · 160 ) = 284.4 calls The manager participate in £ 2 844 / £ 5 000 = 56.88 % of any rise in market ( £ 5 000 = £ 10 000 000 / 2 000 ). The highest guarantee is £ 11 mil. at the 10 % risk – free rate => the participation in any rise is 0 %.

48. Fund size Guarantee Exercise price Calls T-bills cost Participation rate (%) NumberCost e 100.0000-10.000 -100.00 d 10 8.82318004901.978 8.02198.00 c 10 10.00020003640.909 9.09172.72 b 10 10.50021002840.455 9.54556.88 a 10 11.000--- 10.0000.00 Tab. 10

49. 8 140016001800200022002400 9 10 11 12 Index level Portfolio value a d c b Fig. 14

50. Portfolio strategy with respect to insurance: the floor, the participation rate in any rise of the index above the floor. Portfolio value = Floor + max {0, w [ ( g · Index ) - Floor]} g – initial portfolio value per index point, 10.0 / 2 000 = £ 0.005, w – participation rate, Let a floor is £ 10.5 mil. and index level is 2 200 Portfolio value = Floor + max {0, w [ ( g · Index ) - Floor]} = = 10.5 + max { 0, 0.5688 [ ( 0.005 · 2 200 ) – 10.5]} = = 10.5 + max {0, 0.2844} = £ 10.7844 mil.