Ekonomická univerzita, Fakulta hospodárskej informatiky Dolnozemská cesta, 85 35 Bratislava INSURANCE PORTFOLIO Doc. RNDr. Ľudovít t Pinda, CSc. FHI EU, Katedra matematiky mail: pinda@euba.sk Marec 00 Bratislava
Call option ( long and short position ) Investor Maximum profit Maximum loss Profit Loss Writter Fig. Put option ( long and short position ) Investor Profit Loss Maximum profit Maximum loss Writer Fig.
The Generalized Black Scholes Option Pricing Formula c GBS p GBS N( d ) S X Price of Europen call option, Price of Europen put option, The cumulative normal distribution function, Stock price, Strike price of option, T σ r Time to expiration in year, Volatility of the relative price change of the underlying stock price, Risk free interest rate. c GBS = S e ( b r ) T r T N( d ) X e N( ) d ()
p GBS = X e N ( b r ) T ( d ) S e N ( ) r T d () b = r Black-Scholes (973) stock option model, b = r-q Merton (973) stock option model with continuous dividend yield, b = 0 Black (976) futures option model, b = r-r f Garman and Kohlhagen (983) currency option model. The cumulative normal distribution function N ( x) = e dz π x z (3) N ( x) n = N 3 ( x)( a k + a k + a k ), ( x), 3 for x 0, for x < 0, k = + 0,3367 x a = 0.436836, a = 0.0676, a3 = 0.937980
( )., ln, T d d T T b X S d e d n d σ σ σ π = + + = = Portfolio Portfolio insurance insurance Value of no insure portfolio M K L Value of insure portfolio Fig. 3
Pro bab ilit y Preferred distribution Normal distribution F % Expected return Fig. 4 SITUATION British fund - 0 mil. diversified portfolio, FTSE Index 000 index points, index point - 0 Short term interest rate 0 %,
Problem: To protect the fund from a fall in FTSE below 800 index points. The simple solution: To buy put options on the index at an exercise price of 800 index points Investor need: 0 000 000 / ( 0 000) = 500 put contracts If index fall to 700 index poins: The total fund value = ( 700/ 000) 0 mil. + 500 0 00 = = 8.5 mil. + 0.5 mil. Insurance is not free: The price of an 800 put option = 40 index poins Total insurance cost: 500 40 0 = 0. mil. Manager need: 0. mil. to implement the strategy rather than a 0 mil. To rescale by a factor 0/0. = 0.980 then a 0 mil. fund need 490 put options, the fund = insurance + invest in shars = 490 40 0 + 9.804 mil. = = 0.96 mil. + 9.804 mil. The fund would garantee a terminal value: ( 800/000) 9.804 = 8.83 Level All-share Insured portfolio of index portfolio value Value of shares Value of puts at index 800 Total value of fund 400 7 6.863,96 8.83 500 7.5 7.353,47 8.83 600 8 7.843 0,98 8.83 700 8.5 8.333 0,49 8.83 800 9 8.83 0 8.83 900 9.5 9.34 0 9.34 000 0 9.804 0 9.804 00 0.5 0.94 0 0.94 00 0.784 0 0.784 300.5.75 0.75 400.765 0.765 Tab.
3 Por tfol io val ue 0 9 8 Shares only Insured portfolio 400 600 800 000 00 400 Index level Put-call parita p S = c + PV (E) Fig. 5 + (4) p - Price of Europen call option, c - Price of Europen put option, S - Share, PV(E) - Present value of lending equity.
The garanteed minimum - 8.83 The amount at 0 % - 8.83/. = 8.0 and 0-8.0 =.979 mil. 800 c = p + S PV( E) = 40+ 000 = 403.6, The number of calls - 979000 ( 0 403,6) = 490. index points Level of Value of 490 calls Total value of Value of T-bills index at E = 800 fund 600 8,83 0 8,83 700 8,83 0 8,83 800 8,83 0 8,83 900 8,83 0,49 9,33 000 8,83 0,98 9,803 00 8,83,47 0,93 Tab. Assume: Investor was not willing to accept any loss a one year horizont He invested at risk free rate 0 mil /. = 9.09 mil. 0 mil. - 9.09 = 0.909 mil. in calls. Let calls cost an exercise price 000 index points 50 points for one year Buy 909 000 / ( 0 50 ) = 363.3 calls
For each point exceeded 000 the manager would gain 0 363.3 = 3 633 The manager participate in 3 633 / 5 000 = 7.7 % of any rise in market ( 5 000 = 0 000 000 / 000 ). The higher the guaranteed value of the fund => the smaller participation in any rise 5 % gain => 0.5 mil. => equivalent index level 00 => 0 500 000 /. = = 9. 545 000 invested in the risk free assets and 0.455 mil. in calls with price 60 points, 455 000 / ( 0 60 ) = 84.4 calls The manager participate in 844 / 5 000 = 56.88 % of any rise in market ( 5 000 = 0 000 000 / 000 ). The highest guarantee is mil. at the 0 % risk free rate => the participation in any rise is 0 %. Fund Guarantee Exercise Calls T-bills Participation size price Number Cost cost rate (%) e 0 0.000 0-0.000-00.00 d 0 8.83 800 490.978 8.0 98.00 c 0 0.000 000 364 0.909 9.09 7.7 b 0 0.500 00 84 0.455 9.545 56.88 a 0.000 - - - 0.000 0.00 Tab. 3
Portfolio value 0 a b c 9 d 8 400 600 800 000 00 400 Index level Fig. 6 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, 0.0 / 000 = 0.005, w participation rate, Let a floor is 0.5 mil. and index level is 00 Portfolio value = Floor + max {0, w [ ( g Index ) - Floor]} = = 0.5 + max { 0, 0.5688 [ ( 0.005 00 ) 0.5]} = = 0.5 + max {0, 0.844} = 0.7844 mil.
Ďakujem za pozornosť Doc. RNDr. Ľudovít Pinda, CSc. mail: pinda@euba.sk