金融建模Black-Scholes Model V3

1、Black-Scholes Option Pricing Model copyright THC 2003. All rights reserved by Thomas Ho Company, New York, NY. USA www. reference number9classification#40401VersionV3levelbasicInstructionspublication dateJul-031. The risk source of both the call and put options and the underlying asset is the same.a

2、uthorHanyang Financial Engineering Lab.2. Enter the option descriptionsaffiliationHanyang Universityinputoption terms and conditionsemail addressleesbhanyang.ac.kroutputoption price and the sensitivitieslast revised dateJan-04referencesBlack, F. and M. Scholes, 1973, The Pricing of Options and Corpo

3、rate Liabilities, Journal of Political Economy , Vol.81, 637-654.Ch. 4 of The Oxford Guide to Financial Modeling by Thomas S.Y. Ho and Sang Bin Lee, 2004, Oxford University PressDescriptionsfinancial model classequity optionsissuer/modelermarket participantsmodel typecontinuous time modelrisk source

4、sstock pricerisk distributionlognormaleconomic assumptionsperfect capital markettechnical assumptionscontinuous compounding interest ratekey wordsthe law of one price, no abitrage condition, dynamic hedging, hedge ratio, greeksLinksdataN/Afinancial modelsInputsStock Price(S)100Strike Price(K)100Time

5、 to expiration(T)1Stock volitility()0.2Risk-free rate(r)0.04Dividend yieldsN/AOutputsCallPutPrice9.925056.00400 C = S* N(d1) - Ke-rT N(d2)(Delta)0.61791-0.38209 P = -S* N(-d1) + Ke-rT N(-d2)(Gamma)0.019070.01907v (Vega)38.1387838.13878 (Theta)-5.88852-2.04536 (Rho)51.86609-44.21286 (Omega)6.22577-6.

6、36390Interim CalcuationsStep 1Calculate call price and put price.d10.3d20.12 1ln 2 S RTK d Ts s+= Step 2Calculate option greeks.CallPut(Delta)(Gamma)v (Vega) (Theta) (Rho) (Omega)Step 3Show the performance profile of option greeks.Stock Price505152535455Delta0.00080.00110.00150.00200.00270.0036Gamma

7、0.00030.00040.00050.00060.00080.0010Vega0.13290.18460.25230.33970.45080.5901Theta-0.0148-0.0206-0.0282-0.0380-0.0506-0.0664Rho0.03670.05230.07330.10130.13810.185521ddTs=-1()C N dS D =1() 1P N dS D =-1 2( )()2rTS N dC r K eN dt Ts- Q = - 1 2( )()2rTS N dP r K eNdt Ts- Q = -+ -1()N dS STsDG = 1( )N dS

8、 STsDG = 1()ST N dn =1()ST N dn =2()rTK T eN dr-= 2()rTK T eNdr-= - -1()CSSN dSCC W =( )1( ) 1PSSN dSPP W =-Delta of European Call Option50576471788592991061131201271341411480.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Stock PriceDeltaGamma of European Call Option5057647178859299106113120127134141148

9、0.000 0.005 0.010 0.015 0.020 0.025 Stock PriceGamma2 1ln 2 S RTK d Ts s+= Delta of European Call Option50576471788592991061131201271341411480.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Stock PriceDeltaGamma of European Call Option50576471788592991061131201271341411480.000 0.005 0.010 0.015 0.020 0.0

10、25 Stock PriceGamma copyright THC 2003. All rights reserved by Thomas Ho Company, New York, NY. USA www. 1. The risk source of both the call and put options and the underlying asset is the same.option price and the sensitivitiesBlack, F. and M. Scholes, 1973, The Pricing of Options and Corporate Lia

11、bilities, Journal of Political Economy , Vol.81, 637-654.Ch. 4 of The Oxford Guide to Financial Modeling by Thomas S.Y. Ho and Sang Bin Lee, 2004, Oxford University Press56575859606162630.00470.00600.00770.00970.01210.01490.01830.02220.00120.00150.00180.00220.00260.00310.00360.00420.76240.97291.2269

12、1.52971.88682.30312.78373.3327-0.0861-0.1102-0.1394-0.1744-0.2158-0.2644-0.3207-0.38530.24620.32260.41790.53530.67840.85111.05741.3015V ega of European Call Option50576471788592991061131201271341411480.00005.000010.000015.000020.000025.000030.000035.000040.000045.0000Stock PriceVegaTheta of European

13、 Call Option505662687480869298104110116122128134140146(7.0)(6.0)(5.0)(4.0)(3.0)(2.0)(1.0)0.0 stock PriceThetaPut21ddTs=-1() 1P N dS D =-1 2( )()2rTS N dP r K eNdt Ts- Q = -+ -1( )N dS STsDG = 1()ST N dn =2()rTK T eNdr-= - -( )1( ) 1PSSN dSPP W =-Gamma of European Call Option5057647178859299106113120

14、1271341411480.000 0.005 0.010 0.015 0.020 0.025 Stock PriceGammaGamma of European Call Option50576471788592991061131201271341411480.000 0.005 0.010 0.015 0.020 0.025 Stock PriceGammaV ega of European Call Option50576471788592991061131201271341411480.00005.000010.000015.000020.000025.000030.000035.00

15、0040.000045.0000Stock PriceVegaTheta of European Call Option505662687480869298104110116122128134140146(7.0)(6.0)(5.0)(4.0)(3.0)(2.0)(1.0)0.0 stock PriceTheta64656667686970710.02670.03190.03770.04430.05170.05990.06900.07890.00480.00550.00620.00700.00780.00860.00950.01043.95404.65045.42406.27577.20578

16、.21259.293910.4461-0.4589-0.5419-0.6346-0.7373-0.8503-0.9735-1.1068-1.25001.58791.92082.30482.74403.24283.80504.43425.1337Rho of European Call Option50576471788592991061131201271341411480.0 10.0 20.0 30.0 40.0 50.0 60.0 70.0 80.0 90.0 100.0 Stock PriceRhoTheta of European Call Option5056626874808692

17、98104110116122128134140146(7.0)(6.0)(5.0)(4.0)(3.0)(2.0)(1.0)0.0 stock PriceThetaTheta of European Call Option505662687480869298104110116122128134140146(7.0)(6.0)(5.0)(4.0)(3.0)(2.0)(1.0)0.0 stock PriceThetaRho of European Call Option50576471788592991061131201271341411480.0 10.0 20.0 30.0 40.0 50.0

18、60.0 70.0 80.0 90.0 100.0 Stock PriceRho72737475767778790.08970.10140.11400.12750.14180.15700.17300.18980.01130.01210.01300.01390.01480.01560.01640.017211.664512.943014.274615.651417.064618.504619.961421.4244-1.4027-1.5645-1.7347-1.9125-2.0971-2.2876-2.4829-2.68195.90636.75447.67988.68379.766810.929

19、012.169713.487680818283848586870.20730.22550.24440.26380.28370.30410.32490.34590.01790.01850.01910.01970.02020.02060.02090.021222.882924.326125.743227.123828.457729.735330.947732.0866-2.8835-3.0865-3.2896-3.4918-3.6918-3.8884-4.0806-4.267414.880916.347017.883019.485221.149522.871424.646126.468288899

20、091929394950.36720.38870.41030.43190.45350.47490.49630.51740.02140.02150.02160.02160.02150.02140.02120.021033.144634.115334.993135.773436.452737.028437.498937.8636-4.4478-4.6208-4.7859-4.9421-5.0890-5.2260-5.3528-5.469028.332330.232732.163534.118936.092738.079140.072342.0665969798991001011021030.538

21、20.55870.57890.59860.61790.63670.65510.67280.02070.02030.02000.01950.01910.01860.01810.017538.122838.277638.329938.282638.138837.902637.578437.1711-5.5745-5.6692-5.7530-5.8261-5.8885-5.9406-5.9825-6.014844.056146.035948.000649.945551.866153.758155.617657.44111041051061071081091101110.69010.70680.722

22、90.73840.75330.76760.78130.79440.01700.01640.01580.01520.01460.01400.01340.012836.685936.128435.504434.819634.080133.291832.460631.5923-6.0376-6.0516-6.0571-6.0546-6.0448-6.0280-6.0050-5.976159.225460.967762.665464.316565.919167.471668.973070.42231121131141151161171181190.80690.81890.83020.84110.851

23、30.86100.87020.87900.01220.01170.01110.01050.01000.00950.00900.008530.692729.767228.821227.859826.887625.909324.929123.9510-5.9420-5.9032-5.8603-5.8136-5.7638-5.7113-5.6565-5.600071.819073.162774.453375.691076.876078.009179.090880.12201201211221231241251261270.88720.89490.90220.90910.91550.92160.927

24、20.93260.00800.00750.00710.00670.00620.00590.00550.005122.978422.014921.063420.126519.206818.306317.427016.5703-5.5420-5.4830-5.4233-5.3632-5.3031-5.2432-5.1837-5.124981.103882.037382.923683.764284.560485.313786.025486.69731281291301311321331341350.93750.94220.94650.95050.95430.95780.96110.96410.004

25、80.00450.00420.00390.00360.00340.00310.002915.737614.930114.148513.393712.666011.965811.293210.6482-5.0670-5.0101-4.9544-4.9000-4.8471-4.7956-4.7458-4.697687.330787.927388.488689.016189.511589.976290.411790.81951361371381391401411421430.96690.96950.97200.97420.97630.97820.98000.98160.00270.00250.002

26、30.00220.00200.00180.00170.001610.03079.44058.87708.34007.82897.34316.88196.4447-4.6511-4.6063-4.5633-4.5221-4.4825-4.4447-4.4087-4.374391.200991.557591.890492.201292.490992.760893.012093.24581441451461471481491500.98310.98450.98580.98700.98810.98910.99000.00150.00130.00120.00110.00100.00100.00096.0

27、3065.63905.26904.91984.59054.28053.9887-4.3416-4.3105-4.2810-4.2530-4.2265-4.2015-4.177993.463193.664993.852394.026194.187294.336594.4748Black-Scholes Option Pricing Model copyright THC 2003. All rights reserved by Thomas Ho Company, New York, NY. USA www. reference number9classification#40401Versio

28、n1.0levelbasicInstructionspublication dateJul-031. The risk source of both the call and put options and the underlying asset is the same.authorHanyang Financial Engineering Lab.2. Enter the option descriptionsaffiliationHanyang Universityinputoption terms and conditionsemail addressleesbhanyang.ac.k

29、routputoption price and the sensitivitieslast revised dateJan-04referencesBlack, F. and M. Scholes, 1973, The Pricing of Options and Corporate Liabilities, Journal of Political Economy , Vol.81, 637-654.Ch. 4 of The Oxford Guide to Financial Modeling by Thomas S.Y. Ho and Sang Bin Lee, 2004, Oxford

30、University PressDescriptionsfinancial model classequity options: the Black-Scholes modelissuer/modelermarket participantsmodel typecontinuous time modelrisk sourcesstock pricerisk distributionlognormaleconomic assumptionsperfect capital markettechnical assumptionscontinuous compounding interest rate

31、key wordsthe law of one price, no abitrage condition, dynamic hedging, hedge ratio, greeksLinksdataN/Afinancial modelsInputsStock Price (S)100Mean ()0.15Volitility ()0.10time increment (t)0.004Outputsaverage of stock return-0.000314540.00058variance of stock return0.00003758annualized var0.009394992

32、annualized std.0.096927768max return0.017612345min return-0.01672178max price106.2715444min price91.10188274Interim CalcuationstimeZstock pricestock returnexp(return)01001-0.4417779699.77884001 -0.002214050.997788421.065004653100.5114667 0.007315681 1.00734250631.317084752101.4110230.008909975 1.008

33、949787Simulated Stock Price Path30025020015010050082879297102107112DayStock Price40.323764198101.6778475 0.002627665 1.002631125-1.50261258100.774576-0.008923360.99111633960.878653736101.3949407 0.006137094 1.0061559657-0.56469162101.0920787 -0.002991420.99701304680.137865754101.2389649 0.001451941.

34、00145299490.540452447101.6445408 0.003998121 1.004006125100.655950316102.1263148 0.004728594 1.004739792110.186391844102.3060972 0.001758846 1.00176039312-0.01120725102.3581965 0.000509119 1.00050924913-0.66089115101.9903855 -0.003599840.996406629140.345528406102.2728112 0.002765314 1.00276914115-0.

35、42062084102.0602804 -0.002080240.997921922160.676772061102.5575296 0.004860282 1.00487211317-0.53930194102.2676157 -0.002830840.997173158180.762411219102.8215511 0.005401912 1.00541652919-1.43341822101.9527264 -0.008485730.99155016920-0.93449671101.410736-0.005330280.994683905210.510677864101.797828

36、6 0.003809811.003817077220.735775529102.3319793 0.005233453 1.005247171231.185276233103.1617941 0.008076345 1.00810904724-0.45048182102.9279753 -0.00226910.997733475250.788137413103.5023268 0.005564619 1.0055801326-1.1950624102.7825771 -0.006978240.99304605327-0.63522066102.4298698 -0.003437490.9965

37、6841328-1.19023298101.7206848 -0.006947690.99307638529-2.00252543100.4987802 -0.012085080.98798764830-0.39468546100.3063878 -0.001916210.99808562531-0.22263862100.2233594 -0.000828090.99917225332-1.1482575299.55587767 -0.006682220.993340058330.50467406399.93209549 0.003771839 1.00377896134-1.6045851

38、298.98047642 -0.009568290.990477343350.55708142199.38745666 0.004103292 1.00411172236-0.774230198.95935933 -0.004316660.995692642370.40114400799.26830258 0.003117057 1.003121921380.73858700599.79095479 0.005251234 1.005265046390.641034603100.2544851 0.004634259 1.004645014400.390718924100.5608411 0.

39、003051123 1.003055783410.482638143100.9267897 0.003632472 1.00363907742-1.48395429100.0419951 -0.008805350.991233303430.182656095100.2157405 0.001735219 1.00173672544-0.5717038299.91196966 -0.003035770.99696883145-0.1793910199.856577-0.000554570.999445585460.924599135100.5004902 0.006427678 1.006448

40、38470.409032737100.8192747 0.003166951.0031719748-1.35174105100.0190226 -0.007969160.992062509490.037684913100.1009057 0.000818341.000818675502.693050192101.8795343 0.017612345 1.01776835751-1.70957719100.842386-0.010232320.989819857520.245439651101.0576413 0.002132297 1.002134572531.011628683101.76

41、52980.006978102 1.007002505541.320186129102.6780898 0.008929591.00896957855-0.73137016102.2635353 -0.004045590.99596258156-02250385 -0.000376520.999623553570.229701982102.4330492 0.002032763 1.0020348358-0.42949637102.2144475 -0.002136370.99786590759-0.08443976102.2191454.59561E-051.000

42、045957600.354067424102.507740.002819319 1.002823297611.662065188103.6510668 0.011091823 1.01115356662-0.23370944103.5580187 -0.000898110.999102295630.521953325103.9607210.003881123 1.003888664640.86167347104.5894668 0.006029702 1.00604791765-0.96341864104.0144313 -0.005513190.994501975660.4227945281

43、04.3534443 0.003253987 1.00325928767-1.6569129103.325521-0.009899240.990149599681.353078005104.2739968 0.009137617 1.009179492691.049884304105.0295848 0.007220051 1.00724617970-0.91049742104.4870959 -0.005178490.99483489471-2.28663339103.0466337 -0.013881940.98621397720.743216333103.5922119 0.005280

44、513 1.00529447973-0.9815858103.0108229 -0.005628090.99438771474-0.81973667102.5376029 -0.004604470.99540611475-0.79326128102.0836489 -0.004437020.995572804761.300866188102.9867114 0.00880741.008846299770.953850758103.6699864 0.006612682 1.00663459478-0.96329813103.1000848 -0.005512430.994502733792.4

45、4093826104.7648236 0.016017849 1.016146823801.476860234105.8093138 0.009920484 1.009969855810.250884113106.0388226 0.002166731.0021690882-0.0165619106.0892299 0.000475253 1.000475366830.179779818106.2715444 0.001717027 1.00171850284-0.98817054105.6707174 -0.005669740.99434630385-1.29253294104.871220

46、2 -0.00759470.99243407186-1.12996304104.1848379 -0.006566510.99345499987-14929062 -0.006663540.993358614880.06656478103.5965537 0.001000993 1.001001494890.433526566103.9412595 0.003321863 1.00332738690-0.94951929103.3788752 -0.005425290.994589403910.040363375103.4652616 0.000835281.0008

47、35629920.015950263103.5357329 0.000680878 1.0006811193-0.89019068103.0141892 -0.005050060.99496267940.035158791103.0968772 0.000802364 1.000802686951.770022209104.3179769 0.011774603 1.01184419796-0.75201797103.8832348 -0.004176180.99583252997-0.19363597103.8162868 -0.000644660.999355546981.43598754

48、2104.8242195 0.009661983 1.0097088199-0.12513283104.802061-0.000211410.999788613100-0.35797939104.6257165 -0.001684060.9983173571011.569551387105.7307842 0.010506715 1.010562104102-06979601 -0.00031050.999689551103-1.02447302105.076248-0.005899340.994118031104-2.33991159103.5927511 -0.0

49、1421890.985881711105-1.2912983102.8097785 -0.007586890.992441821060.070781425102.9154864 0.001027661 1.0010281891070.33450533103.1932794 0.002695597 1.002699234108-0.82348834102.716784-0.00462820.995382496109-1.01469595102.1189214 -0.00583750.9941795041100.948439265102.7929208 0.006578457 1.00660014

50、2111-1.19334572102.0792126 -0.006967380.993056835112-0.86467253101.5813989 -0.004888670.995123261113-1.69299256100.5578327 -0.010127430.989923685114-1.9946310299.35485717 -0.012035150.9880369791150.36778374199.64400915 0.002906069 1.002910295116-0.9741825199.08941801 -0.005581270.9944342751170.91072

51、934199.71963644 0.006339958 1.0063600981180.2593401499.94128103 0.002220211 1.002222678119-0.4746607499.69951434 -0.002422020.997580913120087458233 0.001754416 1.001755956121-0.5172250999.60615934 -0.002691220.997312399122-2.1393134498.32455547 -0.012950210.987133287123-0.0048578498.378

52、57766 0.000549276 1.0005494271240.56048065798.78520677 0.004124791 1.00413331125-2.7356509297.1470771-0.016721780.983417257126-1.3475573796.37852314 -0.00794270.992088759127-0.9151403995.87790238 -0.005207860.9948056811280.50911012396.24292143 0.003799895 1.0038071241290.79315668796.78304173 0.00559

53、6363 1.005612052130-1.2418263496.0815962-0.0072740.992752392131-0.3764739695.90870606 -0.001801030.998200591320.00187355995.96548637 0.000591849 1.000592025133-0.354800795.80593674 -0.001663960.998337427134-0.0275804395.84480026 0.000405566 1.0004056481350.30976934896.08847414 0.002539153 1.00254238

54、136-0.6895243195.72585584 -0.003780930.996226204137-1.2469695395.02898119 -0.007306530.9927201138-0.1546436694.9911622-0.000398050.9996020271391.09557959195.70714074 0.007509054 1.007537317140-0.46449695.48175489 -0.002357730.997645047141-0.6487562195.14595551 -0.003523090.996483104142-0.0161037495.

55、19146052 0.000478151 1.000478265143-0.7935750594.76984122 -0.004439010.9955708291440.19472736394.94167845 0.001811564 1.001813206145-2.4049586593.56276533 -0.014630290.985476209146-0.3271520793.4235455-0.001489090.998512017147-1.0323901792.86938075 -0.005949410.994068254148-0.8976053392.39723402 -0.

56、005096950.9949160131490.06004938892.48595813 0.000959786 1.000960246150-2.009965191.37068496 -0.012132140.9879411621511.83938027592.49346153 0.012213262 1.012288149152-0.3851926192.321937-0.001856170.99814555153-0.1536375392.28578258 -0.000391690.999608388154-0.5591380292.01336107 -0.00295630.997048

57、066155-0.2411843591.92641423 -0.000945380.9990550631560.07346670892.02249481 0.001044644 1.001045191570.96585608892.64006044 0.006688611.006711029158-1.3621138491.8987017-0.008034760.9919974281590.64762389292.32942010.004675933 1.004686882160-0.5226547792.07811345 -0.002725560.997278152161-0.7090625

58、391.71929494 -0.003904510.9961031071620.03469949791.79265006 0.000799459 1.000799779163-1.2706209491.11077895 -0.007456110.9925716151640.12999180491.23861866 0.001402141.0014031241650.54772499491.60834541 0.004044117 1.0040523061660.76671540292.10705194 0.005429134 1.005443898167-0.5797392191.823193

59、29 -0.003086590.996918166168-1.0916846891.24429736 -0.006324420.993695537169-0.103802991.23731678 -7.6507E-050.999923496170-0.3265870491.10188274 -0.001485520.9985155851711.06459992891.77056394 0.007313121 1.0073399271720.42564124792.07132823 0.003271992 1.00327735173-1.0395865591.52101892 -0.005994920.994023011174-0.2981516991.40160002 -0.001305680.998695175175112198559 0.007850645 1.007881542176026436390.001544348 1.001545541177-0.5288916392.00960489 -0.0027650.997238815178-0.3659022291.85018298 -0.001734170.9982673341791.23131030792.62196290.008367491.00840259518