Black Scholes Option Pricing with Stochastic Returns on Hedge Portfolio


 Sara Blair
 2 years ago
 Views:
Transcription
1 EJTP 3, No Elecronic Journal of Theoreical Physics Black Scholes Opion Pricing wih Sochasic Reurns on Hedge Porfolio J. P. Singh and S. Prabakaran Deparmen of Managemen Sudies Indian Insiue of Technology Roorkee Roorkee 47667, India Received Sepember 006, Acceped 9 Sepember 006, Published 0 December 006 Absrac: The Black Scholes model of opion pricing consiues he cornersone of conemporary valuaion heory. However, he model presupposes he exisence of several unrealisic and rigid assumpions including, in paricular, he consancy of he reurn on he hedge porfolio. There, now, subsiss ample jusificaion o he effec ha his is no he case. Consequenly, several generalisaions of he basic model have been aemped. In his paper, we aemp one such generalisaion based on he assumpion ha he reurn process on he hedge porfolio follows a sochasic process similar o he Vasicek model of shorerm ineres raes. c Elecronic Journal of Theoreical Physics. All righs reserved. Keywords: Econophysics, Financial derivaives, Opion pricing, Black Scholes Model, Vasicek Model PACS 006: Gh, s, 0.50.r, 0.50.Cw, 0.50.Ey Wih he rapid advancemens in he evoluion of financial markes across he globe, he imporance of generalisaions of he exan mahemaical apparaus o enhance is domain of applicabiliy o he pricing of financial producs can hardly be overemphasized for furher progress and developmen of he financial microsrucure. Though a an embryonic sage, he unificaion of physics, mahemaics and finance is unmisakably discernible wih several fundamenal premises of physics and mahemaics like quanum mechanics, classical & quanum field heory and relaed ools of noncommuaive probabiliy, funcional inegraion ec. being adoped for pricing of exan financial producs and for elucidaing on various occurrences of financial markes like sock price paerns, criical crashes ec ]. The Black Scholes formula for he pricing of financial asses 37] coninues o be he subsraum of conemporary valuaion heory. However, he model, alhough of Jainder
2 0 Elecronic Journal of Theoreical Physics 3, No immense pracical uiliy is based on several assumpions ha lack empirical suppor. The academic fraerniy has aemped several generalisaions of he original Black Scholes formula hrough easing of one or oher assumpion, in an endeavour o augmen is specrum of applicabiliy. In his paper, we aemp one such generalisaion based on he assumpion ha he reurn process on he hedge porfolio follows a sochasic process similar o he Vasicek model of shorerm ineres raes. Secion liss ou he derivaion of he BlackScholes formula hrough he parial differenial equaion based on he consrucion of he complee hedge porfolio. Sec 3, which forms he essence of his paper, aemps a generalisaion of he sandard Black Scholes pricing formula on he lines aforesaid. Secion 4 concludes.. The Black Scholes Model In order o faciliae coninuiy, we summarize below he original derivaion of he Black Scholes model for he pricing of a European call opion 37 and references herein]. The European call opion is defined as a financial coningen claim ha enables a righ o he holder hereof bu no an obligaion o buy one uni of he underlying asse a a fuure dae called he exercise dae or mauriy dae a a price called he exercise price. Hence, he opion conrac, has a payoff of max S T E, 0 = S T E + on he mauriy dae where S T is he sock price on he mauriy dae and E is he exercise price. We consider a nondividend paying sock, he price process of which follows he geomeric Brownian moion wih drif S = e µ+σw. The logarihm of he sock price Y = In S follows he sochasic differenial equaion dy = µd + σdw where W is a regular Brownian moion represening Gaussian whie noise wih zero mean and δ correlaion in ime i.e. E dw dw = dd δ on some filered probabiliy space Ω, F, P and µ and σ are consans represening he long erm drif and he noisiness diffusion respecively in he sock price. Applicaion of Io s formula yields he following SDE for he sock price process ds = µ + σ S d + σs dw Le C S, denoe he insananeous price of a call opion wih exercise price E a any ime before mauriy when he price per uni of he underlying is S. I is assumed ha C S, does no depend on he pas price hisory of he underlying. Applying he Io formula o C S, yields dc = µs C S + σ S C S + C + σ S C C d + σsdw, 3 S S The original opionpricing model propounded by Fischer Black and Myron Scholes envisaged he consrucion of a hedge porfolio,, consising of he call opion and
3 Elecronic Journal of Theoreical Physics 3, No a shor sale of he underlying such ha he randomness in one cancels ou ha in he oher. For his purpose, we make use of a call opion ogeher wih C/ S unis of he underlying sock. We hen have, on applying Io s formula o he hedge porfolio,,: d = d ] C S, dc S, C S, S = C d d S d S.dS 4 d where he erm involving d C d S has been assumed zero since i envisages a change in he porfolio composiion. On subsiuing from eqs. & 3 in 4, we obain d d = dc S, d µ + C S, σ S σs C dw S S d = C S, + σ S C S, S 5 We noe, here, ha he randomness in he value of he call price emanaing from he sochasic erm in he sock price process has been eliminaed compleely by choosing he porfolio = C S, S CS,. Hence, he porfolio is free from any sochasic S noise and he consequenial risk aribued o he sock price process. Now d is nohing bu he rae of change of he price of he socalled riskless bond d porfolio i.e. he reurn on he riskless bond porfolio since he equiy relaed risk is assumed o be eliminaed by consrucion, as explained above and mus, herefore, equal he shorerm ineres rae r i.e. d = r 6 d In he original Black Scholes model, his ineres rae was assumed as he risk free ineres rae r, furher, assumed o be consan, leading o he following parial differenial equaion for he call price: d d or equivalenly = r ] C S, = r C S, S = S C S, C S, + σ S C S, S + σ S C S, C S, + rs rc S, = 0 7 S S which is he famous Black Scholes PDE for opion pricing wih he soluion: C S, = SN d Ee rt N d 8 where d = log S E +r+ σ T σ T, d = d σ T = log S E +r σ T σ T N y = π y e x dx and
4 Elecronic Journal of Theoreical Physics 3, No The Black Scholes Model wih Sochasic Reurns on he Hedge Porfolio As menioned earlier, in he above analysis, he ineres rae r, which is essenially a proxy for he reurn on a porfolio ha is devoid of any risk emanaing from any variables ha cause flucuaions and hence risk in sock price process, is aken as consan and equal o he risk free rae. However, his reurn would, neverheless, be subjec o uncerainies ha influence reurns on he fixed income securiies. I is, now, convenional o model hese shor erm ineres raes ha are represenaive of shor erm reurns on fixed income securiies hrough a sochasic differenial equaion of he form 8] dr = ψ r, ] d + η r, ] du 9 where r is he shor erm ineres rae a ime, ψ and η are deerminisic funcions of r, and U is a Wiener Process. In our furher analysis, we shall assume ha his shorerm ineres rae is represened by he Vasicek model 9] viz. dr d where η is a whie noise sochasic process + Ar + B η = 0 0 η = 0, η η = δ The call price process now becomes a funcion of wo sochasic variables, he sock price process S and he bond reurn process ineres rae process r. Hence, applicaion of Io s formula o C S, r, gives dc = C C C d + ds + S r dr + σ S C S d + C d r where ds is given by eq. and dr by eq. 0 respecively. As in Secion, we formulae a hedge porfolio Π consising of a call opion C S, r, and a shor sale of C unis of sock S i.e. = C S C. We hen have, repeaing S S he same seps as in Secion hereof d = dc d d C ds S d = C + C r.dr d + σ S C S + C 3 r Now, using d = r, we obain d C + σ S C C + r S S S r C + C r + C dr = 0 4 r d This equaion defies closed form soluion wih he exan mahemaical apparaus. We can, however, obain explici expressions for he call price C S, averaged over he sochasic par of he ineres rae process, as follows:
5 Elecronic Journal of Theoreical Physics 3, No C S, would, hen, be given by subsiuing T rτ T for he consan risk free ineres rae r in he Black Scholes formula 8. The averaging process happens o be edious wih exensive compuaions so we proceed erm by erm. We have N d = π d e x dx = π H d x e x dx 5 where H x y is he uni sep Heaviside sep funcion defined by 0] 0, x < y H x, y =, x > y On using he inegral represenaion of H x y as H x y = Lim ε 0 πi 0] i.e. we obain H d x = Lim ε 0 dω eiω d x πi ω iε dω eiωx y ω iε 6 N d = Limε 0 π 3 i e x +iω d x ω iε dxdω = Lim ε 0 πi on performing he Gaussian inegraion over x in he second sep. Now where d = log S E + T σ T σ r τ + T σ T d 0 = log S E + σ T σ T = d 0 + T r τ σ T Since he enire sochasic conribuion comes from he expression T N d, we have where I = Proceeding similarly, we have, N d = Limε 0 πi e iω d +i ω ω iε dω r τ in dω eiωd0 ω ω iε I 0 e iω T σ T rτ and ρ denoes he average expecaion of ρ. N d = Limε 0 πi dω eiωd0 ω ω iε I
6 4 Elecronic Journal of Theoreical Physics 3, No d 0 = log S E σ T σ T Similarly he discoun facor e rt will be replaced by e T rτ = I say. To evaluae he expecaion inegrals I, I we make use of he funcional inegral formalism ]. In his formalism, he expecaion I would be given by ]: I = rt Dr exp T r rt Dr exp r drτ T ] + Ar τ + B + iω T σ r τ T drτ ] = P Q 3 + Ar τ + B where Dr = T drτ τ= π is he funcional inegraion measure. We firs evaluae he funcional inegral P. Making he subsiuion x τ = B A r τ, we obain, wih a lile algebra, P = xt Dx exp T x dxτ +Axτ + iω T σ T B A xτ] = { xt Dx exp T x dxτ ] +A x τ A x T x iωbt ] Aσ T iω σ T T } xτ where I 3 = T = T = { xt Dx exp x A x T x ] } iωbt Aσ T I 3 dx τ + A x τ] iω T + σ x τ T dx τ + A x τ + iω ] σ T x τ In order o evaluae I 3, we perform a shif of he funcional variable x τ by some fixed funcion y τ i.e. x τ = y τ+z τ where y τ is a fixed funcional whose explici form shall be defined laer bu wih boundary condiions y = x, y T = x T so ha z τ, hen, has Drichle boundary condiions i.e. z = z T = 0. Subsiuing x τ = y τ + z τ in 5, we obain I 3 = T 4 5 dyτ + dzτ + dyτ dzτ + A y τ + A z τ + A y τ z τ + iω σ iω 6 y τ + T σ z τ T Inegraing he second and hird erm by pars, we ge
7 Elecronic Journal of Theoreical Physics 3, No I 3 = zτ dzτ dyτ + T + T z τ d zτ z τ d yτ + dyτ + A y τ + A z τ + A y τ z τ + iω σ iw y τ + T σ z τ T 7 Now he boundary erms all vanish since z τ has Drichle boundary condiions. Furher, if we define he fixed funcional y τ in erms of he differenial equaion d y τ + A y τ + iω σ T = 0 8 wih boundary condiion y = x, y T = x T we obain I 3 = T { dy τ + A y τ + iω ] σ T y τ + The funcional y τ is fixed and is given by he soluion of eq 8 as ] } z τ d z τ + A z τ 9 y = αe Aτ + βe Aτ γ 30. where γ = iωσ A σ, α = xt eat xe A T e AT e A + γ eat e A e AT e A and β = xt e AT xe A e AT e A Inegraing ou he y τ erms in eq. 9 using eq. 30, we obain + γ e AT e A e AT e A { I 3 = Aα e AT e A β e AT e A Aγ T ]+ ]} T zτ d zτ +A z τ 3 Subsiuing his value of I 3 in eq. 4 we obain, for P, noing ha Dx = Dz since y τ is fixed by eq 8 P =exp A x T x ] iωb T Aσ zt =0 z=0 Dz exp { T On exacly same lines, we obain ] A xt e AT xe A e AT e xt e AT xe A AT e AT e A ω Σ 4 e AT e A A 3 σ T e AT e e AT e A A xt e AT xe A e AT e A iωσ + Aσ e T AT e A xt e AT xe A e AT e A ]} z τ d zτ + A z τ e AT e A e AT e A AT 3 ]
8 6 Elecronic Journal of Theoreical Physics 3, No { ]]} Q = exp A x T x ] A xt e AT xe A e AT e xt e AT xe A AT e AT e A { zt =0 Dz exp ]} T z=0 z τ d zτ + A z τ 33 Hence I = exp iωb T Aσ + ω Σ 4 A 3 σ T iωσ Aσ T e AT e A e AT e e AT e A A xt e AT xe A e AT e A e AT e A xt e AT xe A e AT e A e AT e A ] e AT e A T A 34 which when subsiued in eqs. 0 & shall give he values N d and N d respecively as: and where X = N d = N N d = N e AT e A e AT e e AT e A A log S E + σ σ T B T Σ X A 3 σ T Σ X A 3 σ T Y Aσ Aσ T ] log S E σ σ B T Y T Aσ Aσ T ] e AT e A A T and Y = x T e AT x e A e AT e A x T e AT x e A e AT e A e AT e A e AT e A 37 To evaluae I, we subsiue ω = iσ T in eq. 34 o ge { B T I = exp A Σ 4 A 3 X Σ A ]} Y The closed form soluion for he Black Scholes pricing problem wih sochasic reurn on he hedge porfolio can now be obained by subsiuing he above averages in eq Conclusion In his paper, we have obained closed form expressions for he price of a European call opion by modifying he Black Scholes formulaion o accommodae a sochasic 38
9 Elecronic Journal of Theoreical Physics 3, No reurn process for he hedge porfolio reurns. We have modelled his reurn process on he basis of he Vasicek model for he shorerm ineres raes. The need for his exension of he Black Scholes model is manifold. Firsly, he consrucion of he hedge porfolio in he Black Scholes heory implies ha he flucuaions in he price of he derivaive and ha of he underlying exacly and immediaely cancel each oher when combined in a cerain proporion viz. one uni of he derivaive wih a shor sale of C unis of he underlying so ha he hedge porfolio is devoid of any impac of S such flucuaions. This mandaes an infiniely fas reacion mechanism of he underlying marke dynamics whereby any movemen in he price of one asse is insananeously annulled by reacionary response in he oher asse consiuing he hedge porfolio. This is, obviously srongly unrealisic and here may subsis brief periods or aberraions when he no arbirage condiion may cease o hold and hence, reurns on he hedge porfolio may be differen from he risk free rae. One way of aending o his anomaly is o model he reurns on he hedge porfolio as a sochasic process as has been done in his sudy. The parameers defining he process can be obained hrough an empirical sudy of he marke dynamics. Anoher imporan jusificaion for adoping a sochasic framework for he hedge porfolio reurn process is ha he hedge porfolio by is very consrucion, envisages he neuralizaion of he flucuaions of he wo asses iner se i.e. i assumes a perfec correlaion beween he wo asses. In oher words, he hedge porfolio may be consrued as an isolaed sysem ha is such ha insofar as facors ha influence one componen of he sysem, he same facors influence he oher componen o an equivalen exen and, a he same ime, oher facors do no impac he sysem a all. This is anoher anomaly ha disors he Black Scholes model. The fac is ha while he hedge porfolio of he Black Scholes model is immunized agains price flucuaions of he underlying and is derivaive hrough muual ineracion, oher marke facors ha would impac he porfolio as a whole are no accouned for e.g. facors affecing bond yields and ineres raes ec. Consequenly, o assume ha he hedge porfolio is compleely risk free is anoher aberraion i is risk free only o he exen of risk ha emanaes from facors ha impac he underlying and he derivaive in like manner and is sill subjec o risk and uncerainies ha originae from facors ha eiher do no effec he underlying and he derivaive o equivalen exen or impac he porfolio as a uni eniy. Hence, again, i becomes necessary o model he reurn on he hedge porfolio as some shorerm ineres rae model as has been done here.
10 8 Elecronic Journal of Theoreical Physics 3, No References ] V.I. Man ko e al, Phy. Le., A 76, 993, 73; V.I. Man ko and R.Vilea Mendes, J.Phys., A 3, 998, ] W. Paul & J. Nagel, Sochasic Processes, Springer, ] J. Voi, The Saisical Mechanics of Financial Markes, Springer, 00. 4] JeanPhilippe Bouchard & Marc Poers, Theory of Financial Risks, Publicaion by he Press Syndicae of he Universiy of Cambridge, ] J. Maskawa, Hamilonian in Financial Markes, arxiv:condma/0049 v, 9 Nov ] Z. Burda e al, Is Econophysics a Solid Science?, arxiv:condma/ v, 8 Jan ] A. Dragulescu, Applicaion of Physics o Economics and Finance: Money, Income, Wealh and he Sock Marke, arxiv:condma/ v, 6 July ] A. Dragulescu & M. Yakovenko, Saisical Mechanics of Money, arxiv:condma/00043 v4, 4 Mar ] B. Baaquie e al, Quanum Mechanics, Pah Inegraion and Opion Pricing: Reducing he Complexiy of Finance, arxiv:condma/0089v, Aug 00. 0] G. Bonanno e al, Levels of Complexiy in Financial markes, arxiv:condma/ v, 9 Apr 00. ] A. Dragulescu, & M. Yakovenko, Saisical Mechanics of money, income and wealh : A Shor Survey, arxiv:condma/075 v, 9 Nov 00. ] J. Doyne Farmer, Physics Aemp o Scale he Ivory Tower of Finance, adaporg/ Dec ] F. Black & M. Scholes, Journal of Poliical Economy, 8, 973, ] M. Baxer & E. Rennie, Financial Calculus, Cambridge Universiy Press, 99. 5] J. C. Hull & A. Whie, Journal of Finance, 4, 987, 8. 6] J. C. Hull, Opions, Fuures & Oher Derivaives, Prenice Hall, ] R. C. Meron, Journal of Financial Economics, 976, 5. 8] Paul Wilmo, Quaniaive Finance, John Wiley, Chicheser, ] O. A. Vasicek, Journal of Financial Economics, 5, 977, 77. 0] hp://funcions.wolfram.com/generalizedfuncions/unisep/07/0/0/. ] R. P. Feynman & A. R. Hibbs, Quanum Mechanics & Pah Inegrals, McGraw Hill, 965. ] P. C. Marin e al, Phy. Rev. A, 8, 973, 43.
Stochastic Optimal Control Problem for Life Insurance
Sochasic Opimal Conrol Problem for Life Insurance s. Basukh 1, D. Nyamsuren 2 1 Deparmen of Economics and Economerics, Insiue of Finance and Economics, Ulaanbaaar, Mongolia 2 School of Mahemaics, Mongolian
More informationTerm Structure of Prices of Asian Options
Term Srucure of Prices of Asian Opions Jirô Akahori, Tsuomu Mikami, Kenji Yasuomi and Teruo Yokoa Dep. of Mahemaical Sciences, Risumeikan Universiy 111 Nojihigashi, Kusasu, Shiga 5258577, Japan Email:
More informationOption Pricing Under Stochastic Interest Rates
I.J. Engineering and Manufacuring, 0,3, 889 ublished Online June 0 in MECS (hp://www.mecspress.ne) DOI: 0.585/ijem.0.03. Available online a hp://www.mecspress.ne/ijem Opion ricing Under Sochasic Ineres
More information= r t dt + σ S,t db S t (19.1) with interest rates given by a mean reverting OrnsteinUhlenbeck or Vasicek process,
Chaper 19 The BlackScholesVasicek Model The BlackScholesVasicek model is given by a sandard imedependen BlackScholes model for he sock price process S, wih imedependen bu deerminisic volailiy σ
More informationJournal Of Business & Economics Research September 2005 Volume 3, Number 9
Opion Pricing And Mone Carlo Simulaions George M. Jabbour, (Email: jabbour@gwu.edu), George Washingon Universiy YiKang Liu, (yikang@gwu.edu), George Washingon Universiy ABSTRACT The advanage of Mone Carlo
More informationON THE PRICING OF EQUITYLINKED LIFE INSURANCE CONTRACTS IN GAUSSIAN FINANCIAL ENVIRONMENT
Teor Imov r.amaem.sais. Theor. Probabiliy and Mah. Sais. Vip. 7, 24 No. 7, 25, Pages 15 111 S 949(5)6344 Aricle elecronically published on Augus 12, 25 ON THE PRICING OF EQUITYLINKED LIFE INSURANCE
More informationDYNAMIC MODELS FOR VALUATION OF WRONGFUL DEATH PAYMENTS
DYNAMIC MODELS FOR VALUATION OF WRONGFUL DEATH PAYMENTS Hong Mao, Shanghai Second Polyechnic Universiy Krzyszof M. Osaszewski, Illinois Sae Universiy Youyu Zhang, Fudan Universiy ABSTRACT Liigaion, exper
More informationValuation of Life Insurance Contracts with Simulated Guaranteed Interest Rate
Valuaion of Life Insurance Conracs wih Simulaed uaraneed Ineres Rae Xia uo and ao Wang Deparmen of Mahemaics Royal Insiue of echnology 100 44 Sockholm Acknowledgmens During he progress of he work on his
More informationOptimal Stock Selling/Buying Strategy with reference to the Ultimate Average
Opimal Sock Selling/Buying Sraegy wih reference o he Ulimae Average Min Dai Dep of Mah, Naional Universiy of Singapore, Singapore Yifei Zhong Dep of Mah, Naional Universiy of Singapore, Singapore July
More informationThe Transport Equation
The Transpor Equaion Consider a fluid, flowing wih velociy, V, in a hin sraigh ube whose cross secion will be denoed by A. Suppose he fluid conains a conaminan whose concenraion a posiion a ime will be
More informationOption PutCall Parity Relations When the Underlying Security Pays Dividends
Inernaional Journal of Business and conomics, 26, Vol. 5, No. 3, 22523 Opion Puall Pariy Relaions When he Underlying Securiy Pays Dividends Weiyu Guo Deparmen of Finance, Universiy of Nebraska Omaha,
More informationRandom Walk in 1D. 3 possible paths x vs n. 5 For our random walk, we assume the probabilities p,q do not depend on time (n)  stationary
Random Walk in D Random walks appear in many cones: diffusion is a random walk process undersanding buffering, waiing imes, queuing more generally he heory of sochasic processes gambling choosing he bes
More informationStochastic Calculus, Week 10. Definitions and Notation. TermStructure Models & Interest Rate Derivatives
Sochasic Calculus, Week 10 TermSrucure Models & Ineres Rae Derivaives Topics: 1. Definiions and noaion for he ineres rae marke 2. Termsrucure models 3. Ineres rae derivaives Definiions and Noaion Zerocoupon
More informationLECTURE 7 Interest Rate Models I: Short Rate Models
LECTURE 7 Ineres Rae Models I: Shor Rae Models Spring Term 212 MSc Financial Engineering School of Economics, Mahemaics and Saisics Birkbeck College Lecurer: Adriana Breccia email: abreccia@emsbbkacuk
More informationLIFE INSURANCE WITH STOCHASTIC INTEREST RATE. L. Noviyanti a, M. Syamsuddin b
LIFE ISURACE WITH STOCHASTIC ITEREST RATE L. oviyani a, M. Syamsuddin b a Deparmen of Saisics, Universias Padjadjaran, Bandung, Indonesia b Deparmen of Mahemaics, Insiu Teknologi Bandung, Indonesia Absrac.
More informationOptimal Investment and Consumption Decision of Family with Life Insurance
Opimal Invesmen and Consumpion Decision of Family wih Life Insurance Minsuk Kwak 1 2 Yong Hyun Shin 3 U Jin Choi 4 6h World Congress of he Bachelier Finance Sociey Torono, Canada June 25, 2010 1 Speaker
More informationTime Consistency in Portfolio Management
1 Time Consisency in Porfolio Managemen Traian A Pirvu Deparmen of Mahemaics and Saisics McMaser Universiy Torono, June 2010 The alk is based on join work wih Ivar Ekeland Time Consisency in Porfolio Managemen
More informationPricing Guaranteed Minimum Withdrawal Benefits under Stochastic Interest Rates
Pricing Guaraneed Minimum Wihdrawal Benefis under Sochasic Ineres Raes Jingjiang Peng 1, Kwai Sun Leung 2 and Yue Kuen Kwok 3 Deparmen of Mahemaics, Hong Kong Universiy of Science and echnology, Clear
More informationThe option pricing framework
Chaper 2 The opion pricing framework The opion markes based on swap raes or he LIBOR have become he larges fixed income markes, and caps (floors) and swapions are he mos imporan derivaives wihin hese markes.
More informationPricing FixedIncome Derivaives wih he ForwardRisk Adjused Measure Jesper Lund Deparmen of Finance he Aarhus School of Business DK8 Aarhus V, Denmark Email: jel@hha.dk Homepage: www.hha.dk/~jel/ Firs
More informationWhy we have always used the BlackScholesMerton option pricing formula
Why we have always used he BlackScholesMeron opion pricing formula Charles J. Corrado Deakin Universiy Melbourne, Ausralia April 4, 9 Absrac Derman and aleb (he Issusions of Dynamic Hedging, 5 uncover
More information2.5 Life tables, force of mortality and standard life insurance products
Soluions 5 BS4a Acuarial Science Oford MT 212 33 2.5 Life ables, force of moraliy and sandard life insurance producs 1. (i) n m q represens he probabiliy of deah of a life currenly aged beween ages + n
More informationINTEREST RATE FUTURES AND THEIR OPTIONS: SOME PRICING APPROACHES
INTEREST RATE FUTURES AND THEIR OPTIONS: SOME PRICING APPROACHES OPENGAMMA QUANTITATIVE RESEARCH Absrac. Exchangeraded ineres rae fuures and heir opions are described. The fuure opions include hose paying
More informationHedging with Forwards and Futures
Hedging wih orwards and uures Hedging in mos cases is sraighforward. You plan o buy 10,000 barrels of oil in six monhs and you wish o eliminae he price risk. If you ake he buyside of a forward/fuures
More informationA general decomposition formula for derivative prices in stochastic volatility models
A general decomposiion formula for derivaive prices in sochasic volailiy models Elisa Alòs Universia Pompeu Fabra C/ Ramón rias Fargas, 57 85 Barcelona Absrac We see ha he price of an european call opion
More informationPROFIT TEST MODELLING IN LIFE ASSURANCE USING SPREADSHEETS PART ONE
Profi Tes Modelling in Life Assurance Using Spreadshees PROFIT TEST MODELLING IN LIFE ASSURANCE USING SPREADSHEETS PART ONE Erik Alm Peer Millingon 2004 Profi Tes Modelling in Life Assurance Using Spreadshees
More informationChapter 7. Response of FirstOrder RL and RC Circuits
Chaper 7. esponse of FirsOrder L and C Circuis 7.1. The Naural esponse of an L Circui 7.2. The Naural esponse of an C Circui 7.3. The ep esponse of L and C Circuis 7.4. A General oluion for ep and Naural
More informationA Generalized Bivariate OrnsteinUhlenbeck Model for Financial Assets
A Generalized Bivariae OrnseinUhlenbeck Model for Financial Asses Romy Krämer, Mahias Richer Technische Universiä Chemniz, Fakulä für Mahemaik, 917 Chemniz, Germany Absrac In his paper, we sudy mahemaical
More informationNew Pricing Framework: Options and Bonds
arxiv:1407.445v [qfin.pr] 14 Oc 014 New Pricing Framework: Opions and Bonds Nick Laskin TopQuark Inc. Torono, ON, M6P P Absrac A unified analyical pricing framework wih involvemen of he sho noise random
More informationTechnical Appendix to Risk, Return, and Dividends
Technical Appendix o Risk, Reurn, and Dividends Andrew Ang Columbia Universiy and NBER Jun Liu UC San Diego This Version: 28 Augus, 2006 Columbia Business School, 3022 Broadway 805 Uris, New York NY 10027,
More informationMTH6121 Introduction to Mathematical Finance Lesson 5
26 MTH6121 Inroducion o Mahemaical Finance Lesson 5 Conens 2.3 Brownian moion wih drif........................... 27 2.4 Geomeric Brownian moion........................... 28 2.5 Convergence of random
More informationA Probability Density Function for Google s stocks
A Probabiliy Densiy Funcion for Google s socks V.Dorobanu Physics Deparmen, Poliehnica Universiy of Timisoara, Romania Absrac. I is an approach o inroduce he Fokker Planck equaion as an ineresing naural
More informationResearch Article Optimal Geometric Mean Returns of Stocks and Their Options
Inernaional Journal of Sochasic Analysis Volume 2012, Aricle ID 498050, 8 pages doi:10.1155/2012/498050 Research Aricle Opimal Geomeric Mean Reurns of Socks and Their Opions Guoyi Zhang Deparmen of Mahemaics
More informationSkewness and Kurtosis Adjusted BlackScholes Model: A Note on Hedging Performance
Finance Leers, 003, (5), 6 Skewness and Kurosis Adjused BlackScholes Model: A Noe on Hedging Performance Sami Vähämaa * Universiy of Vaasa, Finland Absrac his aricle invesigaes he dela hedging performance
More informationTHE PERFORMANCE OF OPTION PRICING MODELS ON HEDGING EXOTIC OPTIONS
HE PERFORMANE OF OPION PRIING MODEL ON HEDGING EXOI OPION Firs Draf: May 5 003 his Version Oc. 30 003 ommens are welcome Absrac his paper examines he empirical performance of various opion pricing models
More informationDifferential Equations. Solving for Impulse Response. Linear systems are often described using differential equations.
Differenial Equaions Linear sysems are ofen described using differenial equaions. For example: d 2 y d 2 + 5dy + 6y f() d where f() is he inpu o he sysem and y() is he oupu. We know how o solve for y given
More informationUNIVERSITY OF CALGARY. Modeling of Currency Trading Markets and Pricing Their Derivatives in a Markov. Modulated Environment.
UNIVERSITY OF CALGARY Modeling of Currency Trading Markes and Pricing Their Derivaives in a Markov Modulaed Environmen by Maksym Terychnyi A THESIS SUBMITTED TO THE FACULTY OF GRADUATE STUDIES IN PARTIAL
More informationPRICING and STATIC REPLICATION of FX QUANTO OPTIONS
PRICING and STATIC REPLICATION of F QUANTO OPTIONS Fabio Mercurio Financial Models, Banca IMI 1 Inroducion 1.1 Noaion : he evaluaion ime. τ: he running ime. S τ : he price a ime τ in domesic currency of
More informationA Simple Approach to CAPM, Option Pricing and Asset Valuation
A imple Approach o CAPM, Opion Pricing and Asse Valuaion Riccardo Cesari (*) Universià di Bologna, Dip. Maemaes, viale Filopani, 5 406 Bologna, Ialy Email: rcesari@economia.unibo.i Carlo D Adda Universià
More informationModeling VIX Futures and Pricing VIX Options in the Jump Diusion Modeling
Modeling VIX Fuures and Pricing VIX Opions in he Jump Diusion Modeling Faemeh Aramian Maseruppsas i maemaisk saisik Maser hesis in Mahemaical Saisics Maseruppsas 2014:2 Maemaisk saisik April 2014 www.mah.su.se
More informationOptimal Time to Sell in Real Estate Portfolio Management
Opimal ime o Sell in Real Esae Porfolio Managemen Fabrice Barhélémy and JeanLuc Prigen hema, Universiy of CergyPonoise, CergyPonoise, France Emails: fabricebarhelemy@ucergyfr; jeanlucprigen@ucergyfr
More informationStochastic Calculus and Option Pricing
Sochasic Calculus and Opion Pricing Leonid Kogan MIT, Sloan 15.450, Fall 2010 c Leonid Kogan ( MIT, Sloan ) Sochasic Calculus 15.450, Fall 2010 1 / 74 Ouline 1 Sochasic Inegral 2 Iô s Lemma 3 BlackScholes
More informationA Note on Using the Svensson procedure to estimate the risk free rate in corporate valuation
A Noe on Using he Svensson procedure o esimae he risk free rae in corporae valuaion By Sven Arnold, Alexander Lahmann and Bernhard Schwezler Ocober 2011 1. The risk free ineres rae in corporae valuaion
More informationDynamic Option Adjusted Spread and the Value of Mortgage Backed Securities
Dynamic Opion Adjused Spread and he Value of Morgage Backed Securiies Mario Cerrao, Abdelmadjid Djennad Universiy of Glasgow Deparmen of Economics 27 January 2008 Absrac We exend a reduced form model for
More informationPricing BlackScholes Options with Correlated Interest. Rate Risk and Credit Risk: An Extension
Pricing Blackcholes Opions wih Correlaed Ineres Rae Risk and Credi Risk: An Exension zulang Liao a, and HsingHua Huang b a irecor and Professor eparmen of inance Naional Universiy of Kaohsiung and Professor
More informationWHAT ARE OPTION CONTRACTS?
WHAT ARE OTION CONTRACTS? By rof. Ashok anekar An oion conrac is a derivaive which gives he righ o he holder of he conrac o do 'Somehing' bu wihou he obligaion o do ha 'Somehing'. The 'Somehing' can be
More informationOptionPricing in Incomplete Markets: The Hedging Portfolio plus a Risk PremiumBased Recursive Approach
Working Paper 581 Business Economics Series 21 January 25 Deparameno de Economía de la Empresa Universidad Carlos III de Madrid Calle Madrid, 126 2893 Geafe (Spain) Fax (34) 91 624 968 OpionPricing in
More informationDuration and Convexity ( ) 20 = Bond B has a maturity of 5 years and also has a required rate of return of 10%. Its price is $613.
Graduae School of Business Adminisraion Universiy of Virginia UVAF38 Duraion and Convexiy he price of a bond is a funcion of he promised paymens and he marke required rae of reurn. Since he promised
More informationOptimal Life Insurance, Consumption and Portfolio: A Dynamic Programming Approach
28 American Conrol Conference Wesin Seale Hoel, Seale, Washingon, USA June 1113, 28 WeA1.5 Opimal Life Insurance, Consumpion and Porfolio: A Dynamic Programming Approach Jinchun Ye (Pin: 584) Absrac A
More informationCapacitors and inductors
Capaciors and inducors We coninue wih our analysis of linear circuis by inroducing wo new passive and linear elemens: he capacior and he inducor. All he mehods developed so far for he analysis of linear
More informationForeign Exchange and Quantos
IEOR E4707: Financial Engineering: ConinuousTime Models Fall 2010 c 2010 by Marin Haugh Foreign Exchange and Quanos These noes consider foreign exchange markes and he pricing of derivaive securiies in
More informationINVESTMENT GUARANTEES IN UNITLINKED LIFE INSURANCE PRODUCTS: COMPARING COST AND PERFORMANCE
INVESMEN UARANEES IN UNILINKED LIFE INSURANCE PRODUCS: COMPARIN COS AND PERFORMANCE NADINE AZER HAO SCHMEISER WORKIN PAPERS ON RISK MANAEMEN AND INSURANCE NO. 4 EDIED BY HAO SCHMEISER CHAIR FOR RISK MANAEMEN
More informationGraduate Macro Theory II: Notes on Neoclassical Growth Model
Graduae Macro Theory II: Noes on Neoclassical Growh Model Eric Sims Universiy of Nore Dame Spring 2011 1 Basic Neoclassical Growh Model The economy is populaed by a large number of infiniely lived agens.
More informationLongevity 11 Lyon 79 September 2015
Longeviy 11 Lyon 79 Sepember 2015 RISK SHARING IN LIFE INSURANCE AND PENSIONS wihin and across generaions Ragnar Norberg ISFA Universié Lyon 1/London School of Economics Email: ragnar.norberg@univlyon1.fr
More informationPRICING AND PERFORMANCE OF MUTUAL FUNDS: LOOKBACK VERSUS INTEREST RATE GUARANTEES
PRICING AND PERFORMANCE OF MUUAL FUNDS: LOOKBACK VERSUS INERES RAE GUARANEES NADINE GAZER HAO SCHMEISER WORKING PAPERS ON RISK MANAGEMEN AND INSURANCE NO. 4 EDIED BY HAO SCHMEISER CHAIR FOR RISK MANAGEMEN
More informationMorningstar Investor Return
Morningsar Invesor Reurn Morningsar Mehodology Paper Augus 31, 2010 2010 Morningsar, Inc. All righs reserved. The informaion in his documen is he propery of Morningsar, Inc. Reproducion or ranscripion
More informationLife insurance cash flows with policyholder behaviour
Life insurance cash flows wih policyholder behaviour Krisian Buchard,,1 & Thomas Møller, Deparmen of Mahemaical Sciences, Universiy of Copenhagen Universiesparken 5, DK2100 Copenhagen Ø, Denmark PFA Pension,
More information4. International Parity Conditions
4. Inernaional ariy ondiions 4.1 urchasing ower ariy he urchasing ower ariy ( heory is one of he early heories of exchange rae deerminaion. his heory is based on he concep ha he demand for a counry's currency
More informationCointegration: The Engle and Granger approach
Coinegraion: The Engle and Granger approach Inroducion Generally one would find mos of he economic variables o be nonsaionary I(1) variables. Hence, any equilibrium heories ha involve hese variables require
More informationI. Basic Concepts (Ch. 14)
(Ch. 14) A. Real vs. Financial Asses (Ch 1.2) Real asses (buildings, machinery, ec.) appear on he asse side of he balance shee. Financial asses (bonds, socks) appear on boh sides of he balance shee. Creaing
More informationPrincipal components of stock market dynamics. Methodology and applications in brief (to be updated ) Andrei Bouzaev, bouzaev@ya.
Principal componens of sock marke dynamics Mehodology and applicaions in brief o be updaed Andrei Bouzaev, bouzaev@ya.ru Why principal componens are needed Objecives undersand he evidence of more han one
More informationAPPLICATION OF THE KALMAN FILTER FOR ESTIMATING CONTINUOUS TIME TERM STRUCTURE MODELS: THE CASE OF UK AND GERMANY. January, 2005
APPLICATION OF THE KALMAN FILTER FOR ESTIMATING CONTINUOUS TIME TERM STRUCTURE MODELS: THE CASE OF UK AND GERMANY Somnah Chaeree* Deparmen of Economics Universiy of Glasgow January, 2005 Absrac The purpose
More informationEconomics Honors Exam 2008 Solutions Question 5
Economics Honors Exam 2008 Soluions Quesion 5 (a) (2 poins) Oupu can be decomposed as Y = C + I + G. And we can solve for i by subsiuing in equaions given in he quesion, Y = C + I + G = c 0 + c Y D + I
More informationDOES TRADING VOLUME INFLUENCE GARCH EFFECTS? SOME EVIDENCE FROM THE GREEK MARKET WITH SPECIAL REFERENCE TO BANKING SECTOR
Invesmen Managemen and Financial Innovaions, Volume 4, Issue 3, 7 33 DOES TRADING VOLUME INFLUENCE GARCH EFFECTS? SOME EVIDENCE FROM THE GREEK MARKET WITH SPECIAL REFERENCE TO BANKING SECTOR Ahanasios
More informationDifferential Equations and Linear Superposition
Differenial Equaions and Linear Superposiion Basic Idea: Provide soluion in closed form Like Inegraion, no general soluions in closed form Order of equaion: highes derivaive in equaion e.g. dy d dy 2 y
More informationJumpDiffusion Option Valuation Without a Representative Investor: a Stochastic Dominance Approach
umpdiffusion Opion Valuaion Wihou a Represenaive Invesor: a Sochasic Doance Approach By Ioan Mihai Oancea and Sylianos Perrakis This version February 00 Naional Bank of Canada, 30 King Sree Wes, Torono,
More information12. Market LIBOR Models
12. Marke LIBOR Models As was menioned already, he acronym LIBOR sands for he London Inerbank Offered Rae. I is he rae of ineres offered by banks on deposis from oher banks in eurocurrency markes. Also,
More informationLectures # 5 and 6: The Prime Number Theorem.
Lecures # 5 and 6: The Prime Number Theorem Noah Snyder July 8, 22 Riemann s Argumen Riemann used his analyically coninued ζfuncion o skech an argumen which would give an acual formula for π( and sugges
More informationDifferential Equations in Finance and Life Insurance
Differenial Equaions in Finance and Life Insurance Mogens Seffensen 1 Inroducion The mahemaics of finance and he mahemaics of life insurance were always inersecing. Life insurance conracs specify an exchange
More informationARCH 2013.1 Proceedings
Aricle from: ARCH 213.1 Proceedings Augus 14, 212 Ghislain Leveille, Emmanuel Hamel A renewal model for medical malpracice Ghislain Léveillé École d acuaria Universié Laval, Québec, Canada 47h ARC Conference
More informationABSTRACT KEYWORDS. Term structure, duration, uncertain cash flow, variable rates of return JEL codes: C33, E43 1. INTRODUCTION
THE VALUATION AND HEDGING OF VARIABLE RATE SAVINGS ACCOUNTS BY FRANK DE JONG 1 AND JACCO WIELHOUWER ABSTRACT Variable rae savings accouns have wo main feaures. The ineres rae paid on he accoun is variable
More informationCircuit Types. () i( t) ( )
Circui Types DC Circuis Idenifying feaures: o Consan inpus: he volages of independen volage sources and currens of independen curren sources are all consan. o The circui does no conain any swiches. All
More informationRISKSHIFTING AND OPTIMAL ASSET ALLOCATION IN LIFE INSURANCE: THE IMPACT OF REGULATION. 1. Introduction
RISKSHIFTING AND OPTIMAL ASSET ALLOCATION IN LIFE INSURANCE: THE IMPACT OF REGULATION AN CHEN AND PETER HIEBER Absrac. In a ypical paricipaing life insurance conrac, he insurance company is eniled o a
More information3 RungeKutta Methods
3 RungeKua Mehods In conras o he mulisep mehods of he previous secion, RungeKua mehods are singlesep mehods however, muliple sages per sep. They are moivaed by he dependence of he Taylor mehods on he
More informationIntroduction to Arbitrage Pricing
Inroducion o Arbirage Pricing Marek Musiela 1 School of Mahemaics, Universiy of New Souh Wales, 252 Sydney, Ausralia Marek Rukowski 2 Insiue of Mahemaics, Poliechnika Warszawska, 661 Warszawa, Poland
More information11/6/2013. Chapter 14: Dynamic ADAS. Introduction. Introduction. Keeping track of time. The model s elements
Inroducion Chaper 14: Dynamic DS dynamic model of aggregae and aggregae supply gives us more insigh ino how he economy works in he shor run. I is a simplified version of a DSGE model, used in cuingedge
More informationDependent Interest and Transition Rates in Life Insurance
Dependen Ineres and ransiion Raes in Life Insurance Krisian Buchard Universiy of Copenhagen and PFA Pension January 28, 2013 Absrac In order o find marke consisen bes esimaes of life insurance liabiliies
More informationDynamic Information. Albina Danilova Department of Mathematical Sciences Carnegie Mellon University. September 16, 2008. Abstract
Sock Marke Insider Trading in Coninuous Time wih Imperfec Dynamic Informaion Albina Danilova Deparmen of Mahemaical Sciences Carnegie Mellon Universiy Sepember 6, 28 Absrac This paper sudies he equilibrium
More informationDistributing Human Resources among Software Development Projects 1
Disribuing Human Resources among Sofware Developmen Proecs Macario Polo, María Dolores Maeos, Mario Piaini and rancisco Ruiz Summary This paper presens a mehod for esimaing he disribuion of human resources
More informationUNDERSTANDING THE DEATH BENEFIT SWITCH OPTION IN UNIVERSAL LIFE POLICIES. Nadine Gatzert
UNDERSTANDING THE DEATH BENEFIT SWITCH OPTION IN UNIVERSAL LIFE POLICIES Nadine Gazer Conac (has changed since iniial submission): Chair for Insurance Managemen Universiy of ErlangenNuremberg Lange Gasse
More informationInductance and Transient Circuits
Chaper H Inducance and Transien Circuis Blinn College  Physics 2426  Terry Honan As a consequence of Faraday's law a changing curren hrough one coil induces an EMF in anoher coil; his is known as muual
More informationOptimal Consumption and Insurance: A ContinuousTime Markov Chain Approach
Opimal Consumpion and Insurance: A ConinuousTime Markov Chain Approach Holger Kraf and Mogens Seffensen Absrac Personal financial decision making plays an imporan role in modern finance. Decision problems
More informationABSTRACT KEYWORDS. Markov chain, Regulation of payments, Linear regulator, Bellman equations, Constraints. 1. INTRODUCTION
QUADRATIC OPTIMIZATION OF LIFE AND PENSION INSURANCE PAYMENTS BY MOGENS STEFFENSEN ABSTRACT Quadraic opimizaion is he classical approach o opimal conrol of pension funds. Usually he paymen sream is approximaed
More informationChapter 8: Regression with Lagged Explanatory Variables
Chaper 8: Regression wih Lagged Explanaory Variables Time series daa: Y for =1,..,T End goal: Regression model relaing a dependen variable o explanaory variables. Wih ime series new issues arise: 1. One
More informationTHE STOCHASTIC SEASONAL BEHAVIOR OF ENERGY COMMODITY CONVENIENCE YIELDS (*)
THE STOCHASTIC SEASONAL BEHAVIOR OF ENERGY COMMODITY CONVENIENCE YIELDS () Andrés García Miranes a, Javier Población b and Gregorio Serna c ( ) a IES Juan del Enzina, c/ Ramón y Cajal, 4 León, Spain. email:
More informationBALANCE OF PAYMENTS. First quarter 2008. Balance of payments
BALANCE OF PAYMENTS DATE: 20080530 PUBLISHER: Balance of Paymens and Financial Markes (BFM) Lena Finn + 46 8 506 944 09, lena.finn@scb.se Camilla Bergeling +46 8 506 942 06, camilla.bergeling@scb.se
More informationT ϕ t ds t + ψ t db t,
16 PRICING II: MARTINGALE PRICING 2. Lecure II: Pricing European Derivaives 2.1. The fundamenal pricing formula for European derivaives. We coninue working wihin he Black and Scholes model inroduced in
More informationCredit Index Options: the noarmageddon pricing measure and the role of correlation after the subprime crisis
Second Conference on The Mahemaics of Credi Risk, Princeon May 2324, 2008 Credi Index Opions: he noarmageddon pricing measure and he role of correlaion afer he subprime crisis Damiano Brigo  Join work
More informationAP Calculus AB 2010 Scoring Guidelines
AP Calculus AB 1 Scoring Guidelines The College Board The College Board is a noforprofi membership associaion whose mission is o connec sudens o college success and opporuniy. Founded in 1, he College
More informationPricing Futures and Futures Options with Basis Risk
Pricing uures and uures Opions wih Basis Risk ChouWen ang Assisan professor in he Deparmen of inancial Managemen Naional Kaohsiung irs niversiy of cience & Technology Taiwan TingYi Wu PhD candidae in
More informationVariance Swap. by Fabrice Douglas Rouah
Variance wap by Fabrice Douglas Rouah www.frouah.com www.volopa.com In his Noe we presen a deailed derivaion of he fair value of variance ha is used in pricing a variance swap. We describe he approach
More informationWorking Paper On the timing option in a futures contract. SSE/EFI Working Paper Series in Economics and Finance, No. 619
econsor www.econsor.eu Der OpenAccessPublikaionsserver der ZBW LeibnizInformaionszenrum Wirschaf The Open Access Publicaion Server of he ZBW Leibniz Informaion Cenre for Economics Biagini, Francesca;
More informationThe Path Integral Approach to Financial Modeling and Options Pricing?
Compuaional Economics : 9 63, 998. 9 c 998 Kluwer Academic Publishers. Prined in he Neherlands. The Pah Inegral Approach o Financial Modeling and Opions Pricing? VADIM LINETSKY Financial Engineering Program,
More informationForecasting Sales: A Model and Some Evidence from the Retail Industry. Russell Lundholm Sarah McVay Taylor Randall
Forecasing Sales: A odel and Some Evidence from he eail Indusry ussell Lundholm Sarah cvay aylor andall Why forecas financial saemens? Seems obvious, bu wo common criicisms: Who cares, can we can look
More informationANALYSIS AND COMPARISONS OF SOME SOLUTION CONCEPTS FOR STOCHASTIC PROGRAMMING PROBLEMS
ANALYSIS AND COMPARISONS OF SOME SOLUTION CONCEPTS FOR STOCHASTIC PROGRAMMING PROBLEMS R. Caballero, E. Cerdá, M. M. Muñoz and L. Rey () Deparmen of Applied Economics (Mahemaics), Universiy of Málaga,
More informationFair valuation of participating policies with surrender options and regime switching
Insurance: Mahemaics and Economics 37 25 533 552 Fair valuaion of paricipaing policies wih surrender opions and regime swiching Tak Kuen Siu Deparmen of Acuarial Mahemaics and Saisics, School of Mahemaical
More informationHOW CLOSE ARE THE OPTION PRICING FORMULAS OF BACHELIER AND BLACKMERTONSCHOLES?
HOW CLOSE ARE THE OPTION PRICING FORMULAS OF BACHELIER AND BLACKMERTONSCHOLES? WALTER SCHACHERMAYER AND JOSEF TEICHMANN Absrac. We compare he opion pricing formulas of Louis Bachelier and BlackMeronScholes
More informationEstimating TimeVarying Equity Risk Premium The Japanese Stock Market 19802012
Norhfield Asia Research Seminar Hong Kong, November 19, 2013 Esimaing TimeVarying Equiy Risk Premium The Japanese Sock Marke 19802012 Ibboson Associaes Japan Presiden Kasunari Yamaguchi, PhD/CFA/CMA
More informationThe performance of popular stochastic volatility option pricing models during the Subprime crisis
The performance of popular sochasic volailiy opion pricing models during he Subprime crisis Thibau Moyaer 1 Mikael Peijean 2 Absrac We assess he performance of he Heson (1993), Baes (1996), and Heson and
More informationConceptually calculating what a 110 OTM call option should be worth if the present price of the stock is 100...
Normal (Gaussian) Disribuion Probabiliy De ensiy 0.5 0. 0.5 0. 0.05 0. 0.9 0.8 0.7 0.6? 0.5 0.4 0.3 0. 0. 0 3.6 5. 6.8 8.4 0.6 3. 4.8 6.4 8 The BlackScholes Shl Ml Moel... pricing opions an calculaing
More information