On Option Pricing by Quantum Mechanics Approach

Transcription

1 On Opion Pricing by Quanum Mechanics Approach Hiroshi Inoue School of Managemen okyo Universiy of Science Kuki, Saiama Japan Absrac We discuss he pah inegral mehodology of quanum mechanics for opion pricing and he relevan oher maerials When we come o analyze opion ha depend on a pah-dependen quaniy, he sraighforward approach is inadequae We discuss some opions ha are only weakly pah-dependen in ha hey can be valued using only he curren values of he underlying asse and ime, wihou he need for any variable o represen he pah-dependen quaniy so ha hey saisfy he equaion 1 Inroducion he breakhrough in opion pricing heory came wih he famous Black and Scholes paper in 197hey were he firs o show ha opions could be priced by consrucing a risk-free hedge by dynamically managing a simple porfolio consising of he underlying asse and cash Afer ha many direcions of research relevan o financial derivaives have been produced and conribued o differen areas On he oher hand, quanum heory as a par of physics has been developed during various hisorical findings, and opion pricing has a mahemaical descripion, which corresponds o a quanum sysem In classical mechanics he posiion of a paricle a ime, is a deerminisic funcion of, which is given by Newon s law of moion In conras in quanum mechanics he paricle s evoluion is random, analogous o he case of he evoluion of a sock price having no-zero volailiy Quanum Mechanics Quanum mechanics was discovered in wo differen forms: wave mechanics and mari mechanics In 1900 Planck inroduced he quanum of acion h and in 1905 Einsein posulaed paricles of ligh wih energy E = hν ( ν :frequency) In 194 de Broglie pu he wo formulas E = mc ( m : mas, E = hν ogeher and invened maer waves hen, he wave naure of maer was eperimenally confirmed by he Davisson-Germer elecron diffracion eperimen in 197 and heoreically suppored by he work of Schrӧdinger in 196 Suppose we have a paricle (say an elecron) of mass m in a poenial V where V is a real funcion on R represening he poenial energy Schrӧdinger aemped o describe he moion of he elecron by means of a quaniyφ subec o a wave equaion His hypohesis is ha a saionary sae vibraes according o he equaion h Δφ + ( E V ) φ = 0, m where h is Planck s consan h divided by π, and E (wih he dimensions of energy) plays he role of

2 an eigen value Hamilonian be suiable o be used for opion pricing 4 Pah-dependen Opion Hamilonians are applied o he sudy of sock opions and sochasic ineres rae models, which are characerized by having finie numbers of degrees of freedom he problem of he pricing of derivaive securiies is recas as a problem of quanum mechanics, and he Hamilonians driving he prices of opions are considered for sock prices wih consan and sochasic volailiy Maringale condiion required for risk-neural evoluion is re-epressed in erms of he Hamilonian Poenial erms in he Hamilonian are shown o represen a class of pah-dependen opions By assuming of several condiions such as no-arbirage, consan spo rae r, coninuous rebalancing of he porfolio, no ransacion coss and infinie divisibiliy of he sock, he Black-Scholes equaion is epressed as f 1 f f + σ S + rs = rf S S he equaion above can be ransformed ino a quanum mechanical version by a change of variable, S = e wih a real variable[] his yields f = H f where H is an Hamilonian given by H = 1 1 σ + ( σ r) + r hen,he Hamilonian can be generalized o saisfy maringale condiion, H V = 1 1 σ + ( σ V ( )) +V () (1) where he poenial V () is an arbirary funcion of Noe ha a derivaive evolving wih will yield a risk-free measure and H Use of a pah inegral formulaion has some advanages since i is relaed o Lagrangian descripion of diffusion process so ha i enables he use of quanum mechanical mehods In classical physics, ime evoluion of dynamical sysems is governed by Leas Acion Principle Equaions of moion such as Newon s equaions can be viewed as Euler-Lagrange equaions for a minimum of a cerain acion funcional, a ime inegral of he Lagrangian funcion defining he dynamical sysem hus, heir deerminisic soluions, which are raecories of he classical dynamical sysem, minimize he acion funcional ha is he leas acion principle In quanum physics, we can hink abou probabiliies of differen pahs a quanum dynamical sysem can ake I is possible o define a measure on he se of all possible pahs from he iniial sae o he final sae of he quanum dynamical sysem so ha epeced values of differen quaniies dependen on pahs are given by pah inegrals over all possible pah from 0 o Classical Mechanics and Acion Funcional In classical physics, i is known ha he produc of acceleraion of moion and mass of a paricle equals he force he paricle receives he moion following Newon s mechanics can be described in anoher way By he principle of leas acion a paricular pah ( ou of all he possible pahs can be deermined ha is, here eiss a cerain quaniy A() which can be compued for each pah he pah ( is ha for which A() becomes a minimum he quaniy A() is defined by

3 A ( ) L( &, d = s s (41) where L( ) is he Lagrangian for he sysem For a paricle of mass m moving in a poenial V (, which is a funcion of posiion and ime, he Lagrangian is m L( &, = & V ( (4) he phase of he conribuion from a given pah is epressed as he acion A() for ha pah in unis of he quanum of acion h In oher words, he probabiliy P( b, o go from a poin a he ime o he poin a a a b is P( b, = K ( b, of an ampliude K ( a, b) o go from a o b his ampliude is he sum of conribuionφ [ ( ] from each pah K( b, = φ[ ( ] he conribuion of a pah has a phase proporional o he acion A() i / ha[ ( ] φ[ ( ] = cons e Noe ha in quanum mechanics he dynamics of a paricle follows a wave funcion φ( ), R and, assuming φ ( ) d = 1 (4) R φ() is inerpreed as he densiy of probabiliy for which he paricle is observed a If a wave funcion φ ( a ime s becomes φ( a every differen ime, as basic law of quanum mechanics, he correspondence is linear and epressed φ ( = U ( φ(, where he linear ransformaion U ( is he moion law of quanum mechanics describing dynamics of a paricle hus, i is wrien as inegral ransformaion φ( = K( φ( y, dy R Le Ω be he space consising of whole curves for which a paricle sars y a b ime s and reaches o a ime hen, he inegral kernel is defined as 1 ih A( r) K( = C e Dr Ω where Dr is he volume elemen inegraing wih respec o r over Ω and C is a consan weigh his can be epressed by using Lagrange funcion insead of A(r) K( = 1 i h L( u, r, r& ) du s e Ω Dr (44) which is called Feynman pah inegral 4- Schrӧdinger Equaion Le L( v) be a Lagrange funcion of ime, posiion and velociy v Define dimensional vecor p = ( p1, p, p) so ha p = L( v), = 1,, v which epress he momenum of a paricle Wih Lagrange funcion (41) he epression above is 1 v = p, = 1,, m hen, Hamilon funcion H ( p) can be defined H ( p) =< v p > L( v) which epresses mechanical energy of a paricle In paricular, in our case i becomes 1 H ( p) = p + ev ( m where p = p1 + p + p Now, we replace each elemen p ( = 1,,) of momenum by parial differenial acion ih o obain Hamilonian, H ( ) hus, in his case h H ( ) = + ev ( 1 m = A wave funcion φ ( ) eplaining paricle s sae of quanum mechanics is

4 he soluion o he parial differenial equaion ih φ( = H ( ) φ( he equaion is called Schrӧdinger equaion Wih he funcion K( of (44), ( ih H ( )) K( = 0 and K( f ( y) dy = f ( ) hold for R every funcion f () In oher words, K( is he basic soluion o Schrӧdinger equaion 5 Opion Pricing Consider a complee probabiliy space ( Ω, I, P) wih a filraion { I },which is righ 0 coninuous and conains all he I0 P null ses of I According o he papers ([10],[11]) we can assume he eisence of a risk-neural probabiliy measure Q equivalen o P such ha he discouned prices of he securiies are maringales Under Q,he dynamics of he underlying sock price follows he sochasic differenial equaion ds ( = r d + σ dw We parially follow he noaion and descripion in [6] o eplain pah dependen siuaion relevan o a poenial Suppose we consider a pah dependen opion defined by is payoff a epiraion G F ( ) = F[ )], where F[ )] is a given funcional on price pahs { ), } hen he presen value of his pah dependen opion a he conrac is given by Feynman-Kac formula G rτ ( S, ) = e E[ F( ))], τ = F = e rτ ( F( e ) e A [ ( )] D( )) d where he average is over he risk-neural process Noe ha he acion funcional A [ ( )] defined on pahs { ( ), } defines he pah inegraion measure wih Lagrangian A [ ( )] = L d, 1 L = ( & ( ) μ) σ If F has he following simple form[8] f1[ S ( )] F = f ( S ) e, where f [ )] can be represened as 1 1 )] = V ( ( ), ) f [ d of some poenial V ( ),hen i becomes rτ ( μ / σ )( ) ( μ τ / σ G ( S, = e f ( e ) e F V (, d K, (51) where KV (, is he Green funcion for zero-drif Brownian moion wih killing a rae V ( ) such ha K (, = V ep( ( L ) ) ( ), (5) 0 + V d D where L 0 is he Lagrangian for a zero drif process he represenaion of (5) saisfy 1 KV KV σ + = V ( K V wih iniial condiion KV (, = δ ( ) where δ () is he Dirac dela funcion Also, he opion price (51) saisfies he parial differenial equaion wih poenial 1 G KV GF σ + μ + = F ( r + V ( ) G F and he erminal condiion G S, ) = f ( S ) F (, )

5 hus,i is showed ha he equaion includes risk-free rae, r + V ( 6 Some Remarks Some relaionships beween pah inegral of quanum mechanics and opion pricing were discussed along wih he risk-neural valuaion, including some correspondence of quanum mechanics and financial derivaives he definiion of pah inegrals is originally used o describe quanum phenomena, and his definiion is compleely rigorous and he limi does converge [] (1)In finance, he fundamenal principle comes from he no-arbirage ([],[7]ec) Since i plays a role similar o he leas acion principle and he law of energy conservaion in physics, similar o physical dynamical sysems Lagrangian funcions and acion funcionals can be inroduced for financial models Also, financial models are sochasic ones so ha epeced values of differen quaniies dependen on price pahs are given by pah inegrals, where he acion funcional for he underlying risk-neural price process defines a risk-neural measure on he se of all pahs ()Averages saisfy he parial differenial equaion, which is a finance counerpar of he Schrӧdinger equaion of quanum mechanics, and he risk-neural valuaion formula is inerpreed as he Feynman-Kac represenaion of he PDE soluion hus, he pah-inegral formalism provides a naural bridge beween he risk-neural maringale pricing and he arbirage-free PDE based pricing[6] ()By maringale condiion required for risk-neural evoluion, risk-free rae r in Hamilonian is replaced by poenial V () in (1) Poenial erms in he Hamilonian show o represen a class of pah-dependen opions (4)From he poin of view ha Lagrange and Hamilon physical sysems are characerized as only one funcion, hen moion equaion, definiion of physical quaniy and conservaion law are deermined if eiher Lagrange or Hamilon is given (5)Schrӧdinger equaion is ime reversible since he Hamilonian is Hermiian hough Black-Scholes process is ime-irreversible due o he Hamilonian being non-hermiian (6)In quanum mechanics a paricle in mechanical sysem is considered o be no isolaed bu a member of he whole mechanical sysem In oher words, he value of some physical quaniy is no deermined in a momen a us neighborhood bu he enire sysem is involved for ha However, Newon s sandpoin shows ha he posiion of a paricle is deermined by knowing force which acs on he paricle (iniial values and velociy need o be provided), no aking ino accoun of oher paricles (7)here are several opions whose payoff funcions depend on he pahin paricular, he barrier opion and Asian opion, which are recenly popular for invesors, will be considered o use he mehodology above under various condiions o find he soluions References [1]Black,F,and MScholes(197) he Pricing of Opions and Corporae Liabiliies JPolEcon,81, []Baaquie,BE,CCoriano,MSrikan (004) Hamilonian and Poenials in Derivaive Pricing Models: Eac Resuls and Laice Simulaions Physica A []Duffie,D (1996) Dynamic Asse Pricing nd ed Princeon Univ Press,

6 New Jersey [4]Nelson,E (001) Dynamic heories of Brownian Moion, Princeon Universiy Press [5]Harrison,JM,and SPliska (1981) Maringales and Sochasic Inegrals in he heory of Coninuous rading Sochasic Process and heir Applicaions,11,15 [6]Linesky,V (1998) he Pah Inegral Approach o Financial Modeling an Opion Pricing, Compuaional Economics 11,19-16 [7]Meron,RC (1990) Coninuous ime Finance,Blacwell, Cambridge,MA [8]Prigogine,I (1989) Eploring Compleiy [9]Feynman,RP and ARHibbs(1965) Quanum Mechanics and Pah Inegrals, McGraw-Hill, New York [10]Harrison,JM and D Kreps,(1979) Maringale and Arbirage in Muliperiods Securiies Markes, JEconom heory,0, [11]Kac,M(1951) On some connecions beween probabiliy and differenial and inegral equaions, Proc of nd Berkeley symp Mah Sa and Prob,Univ of Calif Press,