The Mathematics Of Stock Option Valuation - Part Four Deriving The Black-Scholes Model Via Partial Differential Equations

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1 The Mahemaics Of Sock Opion Valuaion - Par Four Deriving The Black-Scholes Model Via Parial Differenial Equaions Gary Schurman, MBE, CFA Ocober 1 In Par One we explained why valuing a call opion as a sand-alone asse using risk-adjused discoun raes will almos always lead o an incorrec value because he value deermined in his manner will mos likely be subjec o arbirage. In Par Two we calculaed he no-arbirage price of a call opion in he one-period economy via parial differenial equaions. In Par III we calculaed he no-arbirage price of a call opion in he one-period economy via risk-neural probabiliies. In his secion we will swich from he one-period economy o he muli-period economy in coninuous ime where we will derive he Black-Scholes opion pricing model via parial differenial equaions. The One Period Economy (From Par One) The coninous ime equivalen o our one-period economy in Pars II and III is... Table 1: Mui-Period Coninuous Time Economy Sock price a ime zero S $1. Call opion exercise price K $1. Annual discoun rae µ.3 Annual reurn volailiy σ.5 Annual risk-free rae r.5 Time o opion expiraion in years T 1. In Par One we esimaed he sock price a ime = o be $1. using a discoun rae of 3% and an approximae volailiy of 5%. We currenly si a ime = where he sae-of-he-world a ime = 1 is unknown. In his secion we will derive he Black-Scholes equaion and use ha equaion o calculae he no-arbirage value of he call opion a ime =. Legend of Symbols = Call opion price a he end of ime P = Hedge porfolio value a he end of ime S = Sock price a he end of ime = Number of shares of sock in he hedge porfolio a ime K = Call opion exercise price T = Time o opion expiraion in years r = Annual risk-free rae of ineres = Curren ime period in years µ = Annual expeced reurn on he sock σ = Annual sandard deviaion of reurns (Volailiy) W = Brownian moion wih mean zero and variance A Coninuous Time Model For Sock Price We will model sock price as a coninuous ime sochasic process. The equaion for sock price a ime as a funcion of a deerminisic reurn and an innovaion is... S = S e (µ 1 σ )+σw (1) 1

2 Per his equaion sock price is a funcion of drif, which is represened by (µ 1 σ ), and a Brownian moion, which is represened by σw. The discoun rae used o value our sock a ime = was 3% so herefore he variable µ in he equaion above is.3. Drif is fully predicable in ha we expec he sock o earn a an annual rae of 3%. The annual reurn volailiy for our sock is 5% and herefore he variable σ in he equaion above is.5. Siing a ime = we don know he value of he Brownian moion W bu we do know ha i is normally-disribued wih mean zero and variance. The coninuous ime equaion for our sock price a any ime is... S = 1. e ( )+.5W () Sock price equaion (1) is once differeniable wih respec o ime and wice differeniable wih respec o he Brownian moion. The equaion for he oal change in sock price (referred o as a sochasic differenial equaion (SDE)) is... = δ δ + δw + 1 δw δ S δw δw = S µδ + S σδw (3) We can view δ as he variance of δ and δw as he variance of δw. Time is a deerminisic variable and herefore he variance of he change in ime is zero, which means ha he second derivaive of equaion (1) wih respec o ime is zero. The Brownian moion is a random variable and herefore he variance of he change in he Brownian moion is non-zero, which means ha he second derivaive of equaion (1) wih respec o he Brownian moion is non-zero. By definiion... ] ] ] E δ = ; E δ δw = ; E δw = δ (4) For he parial differenial equaion developed below we will need an equaion for he square of he change in sock price which is... δs = (S µδ + S σδw ) = S µ δ + S µσδδw + S σ δw (5) Afer noing he definiions in equaion (4) above we can rewrie equaion (5) as... A Coninuous Time Model For Call Price δs = S µ () + S µσ() + S σ (δ) = S σ δ (6) Our ask is o derive an equaion for call price a = and alhough we do no know he exac equaion we do know is general form. We will model call price as a funcion of ime () and sock price (S ). The general form of he equaion for call price is... = C(S, ) (7) Call price is once differeniable wih respec o ime and wice differeniable wih respec o sock price. equaion for he oal change in call price a ime is... δ = δ δ δ + δ + 1 The δ δs δs (8) We can subsiue equaion (3) for and equaion (6) for δs in he equaion above. Afer making hese subsiuions equaion (8) becomes... δ = δ δ δ + δc } {µs δ + σs δw + 1 δ { } δs δ (9) The equaion for he oal change in he discouned call price a ime is... δ(e r ) = (δe r ) + (δ )e r = re r δ + e r δc δ δ + δ = e r r + δ δ + δ µs + 1 {µs δ + σs δw } + 1 δ { }] δs δ δ ] δs δ + e r δ σs δw (1)

3 A Coninuous Time Model For The Hedge Porfolio The hedge porfolio will consis of a long posiion in shares of he underlying sock and a posiion in a money marke accoun. The value of he hedge porfolio a any ime is... X = Sock + Money Marke The equaion for he oal change in value of he hedge porfolio a ime is... = S + (X S ) (11) δx = + r(x S )δ = (µs δ + σs δw ) + r(x S )δ = rx δ + (µ r)s δ + σs δw (1) The equaion for he oal change in discouned value of he hedge porfolio a ime is... δ(e r X ) = (δe r )X + (δx )e r Pricing Derivaives Via PDEs = ( re r δ)x + (rx δ + (µ r)s δ + σs δw )e r = re r X δ + re r X δ + (µ r)e r S δ + σe r S δw = (µ r)e r S δ + σe r S δw (13) Since boh he sock and he call opion on ha sock are driven by he same random process (i.e. he Brownian moion W ) hese wo asses can be combined in one porfolio such ha he randomness of one asse offses he randomness of he oher resuling a porfolio ha is risk-free. The price of he call opion will be deermined via he following seps... 1 Creae he hedge porfolio and derive he PDE Find he equaion ha solves he PDE derived in sep 1 3 Use he equaion in sep o deermine call price a = We will follow hese seps o price he call opion in our one period economy. Sep One - Creae The Hedge Porfolio And Derive The PDE We can hedge a shor posiion in a call opion via a hedging porfolio ha sars wih some iniial capial X and invess in he underlying sock and a money marke accoun. The goal of he hedging sraegy is o have he hedge porfolio value X equal o he call opion value a every ime. The goal of he hedging sraegy in equaion form is... X = for all, T ] (14) The combinaion of he hedge porfolio and he shor posiion in he call is risk-free and herefore he presen value a ime = of he hedge porfolio and he call opion is he value of he hedge porfolio and he call opion a ime > discouned a he risk-free rae. The equaion for he presen value of he hedge porfolio and he call opion a ime = is... e r X = e r for all, T ] (15) The equivalen of equaion (15) above is... X + δ(e ru X u ) = C + δ(e ru C u ) (16) The amoun of capial ha we will deposi ino he hedge a ime = is X, which we will define as being equal o C. Because X = C we can subrac X from he lef side and C from he righ side of equaion (16) above. The revised equaion is... δ(e ru X u ) = 3 δ(e ru C u ) (17)

4 Per equaion (13) he hedge is self-financing afer he iniial capial conribuion (i.e. once he hedge is se up no cash is deposied o he hedge or wihdrawn from he hedge prior o call expiraion). Wha we need is an equaion for call value such ha δ(e r X ) in equaion (16) is equal o δ(e r ) in equaion (16) a every ime. This relaionship in mahemaical form is... δ(e r X ) = δ(e r ) for all, T ] (18) We sar by subsiuing equaion (13) for δ(e r X ) and equaion (9) for δ(e r ) in equaion (18) above such ha he equaion becomes... (µ r)e r S δ + σe r S δw = e r r + δ δ + δ µs + 1 δ ] δs δ + e r δ σs δw (19) We hen muliply boh sides of equaion (19) by e r such ha he equaion becomes... (µ r)s δ + σs δw = r + δ δ + δ µs + 1 δ ] δs δ + δ σs δw () We wan o remove all randomness from equaion () by eliminaing all erms ha involve δw. We do his by seing he number of shares of he underlying sock held by he hedge porfolio equal o he firs derivaive of call price wih respec o sock price. We will make he following definiion... r δ S δ = = δ (1) Afer making his subsiuion a porfolio ha is long he hedge porfolio and shor he call opion is risk-free. Equaion () becomes... δ (µ r)s δ + δc σs δw = r + δ δ + δ µs + 1 δ ] δs δ + δ σs δw r + δc ] δ = δ δs δ + 1 r + δ δ + r δ S + 1 δ ] δs δ () Since δ is common o all erms in equaion () we can remove i. Equaion () above becomes he Black-Scholes parial differenial equaion... r + δ δ + r δ S + 1 δ δs = (3) Sep Two - Find A Soluion To The PDE The soluion o a parial differenial equaion is anoher equaion such ha when you ake he soluion equaion derivaives and drop hem ino equaion (3) above you ge zero. Our PDE has an infinie number of soluions. To ge one unique soluion we mus specify boundary condiions. We will add he condiion ha he value of he call a expiraion (ime T ) mus be equal o... C T,ST = Max(S T K, ) (4) As i urns ou he Black-Scholes PDE is he one-dimensional hea equaion in disguise. Raher han solving he Black-Scholes PDE via he hea equaion we will prove ha he soluion o he PDE is indeed valid. The soluion o he PDE via he one-dimensional hea equaion and subjec o he boundary condiions in equaion (4) above is... C = S N(d1) Ke r(t ) N(d) (5) where he equaion for d1 is... and he equaion for d is... d1 = ln S K + (r + 1 σ )(T ) σ T (6) d = d1 σ T (7) 4

5 In order o prove ha equaion (5) is a valid soluion o he PDE in equaion (3) we need he soluion equaion s derivaives. We can obain The Greeks from any decen exbook on he Black-Scholes model and indeed ha is wha we have done. The derivaives of equaion (5) wih respec o ime (Thea), sock price (Dela) and he square of sock price (Gamma) are... Dela... Gamma... Thea... δc δs = N(d1) (8) δ C δs = N (d1) Sσ We will now drop equaions (8), (9) and (3) ino equaion (3)... (9) δc δ = SN (d1)σ rke r N(d) (3) r + δ δ + rs δ + 1 S σ δ δs = { } { r S N(d1) Ke r N(d) + SN } { } (d1)σ rke r N(d) + rs N(d1) + 1 { N S σ } (d1) Sσ = rs N(d1) + rke r N(d) SN (d1)σ Conclusion: Equaion (5) is a soluion o PDE equaion (3). Sep Three - Deermine Call Price A Time Zero We will now solve for call price a ime =. The value of d1 is... The value of d is... d1 = rke r N(d) + rs N(d1) + S N (d1)σ = (31) S ln K + (r + 1 σ 1. )(T ) σ ln 1. = + ( )(1 ) T.5 =.1464 (3) 1 The cumulaive normal disribuion value of d1 is... The cumulaive normal disribuion value of d is... The value of he call opion a ime = is... Conclusion d = d1 σ T =.1464 (.5)(1) = (33) N(d1) = N(.1464) = (34) N(d) = N(.51464) =.334 (35) C = S N(d1) Ke r(t ) N(d) = (1.)(.49416) (1.)(.951)(.334) = (36) The value of he call opion a ime = via he coninuous ime Black-Scholes equaion is $ The value of he call opion a ime = using he discree ime single ime sep equaions in Par II and Par III was $14.34 and $14.9, respecively. 5