Chapter 7 Additional Readings. The following is the Black-Scholes differential equation:

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1 Chaptr 7 Aitional Raings olving th Black-chols Diffrntial Equation h following is th Black-chols iffrntial quation: whr V nots th valu of th portfolio that rplicats th rivativ such as a call. In orr to solv for V = V,t, on also ns to b givn a bounary valu. his mans that th valu of th rivativ must b known at som fix tim ; that is, V, is a givn function of an. With this information, on can solv for th valu V,t at any tim t. Hr ar th ias of th proof in a nutshll. On changs variabls from V,, an t to th nw variabls u,y, an τ in two sparat procurs so that th Black-chols Diffrntial Equation simplifis to th iffrntial quation:, his quation is th classical hat quation, whos solution is wll-known. On thn changs back to th original variabls to obtain th solution. his solution will b in intgral form. W will thn solv for th spcial cas of a Europan call. In this cas, it will turn out that th intgral solution will b abl to b xprss in trms of th cumulativ normal nsity function. Now for th tails. W start th solution procss by first changing variabls to rplac t, an V with τ, y, an v: h following rivativs follow from th first two lft-han quations abov: ; ; Nxt, w us th Chain Rul to rwrit th Black-chols quation in trms of v an its partial rivativs with rspct to y an τ:

2 Chaptr 7 Aitional Raings Now, w will substitut ths quations into th Black-chols iffrntial quation: implify an ivi both sis by : Howvr, our quation still inclus trms involving v an v/y, an w still n to mploy an aitional changs of variabls in orr to obtain th sir classic hat quation. W will again mploy a chang of variabls, writing v as a function of u: whr α an β ar constants to b chosn shortly. W iffrntiat as follows:, W will substitut ths thr rivativs an th xprssion for v into our alray transform Black-chols quation:

3 Chaptr 7 Aitional Raings Divi by to obtain: Combin lik trms to obtain our scon rvision of th Black-chols iffrntial quation:, Obsrv that if w choos thn th cofficint for th trm u/ y will qual zro, an if w furthrmor choos thn th cofficint for th trm u will qual zro. With ths choics for th constants α an β, w obtain our first main objctiv: Nxt, w solv this classic hat quation. Rcall that w also n to b givn th initial bounary conition. o, w n to know how to transform th bounary conition V, to th function u in trms of y an th propr choic of τ. inc thn whn t =, τ =. inc, vy, = αy uy,, an = y, thn h solution of th hat quation uy,τ in trms of th bounary conition uy, is 3

4 Chaptr 7 Aitional Raings o prov that this solution is corrct, w fin an as w wish to show. W also must show that th solution approachs uy, in th limit as τ. o, w n to show that Mak th chang of variabls so that an h lft han limit abov bcoms h function is an xampl of what is known as an approximat intity, an th intgral is an xampl of what is call th convolution of th approximat intity with th function ux,. It is a wllknown thorm in mathmatics that in th limit as τ approachs, th convolution abov approachs uy,. Howvr, w will prov this rsult for our spcial cas. 4

5 Chaptr 7 Aitional Raings W will assum that th bounary conition grows no fastr than an xponntial of th form A B z whr A an B ar positiv constants. his is a rasonabl assumption sinc no ral-lif scurity will grow fastr than such a rat. hus, th ntir function insi th intgral gts small vry rapily onc z gts larg. o, most of th contribution to th valu of th intgral occurs in th rang of intgration from N to N with N a larg positiv numbr. W can closly approximat th limit abov by h quantity will vary from to as s rangs ovr th limits of intgration from N to N. As τ an vntually gts much smallr than /N, th quantity will b approximatly qual to y as z rangs ovr th limits of intgration from N to N. hus, ovr this rang of intgration as long as u is a continuous function. his mans that But, th intgral sinc N is vry larg an th right-han intgral is th total ara unr th stanar normal curv. h approximations abov gt bttr an bttr as N an τ. W conclu that as w wish to show. Our rsult allows us to fin th valu of any rivativ that is bas on an unrlying scurity that follows a Brownian motion procss with rift. On only ns to substitut th spcific bounary conitions for th rivativ, to obtain its valu at any tim t. As an illustration, w will pric a Europan call option. For a call option, it is customary to not its valu at tim t in trms of th pric of th stock by c,t rathr than V,t. uppos that X is its xrcis pric, an is its xpiration at. If >X at xpiration tim, thn th call option will b worth -X, sinc th option holr can buy th stock for X an sll it for. If X at tim, thn th option holr shoul not xrcis th option, making th valu of th option 5

6 Chaptr 7 Aitional Raings simply qual to. his shows that th bounary conition for th call option at tim is c, = MAX-X,. In orr to valu th pric of th option at any othr tim t, it turns out that it is asir to first solv th problm for u as a function of y an τ. Aftrwars, on can convrt th solution to c as a function of an t. W will n to fin th bounary conition for th function u. inc as w show arlir, thn ubstituting this initial conition into our intgral solution, w obtain: Obsrv that -αx - -αx mans -αx x -, or x -, or x, or x, o, th solution bcoms: h nxt fw stps of th rivation involv som algbraic manipulation an a coupl changs of variabls in orr to xprss th intgral abov in trms of th cumulativ istribution function for th stanar normal curv. Using compltion of th squar an som algbraic manipulations, w can writ: an o, th solution now bcoms: 6

7 Chaptr 7 Aitional Raings In th first intgral, mak th chang of variabls,. In th scon intgral, mak th chang of variabls, his givs:. Rcall that th cumulativ istribution function for th stanar normal curv is whr th scon quality follows bcaus of th symmtry of th stanar normal curv. hus, th solution can b xprss as: W ar now ray to obtain th solution for pric of th option c,t. inc c,t = Xvy,τ = αy+βτ uy,τ, = y, an y = ln/x, thn inc an it is asy to chck that β + -α = an Using ths rsults an th fact that rsults in Using th customary notation: w can finally xprss th solution at tim t = in th form 7

8 Chaptr 7 Aitional Raings Mor on th lf-financing Rplicating Portfolio an th Black-chols Drivation Rcall in th txt that w wr rquir to construct a slf-financing rplicating portfolio in orr to obtain arbitrag-fr pricing of th call. h portfolio consist of taking th positions γ s,t of th stock an γ b,t of th risklss bon at tim t. h valu of th portfolio at tim t was thn V t = γ s,t t + γ b,t B t. h two conitions to b a slf-financing rplicating portfolio ar: I. II. Using Itô s Lmma an th choosing, w saw that conition I ruc to th quation: his quation is th Black-chols iffrntial quation. inc B t = rt, an V t = γ s,t t + γ b,t B t, thn his mans that our two choics for th positions in th portfolio will giv us a slf-financing rplicating portfolio as long as thr is a solution to th Black-chols iffrntial quation such that conition II is also satisfi. But conition II simply stats that th valu of th portfolio must qual th valu of th call at xpiration at ; that is, V = c. W riv in th prvious topic of th aitional raing for this chaptr th solution to th Black-chols iffrntial quation with th bounary conition V = c. hus, w can conclu that w hav a slf-financing portfolio, an that th solution to th Black- chols iffrntial quation an its bounary valu conition givs us an arbitrag-fr pricing for th call. Driving Black-chols Dlta an Gamma Black-chols an Dlta h Black-chols mol is givn by th following: 8

9 Chaptr 7 Aitional Raings 9 whr N* is th cumulativ normal istribution function for *. Option trars fin it vry usful to know how th valus of thir option positions will chang as th various factors us in th pricing mol chang. For xampl, th snsitivity of th call's valu to th stock's pric is givn by Dlta: N c Dlta Applying th Chain Rul to Driv Dlta W obtain this lta by first applying th Chain Rul to th Black-chols mol as follows: N N N c r N N N c r Our xprssion for Dlta is obtain onc w monstrat that th two trms insi th brackts offst ach othr: 3 N N r W r-writ th right si of this quality as follows: r r r N N W continu to r-writ as follows: r r r N ubstituting for from th Black chols mol an thn cancling offstting trms, w hav his trick to liminat offstting trms is ky to fining many of th option Grks snsitivitis.

10 Chaptr 7 Aitional Raings r N r N ln / X r.5 N ln / X X X N ln / N X Now, it shoul b clar that quation 3 an th formula for lta ar tru: Gamma N r an N N c W can riv gamma, th rivativ of lta with rspct to using th chain rul as follows: c N N c