V. Black-Scholes model: Derivation and solution

1 Model: Derivation and solutionBe ta Stehl kov Financial derivatives, winter term 2014/2015 Faculty of Mathematics, Physics and InformaticsComenius University, BratislavaV. Black-Scholes model: Derivation and solution Black-Scholes model: Suppose that stock priceSfollows a geometricBrownian motiondS= Sdt+ Sdw+ other assumptions (in a moment) We derive apartial differential equation for the price ofa derivative Two ways of derivations: due toBlack and scholes due toMerton Explicit solution for European call and put optionsV. Black-Scholes model: Derivation and solution Further assumptions (besides GBP): constant riskless interest rater no transaction costs it is possible to buy/sell any (also fractional) number ofstocks; similarly with the cash no restrictions onshort selling option is of European type Firstly, let us consider the case of anon-dividend payingstockV.

2 Black-Scholes model: Derivation and solution I. - due to black and scholes Notation:S=stock price,t=timeV=V(S, t) =option price Portfolio: 1 option, stocksP=value of the portfolio:P=V+ S Change in the portfolio value:dP=dV+ dS From the assumptions:dS= Sdt+ Sdw,From the It olemma:dV=( V t+ S V S+12 2S2 2V S2)dt+ S V Sdw Therefore:dP=( V t+ S V S+12 2S2 2V S2+ S)dt+( S V S+ S)dwV. Black-Scholes model: Derivation and solution I. - due to black and scholes We eliminate the randomness: = V S Non-stochastic portfolio its value has to be the same asif being on a bank account with interest rater:dP=rP dt Equality between the two expressions fordPandsubstitutingP=V+ S: V t+12 2S2 2V S2+rS V S rV= 0V. Black-Scholes model: Derivation and solution in the Black-Scholes Derivation We considercontinuous divident rateq- holding a stockwith valueSduring the time differentialdtbringsdividendsqSdt In this case the change in the portfolio value equalsdP=dV+ dS+ qSdt We proceed in the same way as before and obtain V t+12 2S2 2V S2+ (r q)S V S rV= 0V.

3 Black-Scholes model: Derivation and solution due to Merton - motivation Problem in the previous Derivation : we have a portfolio consisting of one option and stocks we compute its value and change of its value:P=V+ S,dP=dV+ dS, , treating as a constant however, we obtain = V SV. Black-Scholes model: Derivation and solution II. - due to Merton Portfolio consisting ofoptions, stocks and cashwith theproperties: in each time, the portfolio haszero value it isself-financing Notation:QS=number of stocks, each of them has valueSQV=number of options, each of them has valueVB=cash on the account, which is continuouslycompounded using the risk-free raterdQS=change in the number of stocksdQV=change in the number of options B=change in the cash, caused by buying/selling stocksand optionsV. Black-Scholes model: Derivation and solution II. - due to Merton Mathematical formulation of the required properties: zero valueS QS+V QV+B= 0(1) self-financing:S dQS+V dQV+ B= 0(2) Change in the cash:dB=rB dt+ B Differentiating (1):0 =d(SQS+V QV+B) =d(SQS+V QV) +rB dt+ B dB0 ==0 SdQS+V dQV+ B+QSdS+QVdV+rB dt0 =QSdS+QVdVrB r(SQS+V QV) Black-Scholes model: Derivation and solution II.

4 - due to Merton We divide byQVand denote = QSQV:dV rV dt (dS rS dt) = 0 We havedSfrom the assumption of GBM anddVfromthe It olemma We choose ( , the ratio between the number of stocksand options) so that it eliminates the randomness (thecoefficient atdwwill be zero) We obtain the same PDE as before: V t+12 2S2 2V S2+rS V S rV= 0V. Black-Scholes model: Derivation and solution in the Merton s Derivation Assumecontinuous dividend rateq. Dividents cause an increase in the cash change in thecashisdB=rB dt+ B+qSQSdt In the same way we obtain the PDE V t+12 2S2 2V S2+ (r q)S V S rV= 0V. Black-Scholes model: Derivation and solution PDE: summary Matematical formulation of the model: Find solutionV(S, t)to thepartial differential equation(socalled Black-Scholes PDE) V t+12 2S2 2V S2+rS V S rV= 0which holds forS >0, t [0, T). So far we have not used the fact that we consider an option PDE holds for any derivative that pays a payoff at timeTdepending on the stock price at this time Type of the derivative determinesthe terminal conditionattimeT In general:V(S, T) =payoff of the derivativeV.]

5 Black-Scholes model: Derivation and solution PDE: simple solutionsSOME SIMPLE"DERIVATIVES": How to price the derivatives with the following payoffs: V(S, T) =S it is in fact a stock V(S, t) =S V(S, T) =E with a certainity we obtain the cashE V(S, t) =Ee r(T t)- by substitution into the PDE we see that they are indeedsolutionsEXERCISES: Find the price of a derivative with payoffV(S, T) =Sn,wheren : Look for the solution in the formV(S, t) =A(t)Sn Find all solutions to the Black-Scholes PDE, which areindependent of time, , for whichV(S, t) =V(S)V. Black-Scholes model: Derivation and solution PDE: binary option Let us consider abinary option, which pays 1 USD if thestock price is higher thatEat expiration time, otherwise itspayoff is zero In this caseV(S, T) ={1ifS > E0otherwise The main idea is totransform the Black-Scholes PDE to aheat equation Transformations are independent of the derivative type; itaffects only the initial condition of the heat equationV.}

6 Black-Scholes model: Derivation and solution PDE: transformationsFORMULATION OF THE PROBLEM Partial differential equation V t+12 2S2 2V S2+rS V S rV= 0which holds forS >0, t [0, T). Terminal conditionV(S, T) =payoff of the derivativeforS >0V. Black-Scholes model: Derivation and solution PDE: transformationsSTEP1: Transformationx= ln(S/E) R, =T t [0, T]and anew functionZ(x, ) =V(Eex, T ) PDE forZ(x, ),x R, [0, T]: Z 12 2 2Z x2+( 22 r) Z x+rZ= 0,Z(x,0) =V(Eex, T)STEP2: Transformation toheat equation New functionu(x, ) =e x+ Z(x, ),where the constants , Rare chosen so that the PDE foruis the heatequationV. Black-Scholes model: Derivation and solution PDE: transformations PDE foru: u 22 2u x2+A u x+Bu= 0,u(x,0) =e xZ(x,0) =e xV(Eex, T),whereA= 2+ 22 r, B= (1 + )r 2 2+ 22. In order to haveA=B= 0, we set =r 2 12, =r2+ 28+r22 2V. Black-Scholes model: Derivation and solution PDE: transformationsSTEP3: Solutionu(x, )of the PDE u 22 2u x2= 0is given byGreen formulau(x, ) =1 2 2 e (x s)22 2 u(s,0)ds.]

7 We evaluate the integral and perform backwardsubstitutionsu(x, ) Z(x, ) V(S, t)V. Black-Scholes model: Derivation and solution PDE: binary option (continued) Transformations from the previous slides We obtain the heat equation u 22 2u x2= 0with initialconditionu(x,0) =e xV(Eex, T) ={e xifEex> E0otherwise={e xifx >00otherwise Solutionu(x, ):u(x, ) =1 2 2 0e (x s)22 2 e sds=..=e x+12 2 2N(x+ 2 )whereN(y) =1 2 y e 22d is the cumulative distributionfunction of a normalized normal distributionV. Black-Scholes model: Derivation and solution PDE: binary option (continued) Option priceV(S, t):V(S, t) =e r(T t)N(d2),whered2=log(SE)+(r 22)(T t) T tV. Black-Scholes model: Derivation and solution PDE: call option In this caseV(S, T) = max(0, S E) ={S EifS > E0otherwise The same sequence of transformations; inital condition forthe heat equation:u(x,0) ={e x(S E)ifx >00otherwiseand similar evaluation of the integral Option price:V(S, t) =SN(d1) Ee r(T t)N(d2),whereNis the distribution function of a normalized normaldistribution andd1=lnSE+(r+ 22)(T t) T t,d2=d1 T tV.}}}}

8 Black-Scholes model: Derivation and solution PDE: call optionHOMEWORK:Solve the Black-Scholes PDE for a call option on a stock whichpays continuous dividends and write it in the formV(S, t) =Se q(T t)N(d1) Ee r(T t)N(d2),whereN(x) =1 2 x e 22d is the distribution function of anormalized normal distributionN(0,1)andd1=lnSE+ (r q+ 22)(T t) T t, d2=d1 T tNOTE: The PDE is different, so the transformations have to beadjusted (do the same steps for the new equation)V. Black-Scholes model: Derivation and solution PDE: call optionPayoff ( , terminal condition at timet=T= 1) and solutionV(S, t)for selected timest:204060305070253545556502010 224681214161822stock priceoption pricepayofft = 0t = = = Black-Scholes model: Derivation and solution PDE: put optionFORMULATION OF THE PROBLEM Partial differential equation V t+12 2S2 2V S2+rS V S rV= 0which holds forS >0, t [0, T].

9 Terminal condition:V(S, T) = max(0, E S)forS >0V. Black-Scholes model: Derivation and solution PDE: put optionAPPROACHI. The same sequence of computations as in the case of acall optionAPPROACHII. We use the linearity of the black - scholes PDE and thesolution for a call which we have already foundWe show the application of the latter Black-Scholes model: Derivation and solution PDE: putoption Recall that for the payoffs of a call and a put we have [call payoff] + [put payoff] + [stock price] =E Hence:[put payoff] = [call payoff] S+E Black-Scholes PDE is linear:a linear combination ofsolutions is again a solutionV. Black-Scholes model: Derivation and solution PDE: put option Recall the solutions forV(S, T) =SandV(S, T) =E(page 13):terminal conditionsolutionmax(0, S E)Vcall(S, t)SSEEe r(T t) From the linearity:terminal conditionsolutionmax(0, S E) S+EVcall(S, t) S+Ee r(T t) Since[put payoff] = max(0, S E) S+E, we getVput(S, t) =Vcall(S, t) S+Ee r(T t)V.

10 Black-Scholes model: Derivation and solution for a put option The solutionVput(S, t) =Vcall(S, t) S+Ee r(T t)can be written in a similar form as the solution for a calloption:Vep(S, t) =Ee r(T t)N( d2) SN( d1),whereN, d1, d2are the same as beforeV. Black-Scholes model: Derivation and solution option - examplePayoff ( terminal condition at timet=T= 1) and solutionV(S, t)for selected timest:40605035455565010246812141618stoc k priceoption pricepayofft = 0t = = = Black-Scholes model: Derivation and solution option - alternative computationComics about negative volatilityon the webpage of Espen Haug: Black-Scholes model: Derivation and solution option - alternative computation A nightmare about negative volatility: Not only a according to internet, it really existsand is connected with professor Shiryaev from Black-Scholes model: Derivation and solution option - alternative computationQUESTION: Why does this computation work?