Four Derivations of the Black-Scholes Formula - …

1 four Derivations of the Black-Scholes Formulaby Fabrice Douglas this note we derive in four separate ways the well-known result of Blackand scholes that under certain assumptions the time-tpriceC(St; K; T)of aEuropean call option with strike priceKand maturity =T ton a non-dividend stock with spot priceStand a constant volatility when the rate ofinterest is a constantrcan be expressed asC(St; K; T) =St (d1) e r K (d2)(1)whered1=lnStK+ r+ 22 p andd2=d1 p , and where (y) =1p2 Ry 1e 12t2dtis the standard normalcdf. We show four ways in which Equation (1) can be By straightforward By applying the Feynman-Kac By transforming the black scholes PDE into the heat equation, for whicha solution is known. This is the original approach adopted by black andScholes [1].4. Through the Capital Asset Pricing Model (CAPM).Free code for the Black-Scholes model can be found at Black-Scholes EconomyThere are two assets: a risky stockSand riskless bondB:These assets aredriven by the SDEsdSt= Stdt+ StdWt(2)dBt=rtBtdtThe time zero value of the bond isB0= 1and that of the stock isS0.

2 Themodel is valid under certain market assumptions that are described in John1 Hull s book [3]. By It o s Lemma the valueVtof a derivative written on thestock follows the di usiondVt=@V@tdt+@V@SdS+12@2V@S2(dS)2(3)= @V@tdt+@V@SdS+12@2V@S2 2S2dt= @V@t+ St@V@S+12 2S2t@2V@S2 dt+ St@V@S dWt:2 The Lognormal The Lognormal PDF and CDFIn this Note we make extensive use of the fact that if a random variableY2 Rfollows the normal distribution with mean and variance 2, thenX=eYfollows the lognormal distribution with meanE[X] =e +12 2(4)and varianceV ar[X] = e 2 1 e2 + 2:(5)The pdf forXisdFX(x) =1 xp2 exp 12 lnx 2!(6)and the cdf isFX(x) = lnx (7)where (y) =1p2 Ry 1e 12t2dtis the standard normal The Lognormal Conditional Expected ValueThe expected value ofXconditional onX > xisLX(K) =E[XjX > x]. Forthe lognormal distribution this is, using Equation (6)LX(K) =Z1K1 p2 e 12(lnx )2dx:Make the change of variabley= lnxso thatx=ey,dx=eydyand theJacobian isey:Hence we haveLX(K) =Z1lnKey p2 e 12(y )2dy:(8)2 Combining terms and completing the square, the exponent is 12 2 y2 2y + 2 2 2y = 12 2 y + 2 2+ +12 2:Equation (8) becomesLX(K) = exp +12 2 1 Z1lnK1p2 exp0@ 12 y + 2 !

3 21 Ady:(9)Consider the random variableXwith pdffX(x)and cdfFX(x), and the scale-location transformationY= X+ . It is easy to show that the Jacobian is1 ,that the pdf forYisfY(y)=1 fX y and that the cdf isFY(y) =FX y .Hence, the integral in Equation (9) involves the scale-location transformationof the standard normal cdf. Using the fact that ( x) = 1 (x)this impliesthatLX(K) = exp + 22 lnK+ + 2 :(10)See Hogg and Klugman [2].3 Solving the Stock PriceApply It o s Lemma to the functionlnStwhereStis driven by the di usion inEquation (2). ThenlnStfollows the SDEdlnSt= 12 2 dt+ dWt:(11)Integrating from0tot, we haveZt0dlnSu=Zt0 12 2 du+ Zt0dWuso thatlnSt lnS0= 12 2 t+ WtsinceW0= 0. Hence the solution to the SDE isSt=S0exp 12 2 t+ Wt :(12)SinceWtis distributed normalN(0; t)with zero mean and variancetwe havethatlnStfollows the normal distribution with meanlnS0+ 22 tandvariance 2t.

4 This implies by Equations (4) and (5) thatStfollows the lognor-mal distribution with meanS0e tand varianceS20e2 t e 2t 1 . We can alsointegrate Equation (11) fromttoTso that, analogous to Equation (12)ST=Stexp 12 2 + (WT Wt) (13)andSTfollows the lognormal distribution with meanSte and variance givenbyS2te2 e 2 1 . Bond PriceApply It o s Lemma to the functionlnBt. ThenlnBtfollows the SDEdlnBt=rtdt:Integrating from0totwe havedlnBt dlnB0=Zt0rudu:so the solution to the SDE isBt= exp Rt0rudu sinceB0= 1. When in-terest rates are constant thenrt=randBt=ert. Integrating fromttoTproduces the solutionBt;T= exp RTtrudu orBt;T=er when interest ratesare Discounted Stock Price is a MartingaleWe want to nd a measureQsuch that underQthe discounted stock price thatusesBtis a martingale. WritedSt=rtStdt+ StdWQt(14)whereWQt=Wt+ rt t.

5 We have that underQ, at timet= 0;the stock priceStfollows the lognormal distribution with meanS0erttand varianceS20e2rtt e 2t 1 ,but thatStis not a martingale. UsingBtas the numeraire, the discountedstock price is~St=StBtand~Stwill be a martingale. Apply It o s Lemma to~St,which follows the SDEd~St=@~S@BdBt+@~S@SdSt(15)since all terms involving the second-order derivatives are zero. Expand Equation(15) to obtaind~St= StB2tdBt+1 BtdSt(16)= StB2t(rtBtdt) +1Bt rtStdt+ StdWQt = ~StdWQt:The solution to the SDE (16) is~St=~S0exp 12 2t+ WQt :This implies thatln~Stfollows the normal distribution with meanln~S0 22tandvariance 2t. To show that~Stis a martingale underQ, consider the expectation4underQfors < tEQh~StjFsi=~S0exp 12 2t EQhexp WQt Fsi=~S0exp 12 2t+ WQs EQhexp WQt WQs FsiAt timeswe have thatWQt WQsis distributed asN(0; t s)which is identicalin distribution toWQt sat time zero.

6 Hence we can writeEQh~StjFsi=~S0exp 12 2t+ WQs EQhexp WQt s F0i:Now, the moment generating function (mgf) of a random variableXwith normaldistributionN( ; 2)isE e X = exp +12 2 2 . UnderQwe have thatWQt sisQ-Brownian motion and distributed asN(0; t s). Hence the mgf ofWQt sisEQhexp WQt s i= exp 12 2(t s) where takes the place of ,and we can writeEQh~StjFsi=~S0exp 12 2t+ WQs exp 12 2(t s) =~S0exp 12 2s+ WQs =~Ss:We thus have thatEQh~StjFsi=~Ss, which shows that~Stis a European call option under Black-Scholes makes use of the fact thatunderQ, at timetthe terminal stock price at expiry,ST, follows the normaldistribution with meanSter and varianceS2te2r e 2 1 when the interestratertis a constant value,r:Finally, note that under the original measure theprocess for~Stisd~St= ( r)~Stdt+ ~StdWtwhich is obviously not a SummaryWe start with the processes for the stock price and bond pricedSt= Stdt+ StdWtdBt=rtBtdt:We apply It o s Lemma to get the processes forlnStandlnBtdlnSt= 12 2 dt+ dWtdlnBt=rtdt.

7 Which allows us to solve forStandBtSt=S0e( 12 2)t+ WtBt=eRt0rsds:5We apply a change of measure to obtain the stock price under the risk neutralmeasureQdSt=rSt+ StdWQt)St=S0e(r 12 2)t+ WQtSinceStis not a martingale underQ, we discountStbyBtto obtain~St=StBtandd~St= ~StdWQt)~St=~S0e 12 2t+ WQtso that~Stisa martingale underQ. The distributions of the processes describedin this section are summarized in the following tableStochasticLognormal distribution forSTjFtProcess aProcessmeanvariancemartingaledS= Sdt+ SdWSte S2te2 e 2 1 NodS=rSdt+ SdWQSter S2te2r e 2 1 Nod~S= ~SdWQwith~S=SB~St~S2t e 2 1 Yesd~S= ( r)~Sdt+ ~SdW Ste( r) S2te2( r) e 2 1 also implies that the logarithm of the stock price is normally The Black-Scholes Call PriceIn the following sections we show four ways in which the Black-Scholes call pricecan be obtained.

8 Under a constant interest raterthe time-tprice of a Europeancall option on a non-dividend paying stock when its spot price isStand withstrikeKand time to maturity =T tisC(St; K; T) =e r EQh(ST K)+ Fti(17)which can be evaluated to produce Equation (1), reproduced here for conve-nienceC(St; K; T) =St (d1) Ke r (d2)whered1=logStK+ r+ 22 p andd2=d1 p =logStK+ r 22 p :6 The rst derivation is by straightforward integration of Equation (17); the sec-ond is by applying the Feynman-Kac theorem; the third is by transforming theBlack- scholes PDE into the heat equation and solving the heat equation; thefourth is by using the Capital Asset Pricing Model (CAPM).5 Black-Scholes by Straightforward IntegrationThe European call priceC(St; K; T)is the discounted time-texpected value of(ST K)+under the EMMQand when interest rates are constant.

9 Hencefrom Equation (17) we haveC(St; K; T) =e r EQh(ST K)+ Fti(18)=e r Z1K(ST K)dF(ST)=e r Z1 KSTdF(ST) e r KZ1 KdF(ST):To evaluate the two integrals, we make use of the result derived in Section ( )that underQand at timetthe terminal stock priceSTfollows the lognormaldistribution with meanlnSt+ r 22 and variance 2 , where =T tisthe time to maturity. The rst integral in the last line of Equation (18) usesthe conditional expectation ofSTgiven thatST> KZ1 KSTdF(ST) =EQ[STjST> K]=LST(K):This conditional expectation is, from Equation (10)LST(K) = exp lnSt+ r 22 + 2 2 0@ lnK+ lnSt+ r 22 + 2 p 1A=Ster (d1);so the rst integral in the last line of Equation (18) isSt (d1):(19)7 Using Equation (7), the second integral in the last line of (18) can be writtene r KZ1 KdF(ST) =e r K[1 F(K)](20)=e r K241 0@lnK lnSt r 22 p 1A35=e r K[1 ( d2)]=e r K (d2):Combining the terms in Equations (19) and (20) leads to the expression (1) forthe European call Change of NumeraireThe principle behind pricing by arbitrage is that if the market is complete wecan nd a portfolio that replicates the derivative at all times, and we can nd anequivalent martingale measure (EMM)Nsuch that the discounted stock priceis a martingale.

10 Moreover, the EMMN determines the unique numeraireNtthat discounts the stock price. The time-tvalueV(St; t)of the derivative withpayo V(ST; T)at timeTdiscounted by the numeraireNtisV(St; t) =NtEN V(ST; T)NT Ft :(21)In the derivation of the previous section, the bondBt=ertserves as the nu-meraire, and sinceris deterministic we can takeNT=erTout of the expectationand withV(ST; T) = (ST K)+we can writeV(St; t) =e r(T t)ENh(ST K)+ Ftiwhich is Equation (17) for the call black scholes Under a Di erent NumeraireIn this section we show that we can use the stock priceStas the numeraire andrecover the Black-Scholes call price. We start with the stock price process inEquation (14) under the measureQand with a constant interest ratedSt=rStdt+ StdWQt:(22)The relative bond price price is de ned as~B=BSand by It o s Lemma followsthe processd~Bt= 2~Btdt ~BtdWQt:The measureQturns~S=SBinto a martingale, but not~B.