Four Derivations of the Black-Scholes Formula by Fabrice Douglas Rouah

Transcription

1 Four Derivations of the Black-Scholes Formula by Fabrice Douglas Rouah In this note we derive in four searate ways the well-known result of Black and Scholes that under certain assumtions the time-t rice C(S t ; ; T ) of a Euroean call otion with strike rice and maturity = T t on a nondividend stock with sot rice S t and a constant volatility when the rate of interest is a constant r can be exressed as where C(S t ; ; T ) = S t (d ) e r (d ) () d = ln St + r + and d = d R, and where (y) = y e t dt is the standard normal cdf. We show four ways in which Equation () can be derived.. By straightforward integration.. By alying the Feynman-ac theorem. 3. By transforming the Black Scholes PDE into the heat equation, for which a solution is known. This is the original aroach adoted by Black and Scholes []. 4. Through the Caital Asset Pricing Model (CAPM). Free code for the Black-Scholes model can be found at Black-Scholes Economy There are two assets: a risky stock S and riskless bond B: driven by the SDEs These assets are ds t = S t dt + S t dw t () db t = r t B t dt The time zero value of the bond is B 0 = and that of the stock is S 0. The model is valid under certain market assumtions that are described in John

2 Hull s book [3]. By Itō s Lemma the value V t of a derivative written on the stock follows the di usion dv ds (ds) ds S V dt + S t dw t The Lognormal Distribution. The Lognormal PDF and CDF In this Note we make extensive use of the fact that if a random variable Y R follows the normal distribution with mean and variance, then X = e Y follows the lognormal distribution with mean E [X] = e + (4) and variance V ar [X] = e e + : (5) The df for X is df X (x) = x ex ln x! (6) and the cdf is ln x F X (x) = where (y) = R y e t dt is the standard normal cdf.. The Lognormal Conditional Exected Value The exected value of X conditional on X > x is L X () = E [XjX > x]. the lognormal distribution this is, using Equation (6) (7) For L X () = Z e ( ln x ) dx: Make the change of variable y = ln x so that x = e y, dx = e y dy Jacobian is e y : Hence we have L X () = Z ln e y and the e ( y ) dy: (8)

3 Combining terms and comleting the square, the exonent is y y + y = y : Equation (8) becomes Z L X () = ex + ln 0 y +! A dy: (9) Consider the random variable X with df f X (x) and cdf F X (x), and the scalelocation transformation Y = X +. It is easy to show that the Jacobian is, that the df for Y is f Y (y)= f X y and that the cdf is y FY (y) = F X. Hence, the integral in Equation (9) involves the scale-location transformation of the standard normal cdf. Using the fact that ( x) = (x) this imlies that L X () = ex + ln + + : (0) See Hogg and lugman []. 3 Solving the SDEs 3. Stock Price Aly Itō s Lemma to the function ln S t where S t is driven by the di usion in Equation (). Then ln S t follows the SDE Integrating from 0 to t, we have so that since W 0 = 0. Z t 0 d ln S t = dt + dw t : () d ln S u = Z t 0 du + Z t ln S t ln S 0 = t + W t Hence the solution to the SDE is 0 dw u S t = S 0 ex t + W t : () Since W t is distributed normal N(0; t) with zero mean and variance t we have that ln S t follows the normal distribution with mean ln S 0 + t and variance t. This imlies by Equations (4) and (5) that S t follows the lognormal distribution with mean S 0 e t and variance S0e e t t. We can also integrate Equation () from t to T so that, analogous to Equation () S T = S t ex + (W T W t ) (3) and S T follows the lognormal distribution with mean S t e and variance given by St e e. 3

4 3. Bond Price Aly Itō s Lemma to the function ln B t. Then ln B t follows the SDE d ln B t = r t dt: Integrating from 0 to t we have d ln B t d ln B 0 = Z t 0 r u du: R t so the solution to the SDE is B t = ex 0 r udu since B 0 =. When interest rates are constant then r t = r and B t = e rt. Integrating from t to T R T roduces the solution B t;t = ex r t u du or B t;t = e r when interest rates are constant. 3.3 Discounted Stock Price is a Martingale We want to nd a measure Q such that under Q the discounted stock rice that uses B t is a martingale. Write ds t = r t S t dt + S t dw Q t (4) where W Q t = W t + rt t. We have that under Q, at time t = 0; the stock rice S t follows the lognormal distribution with mean S 0 e rtt and variance S0e e rtt t, but that S t is not a martingale. Using B t as the numeraire, the discounted stock rice is S ~ t = St B t and S ~ t will be a martingale. Aly Itō s Lemma to S ~ t, which follows the SDE d S ~ t S db t S ds t (5) since all terms involving the second-order derivatives are zero. Exand Equation (5) to obtain d ~ S t = = S t Bt S t Bt = ~ S t dw Q t : db t + B t ds t (6) (r t B t dt) + B t r t S t dt + S t dw Q t The solution to the SDE (6) is ~S t = S ~ 0 ex t + W Q t : This imlies that ln ~ S t follows the normal distribution with mean ln ~ S 0 t and variance t. To show that ~ S t is a martingale under Q, consider the exectation 4

5 under Q for s < t h i E Q St ~ jf s = S ~ 0 ex t h E Q ex = ~ S 0 ex t + W Q s W Q t Fs i E Q h ex W Q t W Q s Fs i At time s we have that W Q t Ws Q is distributed as N(0; t s) which is identical in distribution to W Q t s at time zero. Hence we can write h i E Q St ~ jf s = S ~ 0 ex t + Ws Q h i E Q ex W Q t s F0 : Now, the moment generating function (mgf) of a random variable X with normal distribution N(; ) is E e X = ex +. Under Q we have that W Q t s is Q-Brownian motion i and distributed as N(0; t s). Hence the mgf of W Q t s is E hex Q W Q t s = ex (t s) where takes the lace of, and we can write h i E Q St ~ jf s = S ~ 0 ex t + Ws Q ex (t s) = S ~ 0 ex s + Ws Q = S ~ s : h i We thus have that E Q ~St jf s = S ~ s, which shows that S ~ t is a Q-martingale. Pricing a Euroean call otion under Black-Scholes makes use of the fact that under Q, at time t the terminal stock rice at exiry, S T, follows the normal distribution with mean S t e r and variance St e e r when the interest rate r t is a constant value, r: Finally, note that under the original measure the rocess for S ~ t is d S ~ t = ( r) S ~ t dt + S ~ t dw t which is obviously not a martingale. 3.4 Summary We start with the rocesses for the stock rice and bond rice ds t = S t dt + S t dw t db t = r t B t dt: We aly Itō s Lemma to get the rocesses for ln S t and ln B t which allows us to solve for S t and B t d ln S t = dt + dw t d ln B t = r t dt; S t = S 0 e ( )t+w t B t = e R t 0 rsds : 5

6 We aly a change of measure to obtain the stock rice under the risk neutral measure Q ds t = rs t + S t dw Q t ) S t = S 0 e (r )t+w Q t Since S t is not a martingale under Q, we discount S t by B t to obtain ~ S t = St B t and d ~ S t = ~ S t dw Q t ) ~S t = ~ S 0 e t+w Q t so that ~ S t is a martingale under Q. The distributions of the rocesses described in this section are summarized in the following table Stochastic Lognormal distribution for S T j F t Process a Process mean variance martingale ds = Sdt + SdW S t e St e e No ds = rsdt + SdW Q S t e r St e e r No d S ~ = SdW ~ Q with S ~ = S B ~S t S ~ t e Yes d S ~ = ( r) Sdt ~ + SdW ~ S t e ( r) St e e ( r) No This also imlies that the logarithm of the stock rice is normally distributed.. 4 The Black-Scholes Call Price In the following sections we show four ways in which the Black-Scholes call rice can be obtained. Under a constant interest rate r the time-t rice of a Euroean call otion on a non-dividend aying stock when its sot rice is S t and with strike and time to maturity = T t is h C(S t ; ; T ) = e r E Q (S T ) + i Ft (7) which can be evaluated to roduce Equation (), reroduced here for convenience C(S t ; ; T ) = S t (d ) e r (d ) where and d = log St + r + d = d log St + = 6 r :

7 The rst derivation is by straightforward integration of Equation (7); the second is by alying the Feynman-ac theorem; the third is by transforming the Black-Scholes PDE into the heat equation and solving the heat equation; the fourth is by using the Caital Asset Pricing Model (CAPM). 5 Black-Scholes by Straightforward Integration The Euroean call rice C(S t ; ; T ) is the discounted time-t exected value of (S T ) + under the EMM Q and when interest rates are constant. Hence from Equation (7) we have C(S t ; ; T ) = e r E Q h (S T ) + Ft i (8) Z = e r (S T )df (S T ) Z = e r S T df (S T ) e r Z df (S T ): To evaluate the two integrals, we make use of the result derived in Section (3.3) that under Q and at time t the terminal stock rice S T follows the lognormal distribution with mean ln S t + and variance, where = T t is r the time to maturity. The rst integral in the last line of Equation (8) uses the conditional exectation of S T given that S T > Z S T df (S T ) = E Q [S T j S T > ] = L ST (): This conditional exectation is, from Equation (0) L ST () = ex ln S t + r + ln + ln S t + r + A = S t e r (d ); so the rst integral in the last line of Equation (8) is S t (d ): (9) 7

8 Using Equation (7), the second integral in the last line of (8) can be written e r Z df (S T ) = e r [ F ()] (0) 0 = e r ln ln S 3 t r A5 = e r [ ( d )] = e r (d ): Combining the terms in Equations (9) and (0) leads to the exression () for the Euroean call rice. 5. Change of Numeraire The rincile behind ricing by arbitrage is that if the market is comlete we can nd a ortfolio that relicates the derivative at all times, and we can nd an equivalent martingale measure (EMM) N such that the discounted stock rice is a martingale. Moreover, the EMM N determines the unique numeraire N t that discounts the stock rice. The time-t value V (S t ; t) of the derivative with ayo V (S T ; T ) at time T discounted by the numeraire N t is V (S t ; t) = N t E N V (ST ; T ) N T F t : () In the derivation of the revious section, the bond B t = e rt serves as the numeraire, and since r is deterministic we can take N T = e rt out of the exectation and with V (S T ; T ) = (S T ) + we can write h V (S t ; t) = e r(t t) E N (S T ) + i Ft which is Equation (7) for the call rice. 5.. Black Scholes Under a Di erent Numeraire In this section we show that we can use the stock rice S t as the numeraire and recover the Black-Scholes call rice. We start with the stock rice rocess in Equation (4) under the measure Q and with a constant interest rate ds t = rs t dt + S t dw Q t : () The relative bond rice rice is de ned as B ~ = B S and by Itō s Lemma follows the rocess d B ~ t = Bt ~ dt B ~ t dw Q t : The measure Q turns ~ S = S B into a martingale, but not ~ B. The measure P that turns ~ B into a martingale is W P t = W Q t t (3) 8

9 so that d ~ B t = ~ B t dw P t is a martingale under P. The value of the Euroean call is determined by using N t = S t as the numeraire along with the ayo function V (S T ; T ) = (S T ) + in the valuation Equation () " # V (S t ; t) = S t E P (S T ) + F t (4) S T = S t E P [( Z T )j F t ] where Z t = S t. To evaluate V (S t ; t) we need the distribution for Z T. The rocess for Z = S is obtained using Itō s Lemma on S t in Equation () and the change of measure in Equation (3) dz t = r + Z t dt Z t dw Q t = rz t dt Z t dw P t : To nd the solution for Z t we de ne Y t = ln Z t and aly Itō s Lemma again, to roduce dy t = r + dt dwt P : (5) We integrate Equation (5) to roduce the solution Y T Y t = r + (T t) WT P Wt P : so that Z T has the solution ln Zt Z T = e r+ (T t) (W P T W P t ) : (6) Now, since WT P Wt P is identical in distribution to W P, where = T t is the time to maturity, and since W P follows the normal distribution with zero mean and variance, the exonent in Equation (6) ln Z t r + (T t) WT P Wt P ; follows the normal distribution with mean u = ln Z t r + = ln S t r + and variance v =. This imlies that Z T follows the lognormal distribution with mean e u+v= and variance (e v ) e u+v. Note that ( Z T ) + in the exectation of Equation (4) is non-zero when Z T <. Hence we can write this exectation as the two integrals E P [( Z T )j F t ] = Z df ZT = I I Z Z T df ZT (7) 9

10 where F ZT is the cdf of Z T de ned in Equation (7). The rst integral in Equation (7) is I = ln F ZT = v 0 ln + ln S t + = (d ) : u r + A (8) Using the de nition of L ZT (x) in Equation (0), the second integral in Equation (7) is " Z Z # I = Z T df ZT Z T df ZT (9) = E P [Z T ] L ZT ln = e u+v= e u+v= + u + v v 0 3 St ln = e u+v= r A5 = S t e r (d ) since ( d ) = (d ). Substitute the exressions for I and I from Equations (8) and (9) into the valuation Equation (4) V (S t ; t) = S t E P [( Z T )j F t ] = S t [I I ] = S t (d ) e r (d ) which is the Black-Scholes call rice in Equation (). 6 Black-Scholes From the Feynman-ac Theorem 6. The Feynman-ac Theorem Suose that x t follows the rocess dx t = (x t ; t)dt + (x t ; t) dw Q t (30) 0

11 and suose the di erentiable function V = V (x t ; t) follows the artial di erential equation + (x + (x t; r(t; x)v (x t ; t) = 0 (3) with boundary condition V (x T ; T ). The Feynman-ac theorem stiulates that V (x t ; t) has solution h V (x t ; t) = E Q e R i T t r(xu;u)du V (X T ; T ) F t : (3) In Equation (3) the time-t exectation is with resect to the same measure Q under which the stochastic ortion of Equation (30) is Brownian motion. See the Note on for illustrations of the Feynman-ac theorem. 6. The Theorem Alied to Black-Scholes To aly the Feynman-ac theorem to the Black-Scholes call rice, note that the value V t = V (S t ; t) of a Euroean call otion written at time t with strike rice when interest rates are a constant r follows the rv t = 0 (33) with boundary condition V (S T ; T ) = (S T ) +. The Note on exlains how the PDE (33) is derived. This PDE is the PDE in Equation (3) with x t = S t ; (x t ; t) = rs t ; and (x t ; t) = S t. Hence the Feynman-ac theorem alies and the value of the Euroean call is h V (S t ; t) = E Q e R i T t r(xu;u)du V (S T ; T ) F t (34) = e r E Q (S T ) + Ft which is exactly Equation (8). Hence, we can evaluate the exectation in (34) by straightforward integration exactly in the same way as in Section 5 and obtain the call rice in Equation (). 7 Black-Scholes From the Heat Equation In this section we follow the derivation exlained in Wilmott et al. [4]. We rst resent a de nition of the Dirac delta function, and of the heat equation. We then transform the Black-Scholes PDE into the heat equation, aly the solution through integration, and convert back to the original (untransformed) arameters. This will roduce the Black-Scholes call rice.

12 7. Dirac Delta Function The Dirac delta function (x) is de ned as 0 if x 6= 0 (x) = and if x = 0 For an integrable function f(x) we have that and f(0) = f(x) = 7. The Heat Equation Z Z Z f()()d ()d = : f(x )()d: (35) The heat equation is the PDE for u = u(x; ) over the domain fx R; > 0g : The heat equation has the fundamental solution u(x; ) = 4 ex x 4 (36) which is the normal df with mean 0 and variance : The initial value of the heat equation is u(x; 0) = u 0 (x) which can be written in terms of the Dirac delta function as the limit u 0 (x) = lim!0 u(x; ) = (x): Using the roerty (35) of the Dirac delta function, we can write the initial value as u 0 (x) = Z (x )u 0 ()d (37) We can also aly the roerty (35) to the fundamental solution in Equation (36) and exress the solution as with initial value as before. u (x; ) = = = u(x; 0) = Z Z Z Z u(x )()d (38) u(x 4 e (x (x )u 0 ()d ) =4 u 0 ()d; )u 0 ()d = u 0 (x);

13 7.3 The Black-Scholes PDE as the Heat Equation Through a series of transformations we convert the Black-Scholes PDE in Equation (33) into the heat equation. The rst set of transformations convert the sot rice to log-moneyness and the time to one-half the total variance. This will get rid of the S and S terms in the Black-Scholes PDE. The rst transformations are x = ln S so that S = ex (39) = (T t) so that t = T U(x; ) = V (S; t) = V ex ; T = : Aly the chain rule to the artial derivatives in the Black-Scholes PDE. @ = = = @S @x = @x Substitute for the artials in the Black-Scholes PDE (33) to obtain which simli es + rex + x x @ ru = ku = 0 (40) where k = r. The coe cients of this PDE does not involve x or. The boundary condition for V is V (S T ; T ) = (S T ) +. From Equation (39), when t = T and S t = S T we have that x = ln S T which we write as x T, and that = 0. Hence the boundary condition for U is U 0 (x T ) = U(x T ; 0) = V (S T ) + = (ex T ) + = (e x T ) + : 3

14 We make the additional transformation W (x; ) = e x+ U(x; ) (4) where = (k ) and = (k + ). This will convert Equation (40) into the heat equation. The artial derivatives of U in terms of W = e W @U = e W @ = e x W : Substitute these derivatives into Equation (40) to W (x; ) + (k ) W (x; @x +W (x; ) which simli es to the kw (x; ) @x : (4) From Equation (4) the boundary condition for W (x; ) is W 0 (x T ) = W (x T ; 0) = e x T U(x T ; 0) (43) + = e (+)x T e x T = e x T e x + T : since = +. The transformation from V to W is therefore V (S; t) = e x W (x; ): (44) 7.4 Obtaining the Black-Scholes Call Price Since W (x; ) follows the heat equation, it has the solution given by Equation (38), with boundary condition given by (43). Hence the solution is W (x; ) = 4 Z e (x ) =4 W 0 ()d = 4 Z e ( x) =4 e e + d: 4

15 Make the change of variable z = W (x; ) = x Z ex ex so that = z + x and d = dz. z h z + x i h z + x i + dz (45) Note that the integral is non-zero only when the second exonent is greater than zero, that is, when z + x > z + x which is identical to z > x. We can now break u the integral into two ieces Z h i W (x; ) = x/ ex z ex z + x dz Z h i x/ ex z ex z + x dz = I I : Comlete the square in the rst integral I. The rst integral becomes The exonent in the integrand is z + z + x = z + x + : Z I = e x+ x/ e (z ) dz: Make the transformation y = z so that the integral becomes Z I = e x+ x/ = e x+ x x = e x+ + : e y dy The second integral is identical, excet that is relaced with. x I = e x+ + : Hence Recall that x = ln S, k = r ; = (k ) = r = ; = (k + ) = r+ = and = (T t). Consequently, we have that r + x + ln S + = T 5 (T t) t = d ;

16 and that x + = d T t = d : Hence the rst integral is I = ex x + (d ) : The second integral is identical excet that is relaced by and involves d instead of d I = ex x + (d ) : The solution is therefore W (x; ) = I I (46) = e x+ (d ) e x+ (d ) : The solution in Equation (46), exressed in terms of I and I, is the solution for W (x; ): To obtain the solution for the call rice V (S t ; t) we must use Equation (44) and transform the solution in (46) back to V. From (44) and (46) The rst integral in Equation (47) is V (S; t) = e x W (x; ) (47) = e x [I I ] : e x e x+ (d ) = e ( )x (d ) (48) = S (d ) since =. The second integral in Equation (47) is e x e x+ (d ) = e ( ) (d ) (49) = e r(t t) (d ) since = r. Combining the terms in Equations (48) and (49) roduces the Black-Scholes call rice in Equation (). 8 Black-Scholes From CAPM 8. The CAPM The Caital Asset Pricing Model (CAPM) stiulates that the exected return of a security i in excess of the risk-free rate is E [r i ] r = i (E [r M ] r) where r i is the return on the asset, r is the risk-free rate, r M is the return on the market, and i = Cov [r i; r M ] V ar [r M ] is the security s beta. 6

17 8. The CAPM for the Assets h In the time increment dt the exected stock rice return, E [r S dt] is E dst S t i, where S t follows the di usion in Equation (). The exected return is therefore dst E = rdt + S (E [r M ] r) dt: (50) S t h Similarly, the exected return on the derivative, E [r V dt] is E dvt V t i, where V t follows the di usion in (3), is dvt E = rdt + V (E [r M ] r) dt: (5) V t 8.3 The Black-Scholes PDE from the CAPM Divide by V t on both sides of the second line of Equation (3) to obtain which is dv t V t = + dt ds t S t S t V t r V dt V + dt S t r S dt: V t Dro dt from both sides and take the covariance of r V and r M, noting that only the second term on the right-hand side of Equation (5) is stochastic Cov [r V ; r M ] S t Cov [r S ; r M ] V t This imlies the following relationshi between the beta of the derivative, V, and the beta of the stock, S t V V S : t This is Equation (5) of Black and Scholes []. to obtain Multily Equation (5) by V t E [dv t ] = rv t dt + V t V (E [r M ] r) dt (53) = rv t S t S (E [r M ] r) dt: This is Equation (8) of Black and Scholes []. Take exectations of the second line of Equation (3), and substitute for E [ds t ] from Equation (50) E [dv t [rs tdt + S t S (E [r M r]) dt] S dt: (54) 7

18 Equate Equations (53) and (54), and dro dt from both sides. involving S cancels and we are left with rv t = 0: (55) We recognize that Equation (55) is the Black-Scholes PDE in Equation (33). Hence, we can obtain the Black-Scholes call rice by aealing to the Feynman- ac theorem exactly as was done in Section (6.) and solving the integral as in Section (5). 9 Incororating Dividends The Black-Scholes call rice in Equation () is for a call written on a non dividend-aying stock. There are two ways to incororate dividends into the call rice. The rst is by assuming the stock ays a continuous dividend yield q. The second is by assuming the stock ays dividends in lum sums, "lumy" dividends. 9. Continuous Dividends We assume that the dividend yield q is constant so that the holder of the stock receives an amount qs t dt of dividend in the time increment dt. After the dividend is aid out, the value of the stock dros by the dividend amount. In other words, without the dividend yield, the value of the stock increases by rs t dt, but with the dividend yield the stock increases by rs t dt qs t dt = (r q) S t dt. Hence, the exected return becomes r q instead of r, which imlies that he risk-neutral rocess for S t follows Equation (4) but with drift r q instead of r ds t = (r q) S t dt + S t dw Q t : (56) Following the same derivation in Section (3), Equation (56) has solution S T = S t ex r q + W Q where = T t. Hence, S T follows thelognormal distribution with mean S t e (r q) and variance St e e (r q). Proceeding exactly as in Equation (8), the call rice is C (S t ; ; T ) = e r L ST () e r [ F ()] : (57) 8

19 The conditional exectation L ST () from Equation (0) becomes L ST () = ex ln S t + r q + ln + ln S t + r q + A (58) with d rede ned as = S t e (r q) (d ) ln St + r q + d = : Using Equation (7), the second term in Equation (8) becomes 0 e r [ F ()] = e r ln ln S t r q 3 A5 = e r (d ) (59) with d = d as before. Substituting Equations (58) and (59) into Equation (57) roduces the Black-Scholes rice of a Euroean call written on a stock that ays continuous dividends C (S t ; ; T ) = S t e q (d ) e r (d ): Hence, the only modi cation is that the current value of the stock rice is decreased by e q, and the return on the stock is decreased from r to r q. All other comutations are identical. 9. Lumy Dividends To come. Same idea: the current value of the stock rice is decreased by the dividends, excet not continuously. References [] Black, F., and M. Scholes (973). "The Pricing of Otions and Cororate Liabilities." Journal of Political Economy, Vol 8, No. 3, [] Hogg, R.V., and S. lugman (984). Loss Distributions. New York, NY: John Wiley & Sons. [3] Hull, J. (008). Otions, Futures, and Other Derivatives, Seventh Edition. New York, NY: Prentice-Hall. [4] Wilmott, P., Howison, S., and J. Dewynne (995). The Mathematics of Financial Derivatives: A Student Introduction. Cambridge, U: Cambridge University Press. 9