Fin 674 Problems Ch13

8/3/2019 Fin 674 Problems Ch13

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Chapter 13The Black-Scholes-Merton ModelSOLUTIONS TO QUESTIONS AND PROBLEMSProblem 13.1 .

The Black-Scholes option pric ing model assumes that the probability d istribution of the stockprice in I year (or at any other future time) is lognormal. It assumes that the continuouslycompounded rate of return on the stock during the year is normally d istributed .

Problem 13.2.The standard deviation of the percentage price change in time f..t is (5v;Ij where (5.is thevolatility. In this problem (5 = 0.3 and, assum ing 252 trading days in one year, f..t = 1/252 =0.004 so that (5v;I j = 0.3vO .004 = 0.019 or 1.9%.

Problem 13.3.The price of an option or other derivative when expressed in terms of the price of the un-derlying stock is independent of risk preferences. Options therefore have the same value ina risk-neutral world as they do in the real world. We may therefore assume that the worldis risk neutral for the purposes of valuing options. This simplifies the analysis. In a risk-neutral world all securities have an expected return equal to risk-free in terest rate. A lso , in arisk-neutral world, the appropriate discount rate to use for expected future cash flows is therisk-free interest rate.

Problem 13 .4.In this case So = 50, K = 50, r = O.I , (5 = 0.3, T = 0.25, and

dl = In(50/50) + (0.1 +0.09/2)0.25 =0.24170.3vO.25d2 = d1 - 0.3vO .25 = 0.0917

The European put price is50N( -0.0917)e-O.l xO.25- 50N( -0.2417)

= 50 x 0.4634e-0.lxO.25 - 50 X 0 .4 045 = 2.37or $2.37.


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Problem 13.5.In this case we must subtract the present value of the dividend from the stock price beforeusing B lack-Scho les. Hence the appropriate value o f So is

So = 50- 1.50e-O .\6 67xO .1 = 48.52As before K = 50, r = 0.1, (J = 0.3 , and T = 0 .25 . In th is case

d In(48.52/50) + (0.1 +0.09/2)0.251= = 0.04140.3VO.25d2 = d1 - 0.3\1"0.25 = -0.1086

The European put price is50N(0.1086)e-0.\ xO.25- 48.52N(-0.0414)

= 50 x 0.5432e-0.lxO.25 -48.52 x 0.4835 = 3.03or $3.03.Problem 13.6.

The implied vo latility is the volatility that makes the B lack -Scholes price of an op tion equalto its market price . It is calcula ted using an ite ra tive procedure .

Prob lem 13 .7.In this case J1= 0.15 and (J = 0.25. F rom equation (13.7) the probability d istribution for therate of return over a 2-year period with continuous compoundingis:

rfo.(0.15 - 0.252 0.25 )Y 2 ' J2I.e.,

(jJ(0.11875,0.1768)The expected value of the return is 11.875% per annum and the standard deviation is 17.68%per annum.

Problem 13.8 .(a) The required probability is the probability of the stock price being above $40 in six

months' tim e. Suppose that the stock price in six months is ST0352InST rv (jJ(In38+ (0.16 - ~ )0.5, 0.35v(5)2


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84 Chapter 13. The Black-Scholes-Merton ModelI.e.,

InST rv (3.687,0.247)Since In40 = 3 .689 , the required probability is

F rom normal d istribution tables N(0.008) = 0.5032 so that the required probability is0.4968. In general the required probability is N((h). (See Problem 13.22).

J - N (3.689 - 3.687 ) = 1- N(0.008)0.247(b) In this case the required probability is the probability of the stock price being less than

$40 in six months' time. It isI - 0.4968 = 0.5032

Prob lem 13.9.From equat ion (13.3):

95% confidence intervals for InST are therefore')a-InST rv In So + (J.1- - )T , a VT ]2

')a-InSo+ (J.1- -)T - 1.96aVT2and')a-InSo + ( J.1 - -)T + J .96aVT295% confidence intervals for ST are therefore

InSo+(,u-a2/2)T-1.96aVTeand

e ln So +( ,u -a 2 / 2) T + J .96aVTI.e.

Soe(,u-a2 /2)T - J .96aVTand

Soe(,u-a2 /2)T + 1.96aVTProblem 13.10.

The statement is misleading in that a certain sum of money, say $ J 000, when invested for 10years in the fund would have realized a return (with annual compounding) of less than 20%per annum.

The average of the retu rns realized in each year is always g reater than the return pcr annum(with annual compounding) realized over 10 years. The first is an arithmetic average o f theretu rns In each year; the second is a geometric average o f these return s.


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85Problem 13.11.

(a) At time t, the expected value oflnST is, from equation (\3.3),(J-InS+(,u--)(T-t)2

In a risk-neu tral world the expected value of InS r is therefore:,(J-I nS+ (r - - ) (T - t)2

Using risk-neutral v aluation the value of th e security at time tis:e-,(T-'j [Ins + (r - ~2 )(T - t)]

(b) If:

.f = e-,(T-'j [Ins + (r - ~2 )(T - t)]af

[(J2

](J2-=re-r(T-t) InS+(r--)(T-t) -e-r(T-t)(r--. )t 2 2a f e-r(T -t)

as - Sa2 f - e-r(T -t)as2 - S2The left-hand side of the Black Scholes d ifferential equation is

[, , '

](J- (J- (J--r(T-t) rlnS+ r(r- 2 )(T -t) - (r- 2) + r- 2

=re-,(T-'j [Ins+ (r- ~2)(T-t)]= rf

Hence equation (13.16) is sat is fied.


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86 Chapter 13. The Black-Scholes-Merton ModelProblem 13.12.

This prob lem is re lated to P roblem 12.10.(a) If C(S,t) = h (t, T )S II then dC/dt = hlSII, dC/dS = hI1SII-], and d2C/dS2 = hl1(l1-)SII-2 where hI = dh/dt. Substituting into the Black-Scholes differential equation weobtain

I ')hI + rIm + -(rhl1(l1- I) = rh2(b) The derivative is worth SII when t = T. The boundary condition for th is differen tialequation is therefore h(T, T) = 1(c) The equation

h(t, T) = eIO.S(J"211(1I-])+r(II-I)I(T-I)satisfies the boundary condition since it co llap ses to h = 1 w hen t = T. It can also beshow n that it satisfies the differential equation in (a). A lternatively w e can solve thedifferential equation in (a) directly. The differential equation can be written

hI 1 ')- - -r(n-1)--(rl1(n-l)h 2The solution to this is1 ')Inh = [-r(n -1) - -(;-n(11 -1 )]t +k2

where k is a constant. Since In h = 0 when t = T it fol lows that1 ')k ~ [r(n - 1) + 2:cr-n(n- 1)]T

so that1 ')1nh= [r(n - 1)+ -cr-n(n - 1)](T - t)2or

h(t, T) = eIO.5(J"2n(II-I)+r(n-I)I(T-I)

Prob lem 13 .13.In this case So = 52, K = 50 , r = 0.12, cr = 0.30 and T = 0.25.

dl = In(52/50) +(0.12+0.32/2)0.25 =0.53650.30JO.25 .d2 = dl - 0.30JO.25 = 0.3865"


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87The price of the European call is

52N(0.5365) - 50e-0.J2xO.25 N(0.3865)= 52 x 0.7042 - 50e-0.03 x 0.6504= 5.06or $5.06.

Problem 13.14.In this case So = 69, K = 70, r = 0.05,


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88 Chapter 13. The B lack-Scholes-Merton ModelProblem 13.17.

(a) Since N(x) is the cumu lative p robab ility that a variable w ith a standardized normal dis-tribution w ilI be less than x, N' (x) is the probab ility density function for a standard izednormal dis tr ibut ion, that is,

I ,2N'(x) = -e-2J2ii(b) N'(dd =N'(d2+avT-t)

I[

d~ ~]?]= - exp --=- - ad? v T - t - -a-(T - t)v5 2 - 2

= N'(d ,)exp [-ad,~ - ~a'(T -t)]Because

In(S/ K) + (r - a2 /2) (T - t)d? =- avff=tit follow s that

As a result[

I?]

Ke-r(T-t)exp -ad2VT-t-"2(r(T-t) = SSN'(dd = Ke-r(T-I)N'(d2)

which is the requ ired resu lt.(c) 0d 1 = In~ + (r + T) (T - t)avT - t

0InS -InK + (r+ T )(T - t)avT - t

Henceadl 1as = SaJT -t

SimilarlyIn S - InK + (r - a,,2)(T - t)d? = -- avT -tand

ad2 1as - SavT -tTherefore:

'> ad) dd2as - as

J.


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89(d)

From (b):

c = SN(dl) - Ke-r(T -I) N(d2)ac = SN1(dl) adl - rKe-r(T -I) N(d2) - Ke-r(T-I) N ' (d2 ) a(hat at . at

HenceSN'(dI) = Ke-r(T-I)N'(d2)

Sinceac .(T I ) ) , (adl ad., )- = -rKe-1 - N(d2 +SN (dl) - =-t at at

dl -d2 = a~ad, - a{h = i(a~)at at ata- 1~Hence ac = -rKe-r(T-I)N(d2) -SN'(d,) aat 1JT-t(e) From differentiating the BJack -Scholes formula fo r a call p rice we obtain

ac =N (d ) +SN' (d ) adl -Ke-r(T-t)N' (d7 ) ad2as I I as - dSFrom the results in (b) and (c) it follow s thatac = N (d )as I

(f) D ifferen tiating the result in (e) and using the resu lt in (c), we obtaina2c = N' (d ) adlaS2 I as

=N'(dl) ~a T - tFrom the results in (d) and (e)ac ac 1 7 7 a2c -r (T-t) ( ) ' (d ) a+rS- + -a-S-- = -rKe N d2 -SN I . +t as 2 as2 2JT=t1 7 7 , 1+rSN(dl) + -a-S-N (dJ) JT=tSa T-t= r[SN(dl) - Ke-r(T -f) N(d2)]= rc

Th is shows that the Black-Scholes formula for a call option does in de ed s atisfy theBlack~Scholes differential equation.


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90 Chapter 13. The Black-Scholes-Merton Model(g) Consider what happens in the formula for c in part (d) as t approaches T. If S > K, dl

and d2 tend to infinity and N(dI) and N((h) tend to I. If S< K, dl and d2 tend to zero.t follows that the formula for c tends to max(S - K, 0).

Problem 13.18.From the Black -Scholes equations

p + So = Ke-rT N ( -dJ - SoN(-dl) + SoBecause I - N( -dl) = N(dl) th is is

Ke-rT N ( -d2 ) +SoN(dl)Also:

c + Ke-rT = SoN(dl) - Ke-rT N(d2) + Ke-rTBecause I - N(d2) = N( -d2), this is alsoKe-rT N( -d2) + SoN(dJ)

The B lack -Scholes equations a re there fo re cons is tent w ith put-call parity .Rroblem 13.19.

This problem naturally leads on to the material in Chapter 16 on volati li ty smiles. UsingDerivaGem we obta in the fo llowing table of implied volatilities:

Maturity (months)634.9932.7830.77

Strike Price ($)455055

337.7834.1531.98

1234.0232.0330.45The option prices are not exact ly consistent with Black-Scholes. If they were, the impliedvolatil it ies would be all the same. We usual ly find in practice that low strike price options ona stock have significantly h igher implied vola tilities than high strike price options on the same

stock.Prob lem 13.20.

B lack's app roach in effect assumes that the holder o f option must decide at time zero whetherit is a European option maturing at time tn (the final ex-d iv idend date) or a European op tionmaturing a t time T. In fact the holder of the option has more flexibility than this. The holdercan choose to exercise at time tn if the stock price at that time is above some level but nototherwise. Furthennore, if the option is not exerc ised a t time tn, it can still be exercised attime T.>


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91It appears from this argument that Black's approach understates the true option value.

However, the way in which volatility is applied can lead to Black's approach overstating theoption value. B lack applies the volatility to the option price. The binom ial model, as we w illsee in Chapte r 17, applies the vola tility to the stock price less the present value of the d iv idend.This issue is also discussed in Example 13 .9.

P rob lem 13 .21 .IWith the notation in the textDI = D2 = 1.50, tl = 0.3333, t2 = 0.8333 , T = 1.25, r = 0.08 and K = 55

K [1-e-r(T-t2)] = 55(I-e-O.08xOAI67) = 1.80HenceD2 < K [1-e-r(T-t2)]Also:

K [1-e-r(12-IJ)] = 55(1-e-0.08xO.5) = 2.16Hence:

D) < K [1 - e-r(t2-ld]It follows from the conditions established in Section 13.12 that the option should never beexercised early.

.The presen t value of the dividends is1.5e-0.3333xO.08 + 1.5e-0.8333xO.08 = 2.864

The option can be valued using the European pricing fo rmula with:So = 50 - 2.864 = 47.136, K=55, (J = 0 .25, r = 0.08 , T = 1.25

_In(47.136/55)+(0.08+0.252/2)1.25 - 00 4l - - - . 5 50.25V1.25d2 = dl - 0.25v1125 = -0.3340

N(dJ) = 0.4783, N(d2) = 0.3692and the call price is

47.136 x 0.4783 - 55e-0.08x 1.25x 0 .3692 = 4.17or $4.17.

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92 Chapter 13. The Black-Scholes-Merton ModelProblem 13.22.

The probability that the call op tion will be exercised is the probability that ST > K where STis the stock price at time T. In a risk neutral world

InST rv c t>In So + (r - (J2 /2)T, (JVT]The probabil ity that ST > K is the same as the probability that In ST > InK. This is

I -N [InK -lnSo - (r- (J2/2)T

]JVf

= N [ In(So/ K) + (r - (J2/2)T ]JVf= N(d2)The expected value at time T in a risk neutral world of a derivative security which pays off

$100 when ST > K is thereforeIOON(d2)From risk neu tral valuation the value of the security at time t is

IOOe-rT N(d2)Prob lem 13 .23 .

') / 2If 1= s-_r 0- thenaI = - 2r s-2r/o-2-1as (J2a21 = (2r )( 2r +1 )S-2r/o-2-2aS2 (J2 (J2al =0at

aISaI 1 2s2 a2I .s-2r/o-2 It+r as + 2(J aS2 = r = rThis shows that the Black-Scholes equation is sa tisfied. S-2r/o-2 cou ld therefore be the priceof a traded security .

I Problem 13 .24.The answer is no. If markets are efficient they have already taken potential dilution into ac-count in determ ining the stock price. Th is argumen t is explained in Bus in es s Snapsho t 13.3.


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Problem 13 .25.Iffhe Black-Scholes price of the option is given by setting So= 50, K = 50, r = 0.05,

(j = 0.25,nd T = 5. It is 16.252. From an analysis similar to that in Section 13.10 the cost to the~ompany of the op tions is10 x 16.252= 12.510+3or about $12.5 per option. The total cost is therefore 3 m illion times this or $37.5 m illion. If

:themarket perceives no benefits from the options the stock price will fall by $3.75.

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Fin 674 Problems Ch13