In this section we will derive the (Black–Scholes) partial differential equation that is obeyed by options written on a single asset.
Previously,in Sections 8.4 and 8.5,we derived Ito’s lemma,which provides an expression for the change in value of the function (X,t),where X is a stochastic variable. When the stochastic variable,X,follows GBM,the change in the value of was shown to be given by Equation 8.21. Here we will assume that the function(S,t) is the value of a financial option and that the price of the underlying asset,S,follows GBM.
If we denote the value of the financial derivative byf,then its change,df,over the time intervaldtis given by:
df ¼ S@f
@Sþ@f
@tþ2S2 2
@2f
@S2
dtþ@f
@SSdZ; dZNð0;dtÞ The discretized version of this equation is:
f ¼t S@f
@Sþ@f
@tþ2S2 2
@2f
@S2
þ@f
@SSdZ; dZNð0;tÞ ð9:7Þ where the time interval is nowtand the change in derivative value isf.
If we assume that the asset price,S,follows GBM we also have:
S¼StþSZ; ZNð0;tÞ ð9:8Þ
whereis the constant drift and the definition of the other symbols is as before. Let us now consider a portfolio consisting of1 derivative and@f=@Sunits of the underlying 90 Pricing Assets
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stock. In other words we have goneshort(that is sold) a derivative on an asset and have
@f=@Sstocks of the (same) underlying asset. The value of the portfolio,,is therefore:
¼ fþ@f
@SS ð9:9Þ
and the change,,in the value of the portfolio over timetis:
¼ f þ@f
@SS ð9:10Þ
Substituting Equations 9.7 and 9.8 into Equation 9.10 we obtain:
¼ S@f
@Sþ@f
@tþ1
22S2@2f
@S2
tSZ@f
@Sþ @f
@SfStþSZg
;¼ St@f
@St@f
@t1
2t2S2@2f
@S2SZ@f
@S þSt@f
@SþSZ@f
@S ð9:11Þ
Cancelling terms we obtain:
¼ t @f
@tþ1
22S2@2f
@S2
ð9:12Þ If this portfolio is risk neutral then it grows at the riskless interest rate,rand we have:
rt¼
So we have that:
rt¼ t @f
@tþ1
22S2@2f
@S2
ð9:13Þ Substituting forand we obtain:
rt fS@f
@S
¼ t @f
@tþ1
22S2@2f
@S2
ð9:14Þ On rearranging we have:
The Black–Scholes partial differential equation
@f
@tþS@f
@Sþ1
22S2@2f
@S2¼rf ð9:15Þ
Let us now consider put and call options on the same underlying asset. If we letcbe the value of a European call option andpthat of a European put option then we have the following equations:
@p
@tþS@p
@Sþ1
22S2@2p
@S2¼rp ð9:16Þ
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and
@c
@tþS@c
@Sþ1
22S2@2c
@S2¼rc ð9:17Þ
If we now form a linear combination of put and call options,¼a1cþa2p,where botha1 anda2are constants,thenalso obeys the Black–Scholes equation:
@
@t þS@
@Sþ1
22S2@2
@S2 ¼r ð9:18Þ
We will now prove thatsatisfies Equation 9.15.
First we rewrite Equation 9.15 as:
@ða1cþa2pÞ
@t þS@ða1cþa2pÞ
@S þ1
22S2@2ða1cþa2pÞ
@S2 ¼rða1cþa2pÞ ð9:19Þ and use the following results from elementary calculus:
@ða1cþa2pÞ
@t ¼a1@c
@tþa2@p
@t
@ða1cþa2pÞ
@S ¼a1
@c
@Sþa2
@p
@S and
@2ða1cþa2pÞ
@S2 ¼a1@2c
@S2þa2@2p
@S2
If we denote the left hand side of Equation 9.15 by LHS,then we have:
LHS¼a1
@c
@tþS@c
@Sþ1
22S2@2c
@S2
þa2
@p
@tþS@p
@Sþ1
22S2@2p
@S2
ð9:20Þ We now use Equations 9.13 and 9.14 to substitute for the values in the curly brackets in Equation 9.19,and we obtain:
LHS¼a1rcþa2rp ð9:21Þ
which is just the LHS of Equation 9.21; so we have proved the result. It should be noted that this result is also true for American options,since they also obey the Black–Scholes equation.
The above result can be generalized to include a portfolio consisting of n single asset options. Here we have:
¼Xn
j¼1
ajfj; j¼1;. . .;n
wherefjrepresents the value of thejth derivative andajis the number of units of the jth derivative. To prove thatfollows the Black–Scholes equation we simply parti- tion the portfolio into sectors whose options depend on the same underlying asset.
We then proceed as before by showing that the value of each individual sector obeys the Black–Scholes equation and thus the value of the complete portfolio (the sum of the values of all the sectors) obeys the Black–Scholes equation. It should be 92 Pricing Assets
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mentioned that this result applies for both American and European options and it doesn’t matter whether we have bought or sold the options.
In Section 10.3.2 we will use the fact that the difference between the value of a European option and the equivalent American option obeys the Black–Scholes equation. We can see this immediately by considering the following portfolios that are long in an American option and short (that is have sold) a European option:
p¼Pp; c¼Cc
wherePandCare the values of American put and call options.pandcboth obey the Black–Scholes equations,and are the respective differences in value of American/
European put options and American/European call options.
9.3.2 The multiasset option pricing partial differential equation
In this section we will derive the multiasset (Black–Scholes) partial differential equation that is obeyed by options written onnassets. Proceeding as in Section 9.3.1 we will use then-dimensional version of Ito’s lemma to find the process followed by the the value of a multiasset financial derivative. We will denote the value of this derivative byf(S,t), whereSis anelement stochastic vector containing the prices of the underlying assets, Si,i¼1,. . .,n. If we assume thatSfollowsn-dimensional GBM then the change in the value of the derivative,df,is (see Section 8.5,Equation 8.28) given by:
df ¼ @f
@tþXn
i¼1
iSi
@f
@Si
þ1 2
Xn
i¼1
Xn
j¼1
ijSiSjij @2f
@Si@Sj
( )
dtþXn
i¼1
@f
@Si
iSidZi ð9:22Þ The discretized version of this equation is:
f ¼ @f
@tþXn
i¼1
iSi@f
@Si
þ1 2
Xn
i¼1
Xn
j¼1
ijSiSjij @2f
@Si@Sj
( )
tþXn
i¼1
@f
@Si
iSiZi ð9:23Þ where the time interval is nowtand the change in derivative value isf.
Let us now consider a portfolio consisting of1 derivative and@f=@Siunits of the ith underlying stock. In other words we have goneshort(that is sold) a derivative that depends on the price,Si,i¼1,. . .,n,ofnunderlying assets,and have@f=@Siunits of theith asset. The value of the portfolio,,is therefore:
¼ fþXn
i¼1
@f
@Si
Si ð9:24Þ
and the change,,in the value of the portfolio over the time intervaltis:
¼ f þXn
i¼1
@f
@SiSi ð9:25Þ
Since the stochastic variablesSi,i¼1,. . .,nfollown-dimensional GBM the change in theith asset price,Si over the time intervaltis given by:
Si¼iSitþiSiZi; i¼1;. . .;n ð9:26Þ Analytic methods and single asset European options 93
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whereZi¼i ffiffiffiffiffiffi pt
and,as in Section 8.5,we write:
E½2i ¼1; i¼1;. . .;n and
E½i j ¼i;j; i¼1;. . .;n; j¼1;. . .;n; i6¼j
Substituting Equations 9.23 and 9.26 into Equation 9.25 we obtain:
¼ @f
@tþXn
i¼1
iSi
@f
@Siþ1 2
Xn
i¼1
Xn
j¼1
ijijSiSj
@2f
@Si@Sj
( )
t
Xn
i¼1
iSiZi
@f
@Si
þXn
i¼1
@f
@Si
fiSitþSiZig
;¼ Xn
i¼1
iSit@f
@Si
t@f
@t1 2tXn
i¼1
Xn
j¼1
ijijSiSj @2f
@Si@Sj
Xn
i¼1
iSiZi@f
@Si
þXn
i¼1
iSit@f
@Si
þXn
i¼1
iSiZi @f
@Si
ð9:27Þ
Cancelling terms we obtain:
¼ t @f
@tþ1 2
Xn
i¼1
Xn
j¼1
ijijSiSj
@2f
@Si@Sj
( )
ð9:28Þ If this portfolio is to grow at the riskless interest rate,rwe have:
rt¼
So from Equation 9.28 we have that:
rt¼ t @f
@tþ1 2
Xn
i¼1
Xn
j¼1
ijijSiSj @2f
@Si@Sj
( )
ð9:29Þ
Substituting forand we obtain:
rt f Xn
i¼1
Si
@f
@Si
( )
¼ t @f
@tþ1 2
Xn
i¼1
Xn
j¼1
ijijSiSj
@2f
@Si@Sj
( )
ð9:30Þ 94 Pricing Assets
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Rearranging Equation 9.30 gives:
Then-dimensional Black–Scholes partial differential equation
@f
@tþXn
i¼1
Si
@f
@Si
þ1 2
Xn
i¼1
Xn
j¼1
ijijSiSj
@2f
@Si@Sj
¼rf ð9:31Þ
9.3.3 The Black–Scholes formula
In this section we will derive the Black–Scholes formula for pricing European put and call options on a single asset which follows GBM. The approach we will adopt here is to first derive an expression for the value of a European call option,and then use the put/call parity relationships of Section 9.2 to obtain the value of the corresponding European put option. If we denote the current time bytand the expiry time of the option byT,then the duration of the option is ¼Tt. Since the asset is assumed to follow GBM we can use a discretized version of Equations 8.14 and 8.15 in Section 8.3 to write:
log Stþt
St
N r1 22
t; 2t
ð9:32Þ Here we use the following notation:
t¼; St¼S,and Stþt¼ST
whereSis the asset value at the current timet,andST is the asset value at option maturity. We will now introduce the variableXwhich we define as follows:
X¼log ST
S
or equivalently ST¼SexpðXÞ From Equation 9.32 we have that
XNððr2=2Þ; 2Þ
The probability density function ofX,f(X),is thus the Gaussian:
fðXÞ ¼ 1 pffiffiffi ffiffiffiffiffiffi
p2exp ðX ðr2=2ÞÞ2 22
The value of a European call option,c(S,E,),with strike priceE,is the expected value of the option’s payoff at maturity discounted to the current time by the riskless interest rater. That is:
cðS;E; Þ ¼expðrÞE½STE
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This can be rewritten in terms of the probability density function ofST as follows:
cðS;E; Þ ¼expðrÞ Z 1
ST¼E
fðSTÞðSTEÞdST ð9:33Þ Instead of integrating over values of ST,as above,we will useST ¼Sexp (X) and then integrate overX. Equation 9.33 then becomes:
cðS;E; Þ ¼expðrÞ pffiffiffi ffiffiffiffiffiffi p2
Z 1 X¼logðE=SÞ
SexpðXÞ E
ð Þ
exp ðX ðr2=2ÞÞ2 22
dX ð9:34Þ
where we have usedSexp (X)¼E,givingX¼log (E=S),to obtain the lower limit of the integral. This integral is evaluated by splitting it into the two parts:
cðS;E; Þ ¼IAIB ð9:35Þ
where
IA¼SexpðrÞ pffiffiffi ffiffiffiffiffiffi
p2 Z 1
X¼logðE=SÞ
expðXÞexp X ðr2=2Þ2
22
!
dX ð9:36Þ and
IB¼EexpðrÞ ffiffiffi
p ffiffiffiffiffiffi p2
Z 1 X¼logðE=SÞ
exp fX ðr2=2Þg2 22
EdX ð9:37Þ
To evaluate these integrals we will make use of the fact that the univariate cumulative normal functionN1(x) is:
N1ðxÞ ¼ 1 ffiffiffiffiffiffi p2
Z x
u¼ 1
exp u2 2
du
by symmetry we haveN1(x)¼1N1(x) and 1ffiffiffiffiffiffi
p2 Z 1
x
exp u2 2
du¼ 1 ffiffiffiffiffiffi p2
Z x 1
exp u2 2
du¼N1ðxÞ We will first considerIB,which is the easier of the two integrals.
IB¼EexpðrÞ pffiffiffi ffiffiffiffiffiffi
p2 Z 1
X¼logðE=SÞ
exp X ðr2=2Þ2
22
! dX
If we letu¼(X(r2=2))=pffiffiffi
then dX ¼pffiffiffi du. So
IB¼EexpðrÞpffiffiffi ffiffiffiffiffiffi
p2pffiffiffi Z 1
u¼k2
exp u2 2
du
where the lower integration limit isk2¼( log (E=S)(r2=2))=pffiffiffi . 96 Pricing Assets
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We therefore have:
IB¼EexpðrÞN1ðk2Þ ð9:38Þ
We will now consider the integralIA. IA¼SexpðrÞ
pffiffiffi ffiffiffiffiffiffi p2
Z 1 X¼logðE=SÞ
expðXÞexp X ðr2=2Þ2
22
! dX
Rearranging the integrand:
IA¼expðrÞ ffiffiffi
p ffiffiffiffiffiffi p2
Z 1 X¼logðE=SÞ
exp X ðr2=2Þ2
22X 22
!
dX ð9:39Þ Expanding the terms in the exponential:
X ðr2=2Þ
2
22X¼X22ðr2=2Þ
Xþ ðr 2=2Þ2
22X
¼X22ðrþ2=2Þ
Xþ ðr 2=2Þ2
¼X ðrþ2=2Þ2
þ ðr 2=2Þ2
ðr þ2=2Þ2
which results in:
X ðr2=2Þ
2
22X¼X ðrþ2=2Þ2
22r2 ð9:40Þ Substituting Equation 9.47 into the integrand of Equation 9.45 we have:
expðXÞexp X ðr2=2Þ2
22
!
¼expðrÞexp X ðrþ2=2Þ2
22
!
The integralIAcan therefore be expressed as:
IA¼SexpðrÞexpðrÞ ffiffiffiffiffiffi
p2
Z 1 X¼logðE=SÞ
exp X ðrþ2=2Þ2
22
! dX
If we letu¼(X(rþ2=2))=pffiffiffi
then dX ¼pffiffiffi du. So
IA¼ Spffiffiffi ffiffiffiffiffiffi
p2pffiffiffi Z 1
u¼k1
exp u2 2
du
where the lower limit of integration isk1¼( log (E=S)(rþ2=2))=pffiffiffi . We therefore have:
IA¼SN1ðk1Þ ð9:41Þ
Therefore the value of a European call is:
cðS;E; Þ ¼SN1ðk1Þ EexpðrÞN1ðk2Þ
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which gives the usual form of the Black–Scholes formula for a European call as:
The Black–Scholes formula for a European call
cðS;E; Þ ¼SN1ðd1Þ EexpðrÞN1ðd2Þ ð9:42Þ where
d1¼logðS=EÞ þ ðrþ2=2Þ pffiffiffi and d2¼logðS=EÞ þ ðr2=2Þ
pffiffiffi ¼d1pffiffiffi
ð9:43Þ
To gain some insight into the meaning we will rewrite the above equation in the following form:
cðS;E; Þ ¼expðrÞfSN1ðd1ÞexpðrÞ EN1ðd2Þg ð9:44Þ The termN1(d2) is the probability that the option will be exercised in a risk-neutral world,so thatEN1(d2) is the strike price multiplied by the probability that the strike price will be paid. The termSN1(d1) exp (r) is the expected value of a variable,in a risk neutral world,that equalsST ifST >Eand is otherwise zero.
The corresponding formula for a put can be shown using put–call parity,see Section 9.2,to be:
The Black–Scholes formula for a European put
pðS;E; Þ ¼EexpðrÞN1ðd2Þ SN1ðd1Þ ð9:45Þ where
d1¼logðS=EÞ þ ðrþ2=2Þ pffiffiffi and d2¼logðS=EÞ þ ðr2=2Þ
pffiffiffi ¼d1 ffiffiffi p
ð9:46Þ or equivalently,usingN1(x)¼1N1(x) we have
pðS;E; Þ ¼EexpðrÞf1N1ðd2Þg Sf1N1ðd1Þg ð9:47Þ
The inclusion of continuous dividends
The effect of dividends on the value of a European option can be dealt with by assuming that the asset price is the sum of a riskless component involving known dividends that will be paid during the life of the option,and a risky (stochastic) component; see Hull (1997).
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As dividends are paid the stock price is reduced by the same amount,and by the time the European option matures,all the dividends will have been paid leaving only the risky component of the asset price.
This means that,in the case of a continuous dividend yieldq,European put/call options can be priced using Equations 9.42 and 9.45 but with S replaced by Sexp (q).
This results in:
The Black–Scholes formula with continuous dividends
cðS;E; Þ ¼SexpðqÞN1ðd1Þ EexpðrÞN1ðd2Þ ð9:48Þ and the corresponding formula for a put can be shown (using put–call parity) to be:
pðS;E; Þ ¼EexpðrÞN1ðd2Þ SexpðqÞN1ðd1Þ ð9:49Þ or equivalently,usingN1(x)¼1N1(x),we have
pðS;E; Þ ¼EexpðrÞf1N1ðd2Þg SexpðqÞf1N1ðd1Þg ð9:50Þ where
d1¼logðS=EÞ þ ðrqþ2=2Þ pffiffiffi and d2¼logðS=EÞ þ ðrq2=2Þ
pffiffiffi ¼d1 ffiffiffi p
The above values ofd1andd2are obtained by simply substitutingS¼Sexp (q) into Equation 9.43 as follows:
d1¼logðSexpðqÞ=EÞ þ ðrþ2=2Þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ðTtÞ
p ¼logðS=EÞ qþ ðrþ2=2Þ
pffiffiffi d2¼logðSexpðqÞ=EÞ þ ðr2=2Þ
pffiffiffi ¼logðS=EÞ qþ ðr2=2Þ pffiffiffi
The inclusion of discrete dividends
Here we consider n discrete cash dividends Di,i¼1,. . .,n,paid at times ti, i¼1,. . .,n during the life of the option. In these circumstances the Black–Scholes formula can be used to price European options,but with the current asset valueS reduced by the present value of the cash dividends.
This means that instead ofSwe use the quantitySDwhich is computed as SD¼SXn
i¼1
DiexpðrtiÞ
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whereris the (in this case constant) riskless interest rate. The formulae for European puts and calls are then
cðS;E; Þ ¼SDN1ðd1Þ EexpðrÞN1ðd2Þ ð9:51Þ pðS;E; Þ ¼EexpðrÞf1N1ðd2Þg SDf1N1ðd1Þg ð9:52Þ where
d1¼logðSD=EÞ þ ðrþ2=2Þ pffiffiffi and d2¼logðSD=EÞ þ ðr2=2Þ
pffiffiffi ¼d1 ffiffiffi p
ð9:53Þ In Section 10.2.3 we give results for perpetual European options.
The greeks
Now that we have derived formulae to price European vanilla puts and calls it is possible to work out their partial derivatives (hedge statistics). We will now merely quote expressions for the Greeks (hedge statistics) for European options. Here the subscriptcrefers to a European call,and the subscriptp refers to a European put.
Complete derivations of these results can be found in Appendix C.
Gamma
c¼@2c
@S2¼p¼@2p
@S2¼expðqÞ nðd1Þ
Spffiffiffi ð9:54Þ
Delta
c¼@c
@S¼expðqÞN1ðd1Þ; p¼@p
@S¼expðqÞfN1ðd1Þ 1g ð9:55Þ Theta
c¼@c
@t¼qexpðqÞSN1ðd1Þ rEexpðrÞN1ðd2Þ Snðd1ÞexpðqÞ 2pffiffiffi p¼@p
@t¼ qexpðqÞSN1ðd1Þ þrEexpðrÞN1ðd2Þ Snðd1ÞexpðqÞ 2pffiffiffi ð9:56Þ Rho
c¼@c
@r¼EN1ðd2Þ; p¼@p
@r¼ EN1ðd2Þ ð9:57Þ Vega
Vc¼@c
@¼ Vp¼@p
@¼SexpðqÞnðd1Þ ffiffiffi p
ð9:58Þ wheren(x)¼(1= ffiffiffiffiffiffi
p2
) exp (x2=2).
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We now present,in Code excerpt 9.1,a computer program to calculate the Black–
Scholes option value and Greeks given in Equations 9.54 to 9.57. The routine uses the NAG C library macroX02AJCto identify whether the arguments are too small,and also the NAG C library function s15abc to compute the cumulative normal distribution function.
void black_scholes(double *value, double greeks[], double s0, double x,
double sigma, double t, double r, double q, Integer put, Integer *iflag) {
/* Input parameters:
s0 — the current price of the underlying asset
x — the strike price
sigma — the volatility t — the time to maturity
r — the interest rate
q — the continuous dividend yield put
Output parameters:
value — the value of the option
greeks[] — the hedge statistics output as follows: greeks[0] is gamma, greeks[1] is delta greeks[2] is theta, greeks [3] is rho, and greeks[4] is vega
iflag — an error indicator
*/
double one¼1.0,two¼2.0,zero¼0.0;
double eps,d1,d2,temp,temp1,temp2,pi,np;
eps¼X02AJC;
if( (x < eps) || (sigma < eps) || (t < eps) ) { /* Check if any of the the input arguments are too small */
*iflag¼2;
return;
}
temp¼log(s0/x);
d1¼tempþ(rqþ(sigma*sigma/two))*t;
d1¼d1/(sigma*sqrt(t));
d2¼d1sigma*sqrt(t);
/* evaluate the option price */
if (put¼¼0)
*value¼(s0*exp(q*t)*s15abc(d1)x *exp (r *t) *s15abc (d2));
else
*value¼(s0*exp(q*t)*s15abc(d1) þx*exp(r*t)*s15abc (d2));
if (greeks){/* then calculate the greeks */
temp1¼ d1*d1/two;
d2¼d1sigma*sqrt(t);
pi¼X01AAC;
np¼(one/sqrt(two*pi)) * exp(temp1);
if (put¼¼0) { /* a call option */
greeks[1]¼ (s15abc(d1))*exp(q*t); /* delta */
greeks[2]¼ s0*exp(q*t)*np*sigma/(two*sqrt(t))
þq*s0*s15abc(d1)*exp(q*t)r*x*exp(r*t) *s15abc (d2);/* theta */
greeks[3]¼x*t*exp(r*t)*s15abc(d2); /* rho */
}
else { /* a put option */
greeks[1]¼(s15abc(d1)one)*exp(q*t); /* delta */
greeks[2]¼ s0*exp(q*t)*np*sigma/(two*sqrt(t))
q*s0*s15abc(d1)*exp(q*t)þr*x*exp(r*t)*s15abc (d2); /* theta */
greeks[3]¼ x*t*exp(r*t)*s15abc(d2); /* rho */
}
greeks[0]¼np*exp(q*t)/(s0*sigma*sqrt(t)); /* gamma */
greeks[4]¼s0*sqrt(t)*np*exp(q*t); /* vega */
} return;
}
Code excerpt 9.1 Function to compute the Black–Scholes value for European options
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It can be seen in Tables 9.1 and 9.2 that the values for gamma and vega are the same for both puts and calls. We can also demonstrate that the option values are consistent by using put–call parity.
cðS;E; Þ þEexpðrÞ ¼pðS;E; Þ þSexpðqÞ
For example when ¼1:0,we havec(S,E,)¼12:952 andP(S,E,T)¼9.260.
So: c(S,E,)þEexp (r)¼12:952þ100exp (0:1)¼103:436 and p(S,E,)þ Sexp (q)¼9:260þ100exp (0:06)¼103:436.
9.3.4 Historical and implied volatility
Obtaining the best estimate of the volatility parameter, ,in the Black–Scholes formula is of crucial importance. There are many different approaches to volatility estimation. These include:
. Historical estimation . Implied volatility . Time series methods.
Table 9.1 European put: option values and greeks. The parameters are:S¼100.0, E¼100.0,r¼0.10,¼0.30,q¼0.06
Value Delta Gamma Theta Vega Rho
0.100 3.558 0:462 0.042 16:533 12.490 4:971
0.200 4.879 0:444 0.029 10:851 17.487 9:860
0.300 5.824 0:431 0.024 8:298 21.204 14:663
0.400 6.571 0:419 0.020 6:758 24.241 19:377
0.500 7.191 0:408 0.018 5:698 26.832 24:004
0.600 7.720 0:399 0.016 4:909 29.100 28:544
0.700 8.179 0:390 0.015 4:292 31.118 32:997
0.800 8.582 0:381 0.014 3:792 32.935 37:364
0.900 8.940 0:373 0.013 3:377 34.585 41:646
1.000 9.260 0:366 0.012 3:025 36.093 45:843
Table 9.2 European call: option values and greeks. The parameters are:S¼100.0, E¼100.0,r¼0.10,¼0.30,q¼0.06
Value Delta Gamma Theta Vega Rho
0.100 3.955 0.532 0.042 20:469 12.490 4.929
0.200 5.667 0.544 0.029 14:724 17.487 9.744
0.300 6.996 0.552 0.024 12:109 21.204 14.451
0.400 8.121 0.558 0.020 10:508 24.241 19.054
0.500 9.113 0.562 0.018 9:387 26.832 23.557
0.600 10.007 0.566 0.016 8:539 29.100 27.962
0.700 10.826 0.569 0.015 7:863 31.118 32.271
0.800 11.584 0.572 0.014 7:305 32.935 36.485
0.900 12.290 0.574 0.013 6:832 34.585 40.608
1.000 12.952 0.576 0.012 6:422 36.093 44.640
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Here we will consider both historical and implied volatility estimation. Part III of this book deals with the more complex issues connected with time series volatility estimation.
Historical volatility
In this method we calculate the volatility using nþ1 historical asset prices, Si,i¼0,. . .,n,and we assume that the asset prices are observed at the regular time interval, d. Since the asset prices are assumed to follow GBM,the volatility is computed as the annualizedstandard deviation of then continuously compounded returns,ui,i¼1,. . .,n,where
Si¼Si1expðuiÞ or ui¼log Si
Si1
We already know,see Section 8.3,Equation 8.15,that the expected stand- ard deviation of the asset returns over the time interval is ffiffiffiffiffiffi
pd
. This means that we obtain the following expression for,the estimated volatility^
^ ffiffiffiffiffiffi
pd
¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1
n1 Xn
i¼1
ðuiuuÞ2 s
ð9:59Þ or
^ ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1
ðn1Þd Xn
i¼1
ðuiuuÞ2 s
ð9:60Þ The estimated standard error in^is,see for example Hull (1997),given by
^ std¼^
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2ðn1Þ s
ð9:61Þ A computer program to perform these calculation is given below in Code excerpt 9.2.
void hist_vol(double *sigma, double *err, double data[], Integer n, double dt, Integer *ifail) {
/*Input parameters:
data[] — the data, which consists of n asset prices n — the number of data points
dt — the (constant) time spacing between the data points (in years) Output parameters:
sigma — the computed historical volatility
err — the standard error in the volatility estimate sigma iflag — an error indicator
*/
#define DATA(I) data[(I)1]
double mean¼0.0,sum¼0.0;
Analytic methods and single asset European options 103
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double temp,tn;
Integer i;
for(i¼2; i <¼n;þþi)
mean¼meanþlog(DATA(i))log(DATA(i1));
mean¼mean/(double)(n1);
for(i¼2; i <¼n;þþi) {
temp¼log(DATA(i))log(DATA(i1));
sum¼sumþ(tempmean)*(tempmean);
}
sum¼sum/(double)(n2);
*sigma¼sqrt(sum/dt);
tn¼(double)(2*(n1));
*err¼*sigma/sqrt(tn);
return;
}
Code excerpt 9.2 Function to compute the historical volatility from asset data
Implied volatility
The implied volatility of a European option is the volatility which,when substituted into the Black–Scholes formula,yields the market value quoted for the same option.
The routine provided in Code excerpt 9.2 uses Newton’s method to calculate the implied volatility for a European option from its market price. We will now illustrate this technique for a European call option with market value opt value. The implied volatility,,is then that value which satisfies:
KðÞ ¼cðS;E; ; Þ opt value¼0
where c(S,E,,) represents the value of the European call and the other symbols have their usual meaning.
From Newton’s method we have:
iþ1¼i FðiÞ F0ðiÞ where
F0ðiÞ ¼@F
@ ¼@cðS;E; ; Þ
@ ¼ Vc
Therefore the iterative procedure is iþ1¼icðS;E; ; Þ optvalue
Vc
where 0 is the initial estimate,and iþ1 is the improved estimate of the implied volatility based on the ith estimate i. Termination of this iteration occurs when ABS(iþ1i)<tol,for a specified tolerance,tol.
It can be seen that as !0,d1! 1,d2! 1 and,from Equation 9.58 we have Vc!0. Under these circumstances Newton’s method fails.
The same procedure can be used to compute the implied volatility for a European put,in this can we just replacec(S,E,,) byp(S,E,,),the value of a European put; from Equation 9.58Vc¼ Vp.
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