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10.2.4 Perpetual European down and out call
Here we find the value of a perpetual down and out European call barrier option, see Merton (1973).
Let the exercise price beEand the barrier be atBwhereB<E.
Since the Black–Scholes partial differential equation governs the price of the option we can, as before, look for solutions of the form:
cðS;EÞdo¼A1Sm1þA2Sm2 ð10:27Þ subject to the boundary conditions: (i) cdo(B,E)¼0 and (ii)c(1,E)do¼S, see the previous section.
From (i) we have:
cdoðB;EÞ ¼A1Bm1þA2Bm2¼0; so A1¼ A2Bm2m1 Therefore
cdoðS;EÞ ¼ A2Bm2m1Sm1þA2Sm2 From (ii), asS! 1:
cdoðS;EÞ ¼ A2Bm2m1Sm1þA2Sm2¼S
However, sincem2<0, we haveA2Sm2!0, asS! 1, giving cdoðS;EÞ ¼ A2Bm2m1Sm1¼S
So
A2¼ S1m1
Bm2m1 and cdoðS;EÞ ¼S1m1Sm1Bm2m1
Bm2m1 S1m1Sm2 Bm2m1 which results in:
cdoðS;EÞ ¼SS1þm2m1
Bm2m1 ð10:28Þ
When there are no dividends (q¼0) we have already shown in Sections 10.2.1 and 10.2.2 thatm1 ¼1 andm2¼ 2r=2so the value of a perpetual down and out call is (see Merton (1973)):
cdoðS;EÞ ¼S Sm2
Bm21¼SB S B
2r=2
ð10:29Þ
10.3 APPROXIMATIONS FOR VANILLA AMERICAN OPTIONS
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The Roll, Geske, Whaley approximation
This method uses the work of Roll (1977), Geske (1979), and Whaley (1981). LetSbe the current (timet) price of an asset which pays a single cash dividendD1 at timet1. At theex-dividend date,t1, there will be a decrease in the asset’s value from St1 to St1D1. Also the current asset price net ofescroweddividends is:
SD¼SD1expðrðt1tÞÞ ð10:30Þ
whereris the riskless interest rate.
Now consider an American call option, with strike price E and expiry time T, which is taken out on this asset. Att1there will be a given ex-dividend asset price,S , above which the option will be exercised early. This value can be found by solving the following equation:
cðS;E; 1Þ ¼S þD1E ð10:31Þ wherec(S ,E,1) is the Black–Scholes value of a European call option with strike priceEand maturity1¼Tt1, on an asset with current valueS at timet1. If just prior to the ex-dividend date St1 >S , then the American option will be exercised and realize a cash payoff of St1þD1E. On the other hand ifSt1 S then the option is worth more unexercised and it will be held until option maturity at timeT.
We can rewrite Equation 10.31 so thatS is the root of the following equation:
KðSÞ ¼cðS ;E; 1Þ S D1þE¼0 ð10:32Þ whereK(S ) denotes the function in the single variableS .
A well-known technique for solving Equation 10.32 is Newton’s method, which in this case takes the form:
Siþ1¼Si KðSiÞ
K0ðSiÞ ð10:33Þ
whereSi is theith approximation toS , andSiþ1is the improved (iþ1)th approximation.
If we now consider the terms in Equation 10.33 we have that KðSiÞ ¼cðSi;E; 1Þ Si D1þE
and
K0ðSiÞ ¼@KðSiÞ
@Si ¼@cðSi;E; 1Þ
@Si 1 Also from Equation C.14 in Appendix C.3
@cðSi;E; 1Þ
@Si ¼Nðd1ðSiÞÞ
We note that here thecontinuousdividend yield,q¼0.
So
K0ðSiÞ ¼@KðSiÞ
@Si ¼Nðd1ðSiÞÞ 1; where d1¼logðSi=EÞ þ ðrþ2=2Þð1Þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiTt1
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Substituting these results into Equation 10.33 gives:
Siþ1¼Si cðSi;E; 1Þ ðSi þD1EÞ Nðd1ðSiÞÞ 1
On rearrangement this yields
Siþ1¼SiN1ðd1ðSiÞÞ cðSi;E; 1Þ þD1E
N1ðd1ðSiÞÞ 1 ; for i¼0;. . .;max iter ð10:34Þ where a convenient initial approximation is to chooseS0 ¼E, andmax_iteris the maximum number of iterations that are to be used.
We will now quote the Roll, Geske, and Whaley formula for the current value of an American call which pays asinglecash dividendD1at timet1, it is:
CðS;E; Þ ¼SD N1ðb1Þ þN2ða1;b1; ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðt1tÞ=
p Þ
n o
þD1expðrðt1tÞÞN1ðb2Þ EexpðrÞ N1ðb2Þexpðrð1ÞÞ þN2ða2;b2; ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ðt1tÞ=
p Þ
n o
ð10:35Þ whereSD is given by Equation 10.30,E is the exercise price,T is the option expiry date, trepresents the current time, is the option maturity, N1(a) is the univariate cumulative normal density function with upper integral limita, andN2(a,b, ) is the bivariate cumulative normal density function with upper integral limitsaandband correlation coefficient . The other symbols used in Equation 10.35 are defined as
a1¼logðS=EÞ þ ðrþ2=2Þ ffiffiffi
p ; a2¼a1 ffiffiffi p b2¼logðS=S Þ þ ðrþ2=2Þðt1tÞ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðt1tÞ
p ; b2¼b1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðt1tÞ p
and Sis the current (time t) asset price, S is found using Equation 10.34, ris the riskless interest rate,is the asset’s volatility, ¼Ttand1¼Tt1.
To compute the value of an American call option which pays n cash dividends Di,i¼1,. . .,n at times ti,i¼1,. . .,n, we can use the fact that optimal exercise normally only ever occurs at the final ex-dividend date tn, see for example Hull (1997). Under these circumstances Equation 10.35 can still be shown to value the American call but now t1 should be set totn,D1 should be set toDn, andSD is given by:
SD¼SXn
i¼1
DiexpðrðtitÞÞ ð10:36Þ
A program to compute the Roll, Geske, and Whaley approximation for an Ameri- can call option with multiple cash dividends is given in Code excerpt 10.1. Here the NAG C library functions s15abcandg01hacare used to calculate the values of N1(a) andN2(a,b, ) respectively. Code excerpt 10.3 was used to compute the values presented in Table 10.1. These compare the Roll, Geske, and Whaley approximation with the Black approximation, which we will now briefly discuss.
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void RGW_approx(double *opt_value, double *critical_value, Integer n_divs, double dividends[], double Divs_T[], double S0, double X, double sigma, double T, double r, Integer *iflag) {
/* Input parameters:
n_divs — the number of dividends
dividends[] — the dividends: dividends[0] contains the first dividend, dividend[1] the second etc.
Divs_T[] — the times at which the dividends are paid: Divs_T[0] is the time at which the first dividend is paid Divs_T[1] is the time at which the second dividend is paid, etc.
S0 — the current value of the underlying asset
X — the strike price
sigma — the volatility
T — the time to maturity
r — the interest rate
Output parameters:
opt_value — the value of the option critical_value — the critical value iflag — an error indicator
*/
double A_1,A_2,S_star,a1,a2,nt1,t1,S;
double b1,b2,d1,alpha,h,div,beta,temp,temp1,temp2,temp3;
double pdf,b,eur_val,fac,tol,loc_q,err,zero¼0.0;
Boolean iterate;
Integer i,iflagx,putx;
static NagError nagerr;
loc_q¼0.0;
temp¼0.0;
for (i¼0; i < n_divs;þþi) { /Check the Divs_T array */
if ((Divs_T[i] <¼temp) || (Divs_T[i] > T) || (Divs_T[i] <¼zero)) {
*flag¼2;
return;
}
temp¼Divs_T[i];
}
/* calculate the present value of the dividends (excluding the final one) */
temp¼0.0;
for (i¼0; i < n_divs1;þþi) {
temp¼facþdividends[i] * exp(r*Divs_T[i]);
}
t1¼Divs_T[n_divs1];
/* decrease the stock price by the present value of all dividends */
div¼dividends[n_divs-1];
S¼S0-temp-div*exp(r*t1);
iterate¼TRUE;
tol¼0.000001;
S_star¼X;
while (iterate) { /* calculate S_star, iteratively */
/* calculate the Black—Scholes value of a European call */
Table 10.1 A comparison of the computed values for American call options with dividends, using the Roll, Geske, and Whaley approximation, and the Black approximation. The parameters used were:E¼100:0, r¼0:04,¼0:2,¼2:0 and there is one cash dividend of value 5.0 at timet¼1:0. The current stock price,S,
is varied from 80.0 to 120.0. The results are in agreement with those given in Table 1 of Whaley (1981) Stock price Critical price,S RGW approximation Black approximation
80.0 123.582 3.212 3.208
85.0 123.582 4.818 4.808
90.0 123.582 6.839 6.820
95.0 123.582 9.276 9.239
100.0 123.582 12.111 12.048
105.0 123.582 15.316 15.215
110.0 123.582 18.851 18.703
115.0 123.582 22.676 22.470
120.0 123.582 26.748 26.476
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d1¼(log(S_star/X)þ(rþ(sigma*sigma/2.0))*(Tt1))/(sigma*sqrt(Tt1));
putx¼0;
loc_q¼0.0;
black_scholes(&eur_val,NULL,S_star,X,sigma, Tt1,r,loc_q, putx,&iflag);
S_star¼(S_star*s15abc(d1)eur_valþdivX)/(s15abc (d1)1.0);
err¼FABS(eur_val(S_starþdivX))/X;
if (err < tol) iterate¼FALSE;
}
a1¼(log(S/X)þ(rþ(sigma*sigma/2.0))*T)/(sigma*sqrt(T));
a2¼a1sigma*sqrt(T);
b1¼(log(S/S_star)þ(rþ(sigma*sigma/2.0))*t1)/(sigma*sqrt (t1));
b2¼b1sigma*sqrt(t1);
nt1¼sqrt(t1/T);
temp1¼S*(s15abc(b1)þg01hac(a1,b1,nt1,&nagerr));
temp2¼ X*exp(r*T)*g01hac(a2,b2,nt1,&nagerr)(Xdiv)* exp(r*t1)*s15abc(b2);
*opt_value¼temp1þtemp2;
*critical_value¼S_star;
}
Code excerpt 10.1 Function to compute the Roll, Geske, and Whaley approximation for the value of an American call option with discrete dividends
We will now consider the Black approximation.
Black’s approximation
The Black (1975) approximation for an American call with cash dividends is simpler than the Roll, Geske, and Whaley method we have just described. For an American call option which expires at timeT, withndiscrete cash dividendsDi,i¼1,. . .,n, at times ti,i¼1,. . .,n, it involves calculating the prices of European options that mature at timesT, and tn, and then setting the option price to the greater of these two values, see for example Hull (1997).
The Black approximation, CBL, can be expressed more concisely in terms of our previously defined notation as:
CBLðS;E; Þ ¼maxðv1;v2Þ
wherev1 andv2are the following European calls
v1¼cðSD;E; Þ and v2¼cðSþD;E; 1Þ, ¼Tt 1¼Ttn
and
SD¼SXn
i¼1
Di and SþD¼SXn1
i¼1
Di
Code excerpt 10.2 below computes the Black approximation.
void black_approx(double *value, Integer n_divs, double dividends[], double Divs_T[], double S0, double X, double sigma, double T, double r, Integer put, Integer *ifail) {
/* Input parameters:
n_divs — the number of dividends
dividends[] — the dividends, dividends[0] contains the first dividend, dividend[1] the second etc.
Divs_T[] — the times at which the dividends are paid, Divs_T[0] is the time at which the first dividend is paid Divs_T[1] is the time at which the second dividend is paid, etc.
S0 — the current value of the underlying asset
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X — the strike price
sigma — the volatility
T — the time to maturity
r — the interest rate
put — if put is 0 then a call option, otherwise a put option
Output parameters:
value — the value of the option, iflag — an error indicator
*/
double zero¼0.0;
double beta,temp,temp1,temp2,temp3;
double tn,val_T,val_tn,tol,loc_q,err,fac;
Integer i,ifailx;
loc_q¼0.0;
temp¼0.0;
for (i¼0; i < n_divs;þþi) {
if (Divs_T[i] <¼temp ) printf (‘‘Error in Divs_T array, elements not increasing \n’’);
if (Divs_T[i] > T) printf (‘‘Error in Divs_T array element has a value greater than T \n’’);
if (Divs_T[i] <¼zero) printf (‘‘Error in Divs_T array element <¼zero \n’’);
temp¼Divs_T[i];
}
/* calculate the present value of the dividends */
fac¼0.0;
for (i¼0; i < n_divs;þþi) {
fac¼facþdividends[i] * exp(r*Divs_T[i]);
}
temp¼S0 - fac;
/* calculate the value of the option on expiry */
black_scholes(&val_T,NULL,temp,X,sigma,T,r,loc_q, put,&ifailx);
/* calculate the value of the option on last dividend date */
tn¼Divs_T[n_divs1];
temp¼tempþdividends[n_divs1]*exp(r*tn);
nag_opt_bs(&val_tn,NULL,temp,X,sigma,tn,r,loc_q, putx,&ifailx);
*value¼MAX(val_tn,val_T);
}
Code excerpt 10.2 Function to compute the value of the Black approximation for the value of an American call option with discrete dividends
Code excerpt 10.3 below uses the same values as in Whaley (1981) and compares the Roll, Geske, and Whaley approximation with that of Black; the results are presented in Table 10.1.
double q,r,temp,loc_r;
Integer i,m,m2,m_acc;
double S0,E,T,sigma,t1,delta,value,ad_value,put_value;
Integer is_american,ifail,put;
double bin_greeks[5],greeks[5],bin_value,bs_value;
double opt_value, critical_value, E1, E2, crit1, crit2;
double black_value;
double Divs_T[3],dividends[3];
Integer n_divs, put;
E¼100.0;
r¼0.04;
sigma¼0.2;
T¼2.0;
t1¼1.0;
put¼0;
/* check using the same parameters as in Whaley (1981) */
Divs_T[0]¼1.0;
dividends[0]¼5.0;
n_divs¼1;
printf (‘‘\nPrice S RGW Approximation Black Approximation \n\n’’);
for (i¼0; i < 9;þþi) { put¼0;
S0¼80.0þ(double)i*5.0;
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opt_RGW_approx(&opt_value,&critical_value, n_divs, dividends,Divs_T,S0,E,sigma,T,r,&ifail);
printf(‘‘%8.4f ’’,S0);
printf(‘‘%12.3f%12.3f ’’,opt_value,critical_value);
opt_black_approx(&black_value,n_divs,dividends, Divs_T, S0,E,sigma,T,r,put,&ifail);
printf(‘‘%12.3f (%8.4e) ’’,black_value);
}
Code excerpt 10.3 Simple test program to compare the results of functionopt_RGW_approxwith functionopt_black_approx, the parameters used are the same as in Whaley (1981)
We will now consider a more general technique for pricing both American puts and calls.
10.3.2 The MacMillan, Barone-Adesi, and Whaley method
Here we consider a method of pricing American options which relies on an approximation that reduces a transformed Black–Scholes equation into a second order ordinary differential equation, see Barone-Adesi and Whaley (1987) and MacMillan (1986).
It thus provides an alternative way of evaluating American options that can be used instead of computationally intensive techniques such as finite-difference methods.
Although the method prices American options it is really based on the value of an American option relative to the corresponding European option value (which can readily be computed using the Black–Scholes pricing formula).
Since an American option gives more choice its value is always at least that of its European counterpart. This early exercise premium ((S, E,)0) is now defined more precisely for American puts and calls. If at current timetthe asset price isS, then the early exercise premium for an American call which expires at timeT, and therefore has maturity ¼Tt, is:
cðS;E; Þ ¼CðS;E; Þ cðS;E; Þ 0 ð10:37Þ whereC(S,E,) denotes the value of the American call and c(S,E,) denotes the value of the corresponding European call. The early exercise premium of an Ameri- can put option,p(S, E,), is similarly defined as:
pðS;E; Þ ¼PðS;E; Þ pðS;E; Þ 0 ð10:38Þ where P(S,E, ) is the value of the American put, and p(S,E,) is the value of the corresponding European put. The key insight provided by the MacMillan, Barone-Adesi, and Whaley method is that since both the American and European option values satisfy the Black–Scholes partial differential equation so does the early exercise premium, (S,E,); see Section 9.3.1. This means that we can write:
@
@tþ ðrqÞS@
@Sþ2S2 2
@2
@S2¼r ð10:39Þ
where as usualSis the asset price,rthe continuously compounded interest rate,qthe continuously compounded dividend, the volatility, and timet increases from the current time to the expiry timeT.
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We will now introduce the variableh()¼1exp (r) and use the factorization (S,E,)¼h()g(S,E,h). From standard calculus we obtain:
@
@t¼g@h
@tþh@g
@t¼rgðh1Þ þh@g
@h
@h
@t¼rgðh1Þ þhrðh1Þ@g
@h and also
@
@S¼h@g
@S and @2
@S2¼h@2g
@S2
Substituting these results into Equation 10.39 yields the following transformed Black–Scholes equation:
S22h 2
@2g
@S2þ ðrqÞSh@g
@Sþrgðh1Þ þrhðh1Þ@g
@h¼rgh ð10:40Þ which can be further simplified to give:
S22@2g
@S2þ2ðrqÞS 2
@g
@S2rg
h22rð1hÞ 2
@g
@h¼rgh ð10:41Þ or
S2@2g
@S2þS@g
@S
hg ð1hÞ@g
@h¼0 ð10:42Þ
where¼2r=2 and¼(2(rq))=2.
We now consider the last term of Equation 10.42 and note that when is large, 1h()0. Also when!0 the option is close to maturity, and the value of both the European and American options converge; which means that(S,E,)0 and
@g=@h0. It can thus be seen that the last term is generally quite small and, the MacMillan, Barone-Adesi and Whaley approximation assumes that it can be ignored. This results in the following equation:
S2@2g
@S2þS@g
@S
hg¼0 ð10:43Þ
which is a second order differential equation with two linearly independent solutions of the formaS. They can be found by substitutingg(S,E,h)¼aS into Equation 10.43 as follows:
@g
@S¼S1 @2g
@S2¼að1ÞS2¼a2S2aS2 so
S2@2g
@S2¼a2SaS¼2gg and
S@g
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When the above results are substituted in Equation 10.43 we obtain the quadratic equation:
2ggþg=h¼gð2þ ð1Þ=hÞ ¼0 or
2þ ð1Þ=h¼0 ð10:44Þ
which has the two solutions 1¼1
2 ð1Þ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1Þ2þ4ð=hÞ
q
ð10:45Þ and
2¼1
2 ð1Þ þ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1Þ2þ4ð=hÞ
q
ð10:46Þ where we note that since=h>0, we have1<0 and2>0.
The general solution to Equation 10.43 is thus:
gðS;E;hÞ ¼a1S1þa2S2 ð10:47Þ We will now derive the appropriate solutions pertaining to American call options and American put options.
American call options
Here we use the fact that both the value and the early exercise premium (c(S,E,)¼hgc(S,E,h)) of an American call tend to zero as the asset price S!0. This means that asS!0,gc(S,E,h)!0.
However, since1<0, the only way this can be achieved in Equation 10.47 is if a1¼0. Sogc(S,E,h)¼a2S2, and the value of an American call is:
CðS;E; Þ ¼cðS;E; Þ þha2S2 ð10:48Þ An expression fora2can be found by considering the critical asset price (point on the early exercise boundary),S , above which the American option will be exercised.
ForS<S , the value of the American call is governed by Equation 10.48, and when S>S we haveC(S,E,)¼SE.
Now, since the value of the American option is continuous, at the critical asset valueS the following equation applies:
S E¼cðS ;E; Þ þha2S 2 ð10:49Þ Furthermore, since the gradient of the American option value is also continuous, at S we have:
@ðS EÞ
@S ¼ @
@S fcðS;E; Þ þha2S 2g ð10:50Þ
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which gives:
1¼expðqÞN1ðd1ðSÞÞ þ2ha2S ð21Þ ð10:51Þ where we have used the value of the hedge parameter c, see Section 9.3.3, for a European call
c¼@cðS ;E; Þ
@S ¼expðqÞN1ðd1ðS ÞÞ Equation 10.51 can therefore be written as:
ha2S 2¼S 2
f1expðqÞN1ðd1ðS ÞÞg ð10:52Þ
When the left hand side of the above equation is substituted into Equation 10.49 we obtain the following equation forS :
S E¼cðS ;E; Þ þS 2
f1expðqÞN1ðd1ðS ÞÞg ð10:53Þ
This equation can be solved forS using standard iterative methods (see the section on the numerical solution of critical asset values). OnceS has been found Equation 10.52 gives:
ha2¼A2S 2 where A2¼S 2
1expðqÞN1ðd1ðSÞÞ
f g
From Equation 10.48 the value of an American call is thus of the form:
MacMillan, Barone-Adesi, and Whaley method: American call option CðS;E; Þ ¼cðS;E; Þ þA2
S S
2
whenS<S ð10:54Þ CðS;E; Þ ¼SE whenSS ð10:55Þ
American put options
For an American put option we proceed in a similar manner to that for the American call. We now use fact that both the value and early exercise premium, p(S,E,)¼hgp(S,E,h), of an American put tend to zero as the asset price S! 1. So gp(S,E,h)!0 as S! 1. Since 2>0 the only way this can be achieved by Equation 10.47 is if a2 ¼0. This gives gp(S,E,h)¼a1S1 and the value of an American put is:
PðS;E; Þ ¼pðS;E; Þ þha1S1 ð10:56Þ An expression fora1can be found by considering the critical asset price,S , below which the American option will be exercised. ForS>S the value of the American put is given by Equation 10.56, and forS<S we haveP(S,E,)¼ES.
Continuity of the American option value at the critical asset price gives:
ES ¼pðS ;E; Þ þha1S 1 ð10:57Þ 130 Pricing Assets
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and continuity of the option value’s gradient at the critical asset price yields:
@ðES Þ
@S ¼ @
@S fpðS ;E; Þ þha1S 1g ð10:58Þ which can be simplified to:
1¼ N1ðd1ðS ÞÞexpðqÞ þ1a1S ð11Þ ð10:59Þ where we have used the value of hedge parameterpfor a European put (see section on the greeks):
p¼@pðS ;E; Þ
@S ¼ fN1ðd1ðS ÞÞ 1gexpðqÞ ¼ N1ðd1ðS ÞÞexpðqÞ Equation 10.59 can therefore be written as:
ha1S 1¼ S 1
f1N1ðd1ðS ÞÞexpðqÞg ð10:60Þ
When the left hand side of the above equation is substituted into Equation 10.57 we obtain the following equation forS :
ES ¼pðS ;E; Þ þ f1expðqÞN½d1ðS ÞgS 1
ð10:61Þ which can be solved iteratively to yieldS (see the section on the numerical solution of critical asset values). OnceS has been found Equation 10.60 gives:
ha1¼A1S 1 where A1¼ S 1
f1expðqÞN1ðd1ðS ÞÞg
We note here thatA1>0 since,1 <0,S >0 andN1(d1(S )) exp (q)<1.
From Equation 10.56 the value of an American put is thus:
MacMillan, Barone-Adesi, and Whaley method: American put option PðS;E; Þ ¼pðS;E; Þ þA1
S S 2
whenS>S ð10:62Þ
PðS;E; Þ ¼ES whenSS ð10:63Þ
Numerical solution of critical asset values
We now provide details on how to iteratively solve for the critical asset price in Equations 10.53 and 10.61.
American call options
For American call options we need to solve Equation 10.53, which is:
S E¼cðS ;E; Þ þS 2
f1expðqÞN1ðd1ðS ÞÞg
Numeric methods and single asset American options 131
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We denote theith approximation to the critical asset valueS bySi, and represent the left hand side of the equation by:
LHSðSi;E; Þ ¼Si E
and the right hand side of the equation by:
RHSðSi;E; Þ ¼cðSi;E; Þ þSi 2
f1expðqÞN1ðd1ðSiÞÞg
If we let K(Si,E,)¼RHS(Si, E,)LHS(Si, E,) then we want to find the value of Si which (to a specified tolerance) gives K(Si,E,)0. This can be achieved with Newton’s root finding method, in which a better approximation, Siþ1, can be found using:
Siþ1¼Si KðSi;E; Þ
K0ðSi;E; Þ ð10:64Þ
where
K0ðSi;E; Þ ¼ @
@Si fRHSðSi;E; Þ LHSðSi;E; Þg
¼ @
@Si fRHSðSi;E; Þg @
@Si fLHSðSi;E; Þg
¼bi1
Here we have usedbi¼(@=@Si)fRHS(Si,E,)g, and the expression forbiis given by Equation 10.66, which is derived at the end of this section.
Substituting for K(Si,E,) and K0(Si,E,) into Equation 10.64 we therefore obtain:
Siþ1¼Si ðRHSðSi;E; Þ LHSðSi;E; ÞÞ ðbi1Þ
¼Si ðRHSðSi;E; Þ ðSi EÞÞ ðbi1Þ
¼biSi RHSðSi;E; Þ E ðbi1Þ
The final iterative algorithm for the American call is therefore:
Siþ1¼EþRHSðSi;E; Þ biSi
ð1biÞ ð10:65Þ
where we can use S0¼E for the initial estimate of the critical value, see computer Code excerpt 10.4.
The expression for biin an American call
Here we derive an expression for the termbiwhich is used in Equation 10.65.
bi¼@cðSi;E; Þ
@Si þ 1 2
1expðqÞN1ðd1ðSiÞÞ Si
2
@N1ðd1ðSiÞÞ
@d1ðSiÞ
@d1ðSiÞ
@Si 132 Pricing Assets