Post on 18-Apr-2020
CHAPTER 4
BSM Formulas for ML-payoff
Functions
After the success of BSM formulas for plain vanilla, several other types
of option pricing formulas were derived with different payoff functions (see
[19]). In [43], Paul Wilmott has discussed BSM formuas for log payoff func-
tion max{ln(STK ), 0}. In this chapter, we shall discuss the modified log payoff
(ML-payoff) function max{ST ln(STK ), 0}. Also we shall discuss its Geek let-
ters. Then we shall discuss BSM formulas in C++ programme and using
mathematica. Finally, we compare three options; namely, plain vanilla option,
log option, and modified log option using tables and graphs.
The results in this chapter are published in [9] by Prof. H. V. Dedania
and Mr. S. J. Ghevariya.
4.1. Deduction of BSM Formulas and Greek Letters
Our search for a payoff function closely related with Wilmott’s log payoff
function, eliminating some of its deficiencies, and that is closer to the cele-
brated plain vanilla payoff function leads to the modified log payoff function
stated above. Notice that it is closely related with the entropy function x log x
in Information Theory. In this section, we derive the BSM formulas for call
and put options for the ML-payoff functions. We start with listing the option
pricing formulas for log payoff functions [19, p.121].
89
1. Deduction of BSM Formulas and Greek Letters 90
Theorem 4.1.1. [19, p.121] The call option pricing formula for the log payoff
function C(S, T ) = max{ln(STK ), 0} is
C(S, t) = e−r(T−t)[ln(S
K)N(d) + (r − 1
2σ2)(T − t)N(d) + σ
√T − t η(d)],
where
d =ln( SK ) + (r − 1
2σ2)(T − t)
σ√T − t
and η(d) =1√2πe−
d2
2 . (4.1.1.1)
Theorem 4.1.2. The put option pricing formula for the log payoff function
P (S, T ) = max{ln( KST ), 0} is
P (S, t) = e−r(T−t)[σ√T − t η(d)− ln(
S
K)N(−d)− (r − σ2
2)(T − t)N(−d)],
where d and η(d) are given by Equation (4.1.1.1).
Theorem 4.1.3. [9] The call option pricing formula for the ML-payoff func-
tion max{ST ln(STK ), 0} is
C(S, t) = S[ln(S
K)N(d) + σ
√T − t η(d) + (r +
1
2σ2)(T − t)N(d)]
where
d =ln( SK ) + (r + 1
2σ2)(T − t)
σ√T − t
and η(d) =1√2π
e−d2
2 .
Proof. The BSM partial differential equation along with the boundary condi-
tions for a European call option C(S, t) is as follow [12, p.76]:
∂C
∂t+
1
2σ2S2∂
2C
∂S2+ rS
∂C
∂S− rC = 0 (4.1.3.1)
with C(0, t) = 0, C(S, t)→ S when S →∞ and C(S, T ) = max{ST ln(STK ), 0},where K is the striking price. Take S = Kex, t = T− τ
12σ2 , C(S, t) = Kv(x, τ).
These imply
∂C
∂t= K
(∂v
∂x
∂x
∂t+∂v
∂τ
∂τ
∂t
)= −Kσ
2
2
∂v
∂τ
1. Deduction of BSM Formulas and Greek Letters 91
∂C
∂S= K
(∂v
∂x
∂x
∂S+∂v
∂τ
∂τ
∂S
)=
1
ex∂v
∂x
∂2C
∂S2=
1
Ke2x
(∂2v
∂x2− ∂v
∂x
).
Substituting above values in Equation (4.1.3.1), we get
−K2σ2 ∂v
∂τ+
1
2σ2K2e2x 1
Ke2x
(∂2v
∂x2− ∂v
∂x
)+ rKex
1
ex∂v
∂x− rKv = 0
⇒ −1
2σ2 ∂v
∂τ+
1
2σ2
(∂2v
∂x2− ∂v
∂x
)+ r
∂v
∂x− rv = 0
⇒ −1
2σ2
(∂v
∂τ− ∂2v
∂x2+∂v
∂x
)+ r
(∂v
∂x− v)
= 0
⇒ ∂v
∂τ− ∂2v
∂x2+∂v
∂x=
r12σ
2
(∂v
∂x− v)
⇒ ∂v
∂τ=∂2v
∂x2− ∂v
∂x+ k
(∂v
∂x− v), where k =
r12σ
2
⇒ ∂v
∂τ=∂2v
∂x2+ (k − 1)
∂v
∂x− kv. (4.1.3.2)
Again by taking
v(x, τ) = eαx+βτu(x, τ), (4.1.3.3)
where α and β are constants, which are to be determined so that Equation
(4.1.3.2) becomes a heat equation. Now, from Equation (4.1.3.3),
∂v
∂τ= βeαx+βτu(x, τ) + eαx+βτ ∂u
∂τ,
∂v
∂x= αeαx+βτu(x, τ) + eαx+βτ ∂u
∂x,
and
∂2v
∂x2= α
(αeαx+βτu(x, τ) + eαx+βτ ∂u
∂x
)+ αeαx+βτ ∂u
∂x+ eαx+βτ ∂
2u
∂x2.
1. Deduction of BSM Formulas and Greek Letters 92
Substituting above values in Equation (4.1.3.2), we get
eαx+βτ
(βu+
∂u
∂τ
)= eαx+βτ
(α2u+ 2α
∂u
∂x+∂2u
∂x2
)+(k − 1)eαx+βτ
(αu+
∂u
∂x
)− keαx+βτu.
This gives
∂u
∂τ=∂2u
∂x2+ (2α + (k − 1))
∂u
∂x+ (α2 − β + (k − 1)α− k)u.
To make Equation (4.1.3.2) a heat equation, we must have
α2 − β + (k − 1)α− k = 0 and 2α + (k − 1) = 0
which gives
α = −1
2(k − 1) and β = −1
4(k + 1)2.
Hence Equation (4.1.3.3) becomes
v(x, τ) = e−12(k−1)x− 1
4(k+1)2τu(x, τ). (4.1.3.4)
This, in turn, gives
∂u
∂τ=∂2u
∂x2(τ > 0, −∞ < x <∞). (4.1.3.5)
Now we have
C(S, T ) = max{ST ln(STK
), 0}
⇒ Kv(x, 0) = max{Kex ln(Kex
K), 0} (∵ t = T ⇒ τ = 0)
⇒ Kv(x, 0) = max{Kxex, 0}
⇒ v(x, 0) = max{xex, 0}.
Hence the initial condition
u0(x) = u(x, 0)
1. Deduction of BSM Formulas and Greek Letters 93
= e−αxv(x, 0)
= e−αx max{xex, 0}
= e12(k−1)x max{xex, 0}
= max{xe12(k+1)x, 0}
=
xe12(k+1)x if x > 0
0 if x ≤ 0.(4.1.3.6)
The solution of Equation (4.1.3.5) is
u(x, τ) =1
2√πτ
∞∫−∞
u0(s)e−(s−x)2
4τ ds, (4.1.3.7)
where u0(x) is given by the Equation (4.1.3.6). Now we have to calculate the
integral in the Equation (4.1.3.7). To change the variable, take
y =s− x√
2τ⇒ dy =
ds√2τ.
So Equation (4.1.3.7) reduces to
u(x, τ) =1
2√πτ
∞∫−∞
u0(x+ y√
2τ)e−y2
2
√2τdy
=1√2π
∞∫−∞
u0(x+ y√
2τ)e−y2
2 dy. (4.1.3.8)
Now, from Equation (4.1.3.6), Equation (4.1.3.8) becomes
u(x, τ) =1√2π
∞∫−x/√
2τ
(x+ y√
2τ)e12(k+1)(x+y
√2τ)e−
y2
2 dy
=xe
12(k+1)x
√2π
∞∫−x/√
2τ
xe12(k+1)y
√2τ− y
2
2 dy
1. Deduction of BSM Formulas and Greek Letters 94
+
√2τe
12(k+1)x
√2π
∞∫−x/√
2τ
ye12(k+1)y
√2τ− y
2
2 dy
= xe12(k+1)x+ 1
4(k+1)2τ 1√
2π
∞∫−x/√
2τ
e−12(y− 1
2(k+1)
√2τ)2
dy
+√
2τ e12(k+1)x+ 1
4(k+1)2τ 1√
2π
∞∫−x/√
2τ
ye−12(y− 1
2(k+1)
√2τ)2
dy
= xe12(k+1)x+ 1
4(k+1)2τI1 +
√2τe
12(k+1)x+ 1
4(k+1)2τI2. (4.1.3.9)
Now
I1 =1√2π
∞∫−x/√
2τ
e−12(y− 1
2(k+1)
√2τ)2
dy
=1√2π
∞∫− x√
2τ− 1
2(k+1)
√2τ
e−12t2dt
(taking t = y − 1
2(k + 1)
√2τ)
=1√2π
x√2τ
+ 12(k+1)
√2τ∫
−∞
e−12t2dt
=1√2π
d∫−∞
e−12t2dt = N(d),
where d = x√2τ
+ 12(k + 1)
√2τ . Also
I2 =1√2π
∞∫−x/√
2τ
ye−12(y− 1
2(k+1)
√2τ)2
dy
1. Deduction of BSM Formulas and Greek Letters 95
=1√2π
∞∫−x/√
2τ
(y − 1
2(k + 1)
√2τ)e−
12(y− 1
2(k+1)
√2τ)2
dy
+1√2π
∞∫−x/√
2τ
1
2(k + 1)
√2τe−
12(y− 1
2(k+1)
√2τ)2
dy
=1
2√
2π
∞∫( x√
2τ+ 1
2(k+1)
√2τ)2
e−ρ2dρ
(ρ =
(y +
1
2(k + 1)
√2τ
)2)
+1
2(k + 1)
√2τN(d)
=1√2πe−
12( x√
2τ+ 1
2(k+1)
√2τ)2
+1
2(k + 1)
√2τN(d)
= η(d) +1
2(k + 1)
√2τN(d), (4.1.3.10)
where η(d) = 1√2πe−
d2
2 . Substituting values of Equations (4.1.3.10) and (4.1.3.10)
in Equation (4.1.3.9), we get
u(x, τ) = e12(k+1)x+ 1
4(k+1)2τ [xN(d) +
√2τ η(d) + (k + 1)τN(d)]. (4.1.3.11)
Substituting value of Equation (4.1.3.11) in Equation (4.1.3.4), we get
v(x, τ) = ex[xN(d) +√
2τ η(d) + (k + 1)τN(d)]. (4.1.3.12)
Also we have
C(S, t) = Kv(x, τ), S = Kex, τ =1
2σ2(T − t), k =
r12σ
2.
Hence Equation (4.1.3.12) becomes
C(S, t) = Kv(x, τ)
= Kex[xN(d) +√
2τ η(d) + (k + 1)τN(d)]
= S[ln(S
K)N(d) + σ
√T − t η(d) + (r +
1
2σ2)(T − t)N(d)] (4.1.3.13)
1. Deduction of BSM Formulas and Greek Letters 96
where
d =ln( SK ) + (r + 1
2σ2)(T − t)
σ√T − t
and η(d) =1√2π
e−d2
2 .
Equation (4.1.3.13) is the BSM formula for the European call option with
modified log payoff function C(S, T ) = max{ST ln(STK ), 0}. �
Next, we state put option pricing formula without proof; the proof is
similar to the proof of Theorem 4.1.3.
Theorem 4.1.4. [9] The put option pricing formula for the ML-payoff func-
tion max{ST ln( KST ), 0} is
P (S, t) = C(S, t)− S[ln(S
K) + (r +
1
2σ2)(T − t)]
where
d =ln( SK ) + (r + 1
2σ2)(T − t)
σ√T − t
and η(d) =1√2π
e−d2
2 .
Next we derive greek letters and deduce the relation of delta and gamma.
∆C =η(d)
σ√T − t
[ln(
S
K) + (r +
3
2σ2)(T − t)− d σ
√T − t
]+
[ln(
S
K) + (r +
1
2σ2)(T − t) + 1
]N(d)
∆P = ∆C −[
ln(S
K) + (r +
1
2σ2)(T − t)
]− 1
ΓC =η(d)
Sσ√T − t
[ln(
S
K) + (r +
1
2σ2)(T − t) + 1
]+N(d)
S
−d η(d)
S
[ln( SK )
σ2(T − t)− d
σ√T − t
+(r + 1
2σ2)
σ2+ 1
]ΓP = ΓC −
1
S
2. C + + and Mathematica 97
ΘC =S(r + 1
2σ2)η(d)
2σ√T − t
[dσ√T − t− ln(
S
K)− (r +
1
2σ2)(T − t)
]− S
2√T − t
[ση(d)− 2(r +
1
2σ2)√T − tN(d)
]ΘP = ΘC + S(r +
1
2σ2)
νC =1
σ2√T − t
[((σ2
2− r)(T − t)− ln(
S
K))(
Sη(d) ln(S
K)− dσ
√T − t
+(r +1
2σ2)(T − t)
)]+ S√T − t
(η(d) + σ
√T − tN(d)
)νP = νC − Sσ(T − t)
ρC =S√T − tη(d)
σ
[ln(
S
K)− dσ
√T − t+ (r +
1
2σ2)(T − t)
]+ S(T − t)N(d)
ρP = ρC − S(T − t)
Theorem 4.1.5. Let ∆C and ∆P (res. ΓC and ΓP ) be the delta (resp.
gamma) for call and put option formulas for ML-payoff functions. Then
∆C = ∆P + [ln(S
K) + (r +
1
2σ2)(T − t)]− 1 and ΓC = ΓP +
1
S.
Remark 4.1.6. The relations among ∆C ,∆P ,ΓC and ΓP for the BSM for-
mulas for log payoff function are as follows:
∆C = ∆P +1
Se−r(T−t) and ΓC = ΓP −
1
S2e−r(T−t).
4.2. C + + and Mathematica
In this section, we write C + + programmes of option pricing formulas
for log payoff and ML-payoff functions. We also discuss about how to use
mathematica for option formulas?
Program-I: C + + program for the log payoff functions (Theorems 4.1.1
and 4.1.2).
2. C + + and Mathematica 98
#include<iostream.h>
#include<conio.h>
#include<math.h>
double CND(double X)
{double L,K,W, a1, a2, a3, a4, a5;
a1 = 0.319381530, a2 = −0.356563782, a3 = 1.781477937, a4 = −1.821255978,
a5 = 1.330274429;
L = fabs(X);
K = 1/(1 + 0.2316419 ∗ L);
W = 1− (1/sqrt(2 ∗M PI) ∗ exp(−L ∗ L/2)) ∗ (a1 ∗K + a2 ∗ pow(K, 2)
+ a3 ∗ pow(K, 3) + a4 ∗ pow(K, 4) + a5 ∗ pow(K, 5));
if(X < 0)
{W = 1−W ;
}return W ;
}void main( )
{clrscr( );
char CallorPut;
double S,K, r, seg, T, t, d, eta, C, P ;
cout<< ”\nS = ”;
cin>> S;
cout<< ”\nK = ”;
cin>> K;
cout<< ”\nr = ”;
cin>> r;
cout<< ”\nseg = ”;
2. C + + and Mathematica 99
cin>> seg;
cout<< ”\nT = ”;
cin>>T;
cout<< ”\nt = ”;
cin>>t;
d = (log(S/K) + (r − pow(seg, 2)/2) ∗ (T − t))/(seg ∗ (sqrt(T − t)));eta = (1/(sqrt(2) ∗ sqrt(M PI))) ∗ exp(−pow(d, 2)/2);
cout<< ”\nCallorPut=c or p\n”;
cin>> CallorPut;
if(CallorPut ==′ c′)
{C = exp(−r ∗ (T − t)) ∗ (log(S/K) ∗ CND(d)
+ (r − pow(seg, 2)/2) ∗ (T − t) ∗ CND(d) + seg ∗ sqrt(T − t) ∗ eta);
cout<< ”\nC = ” << C;
}else if(CallorPut ==′ p′)
{P = exp(−r ∗ (T − t)) ∗ (seg ∗ sqrt(T − t) ∗ eta
− log(S/K) ∗ CND(−d)− (r − pow(seg, 2)/2) ∗ (T − t) ∗ CND(−d));
cout<< ”\nP = ” << P ;
}getch( );
}
Program-II: C + + program for the ML-payoff functions (Theorems 4.1.3
and 4.1.4).
#include<iostream.h>
#include<conio.h>
#include<math.h>
double CND(double X)
2. C + + and Mathematica 100
{double L,K,W, a1, a2, a3, a4, a5;
a1 = 0.319381530, a2 = −0.356563782, a3 = 1.781477937, a4 = −1.821255978,
a5 = 1.330274429;
L = fabs(X);
K = 1/(1 + 0.2316419 ∗ L);
W = 1− (1/sqrt(2 ∗M PI) ∗ exp(−L ∗ L/2)) ∗ (a1 ∗K + a2 ∗ pow(K, 2)
+ a3 ∗ pow(K, 3) + a4 ∗ pow(K, 4) + a5 ∗ pow(K, 5));
if(X < 0)
{W = 1−W ;
}return W ;
}void main( )
{clrscr( );
char CallorPut;
double S,K, r, seg, T, t, d, eta, C, P ;
cout<< ”\nS = ”;
cin>> S;
cout<< ”\nK = ”;
cin>> K;
cout<< ”\nr = ”;
cin>> r;
cout<< ”\nseg = ”;
cin>> seg;
cout<< ”\nT = ”;
cin>>T;
cout<< ”\nt = ”;
2. C + + and Mathematica 101
cin>>t;
d = (log(S/K) + (r + pow(seg, 2)/2) ∗ (T − t))/(seg ∗ (sqrt(T − t)));eta = (1/(sqrt(2) ∗ sqrt(M PI))) ∗ exp(−pow(d, 2)/2);
cout<< ”\nCallorPut=c or p\n”;
cin>> CallorPut;
if(CallorPut ==′ c′)
{C = S ∗ (log(S/K)∗CND(d)+(r+pow(seg, 2)/2)∗ (T − t)∗CND(d)+seg∗
sqrt(T − t) ∗ eta);
cout<< ”\nC = ” << C;
}else if(CallorPut ==′ p′)
{P = S ∗(seg∗sqrt(T − t)∗eta− log(S/K)∗CND(−d)−(r+pow(seg, 2)/2)∗
(T − t) ∗ CND(−d)); cout<< ”\nP = ” << P ;
}getch( );
}
Program-I: Mathematica program for the log payoff functions (Theorems 4.1.1
and 4.1.2).
d = (Log[S/K] + (r − (1/2)seg∧2)(T − t))/(seg Sqrt[T − t])C = Exp[−r(T − t)](Log[S/K]CDF[NormalDistribution[0, 1], d]
+ seg Sqrt[T − t]PDF[NormalDistribution[0, 1], d]
+ (r − (1/2)seg∧2)(T − t)CDF[NormalDistribution[0, 1], d])
P = Exp[−r(T − t)](seg Sqrt[T − t]PDF[NormalDistribution[0, 1], d]
− Log[S/K]CDF[NormalDistribution[0, 1], d]
− (r − (1/2)seg∧2)(T − t)CDF[NormalDistribution[0, 1], d])
3. Comparisons of Three BSM Formulas 102
Program-II: Mathematica program for the ML-payoff functions (Theorems 4.1.3
and 4.1.4).
d = (Log[S/K] + (r + (1/2)seg∧2)(T − t))/(seg Sqrt[T − t])C = S(Log[S/K]CDF[NormalDistribution[0, 1], d]+(r+(1/2)seg∧2)(T−t)CDF
[NormalDistribution[0, 1], d]+seg Sqrt[T−t]PDF[NormalDistribution[0, 1], d])
P = S(seg Sqrt[T − t]PDF[NormalDistribution[0, 1], d]− Log[S/K]CDF
[NormalDistribution[0, 1],−d]− (r + (1/2)seg∧2)(T − t)CDF[NormalDistribution[0, 1],−d])
Finally, using above programmes, we exhibit a few examples to find option
values for log payoff and ML-payoff functions.
Table - 4.1
S K r σ T t log call ML call log put ML put
42 40 0.1 0.2 0.5 0 0.1061 5.3227 0.0217 0.7535
50 55 0.1 0.2 0.5 0 0.0314 2.0257 0.0840 3.7912
50 50 0.08 0.25 0.5 0 0.0801 5.0899 0.0567 2.3087
50 55 0.1 0.3 1.0 0 0.0910 7.3083 0.1275 4.8238
45 42 0.1 0.4 1.0 0.5 0.1491 9.4369 0.0739 2.2821
4.3. Comparisons of Three BSM Formulas
On almost all stock exchanges, the prices and settlement of various op-
tions are as per the plain vanilla payoff functions. This is in the center of all
other types of options. The options with different payoff functions are normally
used in OTC market. So it is very much logical to compare any option formula
with the plain vanilla. Therefore, in this section, we compare Paul Wilmott’s
formula for log payoff function and our formula for ML-payoff function with
the formula for the plain vanilla payoff function. We list the following nota-
tions and formulas for three payoff functions; namely, plain vanilla option, log
option, and modified log option which will be required in this section later.
3. Comparisons of Three BSM Formulas 103
POC1= C1(S, T ) = max{ST −K, 0}
POC2= C2(S, T ) = max{ln(
STK
), 0}
POC3= C3(S, T ) = max{ST ln(
STK
), 0}
POP1= P1(S, T ) = max{K − ST , 0}
POP2= P2(S, T ) = max{ln(
K
ST), 0}
POP3= P3(S, T ) = max{ST ln(
K
ST), 0}
C1 = SN(d1)−Ke−r(T−t)N(d2)
C2 = e−r(T−t)η(d2)σ√T − t+ e−r(T−t)[ ln(
S
K) + (r − 1
2σ2)(T − t)]N(d2)
C3 = S[ ln(S
K)N(d1) + σ
√T − t η(d1) + (r +
1
2σ2)(T − t)N(d1)]
P1 = C1 − S +Ke−r(T−t)
P2 = C2 − e−r(T−t)[ ln(S
K) + (r − 1
2σ2)(T − t)]
P3 = C3 − S[ ln(S
K) + (r − 1
2σ2)(T − t)],
where
d1 =ln( SK ) + (r + 1
2σ2)(T − t)
σ√T − t
, d2 = d1 − σ√T − t, and η(x) =
1√2πe−
x2
2 .
Next we generate data using the option pricing formulas for the three
payoff functions mentioned above. These data will be useful to compare these
formulas. In the following tables, we fix the current asset price S0 = 100, the
maturity time T = 0.5 and the risk free interest rate r = 0.08; using these,
we compute the call/put option values as well as payoff for different striking
prices K and different volatilities σ.
3. Comparisons of Three BSM Formulas 104
Table - 01 : K = 80, σ = 0.25
C1 C2 C3 ST POC1 POC2 POC3 POC1% POC2% POC3%
23.602 0.244 28.311
80 0 0 0 0 0 0
90 10 0.118 10.602 42.37 48.26 37.45
100 20 0.223 22.310 84.74 91.43 78.80
110 30 0.318 35.035 127.11 130.53 123.75
120 40 0.405 48.660 169.48 166.19 171.88
Table - 02 : K = 90, σ = 0.25
C1 C2 C3 ST POC1 POC2 POC3 POC1% POC2% POC3%
15.424 0.148 17.839
80 0 0 0 0 0 0
90 0 0 0 0 0 0
100 10 0.105 10.540 64.83 71.41 59.08
110 20 0.201 22.077 129.67 135.98 123.75
120 30 0.288 34.524 194.50 194.92 193.53
Table - 03 : K = 100, σ = 0.25
C1 C2 C3 ST POC1 POC2 POC3 POC1% POC2% POC3%
9.041 0.080 10.180
80 0 0 0 0 0 0
90 0 0 0 0 0 0
100 0 0 0 0 0 0
110 10 0.095 10.484 110.60 118.98 102.99
120 20 0.182 21.879 221.21 227.59 214.92
Table - 04 : K = 110, σ = 0.25
C1 C2 C3 ST POC1 POC2 POC3 POC1% POC2% POC3%
4.751 0.039 5.245
80 0 0 0 0 0 0
90 0 0 0 0 0 0
100 0 0 0 0 0 0
110 0 0 0 0 0 0
120 10 0.087 10.440 210.49 222.51 199.04
3. Comparisons of Three BSM Formulas 105
Table - 05 : K = 120, σ = 0.25
C1 C2 C3 ST POC1 POC2 POC3 POC1% POC2% POC3%
2.256 0.017 2.455
80 0 0 0 0 0 0
90 0 0 0 0 0 0
100 0 0 0 0 0 0
110 0 0 0 0 0 0
120 0 0 0 0 0 0
Table - 06 : K = 80, σ = 0.30
C1 C2 C3 ST POC1 POC2 POC3 POC1% POC2% POC3%
24.093 0.244 29.438
80 0 0 0 0 0 0
90 10 0.118 10.602 41.51 48.22 36.01
100 20 0.223 22.310 83.01 91.32 75.78
110 30 0.318 35.035 124.52 130.37 119.01
120 40 0.405 48.660 166.02 165.98 165.29
Table - 07 : K = 90, σ = 0.30
C1 C2 C3 ST POC1 POC2 POC3 POC1% POC2% POC3%
16.409 0.154 19.375
80 0 0 0 0 0 0
90 0 0 0 0 0 0
100 10 0.105 10.540 60.94 68.62 54.40
110 20 0.201 22.077 121.88 130.66 113.94
120 30 0.288 34.524 182.82 187.30 178.18
Table - 08 : K = 100, σ = 0.30
C1 C2 C3 ST POC1 POC2 POC3 POC1% POC2% POC3%
10.388 0.090 11.952
80 0 0 0 0 0 0
90 0 0 0 0 0 0
100 0 0 0 0 0 0
110 10 0.095 10.484 96.26 105.89 87.71
120 20 0.182 21.879 192.53 202.56 183.05
3. Comparisons of Three BSM Formulas 106
Table - 09 : K = 110, σ = 0.30
C1 C2 C3 ST POC1 POC2 POC3 POC1% POC2% POC3%
6.136 0.049 6.923
80 0 0 0 0 0 0
90 0 0 0 0 0 0
100 0 0 0 0 0 0
110 0 0 0 0 0 0
120 10 0.087 10.440 162.97 176.47 150.79
Table - 10 : K = 120, σ = 0.30
C1 C2 C3 ST POC1 POC2 POC3 POC1% POC2% POC3%
3.407 0.025 3.787
80 0 0 0 0 0 0
90 0 0 0 0 0 0
100 0 0 0 0 0 0
110 0 0 0 0 0 0
120 0 0 0 0 0 0
Table - 11 : K = 80, σ = 0.35
C1 C2 C3 ST POC1 POC2 POC3 POC1% POC2% POC3%
24.729 0.246 30.803
80 0 0 0 0 0 0
90 10 0.118 10.602 40.44 47.96 34.42
100 20 0.223 22.310 80.88 90.84 72.43
110 30 0.318 35.035 121.31 129.68 113.74
120 40 0.405 48.660 161.75 165.11 157.97
Table - 12 : K = 90, σ = 0.35
C1 C2 C3 ST POC1 POC2 POC3 POC1% POC2% POC3%
17.475 0.160 21.069
80 0 0 0 0 0 0
90 0 0 0 0 0 0
100 10 0.105 10.540 57.22 65.87 50.03
110 20 0.201 22.077 114.45 125.44 104.79
120 30 0.288 34.524 171.67 179.81 163.86
3. Comparisons of Three BSM Formulas 107
Table - 13 : K = 100, σ = 0.35
C1 C2 C3 ST POC1 POC2 POC3 POC1% POC2% POC3%
11.741 0.099 13.804
80 0 0 0 0 0 0
90 0 0 0 0 0 0
100 0 0 0 0 0 0
110 10 0.095 10.484 85.17 95.87 75.95
120 20 0.182 21.879 170.35 183.40 158.50
Table - 14 : K = 110, σ = 0.35
C1 C2 C3 ST POC1 POC2 POC3 POC1% POC2% POC3%
7.535 0.059 8.688
80 0 0 0 0 0 0
90 0 0 0 0 0 0
100 0 0 0 0 0 0
110 0 0 0 0 0 0
120 10 0.087 10.440 132.71 146.96 120.16
Table - 15 : K = 120, σ = 0.35
C1 C2 C3 ST POC1 POC2 POC3 POC1% POC2% POC3%
4.648 0.034 5.277
80 0 0 0 0 0 0
90 0 0 0 0 0 0
100 0 0 0 0 0 0
110 0 0 0 0 0 0
120 0 0 0 0 0 0
Table - 16 : K = 80, σ = 0.40
C1 C2 C3 ST POC1 POC2 POC3 POC1% POC2% POC3%
25.477 0.248 32.360
80 0 0 0 0 0 0
90 10 0.118 10.602 39.25 47.56 32.76
100 20 0.223 22.310 78.50 90.07 68.94
110 30 0.318 35.035 117.75 128.58 108.27
120 40 0.405 48.660 157.00 163.71 150.37
3. Comparisons of Three BSM Formulas 108
Table - 17 : K = 90, σ = 0.40
C1 C2 C3 ST POC1 POC2 POC3 POC1% POC2% POC3%
18.593 0.166 22.892
80 0 0 0 0 0 0
90 0 0 0 0 0 0
100 10 0.105 10.540 53.78 63.30 46.04
110 20 0.201 22.077 107.57 120.54 96.44
120 30 0.288 34.524 161.35 172.79 150.81
Table - 18 : K = 100, σ = 0.40
C1 C2 C3 ST POC1 POC2 POC3 POC1% POC2% POC3%
13.096 0.108 15.732
80 0 0 0 0 0 0
90 0 0 0 0 0 0
100 0 0 0 0 0 0
110 10 0.095 10.484 76.36 87.91 66.64
120 20 0.182 21.879 152.72 168.17 139.07
Table - 19 : K = 110, σ = 0.40
C1 C2 C3 ST POC1 POC2 POC3 POC1% POC2% POC3%
8.942 0.069 8.688
80 0 0 0 0 0 0
90 0 0 0 0 0 0
100 0 0 0 0 0 0
110 0 0 0 0 0 0
120 10 0.087 10.440 111.84 126.64 120.16
Table - 20 : K = 120, σ = 0.40
C1 C2 C3 ST POC1 POC2 POC3 POC1% POC2% POC3%
5.947 0.043 6.898
80 0 0 0 0 0 0
90 0 0 0 0 0 0
100 0 0 0 0 0 0
110 0 0 0 0 0 0
120 0 0 0 0 0 0
3. Comparisons of Three BSM Formulas 109
Table - 21 : K = 80, σ = 0.45
C1 C2 C3 ST POC1 POC2 POC3 POC1% POC2% POC3%
26.310 0.250 34.099
80 0 0 0 0 0 0
90 10 0.118 10.602 38.01 47.06 31.09
100 20 0.223 22.310 76.02 89.13 65.42
110 30 0.318 35.035 114.03 127.25 102.75
120 40 0.405 48.660 152.03 162.01 142.70
Table - 22 : K = 90, σ = 0.45
C1 C2 C3 ST POC1 POC2 POC3 POC1% POC2% POC3%
19.746 0.173 24.828
80 0 0 0 0 0 0
90 0 0 0 0 0 0
100 10 0.105 10.540 50.64 60.99 42.45
110 20 0.201 22.077 101.28 116.15 88.92
120 30 0.288 34.524 151.93 166.49 139.05
Table - 23 : K = 100, σ = 0.45
C1 C2 C3 ST POC1 POC2 POC3 POC1% POC2% POC3%
14.451 0.117 17.737
80 0 0 0 0 0 0
90 0 0 0 0 0 0
100 0 0 0 0 0 0
110 10 0.095 10.484 69.20 81.45 59.11
120 20 0.182 21.879 138.40 155..81 123.35
Table - 24 : K = 110, σ = 0.45
C1 C2 C3 ST POC1 POC2 POC3 POC1% POC2% POC3%
10.351 0.078 12.461
80 0 0 0 0 0 0
90 0 0 0 0 0 0
100 0 0 0 0 0 0
110 0 0 0 0 0 0
120 10 0.087 10.440 96.61 111.83 83.78
3. Comparisons of Three BSM Formulas 110
Table - 25 : K = 120, σ = 0.45
C1 C2 C3 ST POC1 POC2 POC3 POC1% POC2% POC3%
7.285 0.051 8.633
80 0 0 0 0 0 0
90 0 0 0 0 0 0
100 0 0 0 0 0 0
110 0 0 0 0 0 0
120 0 0 0 0 0 0
Table - 26 : K = 80, σ = 0.50
C1 C2 C3 ST POC1 POC2 POC3 POC1% POC2% POC3%
27.206 0.253 35.980
80 0 0 0 0 0 0
90 10 0.118 10.602 36.76 46.52 29.47
100 20 0.223 22.310 73.51 88.11 62.01
110 30 0.318 35.035 110.27 125.79 97.37
120 40 0.405 48.660 147.03 160..15 135.24
Table - 27 : K = 90, σ = 0.50
C1 C2 C3 ST POC1 POC2 POC3 POC1% POC2% POC3%
20.924 0.179 26.867
80 0 0 0 0 0 0
90 0 0 0 0 0 0
100 10 0.105 10.540 47.79 58.88 39.23
110 20 0.201 22.077 95.59 112.12 82.17
120 30 0.288 34.524 143.38 160.73 128.50
Table - 28 : K = 100, σ = 0.50
C1 C2 C3 ST POC1 POC2 POC3 POC1% POC2% POC3%
15.804 0.125 19.818
80 0 0 0 0 0 0
90 0 0 0 0 0 0
100 0 0 0 0 0 0
110 10 0.095 10.484 63.27 76.24 52.90
120 20 0.182 21.879 126.55 145.84 110.39
3. Comparisons of Three BSM Formulas 111
Table - 29 : K = 110, σ = 0.50
C1 C2 C3 ST POC1 POC2 POC3 POC1% POC2% POC3%
11.761 0.086 14.467
80 0 0 0 0 0 0
90 0 0 0 0 0 0
100 0 0 0 0 0 0
110 0 0 0 0 0 0
120 10 0.087 10.440 85.02 100.69 72.16
Table - 30 : K = 120, σ = 0.50
C1 C2 C3 ST POC1 POC2 POC3 POC1% POC2% POC3%
8.650 0.059 10.472
80 0 0 0 0 0 0
90 0 0 0 0 0 0
100 0 0 0 0 0 0
110 0 0 0 0 0 0
120 0 0 0 0 0 0
Table - 31 : K = 80, σ = 0.25
P1 P2 P3 ST POP1 POP2 POP3 POP1% POP2% POP3%
0.465 0.006 0.043
80 0 0 0 0 0 0
90 0 0 0 0 0 0
100 0 0 0 0 0 0
110 0 0 0 0 0 0
120 0 0 0 0 0 0
Table - 32 : K = 90, σ = 0.25
P1 P2 P3 ST POP1 POP2 POP3 POP1% POP2% POP3%
1.895 0.023 1.741
80 10 0.118 9.423 527.65 514.41 541.25
90 0 0 0 0 0 0
100 0 0 0 0 0 0
110 0 0 0 0 0 0
120 0 0 0 0 0 0
3. Comparisons of Three BSM Formulas 112
Table - 33 : K = 100, σ = 0.25
P1 P2 P3 ST POP1 POP2 POP3 POP1% POP2% POP3%
5.120 0.057 4.617
80 20 0.223 17.851 390.62 393.47 386.61
90 10 0.105 9.482 195.31 185.89 205.36
100 0 0 0 0 0 0
110 0 0 0 0 0 0
120 0 0 0 0 0 0
Table - 34 : K = 110, σ = 0.25
P1 P2 P3 ST POP1 POP2 POP3 POP1% POP2% POP3%
10.437 0.107 9.214
80 30 0.318 25.476 287.43 297.11 276.51
90 20 0.201 18.060 191.62 187.22 196.02
100 10 0.095 9.531 95.81 88.90 103.43
110 0 0 0 0 0 0
120 0 0 0 0 0 0
Table - 35 : K = 120, σ = 0.25
P1 P2 P3 ST POP1 POP2 POP3 POP1% POP2% POP3%
17.551 0.169 15.125
80 40 0.405 32.437 227.91 239.94 214.46
90 30 0.288 25.891 170.93 170.24 171.19
100 20 0.182 18.232 113.95 107.87 120.53
110 10 0.087 9.571 56.98 51.48 63.28
120 0 0 0 0 0 0
Table - 36 : K = 80, σ = 0.30
P1 P2 P3 ST POP1 POP2 POP3 POP1% POP2% POP3%
0.956 0.013 0.874
80 0 0 0 0 0 0
90 0 0 0 0 0 0
100 0 0 0 0 0 0
110 0 0 0 0 0 0
120 0 0 0 0 0 0
3. Comparisons of Three BSM Formulas 113
Table - 37 : K = 90, σ = 0.30
P1 P2 P3 ST POP1 POP2 POP3 POP1% POP2% POP3%
2.880 0.035 2.589
80 10 0.118 9.423 347.16 331.83 363.91
90 0 0 0 0 0 0
100 0 0 0 0 0 0
110 0 0 0 0 0 0
120 0 0 0 0 0 0
Table - 38 : K = 100, σ = 0.30
P1 P2 P3 ST POP1 POP2 POP3 POP1% POP2% POP3%
6.467 0.073 5.702
80 20 0.223 17.851 309.26 304.78 313.05
90 10 0.105 9.482 154.63 143.99 166.28
100 0 0 0 0 0 0
110 0 0 0 0 0 0
120 0 0 0 0 0 0
Table - 39 : K = 110, σ = 0.30
P1 P2 P3 ST POP1 POP2 POP3 POP1% POP2% POP3%
11.823 0.124 10.204
80 30 0.318 25.476 253.74 256.65 249.66
90 20 0.201 18.060 169.16 161.72 176.99
100 10 0.095 9.531 84.58 76.79 93.39
110 0 0 0 0 0 0
120 0 0 0 0 0 0
Table - 40 : K = 120, σ = 0.30
P1 P2 P3 ST POP1 POP2 POP3 POP1% POP2% POP3%
18.702 0.184 15.769
80 40 0.405 32.437 213.88 220.50 205.70
90 30 0.288 25.891 160.41 156.44 164.19
100 20 0.182 18.232 106.94 99.13 115.60
110 10 0.087 9.571 53.47 47.31 60.70
120 0 0 0 0 0 0
3. Comparisons of Three BSM Formulas 114
Table - 41 : K = 80, σ = 0.35
P1 P2 P3 ST POP1 POP2 POP3 POP1% POP2% POP3%
1.592 0.022 1.426
80 0 0 0 0 0 0
90 0 0 0 0 0 0
100 0 0 0 0 0 0
110 0 0 0 0 0 0
120 0 0 0 0 0 0
Table - 42 : K = 90, σ = 0.35
P1 P2 P3 ST PO1 PO2 PO3 PO1% PO2% PO3%
3.946 0.050 3.470
80 10 0.118 9.423 253.41 236.55 271.54
90 0 0 0 0 0 0
100 0 0 0 0 0 0
110 0 0 0 0 0 0
120 0 0 0 0 0 0
Table - 43 : K = 100, σ = 0.35
P1 P2 P3 ST POP1 POP2 POP3 POP1% POP2% POP3%
7.820 0.090 6.741
80 20 0.223 17.851 255.76 246.79 264.80
90 10 0.105 9.482 127.88 116.59 140.66
100 0 0 0 0 0 0
110 0 0 0 0 0 0
120 0 0 0 0 0 0
Table - 44 : K = 110, σ = 0.35
P1 P2 P3 ST POP1 POP2 POP3 POP1% POP2% POP3%
13.222 0.142 11.157
80 30 0.318 25.476 226.89 224.61 228.35
90 20 0.201 18.060 151.26 141.54 161.88
100 10 0.095 9.531 75.63 67.21 85.42
110 0 0 0 0 0 0
120 0 0 0 0 0 0
3. Comparisons of Three BSM Formulas 115
Table - 45 : K = 120, σ = 0.35
P1 P2 P3 ST POP1 POP2 POP3 POP1% POP2% POP3%
19.943 0.200 16.447
80 40 0.405 32.437 200.58 202.55 197.22
90 30 0.288 25.891 150.43 143.71 157.42
100 20 0.182 18.232 100.29 91.06 110.84
110 10 0.087 9.571 50.14 43.46 58.19
120 0 0 0 0 0 0
Table - 46 : K = 80, σ = 0.40
P1 P2 P3 ST POP1 POP2 POP3 POP1% POP2% POP3%
2.340 0.033 2.052
80 0 0 0 0 0 0
90 0 0 0 0 0 0
100 0 0 0 0 0 0
110 0 0 0 0 0 0
120 0 0 0 0 0 0
Table - 47 : K = 90, σ = 0.40
P1 P2 P3 ST POP1 POP2 POP3 POP1% POP2% POP3%
5.064 0.065 4.356
80 10 0.118 9.423 197.47 180.67 216.33
90 0 0 0 0 0 0
100 0 0 0 0 0 0
110 0 0 0 0 0 0
120 0 0 0 0 0 0
Table - 48 : K = 100, σ = 0.40
P1 P2 P3 ST POP1 POP2 POP3 POP1% POP2% POP3%
9.175 0.108 7.732
80 20 0.223 17.851 217.99 205.81 230.87
90 10 0.105 9.482 109.00 97.23 122.63
100 0 0 0 0 0 0
110 0 0 0 0 0 0
120 0 0 0 0 0 0
3. Comparisons of Three BSM Formulas 116
Table - 49 : K = 110, σ = 0.40
P1 P2 P3 ST POP1 POP2 POP3 POP1% POP2% POP3%
14.628 0.160 12.066
80 30 0.318 25.476 205.08 198.69 211.14
90 20 0.201 18.060 136.72 149.68 149.68
100 10 0.095 9.531 68.36 59.45 78.98
110 0 0 0 0 0 0
120 0 0 0 0 0 0
Table - 50 : K = 120, σ = 0.40
P1 P2 P3 ST POP1 POP2 POP3 POP1% POP2% POP3%
21.242 0.218 17.130
80 40 0.405 32.437 188.31 186.18 189.35
90 30 0.288 25.891 141.23 132.09 151.14
100 20 0.182 18.232 94.15 83.70 106.42
110 10 0.087 9.571 47.08 39.94 55.87
120 0 0 0 0 0 0
Table - 51 : K = 80, σ = 0.45
P1 P2 P3 ST POP1 POP2 POP3 POP1% POP2% POP3%
3.173 0.046 2.722
80 0 0 0 0 0 0
90 0 0 0 0 0 0
100 0 0 0 0 0 0
110 0 0 0 0 0 0
120 0 0 0 0 0 0
Table - 52 : K = 90, σ = 0.45
P1 P2 P3 ST POP1 POP2 POP3 POP1% POP2% POP3%
6.217 0.082 5.230
80 10 0.118 9.423 160.84 144.01 180.18
90 0 0 0 0 0 0
100 0 0 0 0 0 0
110 0 0 0 0 0 0
120 0 0 0 0 0 0
3. Comparisons of Three BSM Formulas 117
Table - 53 : K = 100, σ = 0.45
P1 P2 P3 ST POP1 POP2 POP3 POP1% POP2% POP3%
10.530 0.127 8.674
80 20 0.223 17.851 189.94 175.53 205.79
90 10 0.105 9.482 94.97 82.93 109.31
100 0 0 0 0 0 0
110 0 0 0 0 0 0
120 0 0 0 0 0 0
Table - 54 : K = 110, σ = 0.45
P1 P2 P3 ST POP1 POP2 POP3 POP1% POP2% POP3%
16.038 0.180 12.930
80 30 0.318 25.476 187.05 177.44 197.03
90 20 0.201 18.060 124.70 111.81 139.68
100 10 0.095 9.531 62.35 53.09 73.70
110 0 0 0 0 0 0
120 0 0 0 0 0 0
Table - 55 : K = 120, σ = 0.45
P1 P2 P3 ST POP1 POP2 POP3 POP1% POP2% POP3%
22.580 0.236 17.803
80 40 0.405 32.437 177.15 171.53 182.20
90 30 0.288 25.891 132.86 121.70 145.44
100 20 0.182 18.232 88.57 77.11 102.40
110 10 0.087 9.571 44.29 36.80 53.76
120 0 0 0 0 0 0
Table - 56 : K = 80, σ = 0.50
P1 P2 P3 ST POP1 POP2 POP3 POP1% POP2% POP3%
4.069 0.0604 3.416
80 0 0 0 0 0 0
90 0 0 0 0 0 0
100 0 0 0 0 0 0
110 0 0 0 0 0 0
120 0 0 0 0 0 0
3. Comparisons of Three BSM Formulas 118
Table - 57 : K = 90, σ = 0.50
P1 P2 P3 ST POP1 POP2 POP3 POP1% POP2% POP3%
7.395 0.099 6.081
80 10 0.118 9.423 135.23 118.51 154.94
90 0 0 0 0 0 0
100 0 0 0 0 0 0
110 0 0 0 0 0 0
120 0 0 0 0 0 0
Table - 58 : K = 100, σ = 0.50
P1 P2 P3 ST POP1 POP2 POP3 POP1% POP2% POP3%
11.88 0.147 9.568
80 20 0.223 17.851 168.31 152.18 186.57
90 10 0.105 9.482 84.15 71.90 99.10
100 0 0 0 0 0 0
110 0 0 0 0 0 0
120 0 0 0 0 0 0
Table - 59 : K = 110, σ = 0.50
P1 P2 P3 ST POP1 POP2 POP3 POP1% POP2% POP3%
17.448 0.120 13.748
80 30 0.318 25.476 171.93 159.57 185.31
90 20 0.201 18.060 114.62 100.55 131.36
100 10 0.095 9.531 57.31 47.74 69.32
110 0 0 0 0 0 0
120 0 0 0 0 0 0
Table - 60 : K = 120, σ = 0.50
P1 P2 P3 ST POP1 POP2 POP3 POP1% POP2% POP3%
23.944 0.256 18.454
80 40 0.405 32.437 167.05 158.40 175.77
90 30 0.288 25.891 125.29 112.38 140.30
100 20 0.182 18.232 83.53 71.21 98.79
110 10 0.087 9.571 41.76 33.98 51.87
120 0 0 0 0 0 0
3. Comparisons of Three BSM Formulas 119
Next we draw graphs based on the option pricing formulas for the three
payoff functions mentioned above. This graphs will also be useful to compare
the option pricing formulas.
Graph-1 & 2: Graph-1 is for plain vanilla call option and log call option.
The blue colored graph indicates that the price of plain vanilla call option is
much higher than the price of log call option indicated by red color. In fact,
the price of log call option is near to zero which indicates that it is less that
even transaction cost. So that log option is not practically useful.
On the other hand, Graph-2 is for plain vanilla call option and modified
log call option. The closure of blue colored graph and green colored graph
indicates that the modified log call option is almost similar to the plain vanilla
call option. Only one difference is that plain vanilla call option is suitable to
the holder while modified log call option is suitable to the writer.
■ Plain Vanilla Call Option ■ Log Call Option
Graph-1
■ Plain Vanilla Call Option ■ Modified Log Call Option
Graph-2
3. Comparisons of Three BSM Formulas 120
Graph-3 & 4: Graph-3 is for plain vanilla put option and log put option.
The blue colored graph indicates that the price of plain vanilla put option is
much higher than the price of log put option indicated by red color. In fact,
the price of log put option is near to zero which indicates that it is less that
even transaction cost. So, that log option is not practically useful.
On the other hand, Graph-4 is for plain vanilla put option and modified
log put option. The closure of blue colored graph and green colored graph
indicates that the modified log put option is almost similar to the plain vanilla
put option. Only one difference is that plain vanilla put option is suitable to
the writer while modified log put option is suitable to the holder.
■ Plain Vanilla Put Option ■ Log Put Option
Graph-3
P
■ Plain Vanilla Put Option ■ Modified Log Put Option
Graph-4
P
0 80
3. Comparisons of Three BSM Formulas 121
Graph-5 & 6: Graph-5 is for plain vanilla call payoff function and log call
payoff function. The blue colored graph indicates that the plain vanilla call
payoff is much higher than the log call payoff indicated by red color. In fact,
the log call payoff is near to zero which indicates that it is again less than even
transaction cost. So that log option is not useful practically.
On the other hand, Graph-6 is for plain vanilla call payoff function and
modified log call payoff function. The closure of blue colored graph and green
colored graph indicates that the modified log call payoff function is almost
similar to the plain vanilla call payoff function.
■ Payoffs of Plain Vanilla Call Option ■ Payoffs of Log Call Option
Graph-5
■ Payoffs of Plain Vanilla Call Option ■ Payoffs of Modified Log Call Option
Graph-6
3. Comparisons of Three BSM Formulas 122
Graph-7 & 8: Graph-7 is for plain vanilla put payoff function and log put
payoff function. The blue colored graph indicates that the plain vanilla put
payoff is much higher than the log put payoff indicated by red color. In fact,
the log put payoff is near to zero which indicates that it is again less than even
transaction cost. So that log option is not useful practically.
On the other hand, Graph-8 is for plain vanilla put payoff function and
modified log put payoff function. The closure of blue colored graph and green
colored graph indicates that the modified log put payoff function is almost
similar to the plain vanilla put payoff function.
■ Payoffs of Plain Vanilla Put Option ■ Payoffs of Log Put Option
Graph-7
■ Payoffs of Plain Vanilla Put Option ■ Payoffs of Modified Log Put Option
Graph-8
4. Conclusion 123
The following table is based on the analysis of the above graphs.
Graph No. Outcome
1 & 2 modified log call option is much closure to the plain vanilla
call option compare to the log call option. Moreover, the
log call option values are near to zero.
3 & 4 modified log put option is much closure to the plain vanilla
put option compare to the log put option. Moreover, the
log put option values are near to zero.
5 & 6 modified log call payoff function is much closure to plain vanilla
call payoff function compare to the log call payoff function.
Moreover, the log call payoff function values are near to zero.
7 & 8 modified log put payoff function is much closure to plain vanilla
put payoff function compare to the log put payoff function.
Moreover, the log put payoff function values are near to zero.
4.4. Conclusion
The very first BSM formula was derived for this option. As explained
earlier, all of the other options are compared directly or indirectly with plain
vanilla. We also compare our modified log option with the plain vanilla option.
Our first conclusion is the following. From all the sixty tables and eight
graphs above, we can see that the option values and payoff values for the log
option are strictly less than unity. Sometimes, they are even less than the
transaction costs (see [19, Table 4.4]). So this BSM formula does not seem to
be practically useful in the financial market. On the other hand, our modified
log option contract is very close to the plain vanilla (see Tables and Graphs).
Our second important conclusion is that as compared to the plain vanilla,
the writer is more beneficial to enter into a call option using the modified log
payoff whereas the holder is more beneficial to enter into a put option using
the same.