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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


∂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.


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)


= 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


+

√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√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)


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).


#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 = ”;


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)


{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 = ”;


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.


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.


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


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


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


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


Table - 05 : K = 120, σ = 0.25


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


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


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


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


Table - 09 : K = 110, σ = 0.30


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


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


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


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


Table - 13 : K = 100, σ = 0.35


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


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


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


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


Table - 17 : K = 90, σ = 0.40


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


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


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


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


Table - 21 : K = 80, σ = 0.45


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


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


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


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


Table - 25 : K = 120, σ = 0.45


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


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


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


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


Table - 29 : K = 110, σ = 0.50


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


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


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


Table - 33 : K = 100, σ = 0.25


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


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


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


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


Table - 37 : K = 90, σ = 0.30


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


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


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


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


Table - 41 : K = 80, σ = 0.35


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


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


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


Table - 45 : K = 120, σ = 0.35


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


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


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


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


Table - 49 : K = 110, σ = 0.40


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


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


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


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


Table - 53 : K = 100, σ = 0.45


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


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


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


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


Table - 57 : K = 90, σ = 0.50


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


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


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


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


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


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


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


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.

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