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( ST K ), 0}. In this chapter, we shall discuss the modified log payoff (ML-payoff) function max{S T ln( ST K ), 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
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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
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