Option Valuation Chapter - Replicating Portfolio - Risk Neutral - Black Scholes - Put Call Party

1. Binomial
Approach
BINOMIAL
TREE
(Binomial
Lattice)
• Value
of
Vd
and
Vu
changes
depending
if
it’s
a
put
option
or
a
call
option
• This
change
in
value
affects
DELTA
(Δ)
Given:
(CALL
OPTION)
EP=
430
MP=
430
Risk
Free
Rate=
.03
/yr
(.015
per
half
a
yr)
2
possible
assumptions
• Price
will
fall
to
322.5
• Price
will
rise
to
573.33
S
=
Share
price
U
=
upward
price
D
=
downward
price
Vu
=
S
–
U
(Call)
Vd
=
S
–
D
(Put)
U
-­‐>
Vu
D
-­‐>
Vd
S
573.33
-­‐>
143.33
322.5-­‐>
0
430

2. Given:
(PUT
OPTION)
EP=
430
MP=
430
Risk
Free
Rate=
.03
/yr
(.015
per
half
a
yr)
2
possible
assumptions
• Price
will
fall
to
322.5
• Price
will
rise
to
573.33
1. Replicating
Portfolio
• CALL
OPTION
VALUATION
(C)
i. 𝐶 = ∆𝑆 − 𝐵
ii. ∆ =
!!! !!
!!!
iii. 𝐵 =
∆ ! !!!
!
573.33
-­‐>
0
322.5-­‐>
107.5
430

3.
• PUT
OPTION
VALATION
(P)
i. 𝑃 = ∆𝑆 − 𝐵 (1 + 𝑟)
ii. ∆ =
!! ! !!
!!!
*
Delta
is
negative
because
you
need
to
sell
the
shares
*
You
exercise
the
put
option
when
MP<EP
that’s
why
(Vu
=
0,
and
not
Vd
=
0)
iii. 𝐵 =
!∆ ! !!!
!
2. Risk
Neutral
• 𝐶/𝑃 =
!
!
• 𝑝 =
!!!!
!! ! !!
Sd
=
downward
change
Su
=
upward
change
• 𝐵 = 𝑝 𝑉! + (1 − 𝑝)(𝑉!)

4. Given:
(CALL
OPTION)
Expected
Rate
of
Return
of
Google
Stock
=
1.5%
Google
Stock
can
either
rise
by
33.33%
to
$573.33
or
fall
by
25%
to
$322.50
𝑝 =
0.15 − (−0.25)
0.3333 – −0.25
= 0.6857
𝐵 = 0.6857 143.33 + 1 − 0.6857 0 = 98.2813
𝐶 =
98.2813
1.015
= 96.8288
Given:
(PUT
OPTION)
Expected
Rate
of
Return
of
Google
Stock
=
1.5%
Google
Stock
can
either
rise
by
33.33%
to
$573.33
or
fall
by
25%
to
$322.50
𝑝 =
0.15 − (−0.25)
0.3333 – −0.25
= 0.6857
𝐵 = 0.6857 0 + 1 − 0.6857 107.5 = 33.78725
𝑃 =
33.78725
1.015
= 10.4623

5. 3. PUT
CALL
PARTY
• Relationship
between
call
and
put
option
for
a
European
Option:
𝑃 = 𝐶 − 𝑆 + 𝑃𝑉 𝑋
P
=
value
of
put
C
=
value
of
call
S
=
stock/
share
price
PV(X)
=
present
value
of
EP
S
=
X
PV(X)
=
S/r
4. Black
Scholes
• ONLY
FOR
EUROPEAN
OPTION
• (what
you
get
–
what
you
give)
𝐵𝑙𝑎𝑐𝑘 𝑆𝑐ℎ𝑜𝑙𝑒 𝑉𝑎𝑙𝑢𝑒 = 𝑆𝑁 𝑑! − 𝐾𝑒!!"
𝑁(𝑑!)
N(d1)
N(d2)
=
probability
K
=
Strike
price
e-­‐rt
=
discount
factor

6.
To
get
the
probabilities
(d1
and
d2)
𝑑! = 𝑙𝑛
!
!
! !!
!!
!
!
! !
𝑑! = 𝑙𝑛
!
!
! !!
!!
!
!
! !
𝑜𝑟 𝑑! − 𝜎 𝑡
𝐵𝑙𝑎𝑐𝑘 𝑆𝑐ℎ𝑜𝑙𝑒 𝑉𝑎𝑙𝑢𝑒 = ∆𝑆 − 𝐵
Δ
=
N(d1)
S
=
share
price
B
=
N(d2)
×
PV(EX)
N(d1)
-­‐>
look
for
the
value
in
the
table
after
computation.
N(d2)
-­‐>
look
for
the
value
in
the
table
after
computation.