THE GREEKS BLACK AND SCHOLES (BS) FORMULA

[Pages:19]THE GREEKS

BLACK AND SCHOLES (BS) FORMULA

The equilibrium price of the call option (C; European on a non-dividend paying stock) is shown by Black and Scholes to be:

Ct = StN (d1) Xe r(T t)N (d2);

Moreover d1 and d2 are given by

d1

=

ln(

St X

)

+

(r + p

1 2

T

2)(T t

t) ;

d2

=

ln(

St X

)

+

(r p

1 2

T

2)(T t

t)

p

Note that d2 = d1

T

Delta of a (European; non-dividend paying stock) call option:

The delta of a derivative security, , is de...ned as the rate of change of its price with respect to the price of the underlying asset.

For a European (on a non-dividend paying stock) call option is given by

=

#Ct #St

=

N (d1)

+

St

#N (d1) #St

+

Xe r(T t)#N (d2)

(1)

#St

where we have applied the product rule:

#[StN (d1)] #St

=

#St #St

N

(d1)

+

St

#N (d1) #St

=

=

N

(d1)

+

St

#N (d1) #St

Next we apply the chain rule

#N (d1) = #N (d1) #d1

(2)

#St

#d1 #St

Since

N (d1)

=

Z

d1 1

p1 2

e

X2

2 dx

it follows that

#N (d1) #d1

=

N 0(d1)

=

p1 2

e

d21 2

(3)

By using

d1;2

=

ln(

St X

)

+

(r p T

1 2

2)(T t

t)

we have

#d1 = #d2 = p1

(4)

#St #St St T t

Using equations (2)-(4) it can be shown that

St

#N (d1) #St

=

Xe

r(T

t)#N (d2) #St

or

StN 0(d1) = Xe r(T t)N 0(d2)

Thus equation (1) reduces to

=

#Ct #St

=

N (d1)

>

0

(5)

A delta neutral position:

Consider the following portfolio:

A short position in one call and a long position

in

#C #S

=

stocks:

#C

=C

S)

#S

#

#C #C

=

= 0:

#S

#S #S

The delta of the investor's hedge position is therefore

zero.

The delta of the asset position o?sets the delta of the option position.

A position with a delta of zero is referred to as being delta neutral.

It is important to realize that the investor's position only remains delta hedged (or delta neutral) for a relatively short period of time. This is because delta changes with both changes in the stock and the passage of time.

The Gamma of a call option:

The second derivative of the call option with respect to the price of the stock is called the Gamma of the option and is given by

#2Ct #St2

=

# #St

=

#N (d1) #St

(6)

Recall that from equation (2) we have

where and

#N (d1) = #N (d1) #d1

#St

#d1 #St

#N (d1) #d1

=

N 0(d1)

=

1 p

2

e

d21 2

#d1 = #d2 =

1 p

#St #St St T t

and the time to maturity for an at the money call option:

Recall that

=

#Ct #St

=

N (d1)

Thus applying the chain rule gives

# #

=

N

0(d1)

#d1 #

;

where

N 0(d1)

=

1 p

2

e

d21

2 >0

For

an

at

the

money

call

option

(St

=

X ),

since

ln(

St X

)

=

0, we have

d1 = (r +p21 2)

=

(r

+

1 2

2)p

Thus

#d1 = (r +p12 2) > 0

#

2

It

can

be

shown

that

#2 #2

<

0

BLACK AND SCHOLES (BS) FORMULA

The equilibrium price of the call option (C; European on a non-dividend paying stock) is shown by Black and Scholes to be:

Ct = StN (d1) Xe r(T t)N (d2);

where d1 and d2 are given by

d1

=

ln(

St X

)

+

(r + p

1 2

T

2)(T t

t) ;

d2

=

ln(

St X

)

+

(r p

1 2

T

2)(T t

t)

or

p d2 = d1

 ................
................

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