ƒ 242

Call

ƒ

When

σ

becomes 0

, the

stock is virtually riskless, its price will

grow at rate r to Se

rT

at time T

and the

payoff from a call option is

max (S

T

-E, 0) = max (Se

rT

-E, 0)

.

ƒ

Discounting at rate r

, the value of the call today is

ƒ

To show that this is consistent with

the BS formula, consider first the case

where S > Ee

-rT

. This implies ln(S/E) + rT > 0. As

σ

tends to zero, d1 and

d2 tend to +



, so that N(d1) and N(d2) tend to 1 and the BS formula

becomes

ƒ

Next consider the case where S < Ee

-rT

. This implies ln(S/E) + rT < 0. As

σ

tends to zero, d1 and d2 tend to -



, so that N(d1) and N(d2) tend to 0

and the BS formula yields 0.

(

)

(

).

(^0) ,
max
(^0) ,
max
rT
rT
rT
t
Ee
S
E
Se
e
c
−
−
−

−

.
rT
t
t
Ee
S
c
−
−

Derivative securities: Options - Black-Scholes modelProperties of the Black-Scholes prices