ƒ 243

Put

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 an European put

option is max (E - S

, 0) = max (E - SeT

rT

, 0)

.

ƒ

Discounting at rate r

, the value of the European put today is

ƒ

To show that this is consistent with

the BS formula, consider first the case

where Ee

-rT

> S. 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 Ee

-rT

> S. 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
S
Ee
Se
E
e
p
rT
rT
rT
t
−

−

−
−
.t
rT
t
S
Ee
p
−

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