Risk Analysis in Finance and Insurance, 2nd Edition

Analysis of Risks: Continuous Time Models 111

and therefore

X

∗

T

= x exp



σα

∗

W

T

+ α

∗

(μ − r) T −

1

2

σ

2

(α

∗

)

2

T



,

where α

∗

t

≡ α

∗

.

However,

X

∗

T

=

x

Z

∗

T

= x exp



μ −r

σ

W

T

+

1

2



μ −r

σ



2

T

%

.

Comparing these formulas, we deduce the expression for the optimal propor-

tion:

α

∗

=

μ −r

σ

2

,

which is often referred to as Merton’s point.

Worked Example 4.1 Find prices of European call and put options on a

Black-Scholes market if r =0.1,T= 215/365,S

0

= 100($),K= 80($),μ=

r, σ =0.1.

Solution Using Black-Scholes formula, we compute

C

T

= C

T

(K, S

0

,σ)=S

0

Φ(y

+

) − Ke

−rT

Φ(y

−

)

= 100 Φ

!

ln(100/80) +

215

365



0.1+(0.1)

2

/2



0.1



215/365

"

−80 e

−0.1

215

365

Φ

!

ln(100/80) +

215

365



0.1 − (0.1)

2

/2



0.1



215/365

"

= 100 Φ(3.177) ...