Application of Real Options Analysis for R&D Project Valuation in High-Tech Public Corporation - Advanced Research on Computer Education, Simulation and Modeling - page 107

3.2 Random Walk of the Benefit

The forecasted demand is inaccuracy and eventually become an underlined
uncertainty for the R&D project in the decision-making process. Because of the wide
variability of product over time, the demand P is random variable. And the benefits of
the R&D project depend on the future demand P as follows:

N

Bx ()

∑

B =

i

(1)

(1 + r )

i

i = 1

Because the demand P is random variable, the benefit B is modeled as the
following stochastic process:

dB = μ Bdt + σ B dt

(2)

Where

dt follows Wiener process.

3.3 The Black-Scholes Formula

Δ

Π

To evaluate the option, we use denote the value of an investment portfolio;

denote the quantity of B , then:

Π= VBt (,) −Δ B

(3)

The change in the portfolio is denoted as follows:
d V B Π= −Δ

(4)

$

Ito we have

From

2

∂ V ∂ V 1

2

∂ V

(5)

22

dV = dt + dB + σ B dt

2

∂ t ∂ B ∂ B

Then

∂ V ∂ V 1

2

∂

2

V

(6)

22

d Π= dt + dB + σ B dt −Δ dB

∂ t ∂ B ∂ B

2

∂
Δ= ∂

V
B

(7)

If we choose

Then the randomness is reduced to zero. Base on the no-arbitrage principle, we can
write:

(8)

dr Π= Π t

Substituting (6) (7) into (8) we find that

2

∂ V 1 ∂ V ∂ V

(9)

22

+ σ B + rB − rV = 0

2

∂ t 2 ∂ B ∂ B