This model, assumed that there is risk neutrality and diffusion asset dynamics, has been shown how to approximate the underlying continuous diffusion process when there is a series of exercise prices and interactions between different options embedded in a single underlying asset. Valuing its option separately and adding up the individuals results is inappropriate since multiple options could possibly interact. Overall, the log-transformed binomial numerical model could be used for valuing complex multi-option investments. This model is appropriate for the biotechnology sector, since a drug development investment is a multi-option investment.

In this model, the value of the underlying asset (V), is the PV of the expected cash flows from immediately undertaking a real project. To the biotechnology company, V represents the market value of a claim on the future cash flows from starting the project now. V could follow a diffusion process:

dV/V = α dt + σ dz,

where:

α is the instantaneous expected return on the drug project

σ is the instantaneous standard deviation of the drug project value alterations

dz is a standard Brownian motion

X = log V follows an arithmetic Brownian motion, in any differential time interval dt. Under conditions of risk neutrality a = r, with r being the risk-free rate, we have:

dx = log (Vt+dt / Vt) = (r-1/2σ2) dt + σ dz

the increments, dX are independent and normally distributed with u = (r-1/2σ2)dt and var = σ2dt

Within each discrete interval τ, X follows a Markov random process moving up by an amount ΔX = H with a risk-neutral P probability or going down by the same amount (-H) with a 1-P probability. Thus, the mean and variance of this Markov process are:

Exp (ΔX) =2PH – H

Var(ΔX) = H2

[Exp(ΔX)]2

For the discrete-time process to be consistent with the continuous-time process, the mean and the variance of the continuous process should equal to the mean and variance of the discrete process, respectively. Thus, according to these criteria:

2PH – H = µK  =>  P = 1/2(1 + µK/H)

H2 – (µK)2 = K  =>  H =

These transformations of state and time variables guarantee stability as well as consistency of the discrete-time approximation to the continuous process.

By its definition, Var(ΔX) ≥ 0, thus:

-1 ≤ µK/H ≤ 1, which implies that 0 ≤ P ≤1.

Specifically:

P = ½ (1+µK/H) => P = ½ (1 +) =>

P ≤  ½ (1 +) =>

P ≤  ½ (1 + IµKI/ IµKI), since K = VarX) ≥ 0  =>

P = 1

Also:

P = ½ (1 + µK/H) ≥ ½ (1-1) = 0

Thus, 0 ≤ P ≤1 and P + (1 – P) = 0 then the conditions that ensure unconditional stability in this weighted average numeric scheme are fulfilled without external constraints on K and H.