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 (V _{t+dt }/ V_{t}) = (r-1/2σ^{2}) dt + σ dz*

the increments, dX are independent and normally distributed with *u = (r-1/2σ ^{2})dt* and var =

*σ*

^{2}dtWithin each discrete interval τ, *X* follows a Markov random process moving up by an amount Δ*X* = *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) = H ^{2} – *

*[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)*

*H ^{2} – (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(Δ*X*) ≥ 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* + I*µK*I/ I*µK*I), since *K* = *Var*(Δ*X*) ≥ 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*.

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