Uncertainty in Refurbishment Investment - Nearly Zero Energy Building Refurbishment - page 160

S uuu F uuu = max [0,S uuu -K]

Solve
Backward

p

S uu F uu

p 1-p

S u F u

S uud F uud = max [0,S uud -K]

p 1-p p

S S ud F F ud

1-p 1-p

p

S udd F udd = max [0,S udd -K]

S d 1-p p F d

Change in S
with time

S dd 1-p F dd

Example :

S ddd F ddd =max [0,S ddd -K]

F uu ( European ) = [ pF uuu + (1-p) F uud ]/ r
F uu (American ) = max {S uu -K; [pF uuu + (1-p) F uud ]/r}

Fig. 2 Binomial trees for solving European and American call options (Menassa 2011 )

discount cash flows to present time. For an American option, this solution needs to
be repeated at each time period to determine whether it is optimal to exercise the
option earlier or wait until expiration. The option is exercised earlier (say at time
period 2) if the exercise payoff (S 2 - K) is greater than the option value F 2 . There
is an optimal stock price S* below which exercising an American option is not
optimal, and the value of waiting to exercise that option is higher (McDonald
2005 ; Dixit and Pindyck 1994 ).
Perpetual American options are a special case of an American option where the
option does not have an expiration date and lives infinitely (McDonald and Siegel
1986 ). Thus, the investor is continuously evaluating early exercise versus option
value at a given period in time and would only exercise when S is greater than S*.
Several analytical and numerical solutions are available to evaluate perpetual
American options. These solutions form the basis for the framework to evaluate
investments in sustainable refurbishment of existing buildings as discussed in the
subsequent sections.

6.2 Model Assumptions and Parameters

As discussed earlier, evaluating investments in NZER presents a number of
challenges to the decision-maker particularly related to the unexpected future
benefits of such an investment. In this framework, these benefits are therefore
assumed to represent the underlying asset of the investment which will be denoted
by V.
These benefits vary with time due to the uncertainties discussed above. For
capital or real projects (i.e., not traded in financial markets), a number of
researchers have consistently assumed that the change in the value of the under-
lying asset under uncertainty follows the same lognormal or GBM distribution as
that of financial market stocks (Menassa et al. 2009 , 2010 ; Ho and Liu 2003 ;
Schwartz et al. 2000 ; Trigeorgis 1996 ; Dixit and Pindyck 1994 ; and Majd and
Pindyck 1987 ). Therefore, the change in value of the expected benefits V from