A unifying methodology is presented, which jointly considers correlated Brownian motion processes with Poisson jumps in both revenue and policy. The methodology is unique in considering price and quantity as geometric Brownian motion processes with jumps following a Poisson process in revenue from market shocks and policy uncertainty.
The literature is rich in employing real options analysis in determining investment entry and exit. Recent literature includes Agaton (2021) , Araya et al. (2021) , Bakker et al. (2021) , Deeney et al. (2021) , and Ioulianou et al. (2021) . The general assumption is returns follow a Markov property with only small changes. Merton (1976) states the antipathetical process to this continuous stochastic process is a jump stochastic process. This precipitates the composition of returns to investment is of three types: 1) Normal trends in price and quantity with variation around these trends resulting from marginal market change. Modeling of these processes is generally with geometric Brownian motion (GMB). 2) Jumps in price and quantity from market shocks more than a marginal effect. Such shocks may be firm- or industry-specific and they arrive at discrete points in time. Through a Poisson jump process, it is possible to incorporate these shocks into models. 3) Policy uncertainty where policymakers establish mechanisms, which create uncertainty through the prospect of policy change ( Dixit and Pindyck, 1994 ). Changes in incentives to establish a firm or industry that reduces sunk costs can heighten uncertainty. Modeling this switching among policies can also involve an additional Poisson jump.
Lacking is an integrated method, which jointly considers these three processes. Our unique feature is considering both price and quantity following correlated GMB with Poisson jumps. Considering the resulting revenue process, an additional Poisson jump results from government policy uncertainty. The aim is to extend the literature by developing this theoretical integration.
This theoretical result is not without practical applications. One application is technical advancements resulting in the development of a commercially viable new business investment opportunity. Zhao (2018) presents this application for the case study if Artemisinin efficiency improvements can result in the development of a commercially viable U.S. agribusiness investment opportunity. Other applications of the theory are in the renewable energy sector characterized by technical advancements in energy generation and policy shifts. Solar energy is a specific application.
There are real options considering correlated Brownian processes price and quantity ( Price and Wetzstein, 1999 ) and independently considering jump diffusion ( Lin and Huang, 2010 ; Yang and Zhang, 2005 ; Wang, 2007 ). The logical next direction is considering jointly stochastic price and quantity processes with independent Poisson jumps.
The literature is rich in considering GBM with Poisson jumps. Recent investigations include fuel prices ( Agaton, 2022 ; Batac et al., 2022 ) and demand shocks along with investment ( Wu and Hu, 2022 ). Volk-Makarewicza et al. (2022) develop a test to determine if a Poisson jump significantly influences a GBM real option. One closely related effort is by Pimentel et al. (2018) who consider both the number of mass-transportation passengers and investment expenditures as stochastic processes with Poisson jumps.
Independent of this literature is research addressing policy uncertainty ( Anderson and Weersink, 2014 ; David et al., 2000 ; Dimos and Pugh, 2016 ; Fuss et al., 2008 ; Fuss et al., 2009 ; Fuss et al., 2012 ; Hassett and Metcalf, 1999 ; Liu et al., 2018 ; Mauer and Ott, 1995 ; Price et al., 2005 ; Reedman et al., 2006 ; Rodrik, 1991 ; Yang et al., 2008 ; Zhou et al., 2010 ).
The problem is determining at what revenue thresholds is it optimal to invest with and without a government subsidy. The investment expected value, V, is a function of the revenue stream, R, V(R). Consider this revenue as a product of price, P, and quantity, Q, stochastic processes with jumps. The optimal revenue-threshold triggers are also determined under policy uncertainty. Specifically, consider the following GBM processes with Poisson jump processes
d X ( t ) = α X X ( t − ) d t + σ X X ( t − ) d z X + S X X ( t − ) d q X , X = P , Q ,
where dX(t) represents the change in stochastic process X, X ( t − ) is the value just prior to a possible jump at t, α X is the drift, σ X denotes the volatility, dt is an infinitesimal time step, d z X follows a Wiener process with z X ( 0 ) = 0 , E ( d z X 2 ) = d t , d q X is a Poisson variable representing the total number of events in dt, and S X denotes the size of jumps in dt.
Assume E ( d z P z Q ) = ρ P Q d t , where ρ P Q represents the correlation between P and Q. Then the stochastic process of revenue, R = PQ, remains a GBM process with Poisson jumps, regardless of the correlation between dP and dQ. As derived in Dixit and Pindyck (1994) , the general form of the correlated Brownian motion processes for R is
d R = ∂ R ∂ P d P + ∂ R ∂ Q d Q + ∂ 2 R ∂ P ∂ Q d P d Q + 1 2 ∂ 2 R ∂ P 2 d P 2 + 1 2 ∂ 2 R ∂ Q 2 d Q 2 + { R [ ( 1 + S P ) P , Q ] − R [ P , Q ] } d q P + { R [ P , ( 1 + S Q ) Q ] − R [ P , Q ] } d q Q . (1)
The first five terms in (1) result from applying Ito’s lemma ( Bjork, 2009 ). The last term represents the revenue change if a jump occurs in X over time interval dt with probability γ X d t .
Simplifying (1) by omitting higher order dt terms, assuming E ( d q X d t ) = E ( d z X d q X ) = 0 , and given d P d Q = σ P σ Q ρ P Q R , yields
d R = Q d P + P d Q + d P d Q + γ P d t S P R + γ Q d t S Q R = R d t ( α P + α Q + σ P σ Q ρ P Q ) + R ( σ Q d z Q + σ P d z P ) + γ P R S P d t + γ Q d t R S Q d t .
Allowing both the unit price and quantity to fluctuate randomly, two-correlated GBM processes result with jump processes, yielding the stochastic differential equation
d R = α R R d t + σ R R d z R + J R , (2)
where α R = α P + α Q + σ P σ Q ρ P Q + γ P μ P S + γ Q μ Q S , σ R = ( σ Q 2 + σ P 2 + 2 σ P σ Q ρ P Q ) 1 / 2 . The Wiener process d z R represents N ( 0 , d t ) , and J is assumed to be a Chi-square random variable with mean ( μ P S γ P S + μ Q S γ Q S ) d t and variance 2 ( μ P S γ P S + μ Q S γ Q S ) d t , where μ X S is the jump mean, X = P, Q. In (2), R fluctuates as a GBM process, but over each time interval dt, it will change by J times its original value and then continue fluctuating.
As presented in Dixit and Pindyck (1994) , the Bellman equation for F(R), the value of the investment opportunity, is then
r F d t = E ( d F ) , (3)
where r is the discount rate. Transforming the Bellman equation by expanding dF using Ito’s lemma for combined Brownian and Poisson processes yields
d F = F ′ ( R ) d R + 1 2 F ″ ( R ) ( d R ) 2 .
Note that ( d R ) 2 = σ R 2 R 2 d t and dividing by dt, (3) is then
1 2 σ R 2 R 2 F ″ ( R ) + ( α R + μ P S γ P S + μ Q S γ Q S ) R F ′ ( R ) − r F ( R ) = 0 , (4)
where α R = r − δ R with δ R denoting the short-fall rate. Equation (4) is a second-order nonhomogeneous differential equation for determining when to invest. Following Dixit and Pindyck (1994) the solution is
F ( R ) = A 1 R β 1 ,
where A 1 is a constant to be determined and
β 1 = 1 2 − μ P S γ P S + μ Q S γ Q S + r − δ R σ R 2 + [ μ P S γ P S + μ Q S γ Q S + r − δ R σ R 2 − 1 2 ] 2 + 2 r σ R 2 .
The project’s expected present value, V(R), is
V ( R ) = E { ∫ 0 ∞ R ( t ) e − r t d t } .
Equating the normal return on V(R) at rate r to the sum of revenue and the expected capital gain yields
r V ( R ) d t = R ( t ) d t + E ( d V ) ,
where the expected capital gain is
E ( d V ) = E { ( d V d R ) ( d R d t ) d t } = [ α P + α Q + σ P σ Q ρ P Q + 2 ( μ P S γ P S + μ Q S γ Q S ) ] R r .
From (2)
E ( d R ) = α R R d t + ( μ P S γ P S + μ Q S γ Q S ) = α P + α Q + σ P σ Q ρ P Q + γ P μ P S + γ Q μ Q S + 2 ( μ P S γ P S + μ Q S γ Q S ) .
Then,
V ( R ) = R ( t ) d t r d t + E ( d V ) r = R r + [ α P + α Q + σ P σ Q ρ P Q + 2 ( μ P S γ P S + μ Q S γ Q S ) ] R r 2 .
Denote α m = α P + α Q + σ P σ Q ρ P Q + 2 ( μ P S γ P S + μ Q S γ Q S ) . From the optimal switching revenue threshold solve for the boundary conditions
R * = ( β 1 β 1 − 1 ) ( r 2 r + α m ) ( K + C r ) ,
where K is the initial sunk investment cost and C denotes operating costs.
Following Dixit and Pindyck (1994) , consider market uncertainty resulting from time inconsistent subsidy policies on adoption. Suppose the policy instrument is an investment subsidy, θ . When the policy is in effect, a lower sunk cost of investment exists ( 1 − θ ) K . The optimal investment threshold is then
R 1 N R S = ( β 1 β 1 − 1 ) ( r 2 r + α m ) ( C r + ( 1 − θ ) K ) .
The government may switch between the regimes of the discrete subsidy policy (
Assume the initial investment in exercising the option to invest is equal to the revenue difference between implementing subsidy and not. Then, for revenue beyond R 0 , the firm always invests at once. The investment opportunity with subsidy is then
F 1 ( R ) = ( 1 r + α m r 2 ) R − C r − ( 1 − θ ) K . (5)
For the absence of a governmental subsidy, the option to invest yields
F 0 ( R ) = ( 1 r + α m r 2 ) R − C r − K .
Over the revenue range ( R 1 , R 0 ) , the investment is undertaken with subsidies, and it waits to exercise the option, otherwise. Note that within the range of revenues ( R 1 , R 0 ) , we know the expression for the value of the investment opportunity (5) if the investment is subsidized. As derived by Dixit and Pindyck (1994) , the investment opportunity without subsidy is
F 0 ( 1 ) ( R ) = B 1 R β 1 ( 1 ) + B 2 R β 2 ( 1 ) + ( λ 1 λ 1 + r − α m ) ( 1 r + α m r 2 ) R − λ 1 r + λ 1 [ C r + ( 1 − θ ) K ] ,
where B 1 and B 2 are constants
F 0 ( 2 ) ( R ) = λ 0 λ 1 G a R β a − λ 1 D s R β s λ 1 + λ 0 ,
and
F 1 ( 2 ) ( R ) = λ 0 λ 1 G a R β a + λ 0 D s R β s λ 1 + λ 0 ,
where
β a = − ( α m − 1 2 σ R 2 ) + ( α m − 1 2 σ R 2 ) 2 + 2 r σ R 2 σ R 2 ,
β s = − ( α m − 1 2 σ R 2 ) + ( α m − 1 2 σ R 2 ) 2 + 2 ( r + λ 0 + λ 1 ) σ R 2 σ R 2 ,
and G a and D a are parameters to be determined.
Following the policy uncertainty section in Dixit and Pindyck (1994) , results are given by the following six equations.
λ 0 λ 1 G a R 1 β a + λ 0 D s R 1 β s λ 1 + λ 0 = ( 1 r + α m r 2 ) R 1 − C r − ( 1 − θ ) K , (6a)
β a λ 0 λ 1 G a R 1 β a − 1 + β s λ 0 D s R 1 β s − 1 λ 1 + λ 0 = 1 r + α m r 2 , (6b)
B 1 R 0 β 1 ( 1 ) + B 2 R 0 β 2 ( 1 ) + ( λ 1 λ 1 + r − α m ) ( 1 r + α m r 2 ) R 0 − λ 1 r + λ 1 [ C r + ( 1 − θ ) K ] = ( 1 r + α m r 2 ) R 0 − ( C r + K ) , (6c)
β 1 ( 1 ) B 1 R 0 β 1 ( 1 ) − 1 + β 2 ( 1 ) B 2 R 0 β 2 ( 1 ) − 1 + ( λ 1 λ 1 + r − α m ) ( 1 r + α m r 2 ) = 1 r + α m r 2 , (6d)
λ 0 λ 1 G a R 1 β a − λ 1 D s R 1 β s λ 1 + λ 0 = B 1 R 1 β 1 ( 1 ) + B 2 R 1 β 2 ( 1 ) + ( λ 1 λ 1 + r − α m ) ( 1 r + α m r 2 ) R 1 − λ 1 r + λ 1 [ C r + ( 1 − θ ) K ] , (6e)
β a λ 0 λ 1 G a R 1 β a − 1 − β s λ 1 D s R 1 β s − 1 λ 1 + λ 0 = β 1 ( 1 ) B 1 R 1 β 1 ( 1 ) − 1 + β 2 ( 1 ) B 2 R 1 β 2 ( 1 ) − 1 + ( λ 1 λ 1 + r − α m ) ( 1 r + α m r 2 ) . (6f)
This set of six Equation (6) are solved numerically for the two revenue thresholds R 1 and R 0 and four parameters B 1 , B 2 , G a and D s . For an application in solving (6), refer to Zhao (2018) , which provides details on the theory, numerical estimation, and application results.
The unifying methodology considers correlated Brownian motion processes with Poisson jumps in both revenue and policy. An integrated procedure for evaluating investments is now provided as an aid for decision makers. Specifically, the extended methodology considers normal trends in price and quantity with variation around these trends as a geometric Brownian motion and jumps following a Poisson process in revenue from market shocks and policy uncertainty.
Further research should be directed toward applying the theory to other industries. Possible applications include energy with both fuel price and volume uncertainty within an uncertain government policy; firm investment decisions with asset price and output uncertainty along with possible government subsidy uncertainty; and game-theoretic market structure issues of leader’s and followers’ real options in revenue uncertainty. Such applications will provide feedback on the direction of further theoretic advancements. For policy, when considering uncertain revenue, the interplay of price, quantity, and policy uncertainty is important when calculating real options. Failure to consider this interplay may result in erroneous policy.
The authors declare no conflicts of interest regarding the publication of this paper.
Zhao, C., Colson, G., Wetzstein, H., & Wetzstein, M. (2023). Real Options Adoption with Poisson Price, Quantity, and Policy Uncertainty Jumps. Theoretical Economics Letters, 13, 1548-1556. https://doi.org/10.4236/tel.2023.136087