General model and assumptions
The situation of this study is that of a franchisor who operates a business format franchise needs to decide whether to enter a new market with one of two organizational forms: either to expand through a franchise unit (run by the franchisee) or invest as a companyowned unit (run by an employee). At time zero, aligned with ownership redirection theory conjecture, it is assumed that the franchisor chooses to enter the market through a franchise unit to start, but leaves the option to acquire the unit in the future by buying a real option to acquire the unit. The franchisor has the right (not the obligation) to execute the option within the expiration time [0, T], which is divided into N intervals so that \( \Delta t=\frac{T}{N} \).
The difference in the choice of organizational form depends on how the franchisor extracts profit. Due to residual claimancy, it is assumed that the agency costs only exist when the franchisor operates the unit as companyowned. The franchisee has more motivation than a companyowned unit manager as the franchisee will receive the residual profit generated by the franchise establishment. However, in the companyowned mode, the franchisor can extract all the recurring operating profit compared to partial revenue (in the form of royalty fee) in the franchise arrangement. The franchisor total profit is formulated as:
$$ \mathrm{P}\left(\tau, X,A\right)={\displaystyle {\sum}_{t=1}^{\tau}\delta {X}_t+{\displaystyle {\sum}_{t=\tau +1}^T{X}_t\left(1c\right){A}_t}} $$
(1)
Where P(⋅) is the franchisor’s total profit, which includes all profits before the franchise acquisition (franchise mode) and after the unit is converted to companyowned. The time τ, indicates the time the franchisor converts the unit. In the franchise organizational form, the franchisor only gains profits from royalty fees, δ, which are calculated as a percentage of stochastic revenue, X. Whereas in the companyowned mode, the franchisor gains profits after considering operational costs, c, and agency costs, A, of running the unit as companyowned. The operational cost (c) that incur during the companyowned mode is considered to be constant, meaning there is no difference in the operational cost when the unit is run by either the franchisee or the employee in the companyowned unit. Whereas the agency costs (A) is assumed to be impacted by many factors, so it considered as stochastic. In this study, the following assumptions have also been made:

There is usually an initial upfront franchise fee that the franchisor charges the franchisee per period of contract for running a franchise unit, but without loss of generality, it is assumed to be zero.

The period that was modeled in the study only refers to the period before the options mature. Whereas, after the options expire, the possibility that the franchisor will acquire the franchise unit becomes very small due to difficulties in renegotiating the acquisition price (K).

The decision to convert is irreversible. Once it is made, the franchisor is locked into the costs and profits associated with ownership of the unit. This assumption is very important in order to assess the franchise acquisition in the real options model.
Stochastic process
Every real options model relies on the design of a stochastic process, as the value of the underlying assets will depend on this. In this study, two underlying assets are considered: revenue (X) and agency costs (A). In implementing options pricing using the LSM method, it is important to use a discrete, stochastic process. While most financial options use the GBM, in this study, the stochastic process of the franchisor’s profit is assumed to follow a logdiscrete time diffusion (Log DD) model (Kariya & Liu, 2002). The Log DD model considers the stochastic process as a discrete model, which is also needed in the real options valuation using the LSM method and the Log DD model, as shown below.
$$ {X}_t={X}_{t1} exp\left[{\mu}_{X_{t1}}h+{\sigma}_{X_{t1}}\sqrt{h}{\varepsilon}_n\right]\kern1.25em {\varepsilon}_n\sim iid $$
(2)
Where \( {\mu}_{X_{t1}} \) is the drift function and \( {\sigma}_{X_{t1}} \) is the volatility. \( {\mu}_{X_{t1}} \) and \( {\sigma}_{X_{t1}} \) can be setup as time varying factor dependent on X
_{
t − 1} = (X
_{
t − 2}, X
_{
t − 3}, …, X
_{1}, X
_{0}), which in this study will be assumed to be constant for simplicity, so that:
$$ {X}_t={X}_{t1} exp\left[\mu +\sigma {\varepsilon}_n\right]\kern1.25em {\varepsilon}_n\sim iid $$
(3)
The stochastic model of the agency costs is the same as in Eqs. (2) and (3) after replacing X with A. This stochastic process is similar to that in the Kariya et al. (2005) model in valuing a lease agreement in commercial real estate in Japan and to Nugroho (2015) in valuing a franchise revenue guarantee. The difference with the current study is that both studies assumed the stochastic process as path dependent by modeling the drift using an exponential smoothing model.
Real options model
At time zero, neither the franchisor nor the franchisee knows the future of the business. It is assumed that as time progresses, the value of revenue and agency costs will continuously fluctuate in a stochastic manner. Thus, a real options model will be developed to capture the value of the rights to acquire the franchise based on future uncertainty that impacted by stochastic movement. In this study, the real options framework developed by Longstaff and Schwartz (2001) is used because it can value early exercise options such as American and Bermudan options. Additionally, as a simulation based valuation, this method can easily handle more than one stochastic structure, which is important in this model. Unlike financial options valuation, where every state of time reflects the state of the price, in this study, the movement of revenue (X) accumulates in every t. The real options value (ROV) of the franchise acquisition is:
$$ F\left(t,{X}_t,{A}_t\right)=\underset{\tau \in \mathbb{T}\left(t,T\right)}{\underbrace{max}}\;{E}^{*}\left[\left(P\left(t,{X}_t,{A}_t\right)\left.K\right){e}^{r\left(\tau t\right)}\right)\right] $$
(4)
Where F(⋅) is the real options value, E* is the expectation under risk neutrality, and t is the restricted stopping time {\( t \)
_{0} = 0, \( t \)
_{1} = Δ\( t \), …, \( t \) = \( N \)Δ\( t \)}. P(⋅) is the franchisor’s total profit as described in Eq. (1), and K is the cost of acquisition or the amount that needs to be paid as an acquisition cost to the franchisee. Hence, the payoff that the franchisor receives is total operating profit at that point in time less the acquisition cost.
The unique aspect of the real options model in this franchise acquisition case compared to common investments under uncertainty is how the franchisor extracts profit. Profit is not only incurred after the acquisition but also before the acquisition in the form of royalty fee. Combining Eqs. (1) and (4) results in:
$$ F\left(\tau, {X}_t,{A}_t\right)={E}^{*}\ \left[\underset{\tau }{\underbrace{max}}\left({\displaystyle {\sum}_{t=1}^{\tau}\delta {X}_t+}{\displaystyle {\sum}_{t=\tau +1}^{\mathrm{T}}\left[{X}_t\left(1c\right){A}_t\right]K,0}\right){e}^{\tau}\right] $$
(5)
From another perspective, all the franchisor’s income (royalty fee) during the franchise period will reduce the cost of acquisition. Thus, the franchisor’s decision when to convert the unit to companyowned not only considers income after the acquisition, but also the amount the acquisition cost is reduced if the franchisor postpones it.
The key issue also to be handled in this real options formulation is the optimal time to convert the franchise to a companyowned unit, denoted by τ. In valuing options with early exercise features, time (τ) will be referred as the optimal stopping time, which is the main difference in European options that can only be exercised at maturity date. The LSM method was adopted to identify the τ through backward dynamic programming using Monte Carlo simulations. For each path of simulation, it denotes with a superscript ^{(i)} so that \( {P}_{t_n}^{(i)} \) is the profit function in the (i
^{th}) path and time (t). The equation for the optimal stopping problem in this case is:
$$ F\left({t}_n,{P}_{t_n}\right)= max\left\{\Pi \left({t}_n,{P}_{t_n}\right),\Phi \left({t}_n,{P}_{t_n}\right)\right\} $$
(6)
With
$$ \Pi \left({t}_n,{P}_{t_n}\right)=P\left(\tau, {X}_t,{A}_t\right)K $$
(7)
and
$$ \Phi \left({t}_n,{P}_{t_n}\right)={e}^{r\left({t}_{n+1}{t}_n\right)}{E}_{t_n}^{*}\left[F\left({t}_{n+1},{P}_{t_{n+1}}\right)\right] $$
(8)
The Eq. (6) above is basically the Bellman equation of finding optimal stopping time by comparing continuation value, \( \Phi \left({t}_n,{P}_{t_n}\right), \) and \( \Pi \left({t}_n,{P}_{t_n}\right) \) so that:
$$ \mathrm{if}\;\Phi \left({t}_n,{X}_{t_n}\right)\le \Pi \left({t}_n,{X}_{t_n}\right)\;\mathrm{then}\;{\tau}^{(i)}={t}_n $$
(9)
The optimal stopping time will be found by completing Eq. (9) recursively. After the optimal stopping time is updated, the ROV is averaged for all paths:
$$ ROV=F\left(0,x\right)=\frac{1}{M}{{\displaystyle {\sum}_{i=1}^Me}}^{r{\tau}^{(i)}}\Pi \left({\tau}^{(i)},{P}_{\tau^{(i)}}^{(i)}\right) $$
(10)
From Eq. (10), the problem boils down to how to find the continuation value (Ф) in order to apply Eq. (9). The LSM method contributes by approximating the continuation value (Φ) that is the conditional expectation of time (t) (if exercise is still allowed) of future optimal payoffs from the contingent claim. As discussed in Gamba, (2003) and Longstaff and Schwartz (2001), Φ is an element of a vector space, which can be represented as:
$$ \Phi \left({t}_n,{P}_{t_n}\right)={\displaystyle {\sum}_{j=1}^{\infty }{\varphi}_j(t){L}_j\left(t,{P}_t\right)} $$
(11)
With respect to the basis of {Lj}. If only J < ∞ are elements of the basis that can be used to approximate Ф, the continuation value becomes:
$$ {\Phi}^J\left({t}_n,{P}_{t_n}\right)={\displaystyle {\sum}_{j=0}^M{\varphi}_j(t){L}_j\left(t,{P}_t\right)} $$
(12)
Then, \( \varphi j \) (\( t \)) can be estimated by least square regression of Φ(\( t \), \( Pt \)) on the basis that:
$$ {\left\{{\widehat{\Phi}}^J\left({t}_n\right)\right\}}_{j=1}^J= arg\underset{{\left\{{\varphi}_j\right\}}_{j=1}^J}{\underbrace{min}}\left\Vert {\displaystyle {\sum}_{j=1}^J{\varphi}_j\left({t}_n\right){L}_j\left({t}_n,{P}_{t_n}\right){\displaystyle {\sum}_{i=n+1}^N{e}^{r\left({t}_i{t}_n\right)}\Pi \left(t,{t}_i,\tau \right)}}\right\Vert $$
(13)
Thus, the continuation value is:
$$ {\widehat{\Phi}}^J\left({t}_n,{P}_{t_n}\right)={\displaystyle {\sum}_{j=1}^J{\widehat{\varphi}}_j\left({t}_n\right){L}_j\left({t}_n,{P}_{t_n}\right)} $$
(14)
Equation (14) above is used in the comparison rule in Eq. (9), which is calculated backward from T to t = 1.