Yield management is the process of actively managing inventory to maximize revenues. In this paper we present a model demised to apply yield management techniques using real options to the problem of optimal decision making when assigning rooms to hotel customers. Two different methods are proposed to carry out the evaluation: numerical resolution with the PDEs and Monte Carlo simulation. The achieved results using both methods are similar demonstrating the robustness of the simulation in this field and the model lends itself to be a tool for helping the hotel manager in his operational decision of whether or not giving a room to a potential client

Alvarez, Lucia

Ruiz López, Felipe

Solana Perez, Pablo

English

Archivo Digital UPM

Monte Carlo Methods in Finance.

Patents and R&D as Real Options&quot;.

Real options. Managerial flexibility and strategy in resource allocation.

Real Options. Managing strategic investment in an uncertain world.

Yield Management under the Real Options Scheme for Optimal Decision Making in Hotels

Servicio de Coordinación de Bibliotecas de la Universidad Politécnica de Madrid

II International Conference on Industrial Engineering and Industrial Management 
XII Congreso de Ingenieria de Organizacion 
September 3-5, 2008, Burgos, Spain 
Yield management under the real options scheme for optimal decision 
making in hotels* 
Lucia Alvarez1, Felipe Ruiz1, Pablo Solana1 
1
 Dpto. de Ingenieria de Organizacion, Administracion de Empresas y Estadistica. Escuela Tecnica Superior de 
Ingenieros Industrials. Universidad Politecnica de Madrid. Calle Jose Gutierrez Abascal, 2. 28006. Madrid. 
fhiiz@etsii.upm.es, psolana@etsii.upm.es. 
Abstract 
Yield management is the process of actively managing inventory to maximize revenues. In this 
paper we present a model demised to apply yield management techniques using real options to the 
problem of optimal decision making when assigning rooms to hotel customers. Two different 
methods are proposed to carry out the evaluation: numerical resolution with the PDEs and 
MonteCarlo simulation. The achieved results using both methods are similar demonstrating the 
robustness of the simulation in this field and the model lends itself to be a tool for helping the hotel 
manager in his operational decision of whether or not giving a room to a potential client. 
Keywords: real options, optimal decision making, yield management. 
1. Introduction. 
A model for the application of yield management techniques to the management of hotel 
rooms is presented in this article. The growing competitivity in the accommodation market, 
provoked in many cases by the supply excess, shows that a correct management should be 
done in order to maintain profitability. 
Yield management is the perishable inventories managing process to maximize these stocks 
profits. This concept has its origin in the airlines industry where, in the take off moment, 
empty places represent profit opportunities lost. This approach is not only limited to the 
airlines sector, but is applicable to service industries, and in the hotels sector a real time 
tariffication is allowed, adapting the rooms price to the existing demand at every moment and 
maximizing that way sale profits. 
One of the main characteristics that makes the profitability of the hotel industry so dependent 
of its price policy is the impossibility to stock the hotel product. In addition, the difficulty to 
modify the supply to adapt it to the movements of demand, together with the possibility to 
forecast its activity through reservations and historical experience, indicates that yield 
management techniques should be very well suited to optimize hotel operations. 
* This work stems from the participation of the authors hi a research project funded by "CICYT", title "Sistema 
avauzado de ayuda a la toma de decisioues a la gestion hotelera". 
Finances 645 
The problem to be solved in this sector is how to assign the available rooms, depending on the 
current supply and demand, in a way that the profit of the hotel is maximized. Because of the 
intrinsic dynamic characteristics of the system, the techniques used in the past don't take into 
account all the complexities of the problem. This is why the real options theoiy leads to a 
better solution than the static scenarios of the linear programming or queuing techniques. 
The critical decision to be made by a hotel is whether to accept or not a room reservation for a 
future date. Until the moment of occupancy of a room, it will be assumed that its price will 
fluctuate with both a detemrinistic and a random component, which implies that the 
acceptance of a reservation currently involves a hade-off between the present cash flow and a 
possible higher cash flow in the future: should the room be rented today or should we wait for 
a better opportunity?. The exercise decision can occur at any time and every decision affects 
subsequent ones, since any room that gets reserved decreases the number of rooms which may 
be rented in the future. 
A simple model for the hotel reservation process is described in this article. The problem is 
approached in two different ways: on the one hand by solving the partial differential equations 
that govern the process and on the other hand by using simulation techniques. The Least 
Squares MonteCarlo method will be used as well as the variance reduction techniques to 
minimize the computing time. The results obtained through both ways are presented and 
compared, showing the robustness and adequacy of the simulation in this field. 
2. Model. 
The model presented is based on the real options theory and looks forward to implement a 
useful decision making tool for the reservations of a hotel. The business considered has the 
following characteristics: 
• M initially available rooms, all rooms are the same. In case that a hotel with 
different room categories is studied the problem could be broken down in k 
independent cases, each one with a unique type of room. 
• Only one type of client and therefore demand. This demand will be directly related 
to the price evolution. 
• Room's payment will be made upon reservation. 
T is defined as reservation acceptance period. Once the time is over, the non-reserved rooms 
are lost revenue opportunities, because the rooms will be occupied in t=T. This reservation 
period will be divided in N intervals all of the same length AT. This means that we will face 
N times the problem of taking the decision due to the option being exercised on the 
discretized points. These time intervals are given by 0< tj< t2< ... < tN=T where tj =j -AT. 
2.1. Demand model elaboration. 
We consider the maximum number of rooms that can be reserved in an interval (demand in 
that period) as a linear function of the price. If the price of a room in a j period is called P}, 
we could establish the demand expression as follows (1). 
Demand (f>) =a+fi-PJ (1) 
Finances 646 
a, P coefficients could be constant in order to be able to program a simple example, or 
indexing them with time because of the modelizable demand difficulty not only depends on 
price but in factors such as the period of time considered or established prize of the 
competitors. 
2.2. Price model. 
The price will be represented by a random process. This could be justified because a hotel 
fixes his rates mainly looking at the competition and not in a randomised way. What is more 
the reservations done affect the availability and so the prices, conceived as a supply-offer 
relation that will vary with time. It is assumed that the price could be defined with a 
stochastic differential equation. 
dP = H (P, t) dt + oP dX (2) 
Where P=P (t) is the room price in the t reservation moment. Constant c volatility is 
considered to represent the rates constant relative fluctuation. Changing this constant by a 
time dependant function does not make any difference on the problem formulation. We 
choose in (2): 
H(P,t)=y(L(t)-P) (3) 
We consider that the price follows a mean revelling process with a rate y and a time 
dependant mean prize L (t). This decision is made based on the sector study and means a 
good approximation due to the daily price variation being limited from the top and bottom. 
The lower bound exists because prices can not stay significantly below the marginal cost for 
too long or the hotel would have to exit the market. The top level is determined by the market 
competition strength and the clients supply. A growing trend for the mean price evolution will 
be assumed in a way that the nearer the end of the period is the greater is L (t). 
2.3. Problem approach. 
A hotel with T reservation acceptance period is considered. In t=0, M available rooms exist 
for clients. In eachy moment rooms could be rent with a P} price that follows the stochastic 
process shown in (2). In each time step N in which could be make a decision the problem of 
how many room reservations is correct to accept arises. 
A graph that shows the reservation situation in each time step (Figure 1) could be done and it 
could be inteipret as a dynamic matrix of the hotel situation. Each node represents a hotel 
status: at time t,, having m rooms reserved (m vaiying between 0 and M). The developed 
algorithm objective is to obtain the broken red line of the figure, that represents temporal 
succession of the hotel states that arrives to the profit maximum by the comparison of the cash 
flows expected facing with the decision to accept a reservation or wait for potential future 
(higher) revenues. 
Considering the interval fj < f < tJ+j, knowing the fact that the reservations could only be 
done in tJ+l (what implies no reservations at t=0). It is called E]m to the expected value of 
cash flows in case those m reservations exist on the step/ If a room is rented in the following 
instant, tJ+l , the cash flows expected value will be as follows: 
Finances 647 
¥= Ej£ + PJ+1 (4) 
It is quickly deduced that the new value Ej^ is lower than£^ because now there is one less 
room to rent. 
M = 50 
49 
4 
: 
i i — 
II 
II 
II II 
II 
• 1 1 1 1 1 1—, 
to=0 tN-l t K = T 
Figure 1. Reservation matrix. 
In the same way if in the tj+i instant k rooms are reserved the expected value is expressed by: 
W=E^+k-P}+1 (5) 
with tn+k< M because it can not be rent more rooms which are available. What is more, as 
the price is represented as a stochastic differential equation the cash value E3m could be 
considered as an European option with start in t= t, and end in t= tj+i, and which payout could 
be represented by 1> (5). The fact that it is European is because it could only be exercise in 
t= tj+i. 
With the purpose of maximizing revenues the decision criteria for the reseivations acceptance 
will be the one which maximizes the total payout: 
Y=Max(E? , E*+ P]+1 , ..., E» + hP^ 
where m+k <M and k< Demand(Pj+j). 
(6) 
This method is applied for each one of the steps which reservations period is composed. In 
the last inteival (J=N), the payout is the maximum number of rooms that remain available 
taking into account the constraint due to the demand. It is because an empty room at t=T 
represents a lost revenue opportunity. 
Finances 648 
2.4. Partial Derivative Equation Resolution (PDE method). 
The objective is to determine the option value of rent the room in f instant, knowing that if the 
reservation in T is not done the opportunity to rent the room is lost. The value of this option 
is E (P, t;T). The hypothesis that the solution is a risk-neutral measure is done and knowing 
the Bellman principle the next partial derivative equation is obtained: 
^
 + i ^ P ^ + r ( L - P ) ^ - r E = 0 (7) 
dt 2 d2P 7dP 
where r is the risk-free rate discount. For more comfort (7) is transformed substituting the 
time variable by a T local variable defined upon the rectangle (for each level of occupation) 
with extremes on tj+i. and tj. So, the partial derivative equation that rules the process is: 
BE 1
 2j)2d2E , ,dE _. _ . 
— = —aP—-—+a(L-P) rE (8) 
dr 2 d2P v !dP 
The initial condition for this equation is obtained from (6) and is: 
¥=Max (E]>+1 , E^ + PJ+1, ..., E%* + h P}+I) (9) 
withm+k < M and k< Demand (PJ+j). It is necessary to define the boundary conditions. It 
is demonstrated that for P=0 is not necessary a boundary condition because y-L > 0. However 
for P large values, is evident that E will be linearly dependant of P. For the last time interval 
we can write down an exact solution of the form: 
ENm(r)= A(T)+B(T)P (10) 
due to the initial condition^ (0) =K-P, where K is a fixed constant. 
From here a numerical resolution for solving the partial derivative equations based on the 
finite differences method was developed, this procedure yields to a tridiagonal equations 
system solved by the programmed calculus model. 
2.5. Simulating solution. 
MonteCarlo method is a statistical approximation to the problem. N different price paths are 
generated following the model expressed by (2) along the reservations acceptance period. To 
solve the problem is necessaiy to go to the ending instant of time (t=T) in which m 
reservation rooms are considered. For each path of price i we calculate the maximum nmnber 
of rooms which could be reserved in instant T: 
K'T =Min(M-m, Demand (Pj.)) (11) 
where P]. is the price in T for the i path. This procedure is done for each path obtaining the 
decision vector: 
Finances 649 
Dm.T (12) 
Coming back to the previous instant it is calculated again the maximum number of rooms for 
each path that could be reserved considering that m were already rent in past instants. The 
received payout in T-l will be: 
¥=Max (E^ , E™1 + i£_, , ..., E^ + A£_, • P;_,) (13) 
And in a generic step time j the layout can be expressed as follows: 
¥=Max(E^,E% + Pj+1 E£ + K^ J j , ) (14) 
At each instant of time a regression is done by the LSM method. That way the expected cash 
flows payout could be calculated for each price path obtaining a decision vector that 
represents the number of rooms that should be reserved in each instant of time for each i path, 
knowing thatm rooms have been already reserved. That way is proceed until the arrival to the 
initial instant, obtaining a matrix formed by the decisions made in each instant for each path 
and the cash flow expected values. 
3. Results. 
The outputs of the model can be summarized as follows: 
• Present value of the earnings associated to an optimal room management. Here, 
results depart heavily from the standard NPV rule results, mainly due to the value of 
the real options involved in the day to day management of the hotel: basically, the 
option to wait and (potentially) hire rooms at a higher price or for a longer period of 
time. 
• Practical management decision tool. The model confronts the immediate earnings 
associated to rent the room now (exercise value) against the expected earnings 
associated to future inflows. Given the historical data and the demand/price 
function, the hotel manager can actually decide which strategy gives a higher yield. 
The results obtained by both methods for a certain numeric example are presented below. A 3 
month reservation acceptance period divided in 12 steps in a little facility with 50 rooms has 
been considered. The free risk rate is 5% and the mean reverting parameter (y) is 180. In what 
concerns the demand the parameters of equation (1) have the values a=20 and /?=-0.07 
respectively. 
3.1. Partial derivative equation resolution. 
For the indicated values the expected cash flow value at the initial interval is £=1467.67 €. 
Let us focus on the first decision moment behaviour shown in Figure 2, where the number of 
accepted reservations is represented against price. 
Finances 650 
Figure 2. Price and availability at initial booking. PDEs. 
Reseivations are accepted from 28.8€. All the possible rooms for that demand are booked for 
prices between 29.4€ and 30€, and for prices over 30€ the demand decreases. 
This behaviour seems to be reasonable since for low prices there is no advantage at the first 
instant for the hotel manager to accept resei'vations. Due to the fact that the mean reverting 
parameter is high (y=180) a rising evolution of the price during the booking period is granted 
at a low risk. Therefore, if the initial price is low the best decision is to wait as the graphic 
shows. 
Simulation resolution. 
Figure 3. Price and availability at initial booking. MouteCarlo Simulation. 
Finances 651 
hi this case the expected cash flow value at the initial time for the numerical example is 
£'=1455.83€. The results have been obtained with 10,000 simulations. Analysing the 
behaviour at the first decision moment (Figure 3) with this method, it is observed that for 
prices below 28.7€ no booking is accepted. Sixteen rooms are rented for prices between 28.7€ 
and 29.2€, reaching a peak of 18 reservations. The demand decreases for prices over 30€. 
A sensitivity analysis for the mean reverting parameter has been done and it is observed that 
the lower the parameter y is the sooner the rooms are rented, so the reservations are accepted 
for lower prices. The reason is, as formerly commented, that as the value of y decreases the 
growing price evolution uncertainty increases. 
3.2. Results comparison. 
The results obtained with the different methods presented are very similar. For example, when 
the expected value of the hotel business is calculated, the results reached with both methods 
differ in less than 0.8%. In addition, the number of booked rooms in relation to their price 
graphics is almost identical. If we study in more detail the values obtained for y=180, we can 
observe the similarity of results (Table 1). 
Table 1.Comparison between both methods. 
Number of 
Rooms 
0 
15 
16 
18 
Rental Price 
PDEs MonteCarlo Simulation 
28.5 28.478 
28.6 28.542 
28.7 28.566 
28.8 28.717 
28.9 28.762 
29 28.764 
29.1 28.888 
29.2 28.931 
29.3 28.934 
29.047 
29.054 
29.060 
29.123 
29.143 
29.166 
29.181 
29.199 
29.4 29.326 
29.370 
Finances 652 
With the first method we could start booking rooms from a price of 28.8€, according to the 
other one for 28.7176, a really similar result taking into account that we are using iterative 
methods that tend to the exact solution. The transition between sixteen and eighteen rooms 
occurs for a price of 29.4€ and 29.32€ respectively. 
4. Conclusions. 
The objective of the research presented in this paper was to develop a tool that would allow to 
improve the hotel management by means of using the real options theoiy. More specifically it 
was looking forward to being able to apply MonteCarlo simulation techniques, which have 
proved to be very useful when applied in other fields and which have been used for the first 
time to solve a problem of yield management in the hotel sector. 
The resolution of the problem using the real options theory offers robust results. Two 
different methods have been proposed to carry out the evaluation: numerical resolution with 
the PDEs that model the process and MonteCarlo simulation method approximation. The 
achieved results using both methods are similar and thus open the door to the MonteCarlo 
Simulation resolution methods for the decision making on the hotel sector management. 
References 
Amram, H.; Kulatilaka, N. (1999). Real Options. Managing strategic investment in an 
uncertain world. Harvard Business School Press. 
Trigeorgis, L. (1996). Real options. Managerial flexibility and strategy in resource allocation. 
MIT Press. 
Jackel, P. (2002). Monte Carlo Methods in Finance. John Wiley&Sons. 
Schwartz, E. (2004). "Patents and R&D as Real Options". Economic Notes, 33(l):23-54. 
Finances 653 


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Yield Management under the Real Options Scheme for Optimal Decision Making in Hotels

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