Rainfall option impact on profits of the hospitality industry through scenario correlation and copulas

Abstract

This work introduces and empirically tests an option contract designed to manage rainfall risk of hospitality firms. It contributes to the literature on weather derivatives in tourism. We deal with a rainfall option designed to compensate for the lack of revenues caused by too much rain, which could have a negative impact on the hospitality industry business performances in the medium/long run. We concentrate on the typical question of pricing a rainfall option. We introduce a new model with the aim of improving the formulation of the price by considering the highly non-linear relationships between rain and business performances in a multidimensional framework. Such a model results from the integration of three essential elements: scenario correlation, copulas, and Monte Carlo techniques. The model is complemented by an experiment that focuses on Lake Garda, Italy, one of the most important tourist destinations in the summer period for international and Italian visitors. We consider the quantities of rainfall of Lake Garda in the years 2005–2014 and the business performances of 18 hotels operating there during the same period. Based on these data, we obtain multidimensional data. Then, the price of an option contract designed to hedge the decrease in revenues due to rainfall is assessed based on the refined and linked multidimensional data. Finally, an optimal number of contracts to be bought is suggested based on the minimization of the Earnings Before Interests and Taxes variability.

Appendix

1.1 Copulas: definition and Sklar’s theorem

The definition of copula was introduced by Sklar (1959), but the field of finance got to know copulas in 1999 with an article by Paul Embrechts, Alexander McNeil, and Daniel Straumann published in Risk Magazine.¹¹

We report the definition of copula from Nelsen (1999). An n-dimensional copula is a function $C$ with the following properties:

• 1. the domain of $C$ is $\left[ {0,1} \right]^{n}$ and the codomain of $C$ is $\left[ {0,1} \right]$;

• 2. for every $\varvec{u} = \left( {u_{1} , \ldots ,u_{k - 1} ,u_{k} ,u_{k + 1} , \ldots ,u_{n} } \right)$ belonging to $\left[ {0,1} \right]^{n}$, $C\left( \varvec{u} \right) = 0$ if at least one coordinate of $\varvec{u}$ is 0 and $C\left( {\left( {1, \ldots ,1,u_{k} ,1, \ldots ,1} \right)} \right) = u_{k}$, i.e. $C\left( \varvec{u} \right) = u_{k}$ if all coordinates of $\varvec{u}$ are 1 except $u_{k}$, $k = 1, \ldots ,n$;

• 3. for every $\varvec{a} = \left( {a_{1} , \ldots ,a_{n} } \right)$ and $\varvec{b} = \left( {b_{1} \ldots ,b_{n} } \right)$ belonging to $\left[ {0,1} \right]^{n}$, with $a_{k} \le b_{k}$, $k = 1, \ldots ,n$, we have

  $$ \mathop \sum \limits_{{j_{1} = 1}}^{2} \cdots \mathop \sum \limits_{{j_{n} = 1}}^{2} \left( { - 1} \right)^{{j_{1} + \ldots + j_{n} }} C\left( {u_{{1j_{1} }} , \ldots ,u_{{nj_{n} }} } \right) \ge 0, $$

  where $u_{k1} = a_{k}$ and $u_{k2} = b_{k}$, $k = 1, \ldots ,n$.

The previous definition is equivalent to saying that an n-dimensional copula $C$ is the distribution function of an n-dimensional random variable ${\mathbf{U}} = \left( {U_{1} , \ldots ,U_{n} } \right)$, whose components $U_{k}$, $k = 1, \ldots ,n$, are standard uniform random variables, i.e. random variables uniformly distributed on the interval $\left[ {0,1} \right]$.

To complete this brief summary about copulas, we report an important theorem by Sklar (1959).¹² Let $F$ be the distribution function of an n-dimensional random variable ${\mathbf{X}} = \left( {X_{1} , \ldots ,X_{n} } \right)$, whose components $X_{k}$, $k = 1, \ldots ,n$, are random variables defined on a common probability space. Let $F_{1} , \ldots ,F_{n}$ be the distribution functions of $X_{1} , \ldots ,X_{n}$, respectively. Then, there exists an n-dimensional copula $C$ such that

$$ F\left( {x_{1} , \ldots ,x_{n} } \right) = C\left( {F_{1} \left( {x_{1} } \right), \ldots ,F_{n} \left( {x_{n} } \right)} \right) $$

for all ${\mathbf{x}} = \left( {x_{1} , \ldots ,x_{n} } \right) \in {\mathbb{R}}^{n}$. If $F_{1} , \ldots ,F_{n}$ are continuous, then $C$ is unique.

What are the meanings of copulas and Sklar’s theorem in our framework? It is highly instructive to report some features of the tools from Embrechts et al. (1999): “Copulas represent a way of trying to extract the dependence structure from the joint distribution and to extricate dependence and marginal behaviour. … The Sklar marginal-copula representation of a joint distribution shows how we can construct multivariate models with prescribed marginal. We simply apply any copula $C$ … to those marginals. … Copulas represent a useful approach to understanding and modelling dependent risks. … Instead of summarising dependence with a dangerous single number such as correlation, we choose a model [the copula] for the dependence structure that reflects more detailed knowledge of the risk management problem in hand. It is a relatively simple matter to generate random observations from the fully specified multivariate model thus obtained, so that an important application of copulas might be in Monte Carlo simulation approaches to the measurement of risk in situations where complex dependencies are present.” This is exactly what we want to obtain by applying copulas to our data through Sklar’s theorem. In particular, we identify the appropriate (marginal) distributions for the data that we want to generate and, then, we define the appropriate copula to capture the dependence structure of data in a good manner.

1.2 Matlab Script

The following Matlab script lists the Matlab commands used to perform the copula fitting steps. These steps are outlined in Sect. 3.2.2 for the numerical application of the present work. They are discussed in more general terms in Sect. 2.2.2.

The script is composed of six steps. After importing the refined multidimensional data from an Excel file (first step), we estimated the distributions of the eight components by means of a normal kernel smoothing applied to the empirical distributions of these components (function ksdensity, second step). The cumulative distribution functions of the smoothed components were used to fit a $t$ copula (function copulafit, third step), and estimates of the linear correlation matrix of order $8$ and of the degrees of freedom were obtained. The fitted copula was used to extract 900 random numbers for each component (function copularnd, fourth step). The 900-by-8 simulated data were transformed to the original scale by means of the inverse cumulative distribution functions of the smoothed components (function ksdensity, fifth step). In the sixth step, the simulated rainfall and EBIT were exported to Excel files.