Discrete Dynamics in Nature and Society

Volume 2017 (2017), Article ID 2196702, 9 pages

https://doi.org/10.1155/2017/2196702

## Pricing Mining Concessions Based on Combined Multinomial Pricing Model

School of Humanities and Economic Management, China University of Geosciences, Beijing 100083, China

Correspondence should be addressed to Chang Xiao

Received 16 October 2016; Revised 5 December 2016; Accepted 21 December 2016; Published 18 January 2017

Academic Editor: Ricardo López-Ruiz

Copyright © 2017 Chang Xiao and Jinsheng Zhou. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

A combined multinomial pricing model is proposed for pricing mining concession in which the annualized volatility of the price of mineral products follows a multinomial distribution. First, a combined multinomial pricing model is proposed which consists of binomial pricing models calculated according to different volatility values. Second, a method is provided to calculate the annualized volatility and the distribution. Third, the value of convenience yields is calculated based on the relationship between the futures price and the spot price. The notion of convenience yields is used to adjust our model as well. Based on an empirical study of a Chinese copper mine concession, we verify that our model is easy to use and better than the model with constant volatility when considering the changing annualized volatility of the price of the mineral product.

#### 1. Introduction

It is very important for a mining company to set reasonable prices for mining concessions. In China, mining resources belong to the state. To conduct mining operation, a mining company must first purchase the mining concession. In addition, a mining company can profit from transferring mining concessions that they have purchased. Mining concession’s appreciation can have a favorable impact on the valuation of the mining company. A relatively large error in the estimation of the price of the mining concession during the evaluation step can result in a loss of the company’s economic benefits.

Decision tree pricing models represent a method of option pricing models. One of these decision tree pricing models, the binomial pricing model, was first proposed by Cox et al. [1]. Later, numerous researchers contributed to binomial pricing models’ evolution to a multinomial pricing model. Some researchers have conducted both theoretical and empirical studies on the multinomial pricing model. All of these studies have demonstrated that the multinomial pricing model is suitable for option pricing and can be used to obtain an option price that is close to the market price [2–4].

Because a mining concession has the characteristics of a real option, a decision tree pricing model can be used to price a mining concession. When using traditional decision tree pricing models to price mining concessions, it is of particular importance to determine the price volatility of the mineral product, which is a key variable of this model. When calculating the price of the mining concession of an offshore oil field in the Netherlands, Smit first calculates six-year annualized volatility of the price of Brent crude oil between 1988 and 1993. Then, he determines the annualized volatility based on the annualized volatility values of the previous two years and mean annualized volatility of all six years [5]. Gu et al. calculate monthly volatility based on the historical data and assume that the monthly volatility of each of the following months remained the same [6]. The aforementioned studies both assume that volatility was a constant and make their calculations based on this assumption. However, some researchers believe that the use of constant volatility will result in overestimation or underestimation of the price of a mining concession because volatility can experience relatively significant variations from year to year. Therefore, it is more reasonable to assume that the volatility is stochastic (e.g., Ting et al. [7] and Huang et al. [8]). However, models based on this assumption are relatively complex and therefore have rarely been used in practice. One of the reasons, which causes the model to be complex to use, is the distribution assumption of the annualized volatility of the price of mineral products.

Some researchers have studied the distribution of the volatility of the price of underlying assets. For example, Liesenfeld and Jung believe that the volatility of stock prices closely approximates a -distribution [9]. Andersen et al. studied the volatility of foreign exchange rates and stocks return, discovering that both of them approximate a log-normal distribution with a sharp peak and a right-skewed feature [10, 11]. The calculation of the distribution feature of the price volatility in the aforementioned studies is both relatively complicated and relatively difficult to carry out in practice. Therefore, some researchers assume that the volatility of the price of underlying assets follows a multinomial distribution and use multinomial pricing model to price options (e.g., Madan et al. [12] and Florescu and Viens [13]). However, it is also difficult to calculate the price of mining concession if we use multinomial pricing model directly.

In addition, because the owner of a mining concession can obtain spot mineral product by mining, convenience yield must be taken into consideration when calculating the price of the mining concession. Since the concept of convenience yield was introduced by Kaldor [14], numerous researchers have studied the relationship between the futures price and the spot price based on the convenience yield. Lin and Duan estimate the change in the convenience yield under the impact of supply and demand, finding that the convenience yield exhibits seasonal characteristics; that is, the convenience yield decreases in seasons when there is a strong demand and increases in seasons when there is a low demand [15]. By using the convenience yield as the indicator of the supply risk, Stepanek et al. find correlations between the convenience yield and the static inventory of stock on the one hand and the future spot price on the other hand [16].

In this study, because the assumption of constant volatility has its defects and the assumption of stochastic volatility is complex to calculate, we assume that the annualized volatility of the price of mineral products follows a multinomial distribution. In order to take into consideration the notion that the annualized volatility of the price of mineral products may experience relatively significant variations from year to year, we propose a combined multinomial pricing model for pricing mining concessions. The combined multinomial pricing model is obtained as follows: first, calculate the distribution series of the annualized volatility of the price of the mineral product through grouped statistical analysis; then, different values of the annualized volatility are calculated based on binomial pricing models; finally, these binomial models are combined into a multinomial pricing model based on the probability corresponding to each value of the annualized volatility values. This study’s empirical research demonstrates that the combined multinomial pricing model considers the high and low annualized volatility simultaneously. Therefore, this model is more practical.

The rest of the paper is structured as follows. Section 2 introduces traditional decision tree models and focuses on binomial and multinomial pricing models. Section 3 elucidates the pricing method and combined multinomial pricing model used in this study. Section 4 describes an empirical study of a copper mine in China, including the calculation result. Section 5 compares the combined multinomial pricing model with previously known work. Section 6 provides this study’s conclusions.

#### 2. Traditional Decision Tree Model

Ever since a binomial option pricing model (CRR model) was first proposed by Cox et al. [1], decision tree pricing models have been widely used. Over the years, other researchers have popularized the CRR model. The scope of application of decision tree models not only has widened but also has been expanded from binomial and trinomial models to multinomial pricing models.

Decision tree pricing models were initially used to price financial options and were later used to price real options. Because mining concession is a type of tradable right and a type of real option, decision tree pricing models can be used to price a mining concession. When using a decision tree pricing model to price a mining concession, the variables are often defined as follows: : value of the mine : cost of the mine : the time to expiration of the mining concession : volatility of the price of the mineral product : risk-free interest rate

##### 2.1. Binomial Pricing Model

Cox et al. derive a binomial option pricing model using the portfolio replication and risk-neutral approaches [1]. When calculating the value of a mining concession using the binomial approach, we assume that the value of the mine equals the product of the price of the mineral product and the reserve of the mineral product. Therefore, it is necessary to determine the extent of the increase and decrease in the value of the mine in each period of time based on the volatility of the price of the mineral product and establish a corresponding tree diagram of the variation of the value of the mine. Ultimately, calculate the price of the mining concession through reverse deduction based on the tree diagram.

The calculation process of a one-step binomial pricing model is as follows:where represents the time to expiration of the mining concession; represents the annualized volatility of the price of the mineral product; represents the upstream multiplier of the value of the mine; represents the downstream multiplier of the value of the mine; represents the price of the mining concession after it increases; represents the price of the mining concession after it decreases; represents the probability that the value of the mine will rise; represents the price of the mining concession at the current time, that is, the price of the mining concession; represents the risk-free interest rate; represents the value of the mine at the current time; and represents the cost of the mine.

If the time to expiration of the mining concession () is divided into number of periods of time, a multistep binomial pricing model can be obtained by further derivation. The expressions of multistep binomial pricing model are as follows:where represents the number of times that the value of the mine increases; represents the price of the mining concession after the value of the mine increases number of times and decreases number of times, that is, the price of the option when the mining concession expires (there are possibilities); and the meaning of each of the remaining variables is the same as that in the one-step binomial-tree model.

##### 2.2. Multinomial Pricing Model

The calculation process of the multinomial pricing model is established based on the popularization of the binomial pricing model. Cox et al. argue that the price of an option is the mathematical expectation of the discount value of the option at expiration [1]. It is the same with mining concessions.

We assume that a one-step multinomial tree has number of branches; that is, there are possible variations of the value of the mine at the end of the period of time. If is the price of the mining concession at the current time and is the value of the mine at the current time, after the th type of price change, the price of the mining concession changes from to and the value of the mine changes from to , that is, number of times the value of the mine at the current time. Based on the assumptions mentioned above, the expressions of the one-step multinomial pricing model are obtained as follows:where represents the probability that the price of the mining concession will change from to ; represents the time to expiration of the mining concession; represents the risk-free interest rate; and represents the cost of the mine.

Similar to the binomial and trinomial pricing models, if the time to expiration of the mining concession () is divided into number of periods of time, a multistep multinomial pricing model can be obtained by further derivation. Its expressions are as follows:where represents the number of times that the value of the mine undergoes the th type of price change; represents the price of the mining concession after the value of the mine undergoes number of times of the first type of change, number of times of the second type of change,, number of times of the th type of change, that is, the price of the mining concession when it expires; and the meaning of each of the remaining variables is the same as that in the one-step multinomial pricing model.

For the multistep multinomial pricing model, the calculation of is very complicated by algebraic methods. Consequently, most of the studies have not used the multistep multinomial pricing model when pricing mining concessions. Some researchers attempt to solve this problem. For example, Cox and Rubinstein obtain the relationship between the probability of continuous stock trading and the present value selected in a chance event by introducing a complex option and then obtaining the variation probability through reverse derivation by the use of other variables [17]. To obtain a simpler expression when using the prospect theory proposed by Rockenbach [18], Yan takes the reciprocal of the number of branches of the multistep multinomial pricing model for the probabilities of each type of price change [19].

#### 3. Pricing Method and Model

##### 3.1. Calculation of Annualized Volatility

When using a decision tree pricing model to price a mining concession, it is necessary to calculate the annualized volatility of the price of the mineral product. Based on the book of Hull [20], the process of calculating the annualized volatility of the price of the mineral product is as follows.

First, the daily yield needs to be calculated, which is expressed in the logarithmic form:where represents the yield on the th day and represents the price of the mineral product on the th day.

Next, the standard deviation, , of the daily yield is calculated:where represents the number of days in a year on which spot trading of the mineral product occurs ( is usually set to 250) and represents the number-of-day average daily yield.

Finally, the daily standard deviation is converted to an annual standard deviation, that is, the annualized volatility, :

##### 3.2. Calculation of Distribution

In some complex models, the annualized volatility of the price of mineral products follows a log-normal distribution or Student’s -distribution. To simplify the calculation process, this study assumes that the annualized volatility of the price of a mineral product follows a multinomial distribution. Under this assumption, each of the previous years’ annualized volatility values of the price of a mineral product is mutually independent and can only take a limited number of values. In addition, the probability that the annualized volatility has each of the aforementioned values is fixed.

Based on the efficient markets hypothesis, the annualized volatility values of the price of a mineral product are mutually independent. Moreover, the annualized volatility of the price of mineral products has mean-reversion characteristics. Therefore, the volatility fluctuates within a certain range most of the time. Here, copper is taken as an example. Based on the study conducted by Figuerola-Ferretti and Gilbert, the monthly volatility of the spot price of copper between October 3, 1982, and December 30, 2005, was less than 70% and fluctuated around 25% [21]. The same was true for the annualized volatility. Therefore, we can calculate the distribution by using the grouped statistical analysis method. In reality, according to the central-limit theorem, the limits of a multinomial distribution follow a normal distribution. Therefore, if a multistep decision tree pricing model is used, the price of a mining concession priced based on the multinomial distribution assumption will be close to the price of the mining concession priced based on the normal distribution assumption.

After the annualized volatility of the price of the mineral product of each of the previous years is obtained, the distribution series of the annualized volatility of the price of the mineral product can be approximated using the grouped statistical analysis method based on the assumption that the annualized volatility of the price of the mineral product follows a multinomial distribution.

To conduct a grouped statistical analysis of the annualized volatility of the spot price of copper, it is necessary to first calculate the annualized volatility of the spot price of copper of each year over the course of number of years and then conduct a grouped statistical analysis of the annualized volatility (each group has a span of 10%). There are seven groups overall. Table 1 lists the statistical results.