Journal of Probability

Volume 2014, Article ID 839204, 11 pages

http://dx.doi.org/10.1155/2014/839204

## Multivariate Option Pricing with Pair-Copulas

^{1}Group Strategic Planning and Control at Generali, Head Office, Via N. Machiavelli 4, 34132 Trieste, Italy^{2}Department of Computer Science, University of Verona, Strada le Grazie 15, 37134 Verona, Italy

Received 31 July 2014; Accepted 9 December 2014; Published 25 December 2014

Academic Editor: Edward Furman

Copyright © 2014 Anna Barban and Luca Di Persio. 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

We propose a copula-based approach to solve the option pricing problem in the risk-neutral setting and with respect to a structured derivative written on several underlying assets. Our analysis generalizes similar results already present in the literature but limited to the trivariate case. The main difficulty of such a generalization consists in selecting the appropriate vine structure which turns to be of D-vine type, contrary to what happens in the trivariate setting where the canonical vine is sufficient. We first define the general procedure for multivariate options and then we will give a concrete example for the case of an option written on four indexes of stocks, namely, the S&P 500 Index, the Nasdaq 100 Index, the Nasdaq Composite Index, and the Nyse Composite Index. Moreover, we calibrate the proposed model, also providing a comparison analysis between real prices and simulated data to show the goodness of obtained estimates. We underline that our pair-copula decomposition method produces excellent numerical results, without restrictive assumptions on the assets dynamics or on their dependence structure, so that our copula-based approach can be used to model heterogeneous dependence structure existing between market assets of interest in a rigorous and effective way.

#### 1. Introduction

In what follows we will consider a European option written on 4 assets. We will assume that the risk-neutral setting holds true, according to the framework defined in [1]; that is, the price of each of the four considered underlying assets only depends on its history; moreover, it is independent form the others past behaviour. Previous condition allows us to write the price of the above mentioned option as the following discounted value: where is the maturity time of the option, is a real positive parameter usually representing the interest rate given by a bank for our cash deposit and it is assumed to be constant over the whole option’s life, is the payoff of the option written on the four assets , whose prices, at any time , are for . We would like to underline that (1) is given directly under the risk-neutral (martingale) measure and it represents the (fair or no-arbitrage) price of the option with payoff determined by the so-called martingale approach, see, for example, [2, Section 5.4].

Under suitable assumptions, the price determined in (1) can be rewritten according to the expected value definition as follows: where is the joint density probability function of the underlying assets with respect to the risk-neutral probability measure .

Our aim is to apply a pair-copula decomposition approach to reproduce the joint density of the 4 underlying assets as correlated random variables. Latter goal requires first to find a method able to (statistically) describe the behaviour of each underlying asset. In particular, we assume that the underlying assets returns evolve as generalized autoregressive conditional heteroskedastic processes with parameters and , namely, a process. Latter choice is motivated by the fact that the GARCH processes properly describe the evolution of variables that do not have constant volatility over time; see, for example, [3]. Therefore, GARCH processes turn out to be very effective models for heteroskedastic processes, as in the case of financial time series which exhibit structured interrelations.

In particular, in what follows we consider 4 assets, namely, and a discrete set of times whose elements represent a day between day 0, beginning of transactions, and day which can be taken as the end of our daily observations of the asset’s prices. The quantity , for and , stands for the* closing price* of the underlying at the trading day . The one-day log-return of the th asset is defined as
Note that the objective 4-variate distribution of the log-returns is specified conditional to all return information available at time , that is conditional to the sigma algebra .

In order to derive the joint risk-neutral return process from the objective one, we assume that the objective marginal distributions of the assets returns evolve as a process with Gaussian innovations; see, for example, [4]; namely,
where, for every , is the expected daily log-return of the asset , while is the Gaussian innovation under the objective, or* real world*, probability linked to the return . In particular, , conditioned to all returns information available at time , that is conditioned to , has mean zero and variance which evolves as a processes with parameters such that .

Since the marginal distributions, at a given time , are specified conditional to a common set of information, that is, with respect to the -algebra , then we are allowed to exploit copula theory techniques to derive the joint conditional distribution. In what follows, inspired by [5, 6], see also [1, 7], we assume that the objective and the risk-neutral copulas are the same.

The main idea behind our option pricing model is that we can use a convenient map to transform each marginal objective distribution to its risk-neutral counterpart, as in [4]. Then we define a proper 4-dimensional copula function, obtained by the pair-copula construction method (see, for example, [8]) instead of deriving the joint risk-neutral distribution directly. Finally, we determine the fair price of the option by taking the discounted expected value of the option’s payoff under the risk-neutral measure.

In particular, assuming that the risk-neutral probability satisfies a* local risk-neutral valuation relationship* (LRNVR) (see [4, Def.2.1, Th.2.2]), then the law of the returns under is given by
where are the Gaussian innovations under . Exploiting the transform defined in (5) it is possible to model the marginal behaviour of the marginal log-returns under the measure .

##### 1.1. Pair-Copula Decomposition

In this section, we show how to model the interdependence of the underlying assets , under the risk-neutral measure , by a copula-based approach. We would like to underline that in the dimensional case for and differently from the bivariate one, the main difficulty is that of finding the right -dimensional copula which properly reproduces the dependence structure existing between the single pairs of components. We solve such a problem by a multivariate copula construction method, namely, the pair-copula decomposition approach; see, for example, [8].

Let us start introducing some basic concepts for the pair-copula representation in view of the analysis of the three-dimensional case; see [9].

*Definition 1 (tree). *A tree is an acyclic graph, where is its set of nodes and is its set of edges (unordered pairs of nodes).

*Definition 2 (regular vine). *A regular vine tree on variables is a structure of connected trees , with nodes and edges , , such that the following conditions are satisfied: (1) is a tree with a set of nodes and a set of edges denoted by ;(2)for the tree has node set and a set of edges denoted by ;(3)two edges in tree are joined in tree if they share a common node in tree .

Here we focus on two special cases of regular vines, namely, the* D-vine* and the* canonical vine*; see, for example, [8] for details.

Before going into details, we would like to underline that while from the pair-copula decomposition point of view the three-dimensional case can be easily treated (for example, the D-vine and the canonical vine specification coincide), difficulties arise when the number of correlated dimensions increases.

In three dimensions the construction method proceeds as follows: let denote the joint density probability function of three random variables , and then can be decomposed as follows: where, for every , stands for the marginal probability distribution function of , is the corresponding marginal density function, and is the associated bivariate copula density, while represents the bivariate copula density of conditional to the second component.

In order to reproduce a density function through a pair-copula decomposition in dimension , we have first to select an appropriate decomposition structure. In particular, from a financial point of view, previous statement means that we first have to analyze the economic relationship between the variables of interest in order to find the variable that governs the interactions in the data set. If such a variable exists, then a canonical vine specification should be the best choice; otherwise, we will use the D-vine decomposition structure; see, for example, [8].

Once the vine structure has been chosen, we are left with the selection of the order of the variables, the families of pair-copulas, and the specification of their parameters.

The order of the variables is set according to the following rule: first we compute the dependence measure Kendall’s tau for each pair of variables and we order the variables in the first tree on the basis of their dependence structure; this means that the two variables with the strongest dependence are put in the first two nodes of the first tree and so on. The next trees are determined as a consequence; see [10]. Then we select the best-fitting pair-copula family; namely, we have to select the pair-copula specification that guarantees the best fitting with observed data. Such a problem can lead to different solutions depending on different setting. In our specific case, we are justified to consider only the Gaussian, the -Student, the Clayton and the Gumbel copulas since they are the most used types in financial applications; see, for example, [1, 5–7]. The particular copula is selected exploiting standard information criteria such as Akaike information criterion (AIC) and the Bayesian information criterion (BIC) respectively; see, for example, [11–13] and references therein, for each edge in each tree. In particular, the choice is made in order to minimize the AIC and the BIC coefficient, respectively. Finally, the selected copula’s parameters are estimated using the maximum likelihood criterion for each pair.

Once we have obtained the results for the first tree, we need to construct a sample for the conditional bivariate distribution, in order to find the pair-copula associated with the copula density in (6). Let us consider the following definition of bivariate conditional distribution function in terms of copula: where and are uniform random variables and is the set of parameters characterizing the copula of the joint distribution of and .

We would like to underline that the function plays a key role in the pair-copula decomposition approach in a -dimensional setting when , since it allows to reproduce the conditional behaviour of a random vector in terms of a bivariate function. Indeed, by an iterative algorithm, we can rewrite the conditional distribution using a proper choice of and of the copula written on its marginals.

Once the pair-copula method has been theoretically implemented, thus obtaining the joint distribution of interest, we are left with the need to calibrate the model in order to obtain satisfactory numerical results. The calibration procedure will be the main goal of the next section.

##### 1.2. Calibration of the Model

The pair-copula decomposition and the -function described in Section 1.1 can be calibrated starting from the definition of the Gaussian innovations of the model, namely, a , we have chosen for the marginal distributions of the underlying assets.

In particular, if we are in a -dimensional setting, , we define the standardized innovations as follows: where is the standard deviation of the underlying at time . By the well-known results for the GARCH model (see, for example, [14, Part V, Sec.16]) the are , standard Gaussian random variables, even if the stochastic processes are not independent. By the Sklar’s Theorem we have that the joint distribution (under the objective measure ) of the underlying innovations can be written in terms of its marginals; see, for example, [15, Sec.1]. In particular, since are continuous stochastic processes, there exists an unique copula such that for all , . Moreover, we assume that the copula in (9) is a parametric copula of parameter . Latter assumption is justified since the dependence structure existing between the variables is usually given by a copula function that depends on a particular parameter. Given the standardized innovations, we want to infer their dependence structure under using the pair-copula decomposition method. Then, we will use the dependence structure, namely, the copula function that better reproduces the joint behaviour of the underlyings, to price an option written on four indexes which will be considered as underlyings.

In order to use the pair-copula decomposition, and in particular the -function defined in (7), we compute the corresponding estimated standardized innovations in the interval , applying the cumulative distribution function of the standard normal distribution . Thus, the corresponding uniform variables , for are defined as follows:
Hereafter, we will work using the random vector , with , where is the maturity date of our* investment*.

The next step is that of finding the copula on the joint distribution of the random vector . In particular, we have to choose the bivariate copulas that best fit our data. Such a copula structure will be then exploited to reproduce the 4-variate copula density and thus the joint distribution density, for the random vector under the objective measure .

##### 1.3. Option Pricing

In this section, we will consider (2) to solve the associated option pricing problem exploiting results obtained through previous sections. To reproduce the joint density function we use, as for the density under the objective probability , a multivariate copula constructed through the pair-copula approach. In particular, we assume that the copula under belongs to the same family as the one determined under , possibly being characterized by different parameters.

The option pricing problem will be solved in two steps. First we simulate the underlying assets under the risk-neutral measure ; then, we simulate the option price and we calibrate the model to determine the set of the copulas’ parameters under . The set is composed of all the parameters of the pair-copulas used to create the proper vine specification. For example, in the 3-dimensional case, we have , for such a particular vine structure.

In order to simulate the assets under we consider the risk-neutral transformation of the returns defined in (5); see [4] for details. This turns out to be a recursive method to estimate both the returns and the volatility terms under the risk-neutral measure. Moreover, in order to maintain the dependence structure obtained from the market data, we simulate, using the vine specification previously defined, the variables , , according to the algorithm described in [10].

Concerning the option price estimation, let us recall that at timethe option price is given by the following equation:

We first estimate the price, defined as , inserting the prices , observed in the market at time , for , into the option payoff equation. Then we use a Monte Carlo approach to estimate the price, defined as , by (1) and using the observation simulated from the vine structure. Finally, we calculate the set minimizing the sum of the quadratic error; namely, where are the past observations of the option prices. The question about the choice of the parameter set corresponds to the calibration of the pricing model.

#### 2. Numerical Implementation

In our implementation, we work with a dataset that comes from the U.S market. In particular, we consider the following indexes: the Standard and Poor’s (S&P) 500 index, the Nasdaq 100 index, the Nasdaq Composite index, and the New York Stock Exchange (Nyse) Composite index; see, for example, the related Wikipedia occurrences for detailed definitions of these market indexes. We have considered such indexes between January 1, 2012 and December 28, 2012, for a total of 249 days resulting in the same amount of* closing levels* for each index. On these indexes we write two option contracts, and , respectively, which are defined by the following payoffs at maturity :
respectively, where is the strike price of the option and . Note that in our numerical implementation. In Figure 1 we plot the four data sets.