Multivariate Option Pricing with Gaussian Mixture Distributions and Mixed Copulas

: Recently, it has been reported that the hypothesis proposed by the classical black Scholes model to price multivariate options in finance were unrealistic, as such, several other methods have been introduced over the last decades including the copulas methods which uses copulas functions to model the dependence structure of underlying assets. However, the previous work did not take into account the use of mixed copulas to assess the underlying assets' dependence structure. The approach we propose consists of selecting the appropriate mixed copula’s structure which captures as much information as possible about the asset’s dependence structure and apply a copulas-based martingale strategy to price multivariate equity options using monte Carlo simulation. A mixture of normal distributions estimated with the standard EM algorithm is also considered for modeling the marginal distribution of financial asset returns. Moreover, the Monte Carlo simulation is performed to compute the values of exotic and up and out barrier options such as worst of, spread, and rainbow options, which shows that the clayton gumble and clayton gaussian have relatively large values for all the options. Our results further indicate that the mixed copula-based approach can be used efficiently to capture heterogeneous dependence structure existing in multivariate assets, price exotic options and generalize the existing results.


Introduction
The pricing of derivatives using copulas has received extensive attention within the actuarial and finance literature in the past two decades. For instance, Chiou and Tsay (2008) priced multivariate exotic derivatives using copula-based models and the risk neutral representation and demonstrated how copula-based models may be used to determine the value at risk of numerous assets. Kim and Kim (2015) used copula functions to derive correlation coefficients that depend on strike prices between assets rather than the more straightforward correlation coefficients to offer an accurate technique of pricing rainbow options with stochastic simulation. Malgrat (2013) used the copula method to model the underlying dependency structure of assets and price basket options by employing several families of copulas with parameters chosen using the maximum likelihood approach. Church (2011) used a similar idea and devised a copulabased algorithm for pricing path dependent basket options and demonstrated how the marginals selected can significantly affect the price of the options. Barban and Di Persio (2014); Berton and Mercuri (2021) presented an alternative to the Monte Carlo simulation method to price multivariate options by using a copula GARCH model. Hassane et al. (2021) presented an approach to price options that provide for the modeling of financial asset returns to take the impact of extreme values into consideration by using a combination of two gaussian distributions and an extreme copula to model the returns' combined dependence structure.
The Multivariate Geometric Brownian Motion (MGBM) approach is the most commonly used in the literature and in practice for pricing options. This approach requires an unrealistic hypothesis of independence or perfect correlation between the underlying assets. During the last decade, several alternatives have been proposed to solve this problem. The stochastic volatility and copulas methods are two examples. The stochastic volatility approach models volatility as a stochastic quantity. One of the most widely used models which assume nonconstant volatility of assets was proposed by Heston (1993). For a better understanding of the Heston model, (Ball and Roma, 1994;Kouritzin and Mackay, 2020). The copulas method's main goal is to distinguish the marginal distributions of variables from their reliance on one another. Flexible dependence structures are possible, such as tail dependent or nonlinear dependence (Chiou and Tsay, 2008). Another unrealistic assumption that has been abandoned over time is the joint normality of asset returns, which does not account for distribution tails or extreme events. In this case, the use of mixtures of gaussian distributions has been shown to be preferable (Hassane et al., 2021) due to the fact that it approximates all the distributions previously used (gaussian, student, hyperbolic, etc.).
Multivariate option pricing using copulas is a field in finance that has been developing very fast for the last two decades. In many works on the pricing of options with copulas, the joint risk neutral density is considered after marginal risk neutral densities or risk neutral copula (Hassane et al., 2021). However, all this past work did not take into account the use of mixed copulas to assess the underlying assets' dependence structure. The ability of the mixed copula model to nest various copula shapes is by far its greatest benefit. Therefore, compared to an individual copula, a mixed copula is empirically more adaptable in modeling dependence structure and can provide better descriptions of dependence structure (Cai and Wang, 2014).
In the present work, we suggest an approach for multivariate options valuation, allowing the dependent structure to be written as a copula mixture and analyzing how our choice of copula affects the cost of multivariate exotic and barrier choices with a range of strikes. In addition, we price exotic options using a copulas-based method by describing the statistical behavior of each underlying asset using a mixture of gaussian distributions, which have been proven to capture excess kurtosis and asymmetry. In particular, we model the underlying asset returns as a mixture of two gaussian distributions, with two components: One capturing the more volatile "crisis" behavior and the other the "business as usual" stock return behavior with a mean close to zero. Also, the modeling of the dependencies between assets using different mixtures of copulas distributions so as to include a variety of dependency structures.
This study is set up as follows: The main copula principles are briefly reviewed in section two. In the third section, we expose the methodology for the martingale option pricing approach with mixed copulas. Then, in the fourth section, the results of different estimations and simulations are presented, along with their analyses and interpretations. The last section presents a conclusion and discussion.

Theorem 1.2
Sklar theorem for any real valued random variables X1,..., Xn with joint distribution function F and univariate marginal distribution functions F1,..., Fn there exists a copula C such that: In contrast, given a copula C and univariate distribution functions F1,..., Fn, then the function F defined by (1) is a d-dimensional distribution function with univariate margins F1,..., Fn.
There are many proofs of this theorem that arise from different properties of copulas.
A proof by multilinear interpolation was established, first in the bivariate situation, by Schweizer and Sklar (1974), and later in the multivariate case by Carley and Taylor (2002). Moore and Spruill (1975) used the idea of generalized probability integral transformation to prove the theorem; (Durante et al., 2012) showed the existence of a random vector's connection to a copula without presenting analytically or probabilistically its form; and finally, (Benth et al., 2022) have demonstrated a topological demonstration of the theorem in any dimension.
Furthermore, the copula is unique if F1,..., and Fn are continuous functions. Sklar's theorem gives a flexible approach to the construction of multivariate distributions and to derive from any monotonic transformation certain features of the copula, such as invariance, which is particularly useful in financial applications where logarithmic conversions are commonly utilized. Also, every copula C satisfies: 3

Classes and Families of Copulas in Finance
Copulas come in a variety of families, some of which are more suitable for financial modeling; these families are stated in the next few subsections.

Elliptical Copulas
One of the most popular types of copulas in use today is the elliptical one. With the aid of Sklar's theorem, they are created from elliptical distributions and are used to dependence structure these elliptical distributions. The normal and t-copulas are the two most prominent elliptical copulas and they have the following respective density functions: the t-copulas are more efficient at modeling tail dependencies in the distribution with an additional parameter, the degrees of freedom v. As n gets smaller, the dependence gets stronger. The t-copula converges to the normal copula for s → ∞.

Archimedean Copulas
The Archimedean copulas, which may be easily built with univariate generating functions and have closed forms, represent another significant family of copulas. The advantage of these copulas is that they may capture the pattern of positive or negative dependence between the variables (asymmetrical dependence) patterns). The main disadvantage is their small number of parameters, which reduces flexibility. An archimedean copula is a copula of the form: where, u1,..., un ∈ [0,1] for a so called generator ψ: [0,∞] → [0,1] which satisfies ψ(0) = 1, ψ(∞) = limt→∞(ψ(t) = 0 and which is strictly decreasing on [0,inf{t: ψ(t) = 0}]. It follows from (4) that archimedean copulas are exchangeable.
The density of the Archimedean copulas is given by: where, ψ (n) is the n th mixed partial derivative of the inverse generator.
The clayton, frank, and gumbel copulas are among the members of the family of archimedean copulas, which is primarily employed in the area of finance. Table 1 their respective generator functions and distributions. where, θ ≥ 0 for clayton and frank and θ ≥ 1 for gumble.
An increase in the parameter theta suggests a stronger dependence for each of these archimedean copulas.

Estimation and Conformity Test of Copulas
Considering the marginal distributions of F1,..., and Fn is unknown, the copulas C need to be estimated assuming that they are members of a parametric, semiparametric, or nonparametric copula family.

Estimation
If we consider F1,..., Fn to be members of absolutely continuous parametric families of univariate distributions. Assuming that the unknown copula Cθ 0 belongs to the parametric family of distribution functions, the estimation problem of interest entails estimating it. The Inference Functions for Margin Estimator (IFME), a two-stage estimator, is used to achieve this: 1. First, the parameters αi, i = 1,...,n of the marginals distributions are estimated via maximum likelihood: 2. Next, the copula's parameter, θ that, according to maximum likelihood estimation, best describes these marginals: Because it requires less processing, this IFME estimation approach is more effective than the accurate MLE method.

Conformity Test
This is a fit test that fits chosen copula model to the data well. The most effective tests are based on the processes ( ) n C C  − whereĈ and Ĉ  are, respectively, the empirical copula and the parametric copula.
The Akaike Criterion (AIC) is used frequently to select the best copula: where, m is the number of parameters to be estimated, n is the size of data, and l(θ) is the model likelihood for the estimated parameter θ.

Copulas Mixtures
Several different copula groups are combined linearly to form a mixed copula. A mixed copula function is defined mathematically as: Note that sampling from a mixture of copulas is immediate from the sampling of individual components.
The copulas used in finance are mostly from the archimedean and elliptical families of copulas, although the survival copulas are also used. Each of these copulas has its own characteristics that help it model the dependency structure. For example, gaussian copulas better model the linear dependence but are symmetrical, which is not convenient for fat tailed distributions; on the other hand, gumble copulas are asymmetrical and are best for right tail distributions. Hence, mixing these copulas helps to combine their different properties into a single copula.

Fitting Mixtures to Data
The three most commonly used methods are pseudo maximum likelihood, maximum likelihood, and Empirical Cumulative Density Function (ECDF).
A two-step process is used in the pseudo maximum likelihood: • Map each marginal piece of data to its quantile using the ECDF • For the quantile data, determine Kendall's  and use it to determine  For elliptical copulas, θ estimated here is their ρ, the correlation parameter.
As for the maximum likelihood and ECDF, the model and likelihood function can be optimized from copula density c(u1, u2) as the objective function to maximize. This is slow, though, and even for a bivariate copula with arbitrary univariate marginal distributions, it might not be the best solution with many parameters to fit. Hence, in general, we don't use this strategy and ECDFs are instead frequently used.

Valuation Approach for Multivariate Options
The objective is to price options whose underlying assets are risky assets (St) t∈+ , where: where, µt and σt are constants and Wt is a standard geometric Brownian motion. A well-known result from the martingale option pricing theory is the fundamental theorem of option pricing.
Where, T ≥ 0 is the option's maturity time, r is a real positive parameter that usually represents the risk-free rate and f(XT) is the underlying assets' joint risk neutral density probability function.
Recall that a market is without arbitrage opportunity if and only if it admits at least one equivalent risk neutral probability measure .

Proposition 2.2
The price at time t ∈ [0,T] of European call and put options for complete markets with strike price K, risk free rate r, and maturity T are respectively given by: where, Φ is the standard gaussian cumulative distribution function and 0 ≤ t ≤ T.
The study attempts to uncover valuation implications resulting from the use of dependence structures in mixed copulas and mixed marginal distributions. The normality of stock returns is a fundamental assumption in mathematical finance. However, empirical evidence often contradicts this theoretical foundation. As a consequence, the application of alternative models is an important research topic in financial modeling. The limitations of this assumption are.
Stock prices are frequently assumed to follow a geometric Brownian motion process; therefore, financial returns are considered to be independent and normally distributed, but the geometric Brownian motion assumption has many shortcomings.
The theoretical distribution, thus estimated, guarantees the exact reproduction of the mean and the variance. On the other hand, some empirical characteristics of the distribution of returns are not reproduced by the gaussian law: 1. Compared to the tails of the gaussian law distribution, the empirical distribution's tails are thicker. Extreme values of the returns are more often observed, hence producing a higher kurtosis 2. Because the empirical distribution is asymmetric compared to the gaussian distribution, it has a negative skewness coefficient

Finite Gaussian Mixtures
A Gaussian Mixture (GM) distribution model is a multivariate distribution that consists of multivariate gaussian distribution components. The mixture is defined by a vector of mixing proportions and each component is specified by its mean and covariance. It has been shown, for example, by Cha (2010) that the gaussian mixture with two or more normal distributions increases the model fit dramatically and captures the shape of the empirical distribution.
A GM model represents a distribution as: where, (x|µk, Σk) is the density function of multivariate Gaussian distribution.

The EM Algorithm
The maximum likelihood estimate of θ can be obtained by the following iterative process known as the EM algorithm:

Algorithm 1: EM algorithm
Input: Data τ, initial guess θ (0) , and evaluate the log likelihood with these parameters Output: Approximation of the maximum likelihood estimate. 1 t  1 while change in Log-likelihood ≥ ε do Expectation Step: Evaluate the posterior probabilities P (t) (Zi|τ) with θ (t-1) and compute the expectation For more information on the fitting of mixture distributions and the EM algorithm, (Ghojogh et al., 2019).
Several families of distributions have been proposed to overcome the shortcomings of the gaussian law: • Alpha stable distributions • The finite mixtures of distributions, such as gaussian mixtures • Simple and generalized student's t-distributions • Hyperbolic distributions We are interested here in the mixture of gaussian distributions for the following reasons: • It is simple to simulate • It allows us to correctly approach all the alternative distributions cited above • It makes it possible to reproduce various features observed in the data such as average, kurtosis, skewness, and variance • It has several theoretical characteristics that make it simple to manipulate in the frame of a hypothetical asset price valuation model

Results and Discussion
In our application, we work with the dataset downloaded from the Yahoo finance website. In particular, we consider the following returns: BNP Paribas SA (BNP.PA) and Louis Vuitton stocks (LVMUY) Fig. 2. All daily data is extracted from Yahoo finance for a total of 237 days, from January 1, 2020, to December 31, 2020 (Fig. 1), resulting in the same number of closing levels for each return. We write an option contract X based on this data, with the payoffs P in Table 2 at maturity T ≥ 0: First, we correct the database with respect to time synchronization between closed prices and convert them to logarithmic returns (log returns). For each asset, Eq. (14) converts a closed price series S into a log returns r series: where, i ∈ (1,d) is the number of assets, t ∈ (1,T) is a time point, in our case d = 2, T = 237 where K ≥ 0 is the strike price of the option X and B ≥ 0 is the barrier level our approach consists of the following steps: 1. Generate log return data from our historical data returns following Eq. (14) 7 2. Estimate the parameters of the gaussian mixture regime 3. Transform the log return sample into the pseudo sample 4. Estimate the parameters of the copulas mixtures using copulas suitable for financial applications 5. Calculate the option prices using the monte Carlo numerical integration method Visualizing the above graphs, the following observations are made about the stock price for both BNP and LVMUY: • Stock prices never reach the zero value • Stock prices are continuous everywhere • Stock prices are never negative • Stock prices show randomness • stock prices increase in the long run From the quantile plots (Figs. 3-4) of the returns and the value of the kurtosis (greater than 3) in the descriptive statistics in Table 3, we can infer that the returns are not normally distributed.
At first, using the Expectation Maximization (EM) algorithm, the parameters of the two gaussian regimes that make up the gaussian mixture are calculated for each of the two assets. We shall limit ourselves to a mixture of two gaussian distributions. Also, we underline the presence of a fat tail for the returns of returns, in agreement with the non-gaussianity of the results.     In the next step, we determine the parameters of the copula's mixtures, which are shown in Table 4. The estimation of the copula parameters and weights is jointly obtained by the minimization of the negative log likelihood of the weighted densities from the copulas, as in Pfaff (2016). Copula densities are computed as in Hofert et al. (2018). Four commonly used copulas in the finance field, namely gaussian, clayton, gumbel, and frank, consist of our candidate copula families. Indeed, all possible combinations of these four copulas have the ability to capture most of the possible dependence structures. We see from the Coefficient of Upper Tail Dependence (CUTD) that the mixtures that capture dependence in the upper tail are clayton gumble, gaussian gumble, and gumble frank. In accordance with the IFME, the copula mixtures are fitted with a mixture of gaussian distributions and the fitted copula parameters are obtained through a maximum likelihood estimate and displayed in Table 5. Now we run the monte Carlo simulation with 50 000 simulations, which involves simulating a large number L of independent paths, 12 , ,..., l l l n S S S from the simulated return and estimating the option price using the average simulated discounted option payoff across the paths.
In this, we provide the simulation results for the prices of all the alternatives mentioned above based on the basket (BNP, LVMUY). On the basis of the univariate risk-neutral distributions and the calibrated copulas, the theoretical values of their prices are calculated at different values of K. In each case, we normalize the underlying to have their prices at t = 0 and assume a constant risk-free rate of r = 0.03.

Conclusion
This study proposes an approach to price multivariate options taking into account the extreme values and the dependence structure of the assets returns using mixtures of copulas and gaussian marginals. An application is made on a basket of the financial market (BNP.PA and LVMUY) downloaded from the Yahoo finance website (between January 1, 2020, and December 31, 2020).
At first, each asset's log returns are modeled by a mixture of gaussian distributions (Table 3) In all the cases, there exists a gaussian distribution with a negative mean and another one with a positive mean. one explanation is that there exist two regimes: The first one corresponds to potentially significant losses (for example, due to a financial crash), and the second one to the standard evolution of prices. The dependence between the returns is estimated using copulas mixtures.
The choice of the best fitted copulas mixture is determined by the statistic parameter of the MLE which showed that clayton gumble and gumble frank (Table 5) are the most effective at simulating dependency structures of the log returns. Prices of three options are obtained via simulation and the Clayton gumble and Clayton gaussian have relatively large values for all the options meaning that they capture more information than the other structures (Tables 6-8  . This can also be observed from the contour plots  which shows the strength of the dependency. Also, the differences in prices for all the copulas are very small showing that all the mixture captures the dependence structure of the returns.

11
Further research could consider the use of nonparametric or hierarchical archimedean copulas with gaussian mixture marginals. Also, the use of cross validation or numerical algebraic geometry may be used to find the optimal number of gaussian mixture components.

Data Availability
Data are available on request.