domain: stat qfin

text:
Camille Aït-Youcef
email: camille.ait-youcef@univ-nantes.fr

Marc Joëts
email: m.joets@ieseg.fr

The Role of Index Traders in the Financialization of Commodity Markets: A Behavioral Finance Approach

Keywords: JEL Classification: Q4, G41, G15, D81 Keywords: Commodity prices, speculation, agents-based model, nonlinear dynamics

This paper investigates the impact of financialization on commodity prices across various markets, particularly over recent decades. We introduce a groundbreaking theoretical model that incorporates both chartist-fundamentalist traders and institutional investors, targeting trading signals in two distinct commodity markets. In alignment with empirical data, our model enables institutional investors to participate in multiple markets simultaneously through index investing. Our findings indicate that the interactions between traditional traders and index investors create price dynamics that closely mirror observed patterns in commodity markets. Specifically, index investors not only cause prices to diverge from their fundamental values but also substantially influence the trading positions of other market actors. Moreover, we elucidate the crucial role of index investors in amplifying market correlations-both among different commodities and between commodities and equities, especially during periods of intense price fluctuations. Our innovative theoretical model goes beyond conventional chartist-fundamentalist frameworks, offering a robust alternative for understanding the complex pricing dynamics of commodities at large.

Introduction

The role of speculators in financial markets, particularly in commodity markets, has been a subject of intense debate among policymakers, media, and academics in recent years. There are traditionally two conflicting views on the intrinsic nature of speculation. The classical view, as exemplified by [START_REF] Keynes | Some aspects of commodity markets[END_REF] and [START_REF] Friedman | The case for flexible exchange rates[END_REF], posits that irrational speculators cannot survive long-term in the marketplace, as asset prices invariably reflect their fundamentals. 1 In contrast, the behavioral finance perspective contends that bounded rationality-manifested through herding behaviors, noise trading, speculative bubbles, and the like-can destabilize markets and cause prices to deviate from their fundamentals (see, [START_REF] Hirshleifer | Determinants of hedging and risk premia in commodity futures markets[END_REF], Shleifer & Summers (1990), De [START_REF] De Long | Noise trader risk in financial markets[END_REF][START_REF] Shiller | From efficient markets theory to behavioral finance[END_REF] to name fews). This debate has gained renewed urgency with the surge in commodity prices since the early 2000s and the increasing participation of financial institutions in commodity markets. Notable concerns have been directed toward institutional investors, such as hedge funds or index traders, who trade a wide array of financial products and generally operate with limited public disclosure. According to the U.S. Commodity Futures Trading Commission (CFTC), the scale of financial institutions' involvement in commodity index-related instruments expanded dramatically over the past decade. Such developments have stoked fears that the financialization of commodity markets is disrupting price mechanisms and cross-market linkages [START_REF] Tang | Index investment and the financialization of commodities[END_REF] and [START_REF] Büyükşahin | Speculators, commodities and cross-market linkages[END_REF]).

This paper aims to address these concerns by focusing on areas where the impact of financialization has significant social and economic repercussions, as outlined by [START_REF] Myers | On the costs of food price fluctuations in low-income countries[END_REF], [START_REF] Baumeister | Do oil price increases cause higher food prices?[END_REF][START_REF] Baumeister | Do oil price increases cause higher food prices?[END_REF][START_REF] Joëts | Heterogeneous beliefs, regret, and uncertainty: The role of speculation in energy price dynamics[END_REF]. For example, fluctuations in food prices present considerable challenges for low-income countries and prompt questions regarding the role of private financial interests in essential public utilities. The same holds true for the energy supply in low-income nations. Moreover, these issues are frequently discussed in political debates concerning the regulation of commodity derivatives.

A substantial body of research has sought to empirically investigate the role of financial institutions in shaping commodity prices. Some studies have primarily focused on the comovements between commodity and financial assets (see, e.g., [START_REF] Tang | Index investment and the financialization of commodities[END_REF], [START_REF] Silvennoinen | Financialization, crisis and commodity correlation dynamics[END_REF], [START_REF] Creti | On the links between stock and commodity markets' volatility[END_REF]), [START_REF] Büyükşahin | Speculators, commodities and cross-market linkages[END_REF], [START_REF] Brunetti | Speculators, prices, and market volatility[END_REF], [START_REF] Bruno | The financialization of food?[END_REF], and Le Pen & Sévi (2018)), while others have examined markets influenced by various structural shocks (see, e.g., [START_REF] Kilian | Not all oil price shocks are alike: Disentangling demand and supply shocks in the crude oil market[END_REF], [START_REF] Kilian | Why agnostic sign restrictions are not enough: understanding the dynamics of oil market var models[END_REF], and [START_REF] Juvenal | Speculation in the oil market[END_REF]). 2 The results are mixed and often hinge on how speculation is quantified. Some studies indirectly measure speculation through correlations across asset classes, while others capture it through the accumulation of commodity inventories. Remark-ably, none of these studies explicitly model the trading behaviors that operate within these markets. A common thread in earlier research is the limiting assumption of market efficiency, rational investors, and representative agents, often neglecting the heterogeneity among market participants. Although contributions in behavioral finance suggest that heterogeneous, boundedly rational speculators can endogenously generate complex price dynamics3 few theoretical attempts have been made to model commodity price movements. Notable exceptions include the chartist-fundamentalist models developed by [START_REF] He | Commodity markets, price limiters and speculative price dynamics[END_REF], Ter Ellen & Zwinkels (2010), and more recently [START_REF] Joëts | Heterogeneous beliefs, regret, and uncertainty: The role of speculation in energy price dynamics[END_REF]. Despite empirical evidence suggesting the significance of institutional investors, these models do not account for the role of index traders in commodity price dynamics. This paper aims to fill an existing research gap by modeling the role of index traders in commodity markets. We construct an agent-based model featuring two types of traders: boundedly rational chartist-fundamentalist traders and index traders. These actors operate in two distinct markets. Index traders allocate resources in both markets simultaneously, guided by index-based profit considerations. In contrast, chartist-fundamentalist traders engage in one market at a time, utilizing either technical analyses based on market trends or fundamental analyses grounded in supply and demand factors. We establish analytical conditions for the local asymptotic stability of the model's steady-state equilibrium and find that the incorporation of index traders leads to significant deviations of commodity prices from their fundamental values. Our model also produces price dynamics that more accurately reflect empirical statistical features of commodity markets, such as kurtosis, skewness, and clustered volatility, compared to traditional models that only include chartist and fundamentalist traders. Overall, our results underscore the importance of incorporating index traders in agent-based models to capture the financialization dynamics in commodity markets more accurately. We further propose several extensions that assess the role of speculation in influencing asset co-movements. These extensions emphasize the impact of index traders on evolving market correlations. Our model is versatile enough to be applied to virtually any commodity market where institutional investors are prevalent, serving as a robust tool for forecasting commodity price trends, especially in volatile environments.

The remainder of the paper is organized as follows: The next section briefly reviews the traditional chartist-fundamentalist approach, presents our agent-based model featuring index traders, and discusses stability and bifurcation properties. Section 3 outlines the model's calibration and analyzes the role of index traders in commodity price dynamics. Section 4 delves into the impact of index traders on commodity price co-movements, and Section 5 concludes the paper.

The theoretical model

Our objective is to provide an accurate representation of speculative behaviors in commodity markets. The architecture of our model relies on the chartist-fundamentalist framework, in which traders exhibit bounded rationality and operate based on predefined rules. Depending on market conditions, traders may generate trading signals from historical price trends (i.e., trend followers) or opt to capitalize on discrepancies between current prices and market fundamentals. Price dynamics are shaped by the interplay between these two rule-based behaviors, as described by [START_REF] Brock | Heterogeneous beliefs and routes to chaos in a simple asset pricing model[END_REF]. This section is divided into two parts for clarity: First, we offer a brief overview of the benchmark chartist-fundamentalist model as developed by [START_REF] Westerhoff | The effectiveness of keynes-tobin transaction taxes when heterogeneous agents can trade in different markets: a behavioral finance approach[END_REF]; second, we introduce an extension tailored to the specific realities of commodity markets, which includes the role of index traders.

The benchmark model

In the literature on Heterogeneous Agent Models (HAMs), markets are commonly characterized by two types of beliefs: chartists and fundamentalists [START_REF] Boswijk | Behavioral heterogeneity in stock prices[END_REF]). In our study, we build upon the work of [START_REF] Westerhoff | The effectiveness of keynes-tobin transaction taxes when heterogeneous agents can trade in different markets: a behavioral finance approach[END_REF] and posit that trading activities transpoire in two separate speculative markets, labeled as Market 1 and Market 2. Traders employing a chartist (or technical) strategy base their decisions on historical price movements in these assets, while fundamentalists choose to trade based on the asset's price convergence toward an equilibrium value, which is determined by the market fundamentals. The choice to invest in either Market 1 or Market 2 is influenced by the historical and current profitability of each respective strategy.

The aggregate price adjustment for Markets 1 and 2 at time t + 1 is described by the following log-linear equations: ∼ N (0, σ P 1 ) and S P 2 t ∼ N (0, σ P 2 ) for each respective market.

P 1 t+1 = P 1 t + a(W C 1 t D C 1 t + W F 1 t D F 1 t ) + S P 1 t (1) P 2 t+1 = P 2 t + a(W C 2 t D C 2 t + W C 2 t D F 2 t ) + S P 2 t (2) P 1 t+1 , P 2 t+2 , P
At any given moment, market price dynamics adjust in response to the buying and selling behaviors exhibited by each category of traders. Excessive buying or selling will influence the aggregate price of the corresponding market in an upward or downward direction. The demands originating from chartists and fundamentalists in Markets 1 and 2 are defined as

D C i t = c(P i t -P i t-1 ) + S C i t (3) D F i t = f (F i t -P i t ) + S F i t ( 4
)
where i = 1, 2 for markets i; S C i t ∼ N (0, σ C i ) and S F i t ∼ N (0, σ F i ) capture deviations from the first deterministic component. P signifies the log-linear price, while c > 0 quantifies the potency of the price signal in the chartist strategy. For any given asset, a P t > P t-1 condition in Equation (3) implies a positive signal, suggesting a buy action. In Equation ( 4), F i t represents the fundamental value at time t for each market i. The parameter f > 0 measures the strength of the price deviation relative to its fundamental value, thereby signaling either market undervaluation (P < F ) or overvaluation (P > F ), which in turn suggests either a buying or selling rule, respectively.

Trading behaviors are not static and are permitted to transition between rules within each trading cycle. This switching mechanism is a crucial component of Heterogeneous Agent Models (HAMs) as it accounts for non-linear adjustments in pricing. Specifically, the switching mechanism-also referred to as 'fitness'-in market i is predicated on the performance of each trading rule and can be described as follows:

A C i t = (exp[P i t ] -exp[P i t-1 ])D C i t-2 + bA C i t-1
(5)

A F i t = (exp[P i t ] -exp[P i t-1 ])D F i t-2 + bA F i t-1 (6)
The investment timing stipulates that orders submitted during period t -2 are executed at prices from period t -1, and the profits are contingent on prices in period t (as captured in the first part of the equations). In the second part of the equations, the memory parameter b gauges the extent to which current fitness relies on past gains for strategic selection.5 Therefore, the fitness or switching rule for strategies hinges on both current and past performance metrics.

In market i, the market shares for each type of trader-specifically W F i for fundamentalists and W C i ) for chartists-are influenced by their relative appeal. To ascertain these proportions, we adopt the discrete choice approach as laid out by [START_REF] Brock | Heterogeneous beliefs and routes to chaos in a simple asset pricing model[END_REF].

W ij t = exp rA ij t V (7) W 0 t = 1 -W C 1 t -W F 1 t -W C 2 t -W F 2 t ( 8
)
where

V = exp rA C 1 t +exp rA C 2 t +exp rA F 1 t +exp rA F 2 t +exp[0], i = 1
, 2 and j = C, F . The parameter r > 0 gauges the sensitivity of traders when selecting the most viable strategy; specifically, higher values of r lead to more traders opting for the strategy with the highest fitness.

In the benchmark model, chartists and fundamentalists are limited to investing in a single market at any given time. Consequently, at each decision-making step, traders must select both the trading rule and the target market based on performance metrics. However, in commodity markets, a prevalent strategy for institutional investors involves diversifying their portfolio by investing in an index of multiple commodities. Known as index traders, these investors emphasize strategic asset allocation between classes of commodities and other financial assets like equities and bonds. They typically enter and exit positions in all commodities within a given index simultaneously. Our extended model incorporates this nuanced trading behavior, shedding light on the impact it has on both the price dynamics and co-movements within commodity markets.6

Heterogenous agent model with index traders: the HAM-IT

Commodity indices are generally constructed based on the values of nearby futures contracts with a delivery period extending beyond one month. This approach avoids the costs associated with holding physical commodities. At the conclusion of each maturity period, the commodity index outlines a so-called "roll," which involves replacing the current contract (known as the front month) with the subsequent contract. This mechanism facilitates passive long positions in the listed commodities. The two most prevalent commodity indices are the SP GSCI and the DJ-UBSCI. Index traders typically mimic the index by diversifying their portfolios across a basket of commodity markets. Upon reaching maturity, they often roll their positions forward, moving from one futures contract to another. This process creates a "weight-of-money" effect on prices, a phenomenon where large position changes by individual market participants can significantly influence market prices. Index traders are generally indifferent to price movements in individual commodities since they allocate their funds across all markets within the index based on strategic portfolio considerations.

The aim of our extension is to model the trading strategies of index traders and to analyze their impact on the dynamics of commodity prices. Given that these traders participate in commodities futures and options markets for reasons other than hedging against specific commodity risks, our model provides a direct account of speculative activity (see citations [START_REF] Brunetti | Speculators, prices, and market volatility[END_REF] and [START_REF] Bruno | The financialization of food?[END_REF] for reference).

To align with market realities, our model assumes that Markets 1 and 2 are part of a commodity index. Consequently, when index traders take positions in the index, they are effectively investing in both markets simultaneously.

The commodity index at time t is defined as

I t = 1 K K k=1 P kt
In this formulation, K = 2 represents the total number of markets included in the index. The index is calculated as an arithmetic equally weighted average of the two commodity futures prices. It should be noted that the simulations presented in the paper are robust to variations in the weighting scheme. Additional results supporting this claim are available upon request from the authors. Index traders make investment decisions based on the historical performance of the commodity index, as elaborated further in the following sections.

D I i t = 1 2 × d(I t -I t-1 ) + S I i t (9)
where d > 0 is the strength of the signal from t -1 to t, and S I i t ∼ N (0, σ I i ). i is for market 1 and 2 respectively. At each step, the performance of the strategy is evaluated by the following fitness equation

A I t+1 = (exp[P 1 t+1 ] -exp[P 1 t ])D I 1 t-1 + (exp[P 2 t+1 ] -exp[P 2 t ])D I 2 t-1 + bA I t ( 10
)
In accordance with the timing framework set forth by [START_REF] Westerhoff | The effectiveness of keynes-tobin transaction taxes when heterogeneous agents can trade in different markets: a behavioral finance approach[END_REF], the fitness measure incorporates both the historical and most recent market performances for each market i. Orders submitted at time t -1 are executed at t. The parameter b serves as a memory factor, quantifying the influence of past performances on the current trading strategy. Specifically, as b → 1 the fitness is calculated as the cumulative sum of all past profits.

To streamline the discussion, the trading rule for index traders is succinctly outlined as follows:

z t+1 = I t v t+1 = z t A I t+1 = (exp[P 1 t+1 ] -exp[P 1 t ])( 1 2 × d(z t -v t ) + S I 1 t-1 ) + (exp[P 2 t+1 ] -exp[P 2 t ])( 1 2 × d(z t -v t ) + S I 2 t-1 ) + bA I t
The market shares of index traders in market i are exogenous and range from approximately 15% to 50% of the commodity market positions.7 This share varies depending on the composition of the commodity index and shifts in relation to futures markets. According to their trading rules, investors either opt to invest (i.e., take a long position) or abstain from trading in the commodity markets.8 The market shares of index traders (W I t + W -I t ) are calibrated based on the works of [START_REF] Mayer | The growing financialisation of commodity markets: Divergences between index investors and money managers[END_REF] and [START_REF] Gilbert | Agent-based models[END_REF]. Utilizing the average percentage of total open interest held by index investors in commodity futures, we set the market shares of index investors at 30%. We define the market share of index investors as follows:

W I t = 0.3 × exp rA I t V I (11)
with

V I = exp rA I t + Exp [0]
then,

W -I t = 0.3 -W I t
Investment from institutional investors has no direct linkage with the investment from other investors. Thus, we change the weights of chartists and fundamentalists discussed in the baseline model corresponding to 70% and no more 100% of the total market shares.

W C 1 t = 0.7 × Exp rA C 1 t V G , W C 2 t = 0.7 × Exp rA C 2 t V G , W F 1 t = 0.7 × Exp rA F 1 t V G , W F 2 t = 0.7 × Exp rA F 2 t V G , V G = Exp rA C 1 t + Exp rA C 2 t + Exp rA F 1 t + Exp rA F 2 t + Exp[0] W 0 t = 0.7 -W C 1 t -W C 2 t -W F 1 t -W F 2 t

Dynamic and steady state of the HAM-IT

Both the dynamic and steady-state outcomes are shaped by incorporating the behavior of index traders into the benchmark model. The dynamical system of HAM-IT is defined by the following set of eleven equations, where the exponent "I" signifies the behavior of index traders. Appendix A offers further details on the analytical solutions and elaborates on the characteristics of the steady state.

P 1 t+1 = P 1 t + a(W C 1 t (c(P 1 t -x 1 t ) + S C 1 t ) + W F 1 t (f (F 1 -P 1 t ) + S F 1 t ) + W I t 1 2 d( 1 2 (P 1 t + P 2 t ) + S I 1 t - 1 2 (x 1 t + x 2 t ) + S I 1 t-1 ) + S I 1 t ) + S P 1 t x 1 t+1 = P 1 t y 1 t+1 = x 1 t P 2 t+1 = P 2 t + a(W C 2 t (c(P 2 t -x 2 t ) + S C 2 t ) + W F 2 t (f (F 2 -P 2 t )S F 2 t ) + W I t 1 2 d( 1 2 (P 1 t + P 2 t ) + S I t - 1 2 (x 1 t + x 2 t ) + S I t-1 ) + S I 2 t ) + S P 2 t x 2 t+1 = P 2 t y 2 t+1 = x 2 t A C 1 t+1 = (exp[P 1 t+1 ] -exp[P 1 t ])(c(x 1 t -y 1 t ) + S C 1 t-1 ) + bA C 1 t A F 1 t+1 = (exp[P 1 t+1 ] -exp[P 1 t ])(f (F 1 -x 1 t ) + S F 1 t-1 ) + bA F 1 t A C 2 t+1 = (exp[P 2 t+1 ] -exp[P 2 t ])(c(x 2 t -y 2 t ) + S C 2 t-1 ) + bA C 2 t A F 2 t+1 = (exp[P 2 t+1 ] -exp[P 2 t ])(f (F 2 -x 2 t ) + S F 2 t-1 ) + bA F 2 t A I t+1 = (exp[P 1 t+1 ] -exp[P 1 t ])( 1 2 d( 1 2 (x 1 t + x 2 t ) + S I t-1 -( 1 2 (y 1 t + y 2 t ) -S I t-1 ) + S I 1 t )) + (exp[P 2 t+1 ] -exp[P 2 t ])( 1 2 d( 1 2 (x 1 t + x 2 t ) + S I t-1 -( 1 2 (y 1 t + y 2 t ) -S I t-1 ) + S I 2 t )) + bA I t
Assuming that the reaction parameters a, c, d, and f are strictly positive, and the memory parameter b lies in the range 0 ≤ b < 1, the dynamical variables at the unique steady state can be obtained from the system as follows:

P 1 = x1 = ȳ1 = F 1 P 2 = x2 = ȳ2 = F 2 ĀC 1 = ĀF 1 = ĀC 2 = ĀF 2 = ĀI = 0
In the long run, the last expression indicates that both prices and the index converge to their fundamental levels. Agents do not realize any profits, leading to a null average realized profit for all market participants. Consequently, at the steady state, the market share is as follows:

W C 1 = W F 1 = W C 2 = W F 2 = W I = W I0 = W 0 = 1 7
The local stability of the steady state in the HAM-IT model is determined through the derivation of the Jacobian matrix. Further details on the derivation and the eigenvalues of this matrix, as well as the necessary conditions for local stability, are provided in Appendix A. We demonstrate that the model's stability is contingent on the behavior of index traders, specifically the parameter d. When d → 0, the steady state is locally stable in the long run-meaning all eigenvalues have a modulus less than one under the following conditions:

7 4a > c √ 7 √ 7 -4ac > 0
When d → ∞, the eigenvalue λ 6 exceeds one in modulus, leading to explosive price behavior in the long run. This finding underscores the destabilizing role of index traders in the dynamics of commodity prices. Our simulations indicates that the Hopf bifurcation occurs at the threshold level of d = 13.31 assuming that a = 1 and c = f = 0.05.

Dynamics of commodity prices with index traders

After establishing the asymptotic properties of our model, we turn our focus to exploring how Heterogeneous Agent Models (HAMs) enhance our comprehension of the impact of speculation on commodity prices. To achieve this, we carry out extensive numerical simulations across various scenarios and benchmark our results against the statistical attributes of selected 1-month futures contracts for commodities including crude oil, natural gas, copper, gold, corn, and cocoa.9 These simulations yield 5,000 daily price observations, covering a timespan of approximately 20 years from 2000 to 2020.

We begin by outlining the calibration framework for each model (i.e., the HAM-IT and the benchmark) and present some preliminary results. Following this, we engage in a thorough Monte Carlo analysis to assess which models are most effective in aligning with the statistical properties of actual commodity prices. This analysis enables us to delve into how index traders exacerbate price fluctuations and influence trading strategies, thereby illuminating the financialization of commodity markets.

Models' calibration and preliminary results

Our dual simulation experiments rely on the parameter settings outlined in Table 1 for the benchmark model, and in Table 2 for the HAM-IT model. Although both tables are largely similar, they differ in parameters associated with the behavior of index traders-specifically, the strength of the index signal (d), the deviation from the deterministic component (σ I ), and traders' sensitivity to performance (r). Key parameters in the model are those that quantify the impact of each investor type on price-namely, a c , a f , and d for chartists, fundamentalists, and index traders, respectively. While calibration can be a critical step, numerous empirical studies suggest that the value for these parameters typically falls within the range of 0 to 0.1 (see, for example, [START_REF] Westerhoff | Nonlinearities and cyclical behavior: the role of chartists and fundamentalists[END_REF], [START_REF] He | Commodity markets, price limiters and speculative price dynamics[END_REF], and Dieci & He ( 2018)). To generate plausible dynamics and maintain consistency with the benchmark model, we follow [START_REF] Westerhoff | The effectiveness of keynes-tobin transaction taxes when heterogeneous agents can trade in different markets: a behavioral finance approach[END_REF] and [START_REF] Reitz | Commodity price cycles and heterogeneous speculators: a star-garch model[END_REF], selecting a value of 0.05 for each investor category. 10 Another crucial parameter is traders' sensitivity in choosing the optimal strategy, denoted as r. In our study, r is set at 250 for the benchmark model and 60 for our extension. 11

Table 1: Parameters setting for the benchmark HAM a = 1 a c = 0.05 a f = 0.05 b = 0.975

σ P 1 = σ P 2 = 0.01 r = 250 F 1 = 0 F 2 = 0 σ C 1 = σ C 2 = 0.05 σ F 1 = σ F 2 = 0.05
10 Simulation results are robust to other values within this range. Additional results are available upon request from the authors.

11 The value of r was chosen to yield reasonable dynamics, as supported by [START_REF] Westerhoff | The effectiveness of keynes-tobin transaction taxes when heterogeneous agents can trade in different markets: a behavioral finance approach[END_REF].

Table 2: Parameters setting for HAM-IT

a = 1 a c = 0.05 a f = 0.05 σ P 1 = σ P 2 = 0.01 r = 60 F 1 = 0 F 2 = 0 σ C 1 = σ C 2 = 0.05 b = 0.975 σ F 1 = σ F 2 = 0.05 d = 0.05 σ I 1 = σ I 2 = 0.05
Figure 1 presents our simulated log price and return series for market 2, comparing the benchmark model (left panel) with the HAM-IT model (right panel).12 In both models, log prices appear to oscillate around their fundamental levels of zero, exhibiting intermittent phases of boom and bust. For example, prices in the HAM-IT model are predominantly undervalued relative to fundamentals from period 0 to 3000, and overvalued from period 3500 to 4000. The last two panels show log price returns for market 2, derived from each model. These returns serve as proxies for market volatility. As evidenced, fluctuations in price returns are significant throughout the observed period. Intriguingly, while the benchmark model does not display a clear pattern, the HAM-IT model manifests clustering in return volatility. In line with the behavior often observed in commodity markets, this suggests that periods of low volatility alternate with periods of high volatility (see, [START_REF] Creti | On the links between stock and commodity markets' volatility[END_REF]).

Figure 1: Simulated log prices and returns for market 2

Note: These figures report simulated log prices and returns for market 2 using parameters setting for the benchmarck HAM (see Table 1) and HAM with index investors (see Table 2). Black line is the fundamental value.

The dynamics of trading strategies fundamentally drive price behavior over time. We hypothesize that the positions taken by index traders not only influence price dynamics but also shift the trading positions of traditional investors, namely chartists and fundamentalists.

To test this hypothesis, Figure 2 displays radar plots of the trading weights for chartists and fundamentalists, comparing results from the benchmark model (shown in orange) with those from the HAM-IT model (in blue). The top panel represents the average weights across our 5,000 simulations, while the bottom panel illustrates the range between the maximum and minimum weights. Additionally, we present these metrics at various quantile levels (e.g., 0.05, 0.25, 0.5, 0.75, and 0.95) to capture shifts in trading strategies more comprehensively. On average, the trading weights in the benchmark model remain relatively consistent across quantiles. In contrast, the HAM-IT model reveals a pronounced shift toward extreme positions for both chartists and fundamentalists, as seen in the top panel. This indicates that the inclusion of index traders generally pushes traditional investors into taking more extreme positions. Moreover, the gap between the maximum and minimum trading positions also diverges substantially into extreme regions under the influence of index traders. Overall, the presence of index traders in the market prompts traditional investors to significantly amplify their extreme long positions on price, potentially leading to heightened price fluctuations.

Stylized facts on commodity price dynamics

To further assess the accuracy of our model in capturing the statistical properties of real commodity prices, we conduct an exhaustive Monte Carlo analysis using our simulated price data.

Building on the work of [START_REF] Westerhoff | The effectiveness of keynes-tobin transaction taxes when heterogeneous agents can trade in different markets: a behavioral finance approach[END_REF], speculative markets can be characterized by four key features13 : (i) the presence of bubbles and crashes, leading to price distortions; (ii) elevated levels of volatility and clustering; (iii) pronounced kurtosis, making the distribution leptokurtic; and (iv) temporal dependence manifested by strong autocorrelation in returns.

To quantify these characteristics, we employ the following statistical measures14 : minimal and maximal daily returns; return volatility V calculated as the average of absolute returns; distortion D relative to fundamental values; kurtosis K; and the coefficients of autocorrelation for both returns ac i r and absolute returns ac i |r| .

Table 3 presents the estimated statistics for artificial agricultural futures returns in market 2, generated by the HAM-IT model across the 5, 25, 50, 75, and 95 percent quantiles. 15 The statistics from the benchmark model are shown in parentheses for comparison. Each model comprises 1,000 simulations, each with 5,000 observations. Broadly speaking, key statistics suggest that HAM-IT yields higher estimates compared to the benchmark model. The range of extreme negative returns varies between -14.89 and -7.37 percent, and extreme positive returns fluctuate between 7.42 and 14.86 percent. While the models don't differ significantly in terms of volatility, both mispricing and kurtosis estimates from HAM-IT substantially exceed those from the benchmark. For example, mispricing at the 95th percentile is more than tenfold higher in HAM-IT (160.63 percent vs. 10.98 percent). Furthermore, at least 95 percent of our simulations display excess kurtosis of 12.81. Autocorrelation coefficients for returns are not significant in either model. However, the autocorrelation coefficients for absolute returns are both significant and higher in HAM-IT, especially more than 90 percent of the time (0.29 vs. 0.03 at the 95th percentile quantile). The persistence of these autocorrelation functions across the five lags considered reveals long memory in volatility dynamics. In summary, incorporating index traders into the traditional Heterogeneous Agent Model (HAM) appears to significantly amplify price fluctuations and results in an unstable, non-fundamental steady state commonly referred to as Hopf bifurcation (see [START_REF] Brock | A rational route to randomness[END_REF]). Consequently, price movements during certain periods can deviate substantially from their fundamental values, i.e., supply-demand conditions, resembling the behavior of speculative assets.

We now assess the model's efficacy by juxtaposing our estimated statistics with those of real commodity futures prices from 2000 to 2020. We consider a diverse set of commodity future prices at 1-month intervals, including crude oil, natural gas, copper, aluminium, corn, cocoa, gold, and silver. 16 Table 4 summarizes the statistics of these commodity prices. Evidently, extreme returns vary significantly across all markets. For instance, corn price returns fluctuate between -26.36 and 12.72. Such extreme returns, coupled with high volatility-ranging from 1.21 to 2.20-point to substantial price distortions. All markets exhibit either high or extremely high kurtosis, ranging from 7.30 to 20.30. Except for gold, the autocorrelation co-efficients of absolute returns display strong temporal dependence in volatility. A comparison between Table 3 and Table 4 demonstrates that our HAM-IT model more accurately replicates the statistical properties of commodity futures prices, capturing crucial stylized facts of speculative markets. Overall, our findings suggest that incorporating index traders enhances the model's ability to faithfully represent commodity price behaviors when compared to existing models. Index traders' strategies impact not only the behaviors of other traders in the market but also the dynamics of asset prices, making markets more erratic. In the subsequent section, we utilize HAM-IT as a computational laboratory to conduct artificial experiments investigating the role of index traders in commodity price co-movements. Note: Statistics estimated from the benchmark model are shown in parentheses. rmin and rmax respectively represent the minimum and maximum daily returns, measured as log price changes. V denotes volatility, defined as the average absolute return. D is the distortion coefficient, representing the deviation of observed log prices from their fundamental value. K stands for kurtosis. ac i r and ac i |r| where i = 1, .., 5, are the autocorrelation coefficients for daily returns and absolute daily returns, ranging from lag 1 to lag 5. Note: r min and r max represent the minimum and maximum daily returns, respectively, calculated using log price changes. V denotes volatility, defined as the average absolute return. Since the fundamental values of these markets are unknown, distortion cannot be computed. K stands for kurtosis. ac i r and ac i |r| where i = 1, .., 5, are the autocorrelation coefficients for daily returns and absolute daily returns from lag 1 to lag 5.

Index traders and commodity prices co-movements

A salient feature of the financialization of commodities is the heightened co-movements among various market prices. Financial institutions, such as index traders or hedge funds, often trade across a diverse range of asset classes, facing fewer restrictions compared to tradi-tional commodity traders. Their trading activities thus frequently induce significant risk transfer between markets, amplifying price co-movements [START_REF] Basak | A model of financialization of commodities[END_REF], [START_REF] Rahi | Strategic financial innovation in segmented markets[END_REF]). The literature has both theoretically and empirically examined this phenomenon, considering co-movements both within commodities [START_REF] Joëts | Energy price transmissions during extreme movements[END_REF], [START_REF] Büyükşahin | Speculators, commodities and cross-market linkages[END_REF], [START_REF] Brunetti | Speculators, prices, and market volatility[END_REF], [START_REF] Bruno | The financialization of food?[END_REF]) and between equity and commodity markets [START_REF] Creti | On the links between stock and commodity markets' volatility[END_REF], [START_REF] Tang | Index investment and the financialization of commodities[END_REF], [START_REF] Silvennoinen | Financialization, crisis and commodity correlation dynamics[END_REF], [START_REF] Cheng | Convective risk flows in commodity futures markets[END_REF], Le Pen & Sévi ( 2018)). Index traders, particularly when motivated by portfolio rebalancing needs and leverage constraints, appear to markedly enhance the propagation of shocks across markets, especially during periods of extreme volatility [START_REF] Vayanos | Limits of arbitrage: The state of the theory[END_REF], Aït-Youcef ( 2019)).

In the upcoming simulated experiments, our objective is to ascertain the extent to which index traders exacerbate market co-movements. To this end, we extend our HAM-IT model to incorporate a third market.17 We focus on three distinct trading scenarios:

1. Scenario 1 -No Restrictions on All Trading Strategies (Baseline Model): In this case, both traditional traders (fundamentalists and chartists) and index traders have the liberty to invest in all three markets.

2. Scenario 2 -Restrictions on Fundamentalists and Chartists Only: Here, traditional traders are constrained to investments in only the first two markets, whereas index traders remain unrestricted.

3. Scenario 3 -Restrictions on Index Traders Only: In this scenario, fundamentalists and chartists can invest across all markets, but index traders are limited to participating in the first two markets.

Each trading scenario serves as a lens through which we can evaluate the influence of index investors on the co-movements in various contexts: (i) commodity futures prices within the same asset class; (ii) a basket of similar commodity prices in relation to other commodities or asset classes; and (iii) indexed commodities as compared to those that are off-index. The dynamic systems governing these extensions are comprehensively outlined in Appendix C.

No restrictions on trading strategies

In the first extension, each asset is considered as an individual commodity futures price of the same class, and we construct the index as an equally weighted amalgamation of these prices. 18 Aligning with industry standards, the commodity index is formed based on the values of nearby futures contracts that have delivery times exceeding one month. To streamline the model, we abstain from incorporating the 'roll-over' process, which entails substituting an expiring contract with the next available one upon maturity. In this extension, both traditional traders-comprising fundamentalists and chartists-and index traders have the freedom to invest in all available assets. The primary distinction lies in their investment scope: while fundamentalists and chartists concentrate their investments on single assets, index traders are free to diversify their positions across all commodities represented in the index. Consequently, index investors predominantly focus on broader asset allocation and display minimal sensitivity to the fluctuations in individual commodity prices. This model serves as an expansion of the framework delineated in Section 2 and includes three markets. For further elaboration, readers are directed to Appendix C.1.

Table 5 presents estimated correlations of commodity prices across the three scrutinized markets, calculated for different quantiles through the unrestricted HAM-IT model. For comparative purposes, we also furnish-in parentheses-correlations derived from the benchmark model, which includes only chartists and fundamentalists. The objective is to determine whether integrating index traders into the HAM-IT model significantly boosts market correlations, thereby more accurately reflecting the complex dynamics among commodity assets.

Our findings affirm that HAM-IT surpasses the benchmark model in capturing elevated correlations. Importantly, consistent with existing empirical studies [START_REF] Gorton | Facts and fantasies about commodity futures[END_REF], [START_REF] Bhardwaj | The business cycle and the correlation between stocks and commodities[END_REF], [START_REF] Creti | On the links between stock and commodity markets' volatility[END_REF], [START_REF] Joëts | Energy price transmissions during extreme movements[END_REF], and Aït-Youcef ( 2019)), we identify robust correlations for each pair of assets in over 95 percent of our simulationsnamely, 0.632, 0.610, and 0.664-compared to notably weaker correlations in the benchmark model, with values of 0.268, 0.263, and 0.258, respectively.

Restrictions on fundamentalists and chartists

In the second extension, we narrow our focus to two markets that represent commodity futures prices within the same class, such as agriculture, and introduce a third market representing a different asset class-for instance, crude oil. Crude oil is a noteworthy asset in this context due to its substantial weight in popular commodity indices. However, the same framework could be extended to cover other commodities or asset classes, including industrial metals, precious metals, equities, or bonds. In this configuration, traditional traders (fundamentalists and chartists) are confined to investing solely in the two agricultural markets, representing their area of specialization. Conversely, index traders are not subject to these limitations and can freely invest across all available markets via index allocation. The portfolio strategy for index traders could thus incorporate combinations such as agricultural-energy or agricultural-equity blends, offering greater diversification. 19 The primary goal of this extension is to investigate whether the inclusion of index traders amplifies correlations across disparate types of commodities or between commodities and other financial assets. Additional elaborations on this theme can be found in Appendix C.2.

The findings are summarized in Table 6, which highlights the significant increase in correlations across all asset classes in the presence of index traders, as modeled in HAM-IT. Our results are consistent with previous findings for the correlation between P1 and P2, the two agricultural futures prices, as showcased in Table 5. Notably, there is also a substantial rise in correlations between different asset classes, namely P1/P3 and P2/P3. These augmented correlations appear in more than 90 percent of our simulations, with values reaching 0.630 and 0.624, respectively. These findings reinforce the conclusions reached in earlier studies, such as that by [START_REF] Brunetti | Speculators, prices, and market volatility[END_REF], affirming that index traders play a pivotal role in amplifying co-movements across diverse asset classes. Note: This table presents the estimated inter-market correlations of commodity returns across various quantiles, as generated by the HAM-IT model. For comparative purposes, the correlations calculated using the benchmark model are included in parentheses.

Restrictions on index traders

The third extension of the study focuses on understanding how index traders influence the co-movement of prices between commodities that are part of an index (in-index) and those that are not (off-index). The chosen scenario models three commodity futures prices: P1 and P2 are part of an index (in-index), while P3 is not (off-index). The investment behavior of index traders is restricted to P1 and P2, while traditional traders (chartists and fundamentalists) can invest in any of the three markets. Additional details are provided in Appendix C.3.

Table 7 presents the correlations between these assets as determined by the HAM-IT model, with the benchmark model's correlations included in parentheses for comparison. Our findings reveal a pronounced difference in the correlation patterns between in-index and off-index commodities. Specifically, the correlation between P1 and P2 is significantly higher than that between P1 and P3 or P2 and P3. These results are consistent in over 90 percent of our simulations, showing correlations of 0.630 for P1-P2 compared to 0.456 and 0.461 for P1-P3 and P2-P3, respectively. This confirms that the presence of index traders in commodity markets increases the correlation of in-index commodities. Most notably, our extended HAM-IT model more accurately captures the nuanced co-movement patterns of indexed commodity prices compared to the benchmark model. This implies that our approach offers a more appropriate framework for understanding the dynamics and correlations among commodity prices, particularly in the context of financialization and the role of index traders. Note: This table presents the estimated inter-market correlations of commodity returns across various quantiles, as generated by the HAM-IT model. For comparative purposes, the correlations calculated using the benchmark model are included in parentheses.

Conclusion

The dynamics of commodity prices and market co-movements-both within the commodityequity space and across different commodities-have seen a sharp increase over recent decades.

Given the global implications of these fluctuations, understanding their causes has become a topic of critical importance. This is especially true in the current climate of global uncertainty and inflation, driven by factors such as the Russia-Ukraine conflict and the COVID-19 pandemic. The surge in commodity prices has fueled debates among scholars and policymakers alike, questioning the role of financial participants in magnifying market responses to fundamental shifts in supply and demand.

In this paper, we explore the underlying causes of commodity price movements through an agent-based framework, inspired by the works of [START_REF] Reitz | Commodity price cycles and heterogeneous speculators: a star-garch model[END_REF] and [START_REF] Joëts | Heterogeneous beliefs, regret, and uncertainty: The role of speculation in energy price dynamics[END_REF].

We introduce a novel behavioral agent-based model that features three distinct categories of traders: chartists, fundamentalists, and a new class called index traders. Designed to encapsulate the financialization of commodity markets, our theoretical model incorporates index traders alongside traditional market participants. Index traders, in keeping with observed market behavior, engage in multi-commodity investments via index funds and act as catalysts for price changes. The interplay between these various trading strategies and the influence of index traders lead to complex market dynamics. Our model aims to provide a more nuanced understanding of these intricacies, enriching the current academic discourse.

Simulations indicate that our model adeptly replicates the statistical characteristics of commodity markets, capturing features such as volatility clustering, leptokurtic returns, and episodes of bubbles and crashes. When compared to traditional models that only incorporate chartists and fundamentalists, our approach-enhanced by the inclusion of index traders-offers a more accurate representation of the financialization of commodity markets. The model reveals that the actions of index traders not only divert prices from their fundamentals but also influence the trading positions of other traditional market participants. Moreover, our extensions demonstrate that the presence of index traders substantially heightens asset comovements, both within the realm of cross-commodity and between commodity and equity markets. This underscores the limitations of existing literature in adequately capturing the complexities of commodity pricing. The HAM-IT model can be reasonably extended to other commodities and asset classes influenced by institutional investors. As such, it offers a robust, micro-founded alternative to conventional models, providing both industry professionals and scholars with a more nuanced tool for predicting and explaining the future trajectories of commodity prices. Future research could explore this model's implications for finance and operations research, among other potential applications.

Appendix

A Local stability of the steady state in HAM-IT

The Appendix provides both the derivation of the Jacobian Matrix associated with our HAM-IT model and an analysis of the eigenvalues at the steady state.

Simplifying the dynamic system outlined in Section 2.2.1 and omitting the random terms, we obtain the following:

P 1 t+1 = P 1 t + a(W C 1 t (c(P 1 t -x 1 t )) + W F 1 t (f (F 1 -P 1 t )) + W I t 1 2 d( 1 2 (P 1 t + P 2 t ) - 1 2 (x 1 t + x 2 t )))
x

1 t+1 = P 1 t y 1 t+1 = x 1 t P 2 t+1 = P 2 t + a(W C 2 t (c(P 2 t -x 2 t )) + W F 2 t (f (F 2 -P 2 t )) + W I t 1 2 d( 1 2 (P 1 t + P 2 t ) - 1 2 (x 1 t + x 2 t ))) x 2 t+1 = P 2 t y 2 t+1 = x 2 t z t+1 = I t = 1 2 (P 1 t + P 2 t ) v t+1 = z t = 1 2 (x 1 t + x 2 t ) A C 1 t+1 = (exp[P 1 t+1 ] -exp[P 1 t ])(c(x 1 t -y 1 t )) + bA C 1 t A F 1 t+1 = (exp[P 1 t+1 ] -exp[P 1 t ])(f (F 1 -x 1 t )) + bA F 1 t A C 2 t+1 = (exp[P 2 t+1 ] -exp[P 2 t ])(c(x 2 t -y 2 t )) + bA C 2 t A F 2 t+1 = (exp[P 2 t+1 ] -exp[P 2 t ])(f (F 2 -x 2 t )) + bA F 2 t A I t+1 = (exp[P 1 t+1 ] -exp[P 1 t )( 1 2 d(z t -v t )) + (exp[P 2 t+1 -exp[P 2 t )]( 1 2 d(z t -v t )) + bA I t
For brevity, we use a prime symbol to denote variables at time t+1. Specifically P ′i represents the price at time t + 1 for market i where i = 1, 2. We proceed to examine the partial derivatives of P ′i with respect to P i , x ′i , and y i .

∂P ′1 ∂P 1 = 1 + a(W C 1 c -W F 1 f + 1 4 W I d) ∂P 1 ′ ∂x 1 = -a(W 1C c + 1 4 W I d)
which gives at the steady state

∂P ′1 ∂P 1 s.s. = 1 + 1 7 a(c -f + d 4 ); ∂P ′1 ∂x 1 s.s. = - 1 7 a(c + d 4 )
while

∂P ′1 ∂P 2 = - ∂P ′1 ∂x 2 = 1 28 ad ∂P ′1 ∂y 1 = ∂P ′1 ∂y 2 = 0
Similarly, at the steady state,

∂P ′2 ∂P 2 s.s. = 1 + 1 7 a(c -f + d 4 ), ∂P ′2 ∂x 2 s.s. = - 1 7 a(c + d 4 )
and

∂P ′2 ∂P 1 = - ∂P ′2 ∂x 1 = 1 28 ad ∂P ′2 ∂y 1 = ∂P ′2 ∂y 2 = 0
The partial derivatives of P ′i with respect to A K i and, with K = C, F, I and i = 1, 2 are

∂P ′i ∂A K i = a c(P i -x i ) ∂W C i ∂A C i + f (F i -P i ) ∂W F i ∂A K i + 1 2 d( 1 2 (P 1 t + P 2 t ) - 1 2 (x 1 t + x 2 t )) ∂W I ∂A I
At the steady state P i = x i = F i . Therefore, all these partial derivatives vanish (i.e.

∂P ′i

∂A K i = 0).

Let's define U i ≡ (Exp P ′i -Exp P i ) for markets i = 1, 2. This expression vanishes at the steady state because P ′i = P i . Therefore, the partial derivatives of A ′K i with respect P i , x i and y i at the steady state are as follows:

∂A ′C i ∂P i = c(x i -y i ) ∂U i ∂P i = 0; ∂A iF ′ ∂P i = f (F i -x i ) ∂U i ∂P i = 0, ∂A ′C i ∂x i = c (x i -y i ) ∂U i ∂x i + U i = 0; ∂A ′F i ∂x i = f ( P i -x i ) ∂U i ∂x i -U i = 0, ∂A ′C i ∂y i = -cU i = 0; ∂A ′F i ∂y i = 0 For j = 1, 2, i = 1, 2 and i = j ∂A ′C i ∂P j = ∂A ′F i ∂P j = ∂A ′C i ∂x j = ∂A ′F i ∂x j = ∂A ′C i ∂y j = ∂A ′F i ∂y j = 0
It turns out that due to (x i -y i ) and (F i -x i ) in the system, all partial derivatives are null at the steady state with exception of

∂A ′C 1 ∂A C 1 = ∂A ′F 1 ∂A F 1 = ∂A ′C 2 ∂A C 2 = ∂A ′F 2 ∂A F 2 = b
Considering the partial derivatives of A ′I , with respect to P i , x i and y i , with i = 1, 2 and the variable U i ≡ (Exp P ′i -Exp P i ) = 0 at the steady state (with P i ′ = P i ), we find that:

∂A ′I ∂P i = ∂A ′I ∂x i = ∂A ′I ∂y i = ∂A ′I ∂A K i = 0; ∂A ′I ∂A I = b
The Jacobian matrix of partial derivatives considering the following variable ordering P

1 , x 1 , y 1 , P 2 , x 2 , y 2 , A C 1 , A F 1 , A C 2 , A F 2 and A I is 1 28 ad -1 28 ad 0 1 0 0 0 0 0 0 1 0 0 0 0 1 28 ad -1 28 ad 0 1 + 1 7 a(c -f + d 4 ) -1 7 a(c + d 4 ) 0 0 0 0 1 0 0 0 0 0 0 1 0            and B =         b 0 0 0 0 0 b 0 0 0 0 0 b 0 0 0 0 0 b 0 0 0 0 0 b        
The local stability of the steady state is determined if all the eigenvalues of matrices A and B have a modulus less than 1. Matrix B is diagonal, so its four eigenvalues are all equal to b, which by definition satisfies 0 ≤ b < 1. Therefore, the local stability is primarily dependent on matrix A. To simplify the expressions for the eigenvalues of A, we assume that c = f . This assumption does not impact the stability of the steady state, as discussed in [START_REF] Westerhoff | Nonlinearities and cyclical behavior: the role of chartists and fundamentalists[END_REF].

λ 1 = 0 λ 2 = 0 λ 3 = 1 2 (1 - 1 7 √ 7 √ 7 4ac) λ 4 = 1 2 (1 + 1 7 √ 7 √ 7 -4ac) λ 5 = 1 28 ad - 1 28 a 2 d 2 -28ad -112ac + 196 + 1 2 λ 6 = 1 28 ad + 1 28 a 2 d 2 -28ad -112ac + 196 + 1 2
Both eigenvalues λ 1 and λ 2 are equal and inherently stable. For λ 3 and λ 4 the necessary and sufficient conditions for their moduli to be less than 1 are:

7 4a > c (12) √ 7 √ 7 -4ac > 0 (13)
The stability properties of λ 5 and λ 6 are contingent upon the parameter d that characterizes index traders. We can distinguish two cases based on the value of d. On the one hand, as d → 0 both λ 5 and λ 6 have moduli less than 1 if conditions ( 12) and ( 13) are met. This suggests that when index investors place little trust in their trading rule, the model's local stability aligns with that of the baseline model under similar conditions. On the other hand, as d → ∞, λ 6 has a modulus greater than 1. This finding corroborates the destabilizing influence of index traders, pushing the long-term price trajectory from a state of equilibrium to one of explosiveness. Additional simulations-available upon request from the authorsindicate that a Hopf bifurcation occurs at a threshold value of d = 13.31, assuming a = 1 and c = f = 0.05.

B Commodity futures price

C HAM-IT with three

This Appendix delves into the technical nuances of the Heterogeneous Agent Model with Index Traders (HAM-IT), specifically its extensions with three markets. We examine three different scenarios: (i) a setting with no trading restrictions on any category of traders; (ii) a framework where traditional traders face certain market restrictions; and (iii) a scenario where trading limitations are imposed solely on index investors.

C.1 No restrictions on trading strategies

In this initial extension, we examine commodity futures prices within a single asset class. The index is conceived as an equally weighted amalgamation of these individual futures prices. Importantly, there are no constraints placed on traders regarding which assets they can invest in. Traditional traders, such as fundamentalists and chartists, are designed to invest in only one asset at a time. In contrast, index traders have the latitude to take positions in all assets simultaneously via the commodity index. This unrestricted setting serves as a foundational model to analyze the investment strategies and market dynamics associated with different types of traders.

P 1 t+1 = P 1 t + a(W C 1 t (c(P 1 t -x 1 t ) + S C 1 t ) + W F 1 t (f (F 1 -P 1 t ) + S 1 t ) + W I t 1 3 d( 1 3 (P 1 t + P 2 t + P 3 t ) + S I t - 1 3 (x 1 t + x 2 t + x 3 t ) + S I t-1 ) + S I 1 t ) + S P 1 t x 1 t+1 = P 1 t y 1 t+1 = x 1 t P 2 t+1 = P 2 t + a(W C 2 t (c(P 2 t -x 2 t ) + S C 2 t ) + W F 2 t (f (F 2 -P 2 t ) + S F 2 t ) + W I t 1 3 d( 1 3 (P 1 t + P 2 t + P 3 t ) + S I t - 1 3 (x 1 t + x 2 t + x 3 t ) + S I t-1 + S I 2 t ) + S P 2 t x 2 t+1 = P 2 t y 2 t+1 = x 2 t P 3 t+1 = P 3 t + a(W C 3 t (c(P 3 t -x 3 t ) + S C 3 t ) + W F 3 t (f (F 3 -P 3 t ) + S F 3 t ) + W I t 1 3 d( 1 3 (P 1 t + P 2 t + P 3 t ) + S I t - 1 3 (x 1 t + x 2 t + x 3 t ) + S I t-1 ) + S I 3 t ) + S P 3 t x 3 t+1 = P 3 t y 3 t+1 = x 3 t z t+1 = I t = 1 3 (P 1 t + P 2 t + P 3 t ) + S I t v t+1 = z t = 1 3 (x 1 t + x 2 t + +x 3 t ) + S I t-1 A C 1 t+1 = (exp[P 1 t+1 ] -exp[P 1 t ])(c(x 1 t -y 1 t ) + S C 1 t-1 ) + bA C 1 t A F 1 t+1 = (exp[P 1 t+1 ] -exp[P 1 t ])(f (F 1 -x 1 t ) + S F 1 t-1 ) + bA F 1 t A C 2 t+1 = (

C.2 Restrictions on traditional investors only

In the second extension, we expand our focus to include three markets: two of which feature commodity futures prices within the same asset class, and the third represents a commod-ity from a different class, such as crude oil. Traditional traders, namely fundamentalists and chartists, are restricted to investing solely in the first two markets, reflecting their specialization in a given class of commodities. On the other hand, index traders face such restrictions and are free to invest across all available markets through index allocation strategies. This setup allows us to delve deeper into how different trading restrictions influence market dynamics and asset correlations, particularly when commodities from diverse classes are involved.

P

C.3 Restrictions on index investors only

In the third extension, the model focuses on three distinct commodity futures markets. Among these, the first two markets (P1 and P2) are part of a predefined index, while the third one (P3) remains off-index. Traditional traders, as chartists and fundamentalists, are not subject to any asset-specific restrictions; they are free to invest in any of the three markets, though only one at a time. In contrast, index traders are limited to positions in only the two in-index markets, via the index investment strategy. This extension enables us to examine the role of index traders more critically, especially in terms of how their restrictions impact price co-movements and market dynamics between in-index and off-index commodities.

P

Figure 2 :

 2 Figure 2: Weights of chartist and fundamentalist strategies

  1 t , and P 2 t represent the log-linear prices for Markets 1 and 2 at times t + 1 and t. The variable a indicates a positive price adjustment. D C 1

	t , D C 2 t , D F 1 t , and D F 2 t denote the
	trading orders placed by chartists (C) and fundamentalists (F ) in Markets 1 and 2 at time
	t. Similarly, W C 1 t , W C 2 t , W F 1 t , and W F 2 t strategy at time t. Both S P 1 t and S P 2 t are random variables 4 , where S P 1 signify the proportions of traders adhering to each t

Table 3 :

 3 Stylized facts on artificial commodity markets

	Quantile	r min	r max	V	D	K
	0.05 0.25 0.50 0.75 0.95	-14.89 (-6.77) -13.15 (-6.00) -11.88 (-5.65) -9.88 (-5.35) -7.37 (-4.99)	7.42 (4.98) 9.57 (5.32) 11.76 (5.62) 13.06 (5.98) 14.86 (6.68)	0.97 (1.20) 1.23 (1.21) 1.72 (1.23) 2.32 (1.24) 2.85 (1.25)	18.77 (7.67) 26.03 (8.50) 38.34 (9.16) 72.44 (9.94) 160.63 (10.98)	3.52 (2.52) 4.92 (2.87) 5.34 (3.22) 7.71 (3.71) 14.81 (5.40)
	Quantile	ac 1 r	ac 2 r	ac 3 r	ac 4 r	ac 5 r
	0.05 0.25 0.50	0.00 (-0.01) 0.01 (0.01) 0.03 (0.02)	-0.03 (-0.03) -0.01 (-0.01) 0.00 (0.00)	-0.03 (-0.03) -0.01 (-0.01) 0.00 (0.00)	-0.03 (-0.03) -0.01 (-0.01) 0.00 (0.00)	-0.03 (-0.03) -0.01 (-0.01) 0.00 (0.00)
	0.75	0.04 (0.03)	0.01 (0.01)	0.01 (0.01)	0.01 (0.01)	0.01 (0.00)
	0.95	0.07 (0.04)	0.03 (0.02)	0.03 (0.02)	0.03 (0.02)	0.03 (0.02)
	Quantile	ac 1 |r|	ac 2 |r|	ac 3 |r|	ac 4 |r|	ac 5 |r|
	0.05	0.01 (-0.02)	0.00 (-0.02)	0.00 (-0.02)	0.00 (-0.02)	0.00 (-0.02)
	0.25	0.05 (0.00)	0.05 (0.00)	0.05 (0.00)	0.05 (0.00)	0.04 (0.00)
	0.50	0.18 (0.01)	0.17 (0.01)	0.18 (0.01)	0.18 (0.01)	0.18 (0.01)
	0.75	0.24 (0.02)	0.23 (0.02)	0.24 (0.02)	0.23 (0.02)	0.24 (0.02)
	0.95	0.29 (0.03)	0.29 (0.03)	0.28 (0.03)	0.28 (0.03)	0.28 (0.03)

Table 4 :

 4 Stylized facts of real agricultural returns (covering the period 2000-2020)

	Commodity returns	r min	r max	V	D	K
	Crude oil Natural gas Copper Aluminium Corn Cocoa Gold Silver	-10.15 -7.69 -8.36 -6.19 -26.86 -17.09 -11.68 -13.41	15.85 8.75 10.58 7.75 12.75 7.81 11.42 19.32	1.30 1.45 1.50 1.82 2.20 1.72 1.21 1.84	--------	20.30 10.60 10.40 7.30 12.41 7.40 10.82 12.27
	Commodity returns	ac 1 r	ac 2 r	ac 3 r	ac 4 r	ac 5 r
	Crude oil	0.02	0.00	0.01	0.01	0.00
	Natural gas Copper Aluminium	-0.01 -0.02 0.00	0.01 -0.01 0.01	0.00 0.01 0.01	0.01 0.00 0.02	0.03 0.00 0.00
	Corn Cocoa Gold Silver	0.02 0.06 -0.04 0.01	-0.01 0.00 0.00 -0.01	0.01 -0.03 0.01 0.00	0.01 -0.03 0.00 -0.02	-0.01 0.00 0.01 -0.05
	Commodity returns	ac 1 |r|	ac 2 |r|	ac 3 |r|	ac 4 |r|	ac 5 |r|
	Crude oil	0.25	0.18	0.16	0.16	0.08
	Natural gas	0.21	0.15	0.24	0.11	0.15
	Copper	0.18	0.11	0.19	0.09	0.10
	Aluminium	0.19	0.19	0.12	0.09	0.03
	Corn	0.11	0.08	0.06	0.09	0.08
	Cocoa	0.11	0.06	0.03	0.07	0.05
	Gold	0.09	0.09	0.03	0.02	0.05
	Silver	0.14	0.14	0.11	0.11	0.09

Table 5 :

 5 Commodity returns co-movements: no restrictions This table presents the estimated inter-market correlations of commodity returns across various quantiles, as generated by the HAM-IT model. For comparative purposes, the correlations calculated using the benchmark model are included in parentheses.

	Quantile	P1/P2	P1/P3	P2/P3
	0.05	-0.598 (-0.268) -0.587 (-0.281) -0.571 (-0.255)
	0.25	-0.191 (-0.112) -0.191 (-0.121) -0.181 (-0.116)
	0.50	-0.105 (0.007) -0.056 (0.003) 0.016 (0.0002)
	0.75	0.376 (0.115)	0.380 (0.113)	0.298 (0.108)
	0.95	0.632 (0.268)	0.610 (0.263)	0.664 (0.258)

Note:

Table 6 :

 6 Commodity returns co-movements: restrictions on chartists-fundamentalists

	Quantile	P1/P2	P1/P3	P2/P3
	0.05	-0.514 (-0.268) -0.569 (-0.281) -0.538 (-0.255)
	0.25	-0.138 (-0.112) -0.200 (-0.121) -0.200 (-0.116)
	0.50	-0.008 (0.007) -0.006 (0.003) 0.009 (0.0002)
	0.75	0.305 (0.115)	0.399 (0.113)	0.371 (0.108)
	0.95	0.632 (0.268)	0.630 (0.263)	0.624 (0.258)

Table 7 :

 7 Commodity returns correlations: restrictions on index traders

	Quantile	P1/P2	P1/P3	P2/P3
	0.05	-0.397 (-0.268) -0.553 (-0.281) -0.564 (-0.255)
	0.25	-0.080 (-0.112) -0.182 (-0.121) -0.173 (-0.116)
	0.50	0.098 (0.007)	0.011 (0.003)	0.016 (0.0002)
	0.75	0.290 (0.115)	0.173 (0.113)	0.197 (0.108)
	0.95	0.630 (0.268)	0.456 (0.263)	0.461 (0.258)

Table 8 :

 8 Commodity futures prices

This table presents the commodity futures prices selected for our study. NYMEX refers to the New York Mercantile Exchange, LME denotes the London Metal Exchange, and CME represents the Chicago Mercantile Exchange.

Arbitrageurs buying when the price is low and selling when the price is high quickly counter mispricings.

See Fattouh et al. (2013) and[START_REF] Joëts | Heterogeneous beliefs, regret, and uncertainty: The role of speculation in energy price dynamics[END_REF] for a review.

See, e.g., Kirman (1991),[START_REF] Lux | Herd behaviour, bubbles and crashes[END_REF] Brock &[START_REF] Brock | Heterogeneous beliefs and routes to chaos in a simple asset pricing model[END_REF] among others.

These shock variables represent the macroeconomic and financial factors that influence commodity prices.

For a more in-depth discussion on the timing and the influence of past performances, refer to[START_REF] Brock | A rational route to randomness[END_REF] and[START_REF] Westerhoff | The effectiveness of keynes-tobin transaction taxes when heterogeneous agents can trade in different markets: a behavioral finance approach[END_REF].

See, for example,[START_REF] Barberis | Style investing[END_REF],[START_REF] Tang | Index investment and the financialization of commodities[END_REF],Büyükşahin &[START_REF] Büyükşahin | Speculators, commodities and cross-market linkages[END_REF][START_REF] Brunetti | Speculators, prices, and market volatility[END_REF].

This share is based on the average percentage of total open interest held by long commodity index traders (see Capelle-Blancard & Coulibaly (2011) and Irwin & Sanders (2011)).

The proportion of index investors within the overall pool of investors âwhich includes both index and traditional investorsâ is exogenously determined. However, the choice of whether or not to invest in each period is endogenous to index investors, contingent upon their confidence in the success of their trading strategy.

For additional details on the markets under consideration, see Appendix B.

Simulations for market 1 yield materially similar results. These are available upon request from the authors.

For a theoretical discussion, see[START_REF] Cont | Empirical properties of asset returns: stylized facts and statistical issues[END_REF],[START_REF] Sornette | A nonlinear super-exponential rational model of speculative financial bubbles[END_REF][START_REF] Sornette | A nonlinear super-exponential rational model of speculative financial bubbles[END_REF][START_REF] Lux | Herd behaviour, bubbles and crashes[END_REF][START_REF] Lux | Market fluctuations i: Scaling, multiscaling, and their possible origins[END_REF]. In our specific context, existing literature has shown that the properties of commodity markets closely resemble those of traditional financial assets (see[START_REF] Creti | On the links between stock and commodity markets' volatility[END_REF][START_REF] Joëts | Energy price transmissions during extreme movements[END_REF][START_REF] Joëts | Heterogeneous beliefs, regret, and uncertainty: The role of speculation in energy price dynamics[END_REF]).

For additional details, refer to[START_REF] Westerhoff | The effectiveness of keynes-tobin transaction taxes when heterogeneous agents can trade in different markets: a behavioral finance approach[END_REF].

Results for market 1 are closely aligned and are available upon request.

For further details, see Appendix B.

The model can, in theory, be extended to include an infinite number of markets.

The specific weightings assigned to individual futures prices within the index do not have a material effect on this extension, thereby allowing our framework to accommodate various weighting schemes.

See Tang & Xiong (2012) for further discussion on this matter.

28 ad 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0


Dhananjay S Doke

Sonali B Khomane

Swati L Pandhare

Mohan K Dongare

Franck Dumeignil

Shubhangi B Umbarkar
email: sb.umbarkar@ncl.res.in

Vanadium-based highly active and selective

Keywords: Ethyl lactate, Ethyl pyruvate, Vanadium, TBHP, Oxidative dehydrogenation

Introduction

Decreasing availability of fossil fuel is forcing researchers to develop alternate routes for production of fuels and chemicals using renewable resources. Biomass is one of the renewable resources, abundantly available and rich in hydrocarbons. Biomass can be converted to some valuable products. Fermentation processes give access to various platform molecules, which can be further converted to value added products using bio/chemical routes [1,2,3,4]. Lactic acid, one of the very important fermentation products of sugar, has been identified as a platform chemical in biorefineries. The bifunctional nature of lactic acid due to the presence of alcoholic OH and carboxylic group, makes it a very important raw material for conversion to a large number of commodity chemicals including pyruvic acid, acrylic acid and its esters for a wide range of applications. They find uses as intermediates in various industrial products such as perfume, food additives, pharmaceuticals and electronic materials. They are also attractive as raw materials for various bioactive substances such as antivirus and anticancer drugs [5]. Pyruvates are also used in dietary supplements, in food industry. However, the production of pyruvic acid is still based on the conventional reaction scheme through dehydrative decarboxylation of tartaric acid. Although this reaction provides pyruvic acid in good yield, it has a major disadvantage due to the use of excess KHSO4 as the decarboxylation agent leading to an expensive process with environmental hazards along with high energy consumption [6]. Pyruvic acid/pyruvates are also produced by microbial process using stains of yeast and E. Coli. However typical limitations of bioprocesses like very high dilutions and critical control of process parameters during fermentation make those less attractive [7]. Therefore, designing a new process with high atom efficiency for the production of pyruvic acid is essential [8]. Pyruvic acid or esters can be synthesized via catalytic oxidative dehydrogenation of lactic acid/esters, e.g., ethyl lactate to ethyl pyruvate. In this case, the whole process would be based on biomass-derived feedstock, and the reaction could in theory proceed directly from lactic acid to pyruvic acid. As a possible candidate for the synthesis of pyruvic acid, oxidative dehydrogenation of lactates to the corresponding pyruvates in both gas and liquid phases on solid catalysts has received significant attention. Oxidative dehydrogenation of ethyl lactate to ethyl pyruvate either in liquid or gas phase using various binary oxides containing molybdenum such as Fe2O3-MoO3, TeO2-MoO3 and MoVNbOx, vanadium oxide species, and iron phosphate have been investigated by various research groups [1,9,10]. These processes achieve high yields of pyruvate from lactate. In the case of vapor phase dehydration of lactates, besides the requirement for vaporizing lactate at high temperature, a reaction temperature above 473 K is also needed, which increases the running cost of the process. High reaction temperatures may lead to product decomposition and then to low productivity along with high energy demand. There are several reports on liquid phase oxidative dehydrogenation (ODH) of lactates to pyruvates. Kazuhiro et al. have reported VOCl3 catalyst in the presence of various oxidants with almost 50% conversion and <50% selectivity to pyruvate. Water is the by-product of this reaction and the catalyst was not stable in the presence of water. VOCl3 is known to be highly sensitive to hydrolysis in presence of water/moisture leading to the formation of V2O5. Hence it is essential to remove water formed in the reaction to maintain the activity of the catalyst [11]. Zhou and others have used TS-1 as a catalyst and peroxide as an oxidant with excellent conversion (~100%) and maximum 98.6 % selectivity to pyruvate [12]. Shiju and co-workers have reported titanium-based catalysts for ODH using oxygen as an oxidant, but this reaction requires high pressure (1 MPa) and temperature (130 ºC). However, selectivity to pyruvate was an issue: Only at lower conversions (~20%) high selectivity (~70%) to pyruvate was reported; with an increase in conversion to 60%, pyruvate selectivity dropped drastically (~25%) due to polymerisation of pyruvate [13]. In a recent work reported by Shiju et al. on the use of mesoporous vanadia-titania catalysts for atmospheric pressure ODH at 130 ºC using diethyl succinate as a solvent, a pyruvate selectivity of >90% at ~20% lactate conversion was observed. However, in this case also the selectivity dropped to 60% at 70% conversion [14]. The same group has again reported vapor phase ODH of ethyl lactate using vanadia supported on different oxides as catalysts in a fixed bed reactor at 180 ºC with a maximum 48% yield of ethyl lactate [15]. The liquid phase ODH of lactic acid using solid catalysts such as Pd/C modified by Te and Pb has been reported by Hayashi et al. [16]with maximum 80% conversion and 50% pyruvic acid yield at 90 ºC. However, high cost of precious metal is a matter of concern due to high Pd loading (5wt.%) essential for the reaction. Wang et al. have reported ethyl lactate ODH using carbon nanotubes as catalysts and TBHP as an oxidant with maximum yield of 40% at 90 ºC [17]. Hence, development of a catalytic system for efficient oxidative dehydrogenation of lactate to pyruvate under mild reaction conditions using simple and inexpensive catalysts is still highly desired.

Vanadium-based catalysts have been extensively studied for selective oxidation of various organics including ethyl lactate. Various vanadium-based homogeneous catalysts including VOCl3 have been studied for the oxidation of ethyl lactate to ethyl pyruvate using gas-liquid micro flow system [11,14]. A maximum yield of 31% has been reported at room temperature using oxygen as an oxidant. Molecular sieves were essential to remove water generated during the reaction as a by-product to avoid hydrolysis of the sensitive catalyst, VOCl3.

Considering the importance of pyruvate and recognizing the previous efforts in this area, we have attempted oxidative dehydrogenation of ethyl lactate to ethyl pyruvate using vanadium-based catalysts in the liquid phase using t-butyl hydroperoxide as an oxidant. Interestingly, very high conversion with nearly 100% selectivity to ethyl pyruvate was observed, and the results are presented herein.

Experimental section

Materials and instrumentation

Ethyl lactate, vanadyl acetylacetonate VO(acac)2, ethyl pyruvate (Aldrich Chemicals), FeCl3, copper acetate (Avra Chemicals India), V2O5, tert-butyl hydroperoxide (TBHP, 70% aqueous) (Avra Chemicals India) and 5.5 M in decane(Aldrich Chemicals), acetonitrile (Finar Chemicals India) were used as received without further purification. MoO2Cl2(DMSO)2 and MoO2Cl2(DMF)2 were prepared using literature method [18].

Typical reaction procedure

In a typical ODH reaction, a two-necked round bottom flask (50 ml) was charged with ethyl lactate (0.23 g, 2 mmol), 70% aqueous or organic (5.5 M in decane) TBHP (4 mmol), catalyst (0.04 mmol, 2 mol%), and 6 g CH3CN as a solvent. The reaction mixture was stirred at desired temperature for specified time. The reaction was monitored by GC. The samples were taken periodically and analyzed using a GC-Hewlett Packard 6890 equipped with an HP-5 column (50 m length, 0.25 mm internal diameter and 1 µm film thickness) and a flame ionization detector (FID). Conversion of ethyl lactate was calculated based on GC-FID results, where substrate conversion (%) = [moles of substrate reacted]/[Initial moles of substrate used] × 100, and selectivity (%) to products was calculated by [total moles of the product formed]/[total moles of substrate converted] × 100. The produced ethyl pyruvate was quantified using external calibration curve plotted using commercial ethyl pyruvate. Individual yields were calculated and normalized with respect to the GC response factors. The product identification was carried out by injecting commercial samples in the GC.

Results and discussion

Oxidative dehydrogenation of ethyl lactate (Scheme 1) was carried out using various commercial metal salts like vanadyl acetylacetonate, copper acetate, iron chloride as well as simple complexes like MoO2Cl2(DMSO)2, MoO2Cl2(DMF)2 as catalysts in the presence of organic TBHP (5.5 M in decane) as an oxidant in acetonitrile as a solvent at 80 ºC. The results are reported in Table 1S. The acetonitrile solvent was chosen based on our previous experience in oxidation reactions using Mobased complexes as homogeneous catalyst and H2O2 or TBHP as oxidant, where better activity was obtained in acetonitrile compared to other solvents. Oxidative dehydrogenation of ethyl lactate without catalyst at 80 ºC using organic TBHP as an oxidant in acetonitrile solvent did not show any conversion even after 11 h. Under the same conditions, various Mo-, Cu-, and Fe-based catalysts yielded moderate to excellent conversion of ethyl lactate (31-92%) with 100% selectivity to ethyl pyruvate. It is interesting to note that in all cases ethyl pyruvate was formed as a sole product and no other by-products were observed. Mo-based complexes showed 92% ethyl lactate conversion with organic TBHP, however, under identical conditions no reaction was observed with aqueous TBHP, due to catalyst deactivation in the presence of water. All the Mo complexes in presence of water formed blue complexes which were found to be inactive for ODH reaction. It is well documented in the literature that molybdenum blue are the complexes with dimeric structure with very facile electron transfer between both molybdenum centers [19]. Note that, attempts to use hydrogen peroxide as an oxidant were not successful with all the tested catalysts.

VO(acac)2 did not deactivate when using aqueous TBHP as an oxidant. Almost similar ethyl lactate conversion was obtained (82%) in the presence of aqueous TBHP at 80 ºC after 16 h (Table 1 entry 1) compared to organic TBHP (80%) under similar reaction conditions, in both cases at 100% selectivity to ethyl pyruvate. Hence, the use of VO(acac)2 was subsequently explored for optimizing different reaction parameters in order to maximize the ethyl pyruvate yield. Initially, reaction was carried out at different temperatures, and, to our surprise, very high ethyl lactate conversion was obtained even at lower temperatures (40 ºC), and even at room temperature though it needed longer reaction times for achieving maximum conversion. After 16 h, the ethyl lactate conversion was 79% and 72% at 40 ºC and room temperature, respectively (Table 1 Ethyl pyruvate is prone to polymerization. Hence, there is a need to use solvent in order to avoid this undesired side reaction [13]. When a solvent is used for the reaction, it reduces the concentration of pyruvate in the reaction medium and in turn reduces the interaction between pyruvate molecules reducing the extent of polymerisation leading to very high selectivity to pyruvate. We thus studied the effect of the amount of solvent on the kinetics of the reaction by varying the amount of acetonitrile with respect to ethyl lactate (Table 2). When ethyl lactate dehydration was carried out without solvent (Table 2 entry 1), 91% conversion was obtained after 16 h, however the selectivity for ethyl pyruvate was only 80% leading to partial polymerization. When one weight equivalent (0.236 g) solvent was used (Table 2 entry 2) the conversion increased to 97% after 16 h with 80% selectivity for ethyl pyruvate. Even with 2 weight equivalent of solvent (Table 2 entry 3) the conversion (98%) and selectivity (80%) remained almost same with partial polymerization. However, when 10 weight equivalents of solvent (Table 2 entry 4) was used; very high conversion (83%) was obtained with 100% selectivity to ethyl pyruvate. With further increase in the solvent amount to 20, 30 and 50 weight equivalents (Table 2 entry 5-7), the conversion gradually decreased to 72% still with 100% selectivity to ethyl pyruvate. These results clearly indicate the need for the use of a solvent to avoid polymerization and to obtain 100% selectivity to ethyl pyruvate. With a substantial excess of solvent, the conversion decreased as expected without decrease in the selectivity. The decrease in the conversion in very dilute reaction mixture is due to low probability of contact of catalyst, substrate and oxidant. Hence 10 weight equivalent of solvent was used for further optimizations. The effect of the catalyst loading on the ethyl lactate conversion was studied by varying the catalyst loading from 0.5 to 2 mol% with respect to ethyl lactate and the results are given in Table 3. As expected, when the catalyst loading increased, the reaction was faster and higher conversion was obtained in a shorter time. After 16 h, the ethyl lactate conversion increased from 32 to 83% with an increase in the catalyst loading from 0.5 mol% to 2 mol%, respectively, with 100% selectivity to ethyl pyruvate. After 28 h, the conversion increased to 67, 76 and 100% respectively for 0.5, 1 and 2 mol% catalyst loading with corresponding TON values of 134, 76 and 50, respectively. It is well known that peroxides get decomposed in the presence of metal catalysts. The decomposition rate gets accelerated at higher temperature leading to non-availability of sufficient oxidant quantities for the targeted reaction if used in stoichiometric amounts. Hence, excess of oxidants are typically used in catalytic oxidation reactions using metal-based catalysts involving peroxides as oxidants. In the present work, we thus studied as well the effect of the amount of oxidant (TBHP) on ethyl lactate conversion (Table 4). Obviously, for ethyl lactate ODH one molar equivalent of peroxide is needed. However, with 1 equivalent oxidant only 72% conversion was observed indicating partial decomposition of the TBHP. With 2 equivalents TBHP very high conversion of 83% was obtained actually underlining the need for an excess of oxidant to compensate for its parallel undesired decomposition. The conversion of ethyl lactate as well as TBHP were recorded with time (Fig. 1). The results indicated higher conversion of TBHP than ethyl lactate confirming TBHP decomposition. After 16 h TBHP conversion was 100%; however, the corresponding ethyl lactate conversion was only 83 % indicating 58.5% decomposition of TBHP even at room temperature. Initially, the effect of temperature was studied to compare the activities of homogeneous and heterogeneous V-based catalysts using 10 wt.% catalyst loading (Table 5). While at room temperature VO(acac)2 showed very high ethyl lactate conversion, no conversion was observed using heterogeneous V2O5 catalyst, which confirms the lower activity of heterogeneous catalyst compared to homogeneous catalyst. With increasing the temperature to 40 and 80 °C the heterogeneously catalyzed ethyl lactate conversion after 5 h increased to 26 and 60%, respectively. Thus, 80 °C was used for further performances' optimization. The effect of the V2O5 catalyst loading on ethyl lactate conversion and ethyl pyruvate selectivity was studied (Table 6). When the catalyst loading was varied from 2 to 10 wt.% with respect to ethyl lactate, conversion after 1 h showed significant difference. The ethyl lactate conversion after 1 h was 12% for the 2wt.% catalyst loading, which increased to 25 and 35% for 5 and 10wt.% loading, respectively. Surprisingly, the conversion after 5 h was almost similar (~60%) for all the catalyst loadings. This indicated the unavailability of sufficient oxidant during the reaction limiting the maximum conversion to ~60%. This may be due to the relatively high temperature (80 °C) needed for the reaction leading to faster decomposition of TBHP. It is well known that peroxides decompose faster at higher temperatures in presence of metal-based catalysts [22,23]. To confirm this observation, the conversion of TBHP was monitored during the reaction along with ethyl lactate conversion and the results are plotted in Figure 1. The results showed almost complete conversion of TBHP in 5 h with only ~60% ethyl lactate conversion indicating TBHP decomposition under reaction conditions due to higher temperature. To control the TBHP decomposition at higher temperature, addition of TBHP was carried out batch wise (Table 7) until quasi-full conversion. It is very clear from the results (Table 7) that adding TBHP batchwise enabled better control of the reaction, by limiting its decomposition with an increase in ethyl lactate conversion from 60% (Figure 2) to 72 % after 5 h after addition of 2 equivalents aq. TBHP. Further, addition of 1 equivalent TBHP in small portions led to 98% ethyl lactate conversion with 100% ethyl pyruvate selectivity after 10 h with total 3 equivalents of TBHP. The results showed very high efficiency of the V2O5 catalyst for oxidative dehydrogenation of ethyl lactate to ethyl pyruvate.

Catalyst recycling

The main aim of the heterogeneous catalyst (V2O5) for oxidative dehydrogenation was efficient recycle of the catalyst. Indeed, the catalyst could be recycled five times (Fig. 3) without losing activity in terms of both ethyl lactate conversion and ethyl pyruvate selectivity. For this study TBHP (total of 2 equivalents) was added batchwise over a period of 4 h and the reaction was continued further till 5 h, like in the procedure used for Table 7. After reaction, the mixture was decanted leaving behind the catalyst which was dried at 100 ºC before next run. Thus, the catalyst was recycled efficiently five times with 72% ethyl lactate conversion and 100% ethyl pyruvate selectivity. To confirm the integrity of the catalyst during the reaction, FTIR of the fresh and used catalyst was recorded (Fig. 4). The FTIR spectra showed similar pattern with characteristics bands at 481,534 and 597 cm -1 for symmetric and asymmetric V-O-V stretching vibrations of V2O5 as well as at 832 cm - 1 for V-O vibrations. The characteristics band at 1017 cm -1 can be attributed to symmetric stretching of V=O bond [24,[START_REF] Fabbri | Infrared Spectra of Stoichiometric Oxides of Vanadium[END_REF]. The similar FTIR patterns before and after reaction suggested no change in structural characteristic of the catalyst after reaction. The structural integrity was further confirmed by XRD analysis of fresh as well as used catalyst (Fig. 5). XRD pattern of V2O5 showed reflections at 2θ 20.2, 26.24, 30.98, 15.42 and 34.38° corresponding to the 001, 110, 301, 200 and 310 planes, respectively, which matched well with JCPDS data file no.77-2418 for orthorhombic V2O5 phase. In the used catalyst, additional peaks at 2θ of 26 and 37.5 ° were observed. However, these additional peaks also corresponded to orthorhombic phase of V2O5. The XRD pattern of used catalyst showed significant increase in the intensity of the peaks compared to the fresh catalyst indicating increase in the crystallinity. Due to the increased crystallinity of used catalyst, very weak peaks at 2θ of 26, 37.5° of orthorhombic V2O5 which were not detected in fresh catalyst, could be observed in used catalyst. Hence no change in the crystallographic phase of the catalyst after catalytic reaction was observed.

Hot filtration Experiment

In heterogeneous catalysis, leaching of active component may be a cause of the catalytic activity. Hence to confirm the heterogeneous nature of the catalyst hot filtration experiment was carried out. Initially, the reaction was carried out for 2 h (Table 8, entry 1) with catalyst after which the catalyst was separated from the reaction mixture by centrifugation followed by decantation. The same reaction was continued further without catalyst for next 3 h. Initially with catalyst after 2 h, 31% conversion was obtained which increased only marginally to 37% after separating the catalyst even after 3 h reaction time indicating no leaching of catalytically active species in the reaction medium. The metal leaching was additionally confirmed by ICP analysis and the results showed the presence of <1 ppm V in the reaction solution confirming no leaching of metal during the reaction.

Plausible reaction mechanism

Typically, oxidation using peroxides proceeds via radical mechanism. To confirm whether this statement applies to our reaction system, one experiment was carried out by adding TEMPO as a radical quencher after 35% conversion. As expected, the reaction did not proceed further after addition of TEMPO, which confirmed the radical mechanism operating as well in the present case. In the literature, various mechanisms have been proposed for oxidative dehydrogenation of ethyl lactate to ethyl pyruvate After addition of TBHP to V2O5 (A), it forms vanadium peroxy species on the surface (B) (Fig. 6). Then, it forms radical via homolytic cleavage (C) and generates t-butyl peroxy radical, which leads to regeneration of catalyst to its original form. Radical of t-butyl hydroperoxide initiates the oxidation of ethyl lactate and form ethyl pyruvate with tert-butanol and water as a side product.

The obtained results with the present catalytic system for ODH of ethyl lactate to ethyl pyruvate under mild reaction conditions are promising. The comparison of the results with the literature is shown in Table 9. It is obvious that the results obtained in the present work using homogeneous and heterogeneous vanadium-based catalysts are promising in terms of ethyl lactate conversion as well as ethyl pyruvate selectivity.

Conclusion

Very efficient protocol for the oxidative dehydrogenation of ethyl lactate to ethyl pyruvate using aqueous TBHP as an oxidant and V2O5 as a very simple and recyclable catalyst was developed. Very high ethyl lactate conversion (upto 98%) was obtained with 100% selectivity to ethyl pyruvate. Homogeneous VO(acac)2 complex also showed very high efficiency for ethyl lactate dehydrogenation at room temperature with ~83% conversion and 100% selectivity for ethyl pyruvate though the complex could not be recycled. The main highlight of this work is almost complete conversion and 100% selectivity for ethyl pyruvate under mild reaction conditions.

Acknowledgements

This work is carried out as a part of collaboration under International Associated Laboratory program between CSIR (NCL) India and CNRS (UCCS) France.

Graphical abstract

Highlights of the paper 1. Vanadium-based homogenous and heterogeneous catalysts for efficient oxidative dehydrogenation of ethyl lactate to ethyl pyruvate under mild conditions 2. Very high ethyl lactate conversion with 100% ethyl pyruvate selectivity using aqueous TBHP as an oxidant at room temperature 3. V2O5 serves as an efficient and recyclable heterogeneous catalyst without compromising the activity with almost 98% conversion and 100% ethyl pyruvate selectivity

CH 3 CN

2 ACNScheme 1 .

 21 Scheme 1. Oxidative dehydrogenation of ethyl lactate to ethyl pyruvate

Fig. 1 .

 1 Fig. 1. Ethyl lactate and TBHP conversion with time using VO(acac)2 catalyst Reaction conditions: Ethyl lactate 2 mmol; solvent CH3CN 2 g, room temperature, catalyst 0.04 mmol (2 mol%), oxidant aq. TBHP 4 mmol

  Fig. 2.Ethyl lactate and TBHP conversion with time using V2O5 catalystReaction conditions: Ethyl lactate 4 mmol; solvent CH3CN 4 g; oxidant aq. TBHP 8 mmol; catalyst 10wt.% w.r.t. substrate

Fig. 3 .

 3 Fig. 3. Catalyst recycle study for ethyl lactate dehydrogenation over V2O5. Reaction conditions: Ethyl lactate 8 mmol; solvent CH3CN 8 g; oxidant aq. TBHP 16 mmol; catalyst 10wt.% (w.r.t. substrate); time 5 h.

Fig. 4 .

 4 Fig. 4. FTIR spectra of V2O5catalyst before and after ethyl lactate dehydrogenation

Fig. 5 .

 5 Fig. 5. XRD pattern of fresh and used V2O5 catalyst

[ 12 -Fig. 6 .

 126 Fig. 6. Plausible reaction mechanism for oxidative dehydrogenation of ethyl lactate to ethyl pyruvate using V2O5catalyst

Table 1 .

 1 entry 2 & 3) which underlines very high efficiency of VO(acac)2 for ethyl lactate dehydrogenation even at room temperature. Effect of temperature on ethyl lactate dehydrogenation using VO(acac)2 as a catalyst

	Entry	Temperature	Time	Conversion	Selectivity	TON
		(ºC)	(h)	(%)	(%)
	1	80	16	82	100	41
	2	40	16	79	100	40
	3	RT	16	72	100	36

Table 2 .

 2 Effect of amount of solvents on ethyl lactate dehydrogenation using VO(acac)2 as a catalyst

	Entry	Amount of	Time (h)	Conversion (%)	Selectivity (%)	TON
		CH3CN, g
	1	without solvent	16	92	80	46
	2	0.236 g	16	97	80	49
	3	0.472 g	16	98	80	50
	4	2	16	83	100	42
	5	4	16	81	100	41
	6	6	16	72	100	36
	7	10	16	72	100	36
	Reaction conditions: Ethyl lactate 2 mmol; aqueous TBHP 4 mmol; catalyst 0.04 mmol (2 mol%) ;
	room temperature

Table 3 .

 3 Effect of catalyst loading on ethyl lactate dehydrogenation using VO(acac)2 as a catalyst

	Entry Catalyst loading	Time(h)	Conversion (%)	Selectivity (%)	TON
	1	0.5mol.%	16	32	100
			28	67	100	134
	2	1 mol.%	16	39	100
			28	76	100	76
	3	2 mol.%	16	83	100
			24	100	100	50
	Reaction conditions: Ethyl lactate 2 mmol; aqueous TBHP 4 mmol; solvent CH3CN 2 g, room
	temperature

Table 4 .

 4 Effect of oxidant on ethyl lactate dehydrogenation using VO(acac)2 as a catalyst

	Entry	Oxidant,	Time(h)	Conversion	Selectivity	TON
		(mmol)		(%)	(%)
	1	0	16	0	-	-
	2	2	16	72	100	36
	3	4	16	83	100	42
	Reaction conditions: Ethyl lactate 2 mmol; solvent CH3CN 2 g, room temperature, catalyst 0.04 mmol
	(2 mol%), oxidant aq. TBHP

Table 5 .

 5 Effect of temperature on ethyl lactate oxidative dehydrogenation using V2O5 catalyst

	Entry	Temperature,	Time, h	Conversion,	Selectivity, %
		°C		%
	1	RT	5	0	-
			a 16	a 72
	2	40	5	26	100
			a 16	a 79
	3	80	5	60	100
			a 16	a 82
	Reaction conditions: Ethyl lactate 4 mmol; solvent CH3CN 4 g, oxidant aq. TBHP 8 mmol, catalyst
	V2O5 10wt.% (w.r.t. substrate); a catalyst VO(acac)2 2 mol.%

Table 6 .

 6 Effect of catalyst loading on ethyl lactate oxidative dehydrogenation using V2O5

	Entry	Catalyst Loading	Time	Conversion	Selectivity
		(wt %)	(h)	(%)	(%)
	1	2	1	12	100
			5	57
	2	5	1	25	100
			5	60
	3	10	1	35	100
			5	60
	Reaction conditions: Ethyl lactate 4 mmol; solvent CH3CN 4 g, 80°C temperature, oxidant aq. TBHP
	8 mmol catalyst 10wt.% (w.r.t. substrate)

2.Ethyl lactate and TBHP conversion with time using V2O5 catalyst

  Reaction conditions: Ethyl lactate 4 mmol; solvent CH3CN 4 g; oxidant aq. TBHP 8 mmol; catalyst 10wt.% w.r.t. substrate

Table 7 .

 7 Change in ethyl lactate conversion using V2O5 catalyst with batch wise addition of TBHP

	Entry.	TBHP addition,	Time, h	Ethyl lactate
		mmol		conversion, %
	1	2	1	24
	2	2	2	31
	3	2	3	49
	4	2	4	54
	5		4.5	67
	6		5	72
	7	2	6	75
	8	1	7	84
	9	1	8	86
	10		10	98
	Reaction condition: Ethyl lactate -0.502g (4 mmol), Catalyst V2O5 0.05g, CH3CN 4 g, Temp. 80°C

Table 8 .

 8 Hot filtration experiment for metal leaching test

	Entry	Time (h)	Conversion (%)	Selectivity (%)
	1	2 #	31	100
	2	1	37	100
	3	2	37	100
	4	3	37	100
	Reaction Conditions: Ethyl lactate 2 mmol; solvent CH3CN 6 g; oxidant aq. TBHP 4mmol; catalyst
	10wt.% w.r.t. substrate.

# Before filtration

Table 9 .

 9 Comparative results of ethyl lactate conversion with respect to literature

		Catalyst	Substrate	Reaction	Conversion	Selectivity	Ref.
				conditions	%	%
	1	MoVNbOx/TiO2	Ethyl	170-190°C, O2	50	80	10
			Lactate
	2	VOCl3	Ethyl	Room temperature,	50	31	11
			Lactate	O2