Open the textbook [①], the so-called price discovery function refers to the process that the futures market forms a real, predictable, continuous and authoritative price through an open, fair, efficient and competitive futures trading operation mechanism. Merton miller, winner of the Nobel Prize in Economics, said: "The charm of the futures market lies in letting you really know the price." The advantage of futures market in price formation determines that it has the function of price discovery. First of all, the futures price is formed by the centralized trading of exchange participants, which is completely different from the spot price where participants are relatively scattered and privately traded. Centralized trading gathers many traders and bids in a free and open environment, so the futures price is more real and authoritative than the spot price. Secondly, the futures price represents the settlement price of the market at a specific time and place in the future. Many participants trade with different expectations, and the trading results represent the market's views on future prices, so the futures market has the function of discovering prices. This reflects that the price discovery function of futures market must meet three conditions: first, there are many participants in futures trading; Secondly, most futures traders are familiar with a certain commodity market; Third, futures trading is highly transparent.

I. Literature review

Personally, theoretically speaking, studying the price discovery function of futures market can be summed up as the following two questions: First, is there a long-term equilibrium relationship between futures prices and spot prices? Second, if there is a long-term equilibrium relationship between futures prices and spot prices, what is the causal relationship between futures prices and spot prices? That is, is the change of futures price the reason for the change of spot price? Or is the spot price change the cause of the futures price change? In other words, is there a cointegration relationship between futures prices and spot prices? If so, is the change of futures price ahead of the change of spot price? Or does the spot price change before the futures price changes? Therefore, in many empirical studies, scholars combine causality test and cointegration test to analyze the price discovery function of futures market: Zhu (2007) [2] found that there is a long-term equilibrium relationship between domestic and international futures prices of copper and soybeans, but there is no long-term equilibrium relationship between domestic and international equilibrium prices of wheat; Gong Guoguang (2007)[③] found that there is an obvious co-integration relationship between the futures market price of natural rubber and the spot market price; Li Huiru (2006) [4] found that there is a long-term equilibrium relationship between cotton futures price and spot price, and both futures market and spot market play a role in price discovery, and futures market is in a dominant position in price discovery; Liu Xiaoyu (2006)[⑤] found that there is a mutual guiding relationship between soybean meal futures price and spot price, and there is a long-term equilibrium relationship between fire and spot price; As early as 2002, Hua Renhai and Zhong [6] found that there was a cointegration relationship between copper and aluminum futures prices and spot prices, and futures prices had a good price discovery function. It can be seen that there are different views on whether the futures market has the price discovery function we usually define.

Second, about the function of futures price discovery.

For the price discovery function of the futures market, there is a question whether the futures price is the spot price expected by the spot in the future. Chen Rong and Zheng Zhenlong (2007) [7] hold that speculators decide futures prices according to their expectations of future futures and future spot prices, while arbitrageurs decide futures prices according to current spot prices. Therefore, whether the futures price is determined by the future spot price or the current spot price depends on whether the power of speculation or arbitrage ultimately determines the futures price. They believe that in a perfect market where you can borrow freely, buy short and sell short freely, the power of arbitrage is infinite, so the ultimate decisive force of futures price is the arbitrageur. That is, in this market, the futures price is not the expectation of the future spot price, but determined by the current spot price.

In this regard, from the financial engineering risk-free arbitrage strategy to find the answer. Taking the underlying assets that generate a certain dividend interest rate during the futures duration as an example, assuming that the current time is T, the market risk-free continuous compound interest is R, and the continuous compound interest of the underlying assets during the futures duration is Q, we can construct the following two combinations when pricing futures:

Portfolio A: A long-term contract, which stipulates that the underlying assets of a unit can be traded at the delivery price plus cash amount K*EXP[-r(T-t)] on the maturity date T.

Portfolio B: EXP[-q(T-t)] unit securities, and all income is reinvested in the securities.

In portfolio A, cash of K*EXP[-r(T-t)] is invested at risk-free interest rate R, and the investment period is (T-t). At time t, when the forward contract expires, you can get K yuan in cash, which is just used to deliver the long position of the forward contract and get the assets of one unit. Similarly, the number of securities owned by portfolio B also increases with the increase of dividends and reinvestment. At time t, it also becomes a unit target asset, and its value is exactly equal to the value of portfolio A. According to the principle of risk-free arbitrage, two combinations with equal value at time t must also have equal value at time t, namely:

f+K * EXP[-r(T-T)]= S * EXP[-q(T-T)]

According to the definition, the futures price F is the delivery price that makes the value of the futures contract F zero, from which F= F=S*EXP[-q(T-t) (1) can be obtained.

If the formula (1) is not established, the market arbitrage forces will gain risk-free profits by buying the spot to sell futures or buying futures to short the spot, until the relationship between the futures price and the spot price meets the formula (1), and the market reaches an arbitrage-free equilibrium.

From the derivation of formula (1), it can be seen that the decision of futures price depends entirely on the power of arbitrage rather than the prediction of future prices by buyers and sellers.