Application of machine learning algorithm and static model of interest rate curve in futures analysis

Abstract

There are still some errors and instabilities in futures analysis. In order to improve the accuracy of prediction, this study combines image processing technology and applies image processing technology to interest rate wireless to construct a static model based on image processing. The necessary conditions for no arbitrage are realized by adjusting the price of issuing virtual vouchers at a parity, that is, the discount rate is monotonous in discrete cases. In addition, this paper obtains static model statistical graphs through data, and combines image noise processing and image segmentation technology to improve the clarity of statistical graphs and records the results for analysis of this research model. The research shows that the model proposed in this study has certain feasibility in futures analysis and can provide theoretical reference for subsequent related research.

Keywords

Interest rate curve static model image processing futures analysis data processing

1 Introduction

Treasury bond futures is a futures product with embedded options, so the classic cost of ownership theory of futures pricing is no longer applicable, and it is difficult to solve its reasonable price. The reasonable pricing of derivative products is crucial to its normal operation and healthy development. However, at present, China’s research on the reasonable pricing of treasury bonds futures is still relatively small. Based on the above reasons, it is necessary to make some improvements to the classic no-arbitrage interest rate term structure model through qualitative and quantitative analysis, summarizing and comparing the previous research results. On this basis, the Chinese government bond futures pricing model is constructed, and empirical pricing research is carried out.

There is a gradual and in-depth process of understanding the role of futures. Historically, just as people opposed the issuance of paper money, people once thought that issuing futures was not conducive to the operation of the country’s macro economy, so they advocated a fiscal policy without debt. They believe that futures will cause extravagance and waste by those in power, exploitation of people’s wealth, and damage to national credit. However, with the development of economy and politics, the information of the bond market is widely disseminated through high-tech means, and the state is no longer the power machine of a certain person or family. Therefore, these concerns no longer exist. The study of economics has also shifted from whether the country should issue futures, and then to how much the country should issue futures, and what term structure and price should be used to issue futures. Futures, as an important financial instrument, are of great significance to the operation and regulation of the national economy, and to the assets and risk management of various trading entities in the financial market.

For other financial products, futures yield is the price of funds. As the most basic financial instrument and an important substitute product, it will provide a basis for pricing other financial products. This effect is achieved through arbitrage. In terms of yield, if there is a financial product in the market with the same risk as the futures (other conditions such as maturity), but with a lower yield. Then people would buy futures and sell the financial product for a risk-free return. In terms of risk, if there is a financial product in the market with the same rate of return as the futures, but with higher risks. Investors can then sell the financial asset and buy futures to obtain arbitrage gains. Considering both, if there is a financial product with higher risks and lower returns, then investors can also arbitrage. As a result of arbitrage, arbitrage opportunities disappear, and the non-arbitrage products presented on the market meet the principle of high risk and high return. Credit bonds are as unsecured and unsecured as futures. However, this financial instrument has risks, so the interest rate under the same term is higher than the futures yield. The analysis of the credit interest rate curve is generally divided into two parts. One is the risk-free interest rate, and the other is the credit spread.

Interest rates are a reflection of bond prices. The interest rate curve summarizes the information on constructing daily bond prices, and in this sense, its construction is similar to a stock index. Stock index summary stock price information uses a certain algorithm to express the price level of almost all stocks in the overall stock market. The interest rate curve summarizes bond information and uses a certain algorithm to express the price level of almost all bonds in the overall bond market. Therefore, it is necessary to further study the static model construction method of the interest rate curve.

2 Related work

Gordillo [1] proposed that even if there is transaction cost, there is a significant correlation between the spot and futures of the government bond closest to the current maturity date. Killeen P R [2] assumed that the closing prices of futures and stocks were subject to random walks, and based on the no-arbitrage pricing principle, the trading strategy was tested to verify that the government bond futures market and the spot market reached a general equilibrium. Vashishtha [3] proposed a pricing model based on perfect market assumptions, namely the cost-holding model, by modifying the simple arbitrage model. However, because the assumption of the model ignores the market environment factors and violates the reality, there is a significant difference between the actual price of the futures and the theoretical price estimated by the holding cost pricing model. Streit [4] studied the pricing of Treasury bond futures contracts on the Chicago Mercantile Exchange. They found that the spot price movements were highly deviated, and the market lacked transaction efficiency, and found that the function did not really play. Panzarino [5] have reached the same conclusion on the pricing of CME government bond futures contracts. However, Singh [6] research shows that the former research results are not convincing. The reason is that the data collected by Chow & Brophy and Hegde & Branch in the study was inappropriate and they failed to use the appropriate model. Research by Mitchell, Vortelinos [7] shows that arbitrage can accurately price financial products and maintain efficient operation of the market, which is an important concept in the financial economy. This basic principle is true whether it is in the complete market or in the incomplete market. In the complete market, a certain value of the price of the financial asset can be obtained, and in the imperfect futures market, an interval of the price of the financial asset is obtained.

Wang [8] calculated the theoretical price range of Treasury futures in the friction market based on the cost of ownership model, and by testing the transaction data of 13 Treasury futures contracts in the US CBOT and CME markets for the past two years, the following conclusions were drawn: Long-term Treasury futures contracts are far more prone to arbitrage opportunities than medium-term Treasury futures contracts, and the longer the remaining contract period, the more arbitrage occurs. Moreover, 13 treasury bond futures contracts have low-frequency arbitrage opportunities, and limited arbitrage opportunities mostly occur when the contract is just listed, when the contract is nearing maturity, or when the special event occurs. In addition, in the national debt futures market, the price of government bond futures contracts is underestimated much more than the number of overestimations, and the chance of reverse arbitrage far exceeds the number, size and frequency of positive arbitrage. MichÃ [9] measured the data of the treasury bond simulation transaction. The results show that there are multiple arbitrage opportunities in the current trading period, and the arbitrage opportunities increase as the contract expiration time approaches. The calculation of the difference between the contract of the next season and the contract of the current season shows that it is difficult to judge the price difference regression theoretical value of intertemporal contract or the historical experience value (the three standard deviations of the mean value). Investors who participate in inter-temporal arbitrage transactions face greater risks and must set stop-loss positions. Moreover, the setting of the stop loss point requires a longer period of historical data to measure. Seefried [10] analyzed and compared the vertical (China’s treasury bond futures pilot in the 1990 s) and horizontal data (China’s Shanghai and Shenzhen 300 stock index futures, US short-term Treasury bills and China’s 5-year simulation of Treasury futures) and found that the launch of China’s 5-year Treasury futures will have a arbitrage opportunity. Judging from the pilot of China’s treasury bond futures in the 1990 s, there were many arbitrage opportunities through the basis trade, and the treasury bond 316 contract had a maximum arbitrage of 6.95%. In addition, from the perspective of China’s 5-year Treasury futures simulation trading, the arbitrage space of the TF1206 contract period can reach more than 7%.

Schuster [11] tested the CSI 300 stock index futures continuously and the current month’s consecutive contract prices and found that there is a significant cointegration relationship between the two. Moreover, by using the variance ratio to test the mean return phenomenon of the stock index futures spread, it is found that the Intertemporal spread has a significant mean recovery characteristic for the inter-temporal spread at 3 days and 5 days. When the time parameter is greater than or equal to 10 days, the mean recovery characteristics of the Shanghai and Shenzhen 300 intertemporal spreads are not significant. In addition, based on the analysis of variance ratio, they proposed a moving average arbitrage strategy based on the mean return theory. Gibbs [12] selected the 5-minute high-frequency data of the CSI 300 stock index futures contract and used the cost-holding model and the cointegration model to analyze the inter-temporal arbitrage. In terms of the cointegration model, in addition to co-integration testing the price of the two sets of futures contracts, the article establishes an ARMA model for the error term, that is, the price difference term, to predict the trend of the next time spread, and improves the model. The data selected in the article has achieved good results on both models. The cointegration model finds more arbitrage opportunities than the holding cost model, and the ARMA model of equilibrium error selects a more sTab. arbitrage opportunity from the many arbitrage opportunities found by the cointegration model.

Through the above analysis, we can see that there are still some errors and instability in futures analysis. In order to improve the accuracy of prediction, this study combines image processing technology and applies image processing technology to interest rate wireless to build a static model based on image processing. The model can realize the effective analysis of futures and provide a theoretical basis for subsequent related development.

3 No-arbitrage bootstrap method

The yield to maturity is converted to the spot rate of return through the bootstrap method, which satisfies the assumption of the bootstrap method on market prices and theoretical prices. The spot rate has a one-to-one correspondence with the discount rate. Moreover, when the cash flow matrix is given, the yield to maturity corresponds to the market price. However, under the condition of satisfying the bootstrap, the yield to maturity at a certain critical point may not be unique. Under the premise that the yield to maturity at any critical time point is unique, that is, the yield curve of maturity exists and is unique, and the transition forming maturity to spot can be considered. Moreover, the rank of the augmented matrix (cash flow CF and price P) is equal to the rank of the coefficient matrix (cash flow CF). When the cash flow matrix is unchanged, the only solution condition for the maturity to spot is that the rank of the augmented moment is equal to the rank of the coefficient matrix and the coefficient matrix is full rank [13]. Through the coupon stripping method, the spot rate of return is converted to the yield to maturity, which satisfies the assumption of the bootstrap method on market prices and theoretical prices [14].

The condition for the spot transfer to the maturity is that the cash flow matrix CF is given. At this point, we can get the theoretical price of the bond. Under the assumption of the bootstrap method (the discounted price is equal to the market price), the yield to maturity can be obtained by using the cash flow and market price of one bond. The meaning of the only solution to the spot-to-maturity is to require the yield to maturity at a certain remaining time point to be equal. Since there may be two or more bonds under a certain remaining period, the expected yield to maturity is not necessarily equal.

The bootstrap method first constructs the cash flow matrix to calculate the maturity-to-spot, and then uses the interpolation method to connect the spot yield scatter to form the curve.

The first step: constructing a cash flow matrix of bootstrap

There are various methods for the bootstrap method. Although different bootstrap methods have different assumptions, they all require that the market price under the assumption is equal to the theoretical price (discounted price). Market prices under hypothetical conditions are sometimes not observed in real markets. Moreover, some assumptions are given to cause the cash flow matrix CF to be full rank, and the augmented matrix is equal to the rank of the coefficient matrix, so that a unique depreciation DF vector can be obtained [15].

The first type: zero interest debt law

In the foregoing, when describing the definition of spot rate of return, two conditions are mentioned:

There must be a corresponding zero-coupon bond portfolio in the market. (2) These bonds are freely traded, the transactions are more active, and the pricing is accurate (regardless of market costs). Assuming that the pricing on the market is reasonable and arbitrage-free, the no-arbitrage condition for satisfying the coupon stripping method is [16]:

P_{Market, 0} = P_{Theory, 0}

That is, the discounted value is equal to the market value: $P_{Market, 0} = \sum_{t = 1}^{n} {CF}_{t} * exp (- r_{sr, t} * t)$

The analysis of the basic construction method of the application of the interest rate curve can obtain the immediate rate of return.

The first condition is too strict. From the actual transaction situation, it is almost impossible to observe in the bond market that all interest-bearing bonds fully correspond to the zero-coupon bonds at the corresponding time. The reason is that the market size is limited, and the variety is limited. Moreover, assuming that all interest-bearing bonds in the bond market fully correspond to the zero-coupon bonds at the corresponding time, there will be no need to issue interest-bearing bonds. Therefore, this precondition is only an ideal state [17].

First, if there is no corresponding zero-interest bond product, the coupon bond assets cannot be copied, and arbitrage activities cannot be carried out. Second, if the zero-coupon bond products are not priced properly or cannot be traded normally, arbitrage activities cannot be carried out. When either of these two conditions is not met, each bond does not fully satisfy the discounted value equal to the market value [18]: $P_{Market, 0} \neq P_{Theory, 0}$

The second type: the virtual coupon method is issued at a low price.

This method relaxes the conditions of the zero-coupon bond coupon stripping method and no longer requires sufficient zero-coupon bonds. At the same time, the cash flow of the virtual vouchers issued at a low price is used to define the cash flow matrix CF of the basic construction method and to make it rank full [19].

A virtual coupon issued at a fair price means: Assuming that a coupon-bearing bond with a coupon rate equal to the annual compound interest-to-maturity yield is issued, its price should be the face value.

For interest-bearing bonds with annual compound interest, the following expression [20]: $P_{Market, 0} = \frac{FV * CR}{{(1 + YTM)}^{i}} + \frac{FV}{{(1 + YTM)}^{n}}$

The interest rate of the interest-bearing bond of the compound interest in the current year is equal to the yield to maturity under the same remaining period, that is, when CR = YTM: $\begin{matrix} P_{Market, 0} = \sum_{t = 1}^{n} \frac{FV * CR}{{(1 + YTM)}^{i}} + \frac{FV}{{(1 + YTM)}^{n}} = FV \\ (CR * \frac{(\frac{1}{1 + YTM} - \frac{1}{{(1 + YTM)}^{n + 1}})}{1 - \frac{1}{1 + YTM}} + \frac{1}{{(1 + YTM)}^{n}}) \end{matrix}$

After CR = YTM is substituted, the above formula is reduced to: $P_{Market, 0} = FV$

There are a few points to note during the derivation process. First, the time interval for paying interest must be an arithmetic progression, otherwise the derivation will not be obtained. This condition is met for bonds that have just been issued and bonds that have just paid interest. Second, time is not necessarily one year. The same result can be obtained by changing the unit of time. However, this requires a certain adjustment to the quotation in the bond market. For example, the coupon rate for half a year and the compound interest rate for half a year are adjusted. Third, the form of continuous compound interest cannot be used. When using the form of continuous compounding to substitute the formula, the same calculation process can only be used to approximate the market price equal to the face value. However, it is possible to convert the continuous compound interest into the compound interest rate of the compound interest for recalculation.

The matrix of the bootstrap method based on the idea of issuing virtual coupons at a low price is as follows: $\begin{matrix} [\begin{matrix} {CF}_{1, 1} \\ {CF}_{2, 1} \\ ⋮ \\ {CF}_{n - 1, 1} \\ {CF}_{n, 1} \end{matrix} \begin{matrix} 0 \\ {CF}_{2, 2} \\ ⋮ \\ {CF}_{n - 1, 2} \\ {CF}_{n, 2} \end{matrix} \begin{matrix} 0 \\ 0 \\ {CF}_{3, 3} \\ ⋮ \\ {CF}_{n, 3} \end{matrix} \begin{matrix} \dots \\ \dots \\ \dots \\ ⋱ \\ \dots \end{matrix} \begin{matrix} 0 \\ 0 \\ 0 \\ ⋮ \\ {CF}_{n, n} \end{matrix}] \\ * [\begin{matrix} {DF}_{1} \\ {DF}_{2} \\ ⋮ \\ {DF}_{m} \end{matrix}] = [\begin{matrix} P \\ P \\ ⋮ \\ P_{m} \end{matrix}] \end{matrix}$

This is a lower triangular matrix. The first line can find DF₁, and it can be substituted into the second line to find DF₂. The bootstrap method is to iteratively solve this system of equations.

For the first line, there are: ${CF}_{1} * exp (- r_{ytm, 1} * t) = {CF}_{1} * exp (- r_{sr, 1} * t)$

For the second line, there are: $\begin{matrix} {CF}_{2} * exp (- r_{ytm, 2} * t) + {CF}_{1} * exp (- r_{ytm, 2} * t) \\ = {CF}_{1} * exp (- r_{sr, 1} * t) + {CF}_{2} * exp (- r_{sr, 2} * t) \end{matrix}$

For the n-th line, there are: $\sum_{t = 1}^{n} {CF}_{t} * exp (- r_{ytm, n} * t) = \sum_{t = 1}^{n} {CF}_{t} * exp (- r_{sr, t} * t)$

The assumption of this method: The yield to maturity must be fair, which requires a certain screening of the sample coupons. Moreover, the virtual coupon issued at the coupon rate is indeed a face value.

The third is a formal method that assumes a spot curve.

In the general market, bonds with shorter maturities are no longer paid by coupons, and only the last cash flow is left. Moreover, we can use them to calculate the spot rate of return. After that, the next bond is selected, and the spot rate is calculated by linear interpolation in the part of the known period, and the discount rate is calculated. In the part larger than the known period, the spot rate of return at the remaining time limit is assumed to be R_T, and then the intermediate spot rate of return is interpolated one by one, so that the pricing formula contains only one unknown number R_T. The current rate of return is obtained by the Newton iteration method. In addition, Zheng Zhenlong and Lin Hai (2003) adopted this method, and the method can also select other interpolation functions.

Step 2: Hermite interpolation is used to connect the immediate rate of return point pairs to construct the curve

In the case where some spot scatters are known, these scatter points can be connected to construct the curve using the conformal three-time Hermite method.

0 = t₁ < L < t_n, For scatter (t_i, y_i) , (t_i+1, y_i+1) i, j ∈ [1, n], t₁ < t < t_n, Corresponding rate of return y (t), Then choose the Hermite algorithm, the formula is as follows: $y (t) = y_{i} H_{1} + y_{i + 1} H_{2} + d_{i} H_{3} + d_{i + 1} H_{4}$

Among them: $\begin{matrix} H_{1} = 3 \frac{t_{i + 1} - t^{2}}{t_{i + 1} - t_{i}} - 2 \frac{t_{i + 1} - t^{3}}{t_{i + 1} - t_{i}} \\ H_{2} = 3 {\frac{t - t_{i}}{t_{i + 1} - t_{i}}}^{2} - 2 {\frac{t - t_{i}}{t_{i + 1} - t_{i}}}^{3} \\ H_{3} = 3 \frac{(t_{i + 1} - t)^{2}}{t_{i + 1} - t_{i}} - 2 \frac{(t_{i + 1} - t)^{3}}{(t_{i + 1} - t_{i})^{2}} \\ H_{4} = 3 {\frac{(t - t_{i})}{(t_{i + 1} - t_{i})^{2}}}^{3} - 2 {\frac{(t - t_{i})}{t_{i + 1} - t_{i}}}^{2} \end{matrix}$

d_j = y (t_j) , j = i, i + 1 is the slope, t_i is the remaining period, and y is the discount factor. The Hermite is preserved three times and on this basis, a curve monotonic condition is added to construct the polynomial curve.

4 No-arbitrage bootstrap method

For the bootstrap method of issuing virtual coupons at a low price, this paper realizes the necessary condition of no arbitrage by adjusting the price of issuing virtual coupons at a low price, that is, the discount rate is monotonous in discrete cases. From the point of view of the discount rate, if there is a case of DF_i < DF_j, i < j, an easy way to deal with data is to make the discount DF_j forcibly reduce DF_j = DF_i, and then subsequent iterations are continued.

From the perspective of yield to maturity, it is calculated using market bond information at key time points, so it is also adjusTable. A method was published on the official website of the China Foreign Exchange Trading Center. It does not mention the discount rate DF monotony, but only mentions the elimination of unreasonable market-making quotation data and the elimination of data that differs greatly from the reported buy-and-sell yield curve. Moreover, as far as national debt is concerned, the data on the yield to maturity announced by the China Money Network and China Government Securities Depository Trust & Clearing Co., Ltd. are also inconsistent. The one-year maturity yield of China Government Securities Depository Trust & Clearing Co., Ltd. was 3.4127% on August 1, 2017, while the one-year maturity yield of China Money Network was 3.0002% on August 1, 2017. However, in fact, it can be seen from the discount rate calculated by their published spot yield curve that both meet the no-arbitrage condition in the discrete case.

From the point of view of the issuance of virtual vouchers at a low price, it is assumed that it will be issued according to the face value. In practice, if it is not issued according to the face value, there is no theory that it will cause arbitrage. Now, the situation is just the opposite. Because the yield to maturity is not fair, there will be arbitrage in the parity issue, and arbitrage can be avoided in the non-parity issue. Therefore, adjusting the price is also reasonable. Then, the following analysis uses financial logic and mathematical formulas to adjust the meaning of the price.

In fact, for most bonds, the discounted value does not necessarily equal the market value: $P_{Market, 0} + Δ P = P_{Theory, 0}$

That is, $P_{Market, 0} + Δ P = \sum_{t = 1}^{n} {CF}_{t} * exp (- r_{sr, t, i} * t)$

The process of solving the following equation step by step is the no-arbitrage bootstrap process. ${\begin{matrix} P_{Market, 0} + Δ P = \sum t = 1 n {CF}_{t} * exp (- r_{sr, t, i} * t) \\ {DF}_{1} \geq {DF}_{2} \\ Δ P \to 0 \end{matrix}$

There are two reasons for ΔP → 0. First, if ΔP = 0 and it can satisfy the no-arbitrage condition, the current bond market price is reasonable. Second, if ΔP = 0N and it does not satisfy the no-arbitrage condition, the forward interest rate is less than zero. Then, if the market price is unreasonable, the forward interest rate will converge to zero with the arbitrage activity. Under the DF₁ ≥ DF₂ constraint, ΔP is a range. The arbitrage behavior stops just when the ΔP change is minimal, and the one closest to 0 should be taken.

In the process of bootstrap, this method firstly recognizes that the short-term bond pricing is reasonable and constructs the interest rate curve through the gradual arbitrage constraint. For example, the zero-coupon bond is priced at 95 yuan for a one-year period and 96 yuan for a two-year period. It is easy to know that the forward rate is less than zero, so it is obviously arbitrage. The arbitrage process is: a one-year bond is purchased, a two-year bond is sold, and a cash 100 is held one year later to pay off the debt. In the bootstrap method, the price of the 2-year zero-interest bond will be revised to 95 yuan, and the forward rate will just reach the critical state of N0. After that, the theoretical treatment was carried out, and the two-year spot was revised without changing the one-year spot rate of return. However, in actual operation, the 1-year price increase and the 2-year price drop may reduce the one-year spot yield and increase the 2-year spot rate. However, we are not sure which change is more. In this article, we do a simplification process that only changes the 2-year spot yield. In addition, the interest rate curve thus constructed will have no arbitrage opportunities.

It can be seen from the formula that the purpose of adjusting these three is to make the original larger DF_j trend equal to DF_i, and only cause the entire matrix and the jth row of the system group to be affected, and the rest remain unchanged. Of course, the solution vector will be affected from the j item down. In the j + k-th row down the j-th line of the entire cash flow matrix, the coefficient of DF_j is the coupon rate. Therefore, the change of DF_j has little effect on the change of DF_j+k, so the no-arbitrage method has little effect on the overall DF.

2.3 No-arbitrage NS model

Elson assume that the instantaneous forward rate is the solution of the differential equation as follows: $\frac{d^{2} f (t)}{{dt}^{2}} + (\frac{1}{τ_{1}} + \frac{1}{τ_{2}}) \frac{df (t)}{dt} + \frac{f (t)}{τ_{1} τ_{2}} = \frac{β_{0}}{τ_{1} τ_{2}}$

In the case of equal roots, the instantaneous forward rate is: $f (t) = β_{0} + β_{1} exp (- \frac{t}{τ}) + β_{2} \frac{t}{τ} exp (- \frac{t}{τ})$

The spot rate of return is: $\begin{matrix} r (t) = β_{0} + β_{1} \frac{τ}{t} (1 - exp (- \frac{t}{τ})) \\ + β_{2} (\frac{τ}{t} - (\frac{τ}{t} + 1)) exp (- \frac{t}{τ}) \end{matrix}$

The meaning of the parameters:

When t → 0⁺, r (0) = f (0) = β₀ + β₁. Therefore, β₀ + β₁ represents the spot (instantaneous forward) interest rate at time zero.

When t→ + ∞, r (+ ∞) = f (+ ∞) = β₀. Therefore, β₀ represents the spot (instantaneous forward) interest rate at time +∞, which represents the horizontal value of the far end of the TN rate curve.

By the forward interest rate being greater than or equal to 0, the no-arbitrage necessary and sufficient condition can be realized, that is, the no-arbitrage condition under continuous conditions. The domain of f (t) is 0 to +∞. The foregoing considers the case of the endpoint, and now considers the extreme point in the middle. $\begin{matrix} f^{'} (t) = & - \frac{1}{τ} (β_{1} exp (- \frac{t}{τ}) + β_{2} \frac{t}{τ} exp (- \frac{t}{τ}) \\ - β_{2} exp (- \frac{t}{τ})) \end{matrix}$

Assuming f′ (t) = 0, the following formula is obtained. $t = τ * (1 - \frac{β_{1}}{β_{2}})$

f (t) is brought into the formula. $f = β_{0} + β_{2} exp (\frac{β_{1}}{β_{2}} - 1) \geq 0$

There is no analytical solution for the function of β₂, and the analysis is more complicated. However, when β₂ ≥ 0, f must be greater than or equal to zero. Then, the third condition is obtained. $β_{2} \geq 0$

In summary, the no-arbitrage NS model satisfies: $[\begin{matrix} - 1 & 0 & 0 \\ - 1 & - 1 & 0 \\ 0 & 0 & - 1 \end{matrix}] * [\begin{matrix} β_{0} \\ β_{1} \\ β_{2} \end{matrix}] \leq 0$

That is $C_{NS} * β \leq 0$

The mathematical expression for the no-arbitrage NS model is: $min \sum_{j} {(P_{Market, j} - P_{Theory, j})}^{2}$ $\begin{matrix} st . {\begin{matrix} r (t) = β_{0} + β_{1} \frac{τ}{t} (1 - exp (- \frac{t}{τ})) \\ + β_{2} (\frac{τ}{t} - (\frac{τ}{t} + 1)) exp (- \frac{t}{τ}) \\ DF = exp (- t * r) \\ CF * DF = P \\ C_{NS} * β \leq 0 \end{matrix} \end{matrix}$

5 Result

The same variety of futures contracts for different delivery months are based on the same target index, and they are highly correlated with the underlying index. If there is a long-term sTab. linear relationship between the prices of two (or more) assets, it can be considered that there is a cointegration relationship between them. When the price deviates from this equilibrium relationship in the short term, there is a correction mechanism that causes this deviation to return to a reasonable range. The cointegration relationship generally requires two conditions: First, their historical price series are first-order single integer vectors, that is, the price series is non-stationary, but the first-order differential sequence (i.e., the rate of return series) is sTab. Second, some linear combination of the two sequences is stationary, that is, the residual of the linear equation constructed with two sequences is stationary.

Table 1
Regression results of daily frequency data

Independent variable Dependent variable Coefficient Standard deviation t value p value Adjusted R2

Closing price of forward contracts Intercept term 16.53529 0.724613 23.50406 0.0000* 0.988232

Closing price of recent contracts 0.855667 0.007833 112.5134 0.0000*

Settlement price of forward contract Intercept term 16.29994 0.660787 25.40747 0.0000* 0.990321

Settlement price of recent contracts 0.858228 0.007143 123.7501 0.0000*

Independent variable	Dependent variable	Coefficient	Standard deviation	t value	Adjusted R2
Closing price of forward contracts	Intercept term	16.53529	0.724613	23.50406	0.988232
	Closing price of recent contracts	0.855667	0.007833	112.5134
Settlement price of forward contract	Intercept term	16.29994	0.660787	25.40747	0.990321
	Settlement price of recent contracts	0.858228	0.007143	123.7501

We return the forward price of the treasury bond futures frequency and the price of the recent contract, using the close price and the settlement price respectively. The regression formula is as follows: ${Future}_{t} = α + β Re {ncent}_{t} + ɛ_{t}$

From the regression results, it can be seen that the impact of the intercept term and the recent contract price on the forward contract price is significant. The recent contract volatility is large and its regression coefficient is 0.83. The close price reggresion resid obtained on this basis is shown in Fig. 1, and the settlement price reggresion resid is shown in Fig. 2.

Fig. 1

Close price reggresion resid.

Fig. 2

Settlement price reggresion resid.

The image transformation is performed on the Close price reggresion resid, and the obtained result is shown in Fig 3.

Fig. 3

Image conversion of Close price reggresion resid.

The image transformation is performed on Settlement price reggresion resid, and the result is shown in Fig. 4.

Fig. 4

Image transformation of Settlement price reggresion resid.

After the ADF test of the residual sequence, the conclusion is that the two residual sequences are stationary sequences, and the two treasury bond futures price series are first-order sTab.. Moreover, it can be known that there is a cointegration relationship between the daily price of the future government bond futures and the frequency price of the recent government bond futures. The residual sequence is shown below:

As shown in Fig. 5, since the data has more instability, the image directly has a strong ambiguity, which causes the output image to be blurred, and the effective information cannot be obtained from the image. In order to improve the image recognizability, the image denoising processing technology is used to denoise the image, and the obtained image is shown in Fig. 6.

Fig. 5

Original image of the residual sequence diagram.

Fig. 6

Preliminary processing image of the residual sequence diagram.

From Fig. 6, we can obtain the basic shape of the waveform and the law of data distribution. In order to further enhance the image effect, the image is further purified, and image segmentation and background separation methods are used, and the results are shown in Fig. 7.

Fig. 7

Clear processing results of residual images.

The transaction cost is set to 0.0326 / lot. Using 80 data points and a moving average with a price range of μ ± 2σ to show the return trend. It can be calculated that the average value of the inter-period spread and the moving average difference μ=3.9595e-4 and the standard deviation σ = 0.0258. Figure 8 shows the benefits of Strategy 1, and Fig. 9 shows the benefits of Strategy 2. The size of the return is an absolute value calculated by buying and selling at a face value of one hundred yuan. Since the difference between the value of the Treasury futures contract and the face value is 10,000 times, the actual return should be the return data * 10000.

Fig. 8

Moving average model-strategy 1 benefits.

Fig. 9

Moving average model-strategy 2 benefits.

The difference between realized income and floating income is that realized income is calculated based on cash receipts and expenditures, regardless of the value of the held position, that is, the value of the held position is considered to be 0; floating income is calculated based on the current value, and held The value of the position, taking into account the impact of position value changes.

6 Analysis and discussion

This paper first analyzes the operation status of treasury bond futures and treasury bonds in China and believes that the transaction volume of China’s treasury bonds futures and spot is not large, and the liquidity is not good. China’s treasury bond futures and spot prices are consistently fluctuating. This paper attempts to use the Granger causality test to find the price discovery function of treasury bonds futures and combines image processing technology to process data images to find data rules.

Figure 7 shows the result of clear processing of the residual image. The image results can be clearly observed from the Fig., and the distribution of futures can be summarized from the image. The residual sequence was subjected to ADF test, and it was concluded that the residual sequence is a stationary sequence. Moreover, there is a cointegration relationship between the minute frequency price of the forward treasury bond futures and the minute frequency of the recent treasury bond futures. In addition, the residual sequence results obtained by the ADF test are stationary, and the intertemporal arbitrage based on the cointegration model seems to be feasible. However, by careful observation, we can find that the residual sequence is indefinite, the speed of returning to zero is slow, the number of arbitrages is extremely limited, and there is a possibility that the futures contract will expire but the residual will not return to zero. In addition, the current residual sequence is derived from the regression of the original data, and whether the intercept term size has an accurate internal meaning is debaTab., and the regression results are suspected of data mining, and the future residual sequence trend may be greatly deviated. Due to these factors, the statistical arbitrage model based on the cointegration model is difficult to apply to inter-temporal arbitrage of treasury bonds.

Judging from the actual price of the inter-period spread, the short-term return volatility of the Treasury bond futures spread is large, and the long-term return volatility is small. Therefore, the inter-period spread of the Treasury bond futures has obvious mean return characteristics (except for the moving average of 100 data points). From the difference between the interim spread of the national debt futures and the moving average, it is a stationary sequence and fluctuates around zero, and the distribution is concentrated, and it is characterized by a peak thick tail. In addition, from the theoretical price of the inter-period spread, the inter-period spread is determined by the short-term financing rate. In the absence of major changes in the market, the short-term financing rate has not changed much in a certain period of time. The main reason for the fluctuation of the inter-period spread in the short term is the change of investor sentiment, the change of short-term and long-term power and the emergence of some abnormal points. Finally, from the analysis of the actual spread and the theoretical spread, the moving average arbitrage method based on the mean return theory is feasible for the interim arbitrage of the national debt futures.

7 Conclusion

This paper uses the moving average model to carry out an empirical simulation of the inter-temporal arbitrage strategy of the national debt futures and combines the image processing technology to clear the model and confirms that the profit of the strategy rises steadily. At the same time, this paper analyzes the impact of transaction cost, moving average data volume, arbitrage interval width and time selection on the results of arbitrage trading, and considers the actual market conditions, and draws the following conclusions: (1) The trading strategy of returning to the mean value, that is, closing the position, is better than the trading strategy of exceeding the price range and closing the position. (2) The moving average data volume size and price range setting have a great influence on the number of arbitrage transactions. (3) In the ideal state where the transaction cost is small, the more trading opportunities are captured, the more transactions are made, and the higher the profit. However, the result is not the case in the case of high transaction costs. (4) Since the actual transaction costs are difficult to determine, in practice which trading strategy is optimal, it needs to be tested in the actual market. Moreover, the most accurate arbitrage trading simulation should use the data of the buy and sell orders and can consider increasing the stop loss mechanism to reduce the risk of intertemporal arbitrage.

Footnotes

Acknowledgment

This paper was supported by (Grant from) The National Natural Science Funds NO 71774101. In addition, we are grateful to Dr. Noura Metawa for his advice on our work.

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