At present, a tool often used to make arbitrage decisions in the world financial futures market is to use the theoretical pricing model of stock index futures to judge whether there is arbitrage opportunity by comparing the actual price and theoretical price of stock index futures:
When the actual price is higher than the theoretical price, short the index futures and buy the index spot;
When the actual price of futures is lower than the theoretical price, buy index futures and short the index spot.
So how did this theoretical pricing model come into being?
There is a basic law in economics called "law of one price". This means that two identical assets must be quoted at the same price in two markets, otherwise a market participant can carry out so-called risk-free arbitrage, that is, buy at a low price in one market and sell at a high price in another market; Eventually, due to the increase in demand for the asset, the price of the asset in the original low-priced market will rise, while the price of the asset in the original high-priced market will fall until the two quotations are equal. Therefore, the supply and demand forces will produce a fair and competitive price, so that arbitrageurs cannot obtain risk-free profits.
According to the above laws, the difference between the contract price of stock index futures and the spot index price should be within a reasonable range in theory. In other words, investors can calculate the theoretical reasonable price of stock index futures price according to the spot price of stocks. Therefore, the theoretical pricing model of stock index futures came into being.
Derivation of theoretical pricing model of 0 1 stock index futures
Because there are too many related factors that affect the price of stock index futures, it is impossible to establish a mathematical model that includes all the influencing factors. Therefore, we can only select the main influencing factors to establish a mathematical model.
1. Stock index futures are derived from stock indexes, so its price (f) is closely related to the price (i) of stock spot index. The rise and fall of stock spot index price will inevitably affect the rise and fall of stock index futures price. From this, the following rough mathematical model can be established:
F~I
Where: symbol ~ stands for correlation.
2. We need to examine the main factors that affect the price of stock index futures in the process of buying and selling. We find that because futures have a delivery period, it takes a while for the seller to deliver the spot to get the cash, and the buyer pays the cash to deliver the spot at delivery, which is equivalent to the buyer financing the seller. This financing cost is expressed by the risk-free interest rate (market interest rate can be considered instead) R. In this way, when selling futures contracts, the seller needs to obtain the income I R equivalent to the "financing" of the buyer before delivery. This income should be added to the futures price F, so we further refined the original mathematical model:
F≈I+I R
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risk-free interest rate
Risk-free interest rate refers to the interest rate that can be obtained without any risk by investing funds in an investment object. This is an ideal return on investment. In the securities market, the interest rate of risk-free securities is the risk-free interest rate. Risk-free securities refer to fixed-income securities that can perform on time, that is, securities without any risk.
In the securities industry, short-term discounted treasury bonds with a maturity of less than 3 months are usually regarded as risk-free securities. This is because the term of three months is very short, during which the fluctuation of market interest rate has little influence on bonds, and its income is basically unchanged, which can be considered as risk-free. At present, there is no three-month national debt in China, so the risk-free interest rate can refer to the one-year deposit rate.
3. We also find that there are dividend opportunities in stocks, which is the third biggest factor affecting the price of stock index futures. Although the futures seller sold the stock index contract, which is equivalent to selling the stock portfolio in advance, he still holds the stock portfolio before the delivery date and can get dividends, thus reducing his position cost, which is unfair to the futures bulls who have already bought this stock portfolio, so the futures price should be lowered by the amount equivalent to dividends. Let the dividend yield be d, and as a result, the futures price will also be deducted from the corresponding asset income I d:
F=I+I R-I D
4. The delivery time of futures contracts is the fourth factor that affects futures prices. Because the longer the delivery time, the longer the financing time, just like regular savings, the longer the money is saved, the higher the interest, so the listing price of futures contracts in different months should have different time premiums. Assume that the delivery period of the futures contract is T-t day and the conversion year is (T-t)/365. The final theoretical pricing model of stock index futures is:
F=I+I R (T-t)/365-I D (T-t)/365
Namely:
F=I+I (R-D) (T-t)/365
Among them: f-theoretical price of stock index futures;
I- spot stock index price;
R- risk-free interest rate;
D- annual dividend yield;
T- delivery time;
A certain time.
5. What about other influencing factors? All these factors are assumed under ideal conditions, so they are not considered in the model. These secondary factors are assumed to be:
① The portfolio constructed by investors is completely consistent with the stock market index in terms of portfolio proportion, stock index value and stock portfolio market value;
(2) investors can easily borrow money in the financial market to invest;
③ No transaction cost;
④ When arbitrage opportunities appear, market participants will participate in arbitrage activities;
⑤ No psychological panic factors;
⑥ Cash settlement of stock index futures contracts.
6. If R and D in the form of continuous compound interest are considered, the theoretical pricing model of the above stock index futures will be more accurate:
Where: r- risk-free interest rate;
D—— Annual dividend yield calculated by continuous compound interest (%);
T-the expiration time of the futures contract;
T- current time;
The number of days the futures contract expires.
Derivation of theoretical pricing model of stock index futures under continuous compound interest
First of all, we must solve what is compound interest.
situation
We invest 100 yuan a year, and the annual interest rate is 10%. If the bonus is paid in one lump sum at the end of the year:
Principal and interest =100× (1+10%) =110.00 Yuan.
If the bonus is paid every six months, then:
Principal and interest for the first half of the year =100× (1+10%/2) =105 (yuan)
Year-end principal and interest =105× (1+10%/2) =1/0.25 (yuan).
Change to continuous:
Year-end principal and interest =100× (1+10%/2 )× (1+10%/2)
=100×1.05×1.05 =110.25 (yuan)
The extra 0.25 yuan is brought by rolling interest, which is called "compound interest".
Then what is continuous compound interest? Let's take a look at the impact of increasing the frequency of calculating compound interest on the year-end value 100 yuan (see table 1).
Table 1 Influence of increasing the frequency of compound interest calculation on the value of 100 yuan at the end of one year.
(Let the interest rate be 10% per year)
When m tends to infinity, it is called continuous compound interest.
We find that when the interval of calculating interest reaches once a week and once a day, when m approaches infinity, the total principal and interest at the end of the year is almost the same. From a practical point of view, it is generally believed that continuous compound interest is equivalent to daily compound interest.
According to the concept of continuous compound interest, we derive the following formula.
Formula derivation: basis
F=I+I (R-D) (T-t)/365
=I [ 1+(R-D) (T-t)/365]
Assumption: I = (r-d) (t-t)/365
Then: f = I (1+I)
With the above theoretical pricing model, we can calculate the theoretical price of stock index futures contracts at any time, and then compare it with the actual price of stock index futures contracts. When the actual price of stock index futures contract is higher or lower than the theoretical price of stock index futures contract, arbitrage trading can make a profit. But in fact, transactions need costs, such as transaction costs, capital interest rate costs and so on. This causes the reasonable price of forward arbitrage to move up and the reasonable price of reverse arbitrage to move down, forming an interval. In this range, arbitrage will not only lead to no profit, but also lead to losses, because the cash profit is not enough to pay the transaction costs. This interval is called "no arbitrage opportunity interval" (also called "arbitrage cost interval"). Only when the actual transaction price of futures index is higher than the upper limit of the interval can forward arbitrage be carried out; On the other hand, when the actual transaction price of the futures index is lower than the lower limit of the interval, reverse arbitrage can be carried out.
The key now is how to determine the upper and lower limits of this arbitrage-free opportunity interval, which is an important part of the success of spot arbitrage.
Determination of upper and lower limits of arbitrage-free opportunity interval
situation
On August 22nd, the spot Shanghai and Shenzhen 300 Index in the stock market was 1224. 1, and the annual dividend yield of the A-share market in that year was about 2.6%. Assuming that the annual interest rate of financing (loan) is 6%, the current theoretical price of the 65438 10 contract due for delivery on October 22nd should be:
F =1224.1+1224.1× (6%-2.6%) ×1/6 =1231.
Based on this, the arbitrage-free interval of stock index futures contracts is calculated.
It is also assumed that the spread between the rate of return required by investors and market financing is1%; The bilateral handling fee for futures contracts is 0.2 index points; The market impact cost is 0.2 index points; The bilateral handling fee and market impact cost of stock trading are 1%. The number of points converted into an index is:
Loan spread cost =1224.1×1%× 2/12 (year) =2.04 (point)
Bilateral handling fee and market impact cost of stock trading =1224.1×1%=12.24 (point)
Bilateral handling fee and market impact cost of futures trading =0.2+0.2=0.4 (points)
Total points =12.24+2.04+0.4 =14.68 (points)
If it is found that the reasonable price of the current futures contract should be 123 1.04, then:
Upper limit of arbitrage interval =1231.04+14.68 =1245.72 (point)
Lower limit of arbitrage interval =1231.04-14.68 =1216.36 (point)
No arbitrage opportunity interval = [12 16.36, 1245.72]
That is to say, when the spot index of the Shanghai and Shenzhen 300 is 1224. 1 on August 22nd, the price of the stock index futures contract 10 is only forward arbitrage above 1245.72 or16.36.
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Market impact cost
Market impact cost refers to the extra cost of failing to make a deal at a predetermined price when it is necessary to buy and sell stocks quickly and on a large scale in arbitrage trading. For example, investors bought the small-cap stock "Seagull Bathroom" at a price of 15.50 yuan per share, but failed to make a deal. At this time, the selling price of the market is 15.70 yuan/share, and the whole market is rising. Investors are worried that they can't buy it, and finally they have to buy it at the price of 15.70 yuan/share, which makes the purchase cost pay 0.20 yuan more per share, which is the impact cost. For another example, the current price of a fund is 1. 10 yuan, and in order to ensure the success of buying transactions, investors are likely to trade at1.1yuan, and the cost higher than the current price of 0.0 1 yuan is the market impact cost.
The data of securities markets in various countries show that the cost of market shocks is generally more than twice that of commissions and taxes. Impact cost is considered as the fatal wound of large institutions. For example, if a large institution is optimistic about a group of stocks, it will take a long time to realize the purpose of opening positions. If you are eager to open a position, buying in large quantities in a short time will drive up the stock price, which will inevitably make the cost of opening a position much higher than expected. Similarly, if you are eager to sell the stock, it is equivalent to depressing the stock price, and the final selling price is lower than the original expected price. For retail investors, the impact cost is almost zero because of the small transaction volume.
The impact cost is not only related to the number of entrusted transactions, but also related to liquidity. The smaller the market liquidity, the greater the impact cost. The daily turnover of the stock market is tens of billions, which seems to be large and liquid. But that's the result of thousands of stocks trading together. If it is subdivided into different stocks, it is not difficult to find that some of them are still very illiquid. Relatively speaking, the impact cost of buying and selling stock index futures is very small. This is because there are only a few corresponding stock index futures contracts, and the concentration of transactions leads to very strong liquidity, even for large institutions, it is very convenient to enter and exit.