SBF and Paradigm research partners conceive a new derivative: eternal option

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Perpetual options provide traders with long-term options exposure, simple operation, and no need to pay the cost of rolling positions.

Written by: Dave White and Sam Bankman-Fried, the former is a research partner of Paradigm, a crypto asset investment fund, and the latter is a co-founder of market maker Alameda Research and crypto derivatives exchange FTX. Compilation: Perry Wang

Dave White, research partner of Paradigm, a crypto asset investment fund, and legendary trader, market maker Alameda Research, and crypto derivatives exchange FTX co-founder Sam Bankman-Fried (SBF) jointly published a paper introducing a new type of finance Derivatives, that is, everlasting option (everlasting option). According to the paper, “Perpetual options provide traders with long-term options exposure, simple operation, no risk, and no cost of rolling positions.”

Dave White and Sam Bankman-Fried also derived a simple no-arbitrage pricing model for perpetual options, which can be applied to all perpetual derivatives based on the funding fee, including perpetual futures.

Basic knowledge of options

Option type

At the beginning of the article, let us briefly introduce the simplest type of option: European option. There are two types of European options: call option and put option.

A call option gives the option holder the right to buy a specific asset (subject) at an agreed price (exercise price) at a specified time on an agreed date.

A put option gives the option holder the right to sell a specific asset (subject) at an agreed price (exercise price) at the time of expiry.

for example

For example, a $3,000 ETH put option on May 15 means that the option holder has the right to sell 1 ETH at a price of $3,000 at a specific time on May 15.

If the market transaction price of ETH at the expiration of the put option on May 15 is US$2900, the holder of this option has the right to buy 1 ETH from the market for US$2900, and then sell it immediately at the price of US$3000 through this put option. Lock in a profit of $100. This money is called payoff.

On the contrary, if the market transaction price of ETH at the expiration of the put option on May 15 is $3,100, the trader’s price of selling ETH in the spot market is higher than executing the put option contract. In this case, the execution of this option contract will not bring any profit, and we call the return of this option zero.

Income calculation

Although European options can only be exercised at a specific time on the expiry date, or exercised, we can calculate their returns at any time. Option return is a measure of how much the option would be worth if it were executed immediately.

Generally, the return of a put option is max(strike-spot, 0). When the spot market price of an ETH transaction is lower than the exercise price, the more money is made by selling ETH through put options. However, if the spot market price of ETH is higher than the exercise price at expiration, then selling ETH directly in the market is better than using put options, which are worthless.

See https://colab.research.google.com/drive/1nehkZjTh_Kloz_vC –e1h7W_s-yGzh9b?usp=sharing

Similarly, the return of a call option is max(spot – strike, 0). If the spot transaction price of ETH is 3100 USD, and we have a call option to buy ETH at the expiration price of 3000 USD, we can exercise the call option to buy ETH at 3000 USD, and then immediately at 3100 USD The price is sold in the market and a gain of $100 is obtained. However, if the transaction price of ETH is 2900 US dollars by then, and the ETH call option with an exercise price of 3000 US dollars in our hand, the return is 0 US dollars.

Option pricing

Before expiration, the price of an option contract is usually higher than its return (excluding some special circumstances).

Assume that an ETH put option with an exercise price of $3,000 expires tomorrow. If the current market price of ETH is $3000, the current return of the put option is $0. But the price of ETH may fall tomorrow, in which case the expiration value of the put option will exceed $0. Therefore, the current value of the put option must be greater than $0 to cover this possibility.

A basic and widely used model for option pricing is the Black-Scholes model. The figure below shows the Black-Scholes price of the day before the expiration of the ETH put option with an exercise price of $3,000, compared with the return of various spot ETH prices.

SBF and Paradigm research partners conceive a new derivative: eternal option See https://colab.research.google.com/drive/1nehkZjTh_Kloz_vC –e1h7W_s-yGzh9b?usp=sharing

Rolling position

definition

The main use case of options is to hedge or prevent risks. For example, if an investor holds a large amount of ETH investment positions, she can choose to buy enough $3000 ETH put options to ensure that no matter how the market price changes, she can always sell the position at a price of at least $3000/ETH.

However, these put options will eventually expire. If an investor wants to maintain a hedge, she will have to roll her options position. In the above circumstances, it means closing the expiring position in the put option and opening a new put option position with the same strike price, but the expiration time is later.

for example

For example, the investor may initially buy a put option of $3,000/ETH that expires on May 15. When these option contracts are about to expire on May 15, she may sell these option contracts and buy the same number of ETH put options on June 15 with the exercise price still at $3,000. As long as she wants to maintain her position, she must repeat this process once a month.

problem

When an investor rolls a position in the market, she is likely to trade with market participants known as market makers.

Market makers make money when market participants with poor information (such as those who roll option positions) trade with market makers. However, market makers lose money when trading with well-informed market participants (such as those who understand the news of ETH prices).

Because market makers do not know who is well-informed and who is not well-informed among the counterparties, they must charge a certain fee for each transaction, which is called spread. Spreads in the options market are often particularly high, because in this case, the informed transaction costs of the market maker may be high. This makes the cost of rolling positions quite expensive.

Rolling positions also involve work and risks. Traders may simply forget to roll, causing their positions to lose their hedge. Or she may click by mistake or execute the transaction by mistake, which can be expensive and dangerous. Even if everything goes well, the whole process is still stressful and takes time, preventing investors from focusing on more productive work.

Existing solutions

There is a product called perpetual American option in the market, which can be exercised at any time and has no expiry date. Selling perpetual American options requires market makers to assume a lot of risk and uncertainty in advance, which makes them expensive and difficult to price. As a result, they have never actually been traded. It is precisely because of the existence of this product that we call the new alternative “eternal option”.

Liquidity fragmentation

If there are many different option expirations, it will cause another problem: liquidity fragmentation. If market makers must make the market not only for options that expire this week, but also options that expire every week for the next three months, they will be forced to diversify their capital, which will enable other participants It is more difficult to make large transactions or to determine a fair price. Since participants must decide which options to trade on expiry dates, this fragmented market also makes option trading more chaotic.

Futures market analogy

For futures contracts that traditionally have expiration dates, all of these problems will also be encountered.

If a trader wants to use traditional expiring futures to hold ETH for a long time, then she will have to roll her position like an option. For example, she might buy an ETH futures contract that expires on May 15. Then before it expires on May 15th, she may sell the contract and buy an ETH futures contract that expires on June 15th, and so on.

Just like choosing an option, rolling her futures position takes time, brings risks, and requires her to continue to pay the market maker the spread. The existence of multiple expiration date cargo contracts has also led to the fragmentation of liquidity in the futures market.

Perpetual contract

BitMEX’s Perpetual Futures (Perpetual Futures) launched for cryptocurrencies in 2016 solved these problems. They provide traders with exposure to futures risks without the need for rolling, and they can be held for any time. They also concentrate all futures liquidity of a specific underlying subject matter in a single product on a specific exchange.

Perpetual contracts have become very popular, trading tens of billions or even hundreds of billions of dollars every day.

Working Mechanism

Simply put, the working principle of a perpetual contract is as follows: Every day, those who are long (already bought) must pay a financing fee to those who are short (already sold).

The calculation method of this financing fee is (mark-index): the difference between the mark price (the transaction price of the perpetual contract) and the index price (the market price of the subject matter, such as ETH).

This financing cost mechanism keeps the pricing of the perpetual contract in line with the price trend of the subject matter. Roughly speaking, if the price of the perpetual contract is much higher than the market price of the underlying material, then the bulls will have to pay high financing costs, which will encourage them to sell the perpetual contract, thereby reducing its price.

Facts have proved that we can get more accurate results than this. For the working mechanism of perpetual contracts, please refer to The Cartoon Guide to Perps article, or refer to our formula for accurate valuation listed below.

for example

If the current ETH perpetual contract is 3100 USD and the current market price of ETH is 3000 USD, the long must pay the short “mark-index= 3100 USD – 3000 USD = 100 USD per day”.

If the price of the ETH perpetual contract is 2900 US dollars and the market price of ETH is 3000 US dollars, then mark-index = 2900 US dollars-3000 US dollars = -100 US dollars, which means that the shorts must pay 100 US dollars to the longs every day.

Eternal option

Perpetual options are equivalent to perpetual contracts in the options market.

A trader holding a $3,000/ETH eternal option can always sell her ETH at an exercise price of $3,000. She will have to pay financing fees to support her position, but since she does not have to continue to trade with market makers, she does not need to pay spreads or incur operational risks unless she enters and exits her position.

Since option contracts with different expiration times are no longer needed, the degree of dispersion of liquidity will be reduced, although in the basic version, there will still be different eternal futures for different exercise prices.

Working Mechanism

Perpetual options work in exactly the same way as perpetual contracts, with one difference: the financing fee is calculated by marking the difference between the price and the current return of the option. Therefore, the funding fee is (mark-payoff) instead of (mark-index).

for example

Take an ETH put eternal option with an exercise price of US$3,000 as an example. Funds are paid once a day.

If the current trading price of ETH is 2900 USD, the current return of the put option is 3000 USD – 2900 USD = 100 USD. If the put perpetual option is traded at $150 immediately before the financing fee is paid, the long will pay the short mark–payoff = $150-$100 = $50 per day.

If the current trading price of ETH is 3100 USD, which is higher than the price of the eternal option, the return of the put option is 0 USD. If the put perpetual option is traded at $50 immediately before the financing fee is paid, the long will pay the short mark–payoff = $50-$0 = $50 per day.

Please note that the return of an ETH call option with an exercise price of 0 is the market price of ETH, in other words, payoff=index. This means that an eternal option with a strike price of 0 is equivalent to an ETH futures. Correspondingly, the daily financing cost of a perpetual option with an exercise price of 0 is “mark-payoff = mark-index”, which is the same as the financing cost of a perpetual contract.

Pricing

If we don’t know the value of perpetual options, perpetual options are useless. Fortunately, through the no-arbitrage argument listed below, we have explored the value of perpetual options: they are equivalent to a specific, continuously rolling option portfolio, so their pricing will be the same as that portfolio. If the price difference between the two is too large, arbitrageurs will step in to bring it back to consistency.

Assuming that the fee is paid once a day, half of the portfolio at the same price is the option contract that expires today, one quarter is the option contract that expires tomorrow, one-eighth is the option contract that expires the day after tomorrow, and so on. The strike price of all these option contracts is the same as the strike price of the eternal option.

We can also create an eternal option that pays smaller fees multiple times per unit time (for example, paying 1/24 fee per hour), thereby changing the composition of the equivalent investment portfolio. For details, see Appendix B of the report .

In either case, we can price this basket of options contracts to achieve the pricing of eternal options. This can be done simply by obtaining the weighted sum of each option price (estimating the contribution of less than 1/1024 positions in the portfolio). Option market makers are fully capable of pricing these individual expiring options.

If we use the simple Black-Scholes hypothesis, it does not match the real-world trajectory, but it is very close. The trajectory of an eternal option that pays twice a day is almost like a regular expiration option that expires one day after the same exercise price.

SBF and Paradigm research partners conceive a new derivative: eternal option See https://colab.research.google.com/drive/1nehkZjTh_Kloz_vC –e1h7W_s-yGzh9b?usp=sharing

Equivalent option portfolio

The price impact caused by the payment of financing fees

It is very difficult to make reasonable price estimations for perpetual derivatives based on financing costs, such as eternal options, because their pricing is naturally discontinuous. The financing fee is paid at a specific and accurate time (for example, midnight). Just as stocks pay dividends, we predict that the price of perpetual derivatives will naturally rise immediately after paying financing costs.

Therefore, although it is natural to think that “financing fee payment” will happen at the same time other things, but the result is not satisfactory. When reasoning about the behavior of perpetual derivatives during the financing fee payment period, it is best to consider what will happen before or after the financing fee is paid.

By the way, to a certain extent, the current perpetual product exchanges will not automatically update their order books after financing fees are paid, just like stock exchanges after stock dividends, which means that the market faces the risk of arbitrage losses. For example, if the longs are to pay financing fees to the shorts, a rational trader should short their perpetual derivatives one millisecond before expiration, get the financing fees, and then buy within one millisecond after expiration to close them Short position, so as to collect profit with minimal risk.

Equivalent portfolio intuition

description

As mentioned above, a perpetual option that pays financing costs once a day is equivalent to a portfolio of fixed-expiring options: half is the option that expires on the next payment, one quarter is the option that expires on the next payment, and eight One part of the option that expires on the next payment, and so on. The total number of option contracts in the portfolio is one.

This means that when financing costs are paid, option contracts that account for half of the total investment portfolio have expired. The financing fee payment corresponds to the cost of the rolling portfolio: buy new options with half the total value of the option contract to fill half of the contract that has just expired. But unlike the case of manually rolling positions, these new option contracts are distributed over multiple expiration dates, without paying any spreads, and without execution risk.

SBF and Paradigm research partners conceive a new derivative: eternal option

SBF and Paradigm research partners conceive a new derivative: eternal option

Argument

Suppose, Alice makes a long eternal option and pays a financing fee at midnight every day.

Alice must pay the mark-payoff financing fee at midnight tonight. Let us consider the meaning of cash flow. The Mark price is the cost of buying the eternal option indefinitely before paying the financing cost, so Alice pays the cost equivalent to the amount she needs to double her position. On the other hand, since the payoff value is negative, she will get the payoff, which is equivalent to the gain she would get by doing an equivalent regular option that expires at midnight.

To repeat, if long options are equivalent to a portfolio of regular options, it means that Alice immediately doubles her position in each option before expiration, and then gets the equivalent of a contract value at expiration payoff. This means that until her position is doubled, Alice has exactly half of the contract’s assets long for options that expire at midnight.

Extending this line of thinking, if we want Alice’s perpetual option equivalent portfolio to continue to operate after tonight, her position in regular options that will expire at midnight tomorrow is exactly half of the contract tonight. This is only guaranteed if its value is equal to a quarter of the option contract before doubling…and so on.

Please note that this argument applies to any expiry derivative with a definite return, and is not limited to European options.

argument

Please refer to Appendix B of the report .

Extended application

By pricing expiry derivatives, this framework is applicable to the pricing of any perpetual derivatives based on financing costs, not just limited to European call options and put options. These include perpetual contracts.

It also includes binary put options. If the price of the underlying material is higher than the given exercise price, its return is 0; if it is lower than the exercise price, you can get a return of $1, so it can be used to prevent DeFi protocol failures ’S buffer.

Floating strike price eternal option

This framework can also be applied to pricing floating options, where the strike price of the latter is an exponentially weighted moving average of the price of the underlying material over time. This is because maturity-type equivalent portfolios (Asian floating options) can also be priced, although the pricing is extremely difficult.

Having such a put option will always be effective for ETH holders to sell their tokens at the exponentially weighted average price of ETH (with a half-life of one day), protecting them from the sudden drop in the price of ETH.

Because its exercise price will automatically follow the price trend of ETH, a single such product may meet the hedging needs of most ETH holders. It is possible to converge the liquidity and trading volume of many ETH options into one market.

future career

The future work is mainly in the application field.

  • Do eternal options or other new perpetual derivatives based on financing costs really own the market?
  • Which type is most useful?
  • How to best parameterize them?
  • How can exchanges and traders best manage their risks? For traders who conduct margin trading, what are the appropriate clearing standards?

If you have any thoughts on these issues, or need to ask your own questions, we will be happy to hear your opinions.

You can contact Dave via email address [email protected] , or send him a private message via Twitter , or contact SBF .

Acknowledgements: Thanks to Dan Robinson for his direct and indirect contributions to this article in many conversations; thanks to Hasu for his extensive feedback to help clarify the concept and structure of this article; thanks to Georgios Konstantopoulos for his pertinent suggestions on image selection.

Source link: www.paradigm.xyz

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