Partnered With:

Paradigm's new derivative concept: a perpetual contract

Read this article in 14 Minutes

This paper introduces a new type of derivative - Multiplier perpetual contract.

原文标题：《 Power Perpetuals 》

原文作者： Dave White, Dan Robinson, Zubin Koticha, Andrew Leone, Alexis Gauba, Aparna Krishnan

Zhang Luyao

Introduction to the

This paper introduces a new type of derivative - Multiplier perpetual contract.

If the price of ETH doubles, the 2nd power of ETH contracts multiply by 4, the 3rd power of ETH contracts multiply by 8, and the 5th power of ETH contracts multiply by 32.

Of course, this asymmetric rise is not free. Those on the long side of the perpetual contract need to regularly pay premium returns to those on the short side.

Multiplier perpetual contracts provide exposure similar to global options without the need for a strike price or expiration date, making it possible to consolidate much of the liquidity of the options market into a single instrument.

In many ways, a perpetual contract is.Sustainable options A reasonable next step. As far as we know, other than us, researchers.Wayne Nilsen And & have spentlllvvuu They discovered the product independently.

Mechanism Prerequisites

A perpetual contract is aSustainable options This paper introduces a special series of perpetual derivatives.

For the rest of this article, we'll assume that the reader is already familiar with perpetual options and & NBSP;Sustainable futures The basic mechanism of.

define

A power perpetual contract is a perpetual derivative linked to the power of some underlying asset price. In this article, we will assume that the underlying asset is Ether.

For any power P, the ETH^ P power perpetual contract is maintained through periodic (e.g., daily) capital payments. If, at the time of the capital injection, the current price of the power perpetuity contract is $MARK, the long power perpetuity contract holder must pay $(mark-index)=$(mark-eth ^p) to the short seller.

In a perpetual contract, we call this capital cost Premium yield. Because such fees are typically a premium paid by the long to the short in exchange for exposure to similar options.

example

Consider ETH^2 power perpetual contract.

For simplicity, assume that ETH is trading at $3 and ETH^2 power perpetuation contract is trading at $9.09 when the capital is paid. As per contract bull will have to pay $(MARK - INDEX) to short = $(MARK - the ETH ^ 2) = $9.09 3 ^ (2) = $9.09 - $9.00 = $0.09.

Summary of the pricing

A perpetual contract with a power greater than 1 has a positive convexityThis means that holders make money more quickly when prices are in their favor and lose money more slowly when prices are against them. In terms of options, we say they have a positive Gamma value (gamma).

As with options, they are often known for their & NBSP;The intrinsic value The power perpetuity contract for ETH^p typically trades at a premium to the p power of the ETH price.

derivation

According to & have spentPerpetual options paper We can first price the power derivatives that are about to expire, and then price the combination of these derivatives that corresponds to the required perpetual contract.

Below we will use the Black-Scholes hypothesis to derive our price. These are not, of course, the most appropriate assumptions, but they should serve as an example of how a market maker might value a multiplier perpetual contract.

Pricing an expiring power derivative under the Black-Scholes hypothesis is much simpler than pricing an option. Interested readers can click on the StackExchange here Find a quick derivation. The price for

S^pe^{t\frac {p-1}{2}(2r+pv^2)}Spet2p - 1 (2 r + p2v)

Where S is the spot price, P is the power number, T is the time to maturity, R is drift or risk-free interest rate, and V is the annualized volatility.

In combination with & have spentPerpetual options paper The perpetual option pricing method in Appendix B, and by summing the resulting geometric sequence, we can obtain the following price expression of the multiplier perpetual contract, which pays the capital once per installment (assuming convergence of the sequence -- see below) :

S^p\frac {1}{2e^{-f\frac {p-1}{2}(2r+pv^2)}-1}Sp2e - f2p - 1 (2 r + p2v) - 11

Where f is the capital injection period in years.

This can be interpreted as the exponent S^p times the adjustment coefficient.\frac {1}{2e^{-frac {p-1}{2}(2r+pv^2)}-1}2e−fracp−12(2R + Pv2)−11, which takes into account the embedded optionability of the power perpetual contract. It should be noted that when the injection period is close to 0, this adjustment factor is close to 1.

Premium return (our new term for financing rate) can be calculated as:

\text {MARK}-\text {INDEX} = S^P (\frac {1}{2e^{-f\frac {p-1}{2}(2r+pv^2)}-1}-1)MARK - INDEX = SP (2 e - f2p - 1 (2 r + p2v) - 11-1)

The convergence

Unlike equity perpetuity options, which can always be priced, poorly deployed multiplier perpetuity contracts may not converge. In particular, we can only set a price for the power party's perpetual contract if:

\frac {e^{f\frac {p-1}{2}(2r+pv^2)}}{2} < 12 ef2p - 1 (2 r + p2v) & lt; 1

Intuitively, the higher the power and volatility, the more valuable the long-term, temporary power futures are; The longer the capital injection period, the more the value of the power perpetual contract is concentrated in the long-term power futures. For some portfolios, equal portfolios can become infinitely valuable.

This problem can easily be avoided in practice by choosing a small enough period of capital injection.

Example & have spent

ETH^2 is a perpetual contract

See & have spenthttps://github.com/para-dave/powerperps/blob/master/power_perp_prices.ipynb.

Under the Black-Scholes hypothesis, the ETH^2 power perpetual contract price is S^2\frac {1}{2e^{-f (r+v^2)}-1} S^2\frac {1}{2e^{-f (r+v^2)}-1}S22e− F (r+v2) −11.

Other things being equal, when ETH is 4 times as expensive, it will be 16 times as expensive.

It has a convenient property, which is that it has a constant gamma value, which is \frac {2}{2e^{-f (r+v^2)}-1}2E − F (R +v2)−12, which means that it provides constant optionability regardless of the price of ETH.

We affectionately called it "Squeeth", short for "ETH Square".

ETH^3 is under perpetual contract

See & have spenthttps://github.com/para-dave/powerperps/blob/master/power_perp_prices.ipynb.

The ETH^3 power perpetual contract price is S^3\frac {1}{2e^{-f (2r+pv^2)}-1}S32e− F (2R + PV2) −11.

Other things being equal, when the price of ETH is quadrupled, it will increase 64 times.

You can clearly see from the graph that the perpetual contract trades at a higher price than its index ETH^3 because it gives the holder an option.

Python pricing implementation

You can check it out at thehttps://github.com/para-dave/powerperps/ See a Python implementation of power perpetual contract pricing, including tests that empirically demonstrate correctness.

conclusion

The Power of perpetuity contract is still in its infancy, but we learned from & NBSP;At the beginning Has studied it in depth and is still very excited about its potential.

If you're as curious about this new stuff as we are, we'd love to hear your thoughts. You can email dave@paradigm.xyz, or & have spentSend me a private message on Twitter, or through squeeth@opyn.co, contact Opyn.

Thanks: Lllvvuu, Wayne Nilsen, Wade Prospere, Grug, Lily Francus, Dr. Benn Eifert, Jeff Wang, Mewn