# Gamma Swap by Deri Protocol

## Introduction

In [the previous paper](https://docs.deri.io/library/academy/gamma-swap/introducing-gamma-swap), we introduced a new type of derivative, Gamma Swap. As discussed, the trading of Gamma swap depends on a composite trading venue consisting of two (virtual) primitive trading venues for power perps and perpetual futures. This paper explains the implementation of such a composite trading venue by the [DPMM of Deri Protocol](https://github.com/deri-protocol/whitepaper/blob/master/deri_v3_whitepaper.pdf).

## Composite DPMM of Gamma Swap

Following the terminology in the previous paper, in this paper, $$i$$  refers to the index price while $$M$$  refers to the different kinds of mark prices. **The core role of DPMM is to determine the mark prices based on the index price input and the trading activities.** Everything else (e.g., PnL, funding fees) is then based on the mark prices.

Internally, 1 unit of Gamma swap is treated as a portfolio of powers and perps. DPMM simultaneously keeps track of the power and perp positions and calculates their respective mark prices. When 1 unit of long Gamma swap is entered, the DPMM adds 1 unit of power and $$-\frac{2i\_0}{1-h\_0T}$$  units of perp to the two positions, respectively. One could also enter a Gamma swap position with a specified entry price $$i\_s$$ . In that case, DPMM would add 1 unit of power and $$-\frac{2i\_s}{1-h\_0T}$$  units of perp to the two positions, respectively. Denote the total net position of power as $$n$$  and total net position of perp as $$m$$ , then DPMM calculates the mark price of perp as follows:

$$
\frac{M\_{perp}-i}{i}=k\_{perp}\left(n\cdot\frac{2i}{1-hT}+m\right)=k\_{perp}m'
$$

where $$i$$ is the index price of the underlying, and $$m'=n\cdot\frac{2i}{1-hT}+m$$. Please note that:

* This is just the classical DPMM for perp with $$m'=\left(n\cdot\frac{2i}{1-hT}+m\right)$$  as the total net position, instead of just $$m$$.
* Notice that $$m'=0$$   at a “balanced point”, at which the liquidity pool has no Delta exposure. For example, immediately after the first position was opened, we have $$n =1$$   and $$m=-\frac{2i\_0}{1-h\_0T}$$  , hence $$m'=0$$ . A “balanced point” is equivalent to the scenario of zero net position in a regular DPMM of perp.
* A trading action reducing $$|m'|$$ (and thus bringing the system to the “balanced point”) enjoys a negative slippage of $$M\_{perp}$$  . However, since opening a new position at the current price (without specifying entry price) does not change $$m'$$  , it does not enjoy a negative slippage. This is a bit different from the case of perp DPMM, in which a new position reducing the total net position always enjoys negative slippage.

And the mark price of power is:

$$
\frac{M\_{power}-i\_{power}}{i\_{power}}=k\_{power}n
$$

where $$i\_{power}$$  is the theoretical price of the power, calculated just like that in the [power perp DPMM](https://github.com/deri-protocol/whitepaper/blob/master/Pricing_Continuously_Funded_Power_Perpetuals.pdf):

$$
i\_{power}=\frac{i^2}{1-hT}
$$

Now we have a composite DPMM that, at any point, takes the state variables $$(n, m, i, \sigma)$$  to determine $$(M\_{perp}, M\_{power})$$  and hence gives$$M\_{gamma}$$  as

$$
M\_{gamma}=M\_{power}-\frac{2i\_0}{1-h\_0T}M\_{perp}+\frac{i\_0^2}{1-h\_0T}
$$

$$M\_{gamma}$$determines the trading cost of every trade and consequently determines PnL.

## Funding Calculation 

1 unit of Gamma swap has composite funding based on the power funding and the perp funding:

$$
F\_{gamma}=F\_{power}+\left(\frac{2i}{1-hT}-\frac{2i\_0}{1-h\_0T}\right)F\_{perp}
$$

where $$(i\_0, h\_0)$$  are the $$(i,h)$$   as of the position entry point, and

$$
\begin{align\*}
F\_{power}=&\frac{M\_{power}-i^2}{T}\\
F\_{perp}=&\frac{M\_{perp}-i}{T}=i\cdot\frac{k\_{perp}}{T}\left(n\cdot\frac{2i}{1-hT}+m\right)
\end{align\*}
$$

$$F\_{power}$$   is simply handled the same way as power perps.

Whereas the handling of $$F\_{perp}$$ is somewhat tricky, as the coefficient of the $$F\_{perp}$$ term is constantly changing with $$i$$ . Essentially, this can be understood as every unit of Gamma swap has $$-\frac{2i\_0}{1-h\_0T}$$   “real futures” as well as $$\frac{2i}{1-hT}$$   “virtual futures”, and hence totally $$\left(\frac{2i}{1-hT}-\frac{2i\_0}{1-h\_0T}\right)$$   effective futures. The tricky part is the constantly varying virtual part. Because of this constantly varying virtual part, the perp funding should be calculated **per Gamma swap, instead of per futures**.

By definition, the exact funding fee of 1 unit of Gamma swap accumulated over the time period $$(t\_1, t\_2)$$   should be the integral of $$F\_{perp}$$  over this period.

$$
\begin{align\*}
\text{Funding}*{perp}(t\_1,t\_2)=&\int*{t\_1}^{t\_2}\left(\frac{2i}{1-hT}-\frac{2i\_0}{1-h\_0T}\right) F\_{perp}dt
\\
\=&\int\_{t\_1}^{t\_2}\left(\frac{2i}{1-hT}\right) F\_{perp}dt-\frac{2i\_0}{1-h\_0T}\int\_{t\_1}^{t\_2} F\_{perp}dt
\\
\=&
A|*{t\_1}^{t\_2}-
\frac{2i\_0}{1-h\_0T}B|*{t\_1}^{t\_2}
\end{align\*}
$$

where $$A|*{t\_1}^{t\_2}$$  and $$B|*{t\_1}^{t\_2}$$  are the accumulated increment of $$A$$   and $$B$$   over $$(t\_i,t\_2]$$  :

$$
\begin{align\*}
A|*{t\_1}^{t\_2}=&\int*{t\_1}^{t\_2}\frac{2i}{1-hT}\cdot F\_{perp}dt=\int\_{t\_1}^{t\_2}\frac{2i}{1-hT}\cdot i\cdot\frac{k\_{perp}}{T}\left(n\cdot\frac{2i}{1-hT}+m\right)dt\\
B|*{t\_1}^{t\_2}=&\int*{t\_1}^{t\_2} F\_{perp}dt=\int\_{t\_1}^{t\_2} i\cdot\frac{k\_{perp}}{T}\left(n\cdot\frac{2i}{1-hT}+m\right)dt\\
\end{align\*}
$$

Please note that the two accumulating variables $$A$$   and $$B$$   are universal to all positions while $$\frac{i\_o}{1-h\_0T}$$   is position-specific. We need to track the accumulations of $$A$$ and $$B$$ separately and then for any position we can calculate its accumulated funding fee with its own value of $$\frac{i\_0}{1-h\_0T}$$  . That is, for a position of $$x$$ gamma swaps entered at $$t\_a$$   with entry values $$(i\_0, h\_0)$$  , its accumulated funding fee from $$t\_1$$ to $$t\_2$$ is:

$$
\left\[(A\_{t\_2}-A\_{t\_1})-(B\_{t\_2}-B\_{t\_1})\frac{2i\_0}{1-h\_0T}\right]\cdot x
$$

In practice, it is impossible to rigorously calculate $$A$$ and $$B$$ due to their path-dependencies. Hence, we need some numerical approximations.

#### Approximation Method 1

We assume the funding over the time period $$(t\_{i-1},t\_i]$$   is accumulated at the speed as of $$t\_i$$  . Essentially this assumes $$i=i(t\_2), \forall i \in(t\_{i-1}, t\_i]$$ . With this approximation, the two integrals over $$(t\_i,t\_2]$$   degenerate into simple multiplications:

$$
\begin{align\*}
A|*{t\_1}^{t\_2}=&\frac{2i}{1-hT}\cdot i \cdot\frac{k*{perp}}{T}\left(n\cdot\frac{2i}{1-hT}+m\right)\Delta t\\
B|*{t\_1}^{t\_2}=&
i \cdot\frac{k*{perp}}{T}\left(n\cdot\frac{2i}{1-hT}+m\right)\Delta t
\end{align\*}
$$

#### Approximation Method 2

A more reasonable approximation is to assume the index price $$i$$ changes linearly over $$(t\_1, t\_2]$$  :

$$
i(t)=i\_1+(i\_2-i\_1)\frac{t-t\_1}{t\_2-t\_1}, \forall i \in(t\_{i-1}, t\_i]
$$

Then we can calculate the accumulating variables $$A$$  and $$B$$   as follows:

$$
\begin{align\*}
A|*{t\_1}^{t\_2}=&\frac{2}{1-hT}\cdot\frac{k*{perp}}{T}\left(n\cdot\frac{2}{1-hT}\cdot\overline {i^3}+m\cdot\overline {i^2}\right)\Delta t\\

B|*{t\_1}^{t\_2}=&
\frac{k*{perp}}{T}\left(n\cdot\frac{2}{1-hT}\cdot\overline {i^2}+m\cdot\overline {i}\right)\Delta t
\end{align\*}
$$

where $$\overline {i^3}$$, $$\overline {i^2}$$ and $$\overline i$$   are the time-weighted average value of $$i^3$$  , $$i^2$$   and $$i$$   over $$(t\_1, t\_2]$$  , respectively:

$$
\begin{align\*}
\overline{i^3}&=\frac{1}{\Delta t}\int\_{t\_1}^{t\_2}i^3dt=\frac{(i\_1+i\_2)(i\_1^2+i\_2^2)}{4}\\

\overline{i^2}&=\frac{1}{\Delta t}\int\_{t\_1}^{t\_2}i^2dt=\frac{i\_1^2+i\_1i\_2+i\_2^2}{3}\\

\overline{i}&=\frac{1}{\Delta t}\int\_{t\_1}^{t\_2}idt=\frac{i\_1+i\_2}{2}\\
\end{align\*}
$$

With this approximation, the power funding component can be calculated similarly:

$$
\text{Funding}*{power}(t\_1,t\_2) = \frac{(k*{power}n+hT)}{(1-hT)T}\cdot \overline{i^2} \Delta t= A\_{power}|\_{t\_1}^{t\_2}
$$

where $$A\_{power}|\_{t\_1}^{t\_2}$$   is the accumulating variable tracking the accumulation of power funding. In practice, the two  $$A$$ variables of power and futures can be combined into a single variable.

## Summary

The [previous paper](https://docs.deri.io/library/academy/gamma-swap/introducing-gamma-swap) introduces the concept of Gamma Swap, a new type of derivative depending on a composite trading venue consisting of two primitive trading venues for power perps and perpetual futures. This paper explains the implementation of such a composite trading venue by the DPMM of Deri Protocol. With the DPMM for Gamma Swap constructed, now we are ready to roll out a comprehensive solution for Gamm Swap.