Delta Neutral Strategies

The Greeks (Delta, Gamma), delta/gamma neutral portfolios, True Delta Neutral (leveraged stable pools), Pseudo Delta Neutral (2x leveraged volatile pools), with full mathematical derivations of Uniswa

Source: Arcadia Finance Blog

Up till now we used a strategy that is both delta neutral and gamma neutral, where we leverage stable pools, by borrowing the same asset. Since these strategies got saturated during the last weeks, yields declined and we worked on a new type of delta neutral strategy with higher returns.

In this article we introduce the new strategy: The Pseudo delta neutral strategy!

This strategy uses a 2x leveraged volatile pool, a big difference however is that this strategy is not Gamma neutral. Practically this means that the strategy has to be rebalanced more often, and that very big market moves still have an impact on the portfolio value.

At the end of this article we will discuss in depth how both the old and new delta neutral strategy work, but first we will give a short refresher what Delta, Gamma and Delta Neutral Portfolios actually mean.

1. The Greeks

The Greeks are variables that are used to assess risks of financial instruments/portfolios. Each greek variable expresses how the value of the financial instrument/portfolio is influenced by a small change of a certain underlying parameter. It is a measure how sensitive a portfolio is to said underlying parameter.

The value of a Greek is not static but changes over time, or with big market movements. Investors aiming to keep a certain Greek value of their portfolio fixed need to periodically rebalance their portfolio.

1.1 Delta

Delta expresses the rate of change between a financial instrument/portfolio and an underlying asset price.

If a portfolio has a positive delta of +0.2 with the price of Ethereum, then when the Ethereum price increases with 1%, the value of the portfolio will increase with 0.2%.

Mathematically, the Delta ($\Delta$) can be expressed as the first-order partial derivative of the portfolio value ($V$) with respect to the price of an underlying asset ($S$).

$$\Delta = \frac{\partial V}{\partial S}$$

1.2 Gamma

Gamma expresses the rate of change between the Delta of a financial instrument/portfolio and the underlying asset price. Gamma measures how sensitive delta itself is to changes in the price of the underlying asset. It gives a measure how hard a portfolio will be influenced by big market movements, or how often a portfolio has to be rebalanced to keep its delta fixed.

Mathematically, Gamma ($\Gamma$) can be expressed as the first-order partial derivative of the Delta value with respect to the price of an underlying asset ($S$), hence it equals to the second-order partial derivative of the portfolio value.

$$\Gamma = \frac{\partial \Delta}{\partial S} = \frac{\partial^2 V}{\partial S^2}$$

2. Portfolios

Investors can use the Greeks to quantify or mitigate certain risks when building portfolios. Two popular strategies are delta-neutral portfolios and gamma-neutral portfolios.

2.1 Delta neutral portfolios

With a Delta neutral portfolio, the overall delta of the portfolio is 0. This means that the total value of the portfolio does not change due to small increases/decreases in the price of an underlying asset.

Important to note is that this is only valid for small changes in price.

2.2 Gamma neutral portfolios

With Gamma neutral portfolios, typically both the delta and the gamma of the portfolio are 0. Since the gamma is 0, the delta itself is very insensitive to price movement, hence even with significant changes of the price of the underlying asset, the portfolio value remains constant.

In general you can say that every Gamma neutral portfolio is also Delta neutral. But the opposite is not necessarily true, not every Delta neutral portfolio is Gamma neutral.

3. Arcadia's Delta-neutral Strategies

3.1 True Delta Neutral: Leveraged Stable pools

For this strategy we use two pegged assets in an LP-pool and borrow an asset that is either equal to one of the two assets, or to a third asset that is also pegged to the two other assets.

Examples are:

• cbETH - WETH pool with WETH debt

• DAI - USDC with USDC debt

• cbETH - wstETH pool with WETH debt

In these strategies all assets and debt are either USDC based, or WETH based. Hence the portfolio value is completely independent from the WETH/USD price.

Since the portfolio value $V$ is not a function of the WETH/USD price $S$, its partial derivatives with respect to $S$ are zero.

$$\Delta = \frac{\partial V}{\partial S} = 0$$

$$\Gamma = \frac{\partial^2 V}{\partial S^2} = 0$$

This strategy is truly delta neutral, since the portfolio is both Delta neutral and Gamma neutral.

If we plot the normalised* portfolio value $V$ (vertical axis) in function of the normalised* WETH/USD price $s$ (horizontal axis) we see that V is independent of $s$.

* Since we normalise the axis, 1 on the vertical axis corresponds to the initial portfolio value and 1 on the horizontal axis corresponds to the initial price.

3.2 Pseudo delta neutral: 2x Leveraged symmetric Volatile pools

This strategy uses a symmetric volatile LP position as underlying asset, and one of the two underlying assets is borrowed.

Intuitively it can be shown why this portfolio is delta neutral with an example. Lets use a WETH/USDC pool, the initial WETH/USD price equals $3000 and we borrow WETH:

• We start with 3000 USDC and borrow 1 WETH which we both deposit in a WETH/USDC pool.

• We now have $6000 worth of assets and $3000 debt

  • Our portfolio value is $3000

  • Our leverage is 2x

• If the WETH/USD increases with $1, both the WETH in our LP position as our WETH debt will increase with $1

  • We now have $6001 worth of assets and $3001 debt

  • Our portfolio value is still $3000

  • Our portfolio is delta neutral!

Unfortunately when the underlying price changes, the LP position will no longer be perfectly symmetric (Impermanent loss!!). The bigger the underlying price changes from the initial price, the less balanced our LP position is and the less delta neutral.

Our position is not gamma neutral.

We can also show this graphically. The value of a generic LP position with relation to the underlying price of the assets is quite complex and non-linear (depending on the AMM bonding curve, available liquidity, liquidity ranges...).

Luckily the relation for a Uniswap V2 pool ($k = x \cdot y$) is still quite simple and we can plot the normalised portfolio value $V(s)$ with respect to the normalised underlying price $s$ (see for derivation in the appendix below).

We see that for a "normal LP position" (without debt) both delta (first derivative V') and gamma (second derivative V'') are non-zero at the initial point (1, 1).

If we now use our 2x Leveraged strategy (see for derivation in the Appendix below), we see that Delta (V') is indeed 0 at the initial price, but Gamma (V'') is non-zero.

• Portfolio is initially Delta neutral

• Portfolio is not Gamma neutral

For Uniswap V3 (and other CLAMMs) the general principle holds but we have another degree of freedom, the liquidity ranges.

• We can use asymmetric positions, and for each asymmetric position there is exactly one amount of leverage such that the initial position is delta neutral. For symmetric positions delta is again 0 when we use 2x leverage.

• In general the smaller the liquidity range, the bigger the impermanent loss and hence the bigger Gamma is (which is bad for our Delta neutral strategy).

Appendix

1. Derivation portfolio value Uniswap V2

A Uniswap V2 position consists of two tokens: let's call the amount of the token0 $x$ and the amount of token1 $y$.

The total value of the LP-position equals (note that the amounts of x and y depend on the relative prices of both tokens):

$$v(p_x, p_y) = x(p_x, p_y) \cdot p_x + y(p_x, p_y) \cdot p_y$$

If we define the portfolio value in units of token0, then the previous equation becomes:

$$v(s) = x(s) + y(s) \cdot s$$

With $s$ defined as the price of token1 in units of token0:

$$s := \frac{p_y}{p_x}$$

And the initial value of the position is then equal to:

$$V_0 = X_0 + S_0 \cdot Y_0$$

For a Uniswap V2 pool in equilibrium with external markets, the value of the reserves of token0 should equal the value of reserves of token1. If this would not be true, arbitrageurs could make a profit until the pool is in equilibrium:

$$x \cdot p_x = y \cdot p_y \implies \frac{x}{y} = \frac{p_y}{p_x} = s$$

$$\implies x = y \cdot s \text{ And } X_0 = Y_0 \cdot S_0$$

So for a LP position of a pool in equilibrium, the value equals:

$$v(s) = 2 \cdot x(s) \text{ And } V_0 = 2 \cdot X_0$$

For Uniswap V2 pools, the relation between $x$ and $y$ is defined via the bonding curve:

$$k = x \cdot y = X_0 \cdot Y_0$$

Which we can rewrite as:

$$x \cdot \frac{x}{s} = X_0 \cdot \frac{X_0}{S_0} \implies x = X_0 \sqrt{\frac{s}{S_0}}$$

Now we can finally rewrite our LP value as only a function of $s$:

$$v(s) = 2 \cdot X_0 \sqrt{\frac{s}{S_0}} = V_0 \sqrt{\frac{s}{S_0}}$$

If we define the normalised portfolio value in function of the normalised underlying price s as:

$$v_n := \frac{v}{V_0} \text{ And } s_n := \frac{s}{S_0}$$

Then we find our relation:

$$v_n(s_n) = \sqrt{s_n}$$

2. Derivation portfolio value 2x Leveraged symmetric Volatile pools

In this strategy we start with the full portfolio value in token0, and we borrow an equal value in token1, we again deposit the initial token0 and borrowed token1 in a Uniswap V2 pool. The total portfolio value in units of token0 is then given as:

$$v(s) = x(s) + y(s) \cdot s - Y_0 \cdot s$$

And the initial value of the portfolio, denominated in token0, is equal to:

$$V_0 = X_0 + S_0 \cdot Y_0 - S_0 \cdot Y_0 = X_0$$

For a pool in equilibrium, the previous defined relationship still holds ($x = y \cdot s$):

$$\implies v(s) = 2 \cdot x(s) - \frac{X_0}{S_0} \cdot s$$

And also the bonding curve for Uniswap V2 AMMs is still valid ($x = X_0 \sqrt{\frac{s}{S_0}}$)

$$\implies v(s) = 2 \cdot X_0 \sqrt{\frac{s}{S_0}} - \frac{X_0}{S_0} s = 2 \cdot V_0 \sqrt{\frac{s}{S_0}} - V_0\frac{s}{S_0}$$

Which gives after normalisation:

$$v_n(s_n) = 2 \sqrt{s_n} - s_n$$