---
title: Leveraged strategies (slides)
author: Alexis Direr
tags: general
type: slide
slideOptions:
theme: solarized
---

# Leveraged strategies (slides)

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- Basics of leverage strategies
- What is volatility decay
- How to mitigate it ?
- Implications for market making

---

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- There is a huge demand for leveraged products, as demonstrated by the popularity of perpetual futures proposed by CEX (Binance, FTX, ...) and on-chain protocols like dYdX or GMX.

- Leveraged return can also be obtained by borrowing assets in lending protocols but is complex to manage.

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- A limited offer exists that assists retail investors with the financial engineering and daily management of borrowing-based leveraged strategies.

- The [Ethereum Flexible Leverage Index](https://medium.com/indexcoop/introducing-the-flexible-leverage-index-fli-by-pulse-inc-a369bd422ef) (ETH2x-FLI), which market cap is $12m, leverages a collateralized debt position on ETH by targeting a long 2x exposure to ETH.

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- The management of the strategy is abstracted away into an index fund


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- A persitent difficulty with leveraged return is [volatility decay](https://indexcoop.com/blog/fli-volatility-drift) (VD).

- VD is a small discount on return which compounds with price volatility.

- It is well documented in traditional finance since the launch of the first [Leveraged ETFs](https://www.investopedia.com/articles/exchangetradedfunds/07/leveraged-etf.asp) in 2006.

---

Example of UPRO ($3 \times$ leveraged S&P 500 ETF)

| | S&P 500 | UPRO (3x) |
| ---------------- | ------: | --------: |
| 12 January 2020 | 3329 | 36.19 |
| 15 March 2020 | 2300 | 10.15 |
| 21 December 2020 | 3709 | 38.43 |
| 2020 performance | 11.4% | 6.2 % |

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## Origin of volatility decay

Variation rates do not add up when the price loops:

)

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What happens when a fund targets a $3 \times$leverage:


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The same for short leveraged positions:


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### Second cause of volatility decay

- the way a leveraged fund rebalances its debt to guarantee a fixed leverage ratio.

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### Leveraged long strategy

- deposit $x_0$ ETH in a lending protocol
- borrow $\delta y_0$ USDC
- with the borrowed amount, buy more ETH

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- The price goes from $p_0$ to $p_1$
- the leveraged return:
$$
\Big( 1+\dfrac{ \delta y_0}{p_0 x_0} \Big) \Big( \dfrac{p_1}{p_0} - 1 \Big)
$$

- is amplified by the leverage ratio $\delta y_0/p_0x$.

- For a $3 \times$ long strategy, the manager (or smart contract) borrows twice the collateral's value.

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- Suppose the amount borrowed is not changed.

- Next period, if:
-- the price has increased (profit): the leverage ratio of next period will be *lower*
-- the price has decreased (loss): next period leverage ratio will be *higher*.

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To maintain a fixed leverage ratio, the fund needs to

- borrow more USDC to buy more ETH after a price increase
- sell some ETH, repay part of the debt in case of price decrease.

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- In order to reset the appropriate amount of leverage, the manager is led to follow a *pro-cyclical* investment strategy.
- This is is appropriate when the price is heading to a single direction.
- When it fluctuates back and forth, the return is affected by volatility decay.

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### Volatility decay

- In the ETH2x-FLI [documentation](https://help.tokensets.com/en/articles/5049454-eth2x-fli-faqs):

- *The index is rebalancing as the price is decreasing and thus selling as the price goes down. That means that if the price drops and comes back up, the Net Asset Value at the end will be less than the Net Asset Value at the beginning.*

$\rightarrow$ Explains why investors are warned against holding their position over long periods of time.

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[Investopedia](https://www.investopedia.com/articles/exchangetradedfunds/07/leveraged-etf.asp):

- *Maintaining a constant leverage ratio (...) is how the fund is able to provide double the exposure to the index at any point in time, even if the index has gained 50% or lost 50% recently. Without rebalancing, the fund's leverage ratio would change every day.*

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- VD also affects perpetual futures.
- If a trader is leveraged long and the price increases (profit), without rebalancing, next period leverage will be lower.

- FTX proposes ERC20 [leveraged tokens]() invested in FTX [perpetual futures](https://ftx.com/markets/futures).

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- From their [documentation](https://help.ftx.com/hc/en-us/articles/360032509552-Leveraged-Token-Walkthrough):

- *[BULL](https://ftx.com/trade/ETHBULL/USD)/BEAR/[HEDGE](https://ftx.com/trade/ETHHEDGE/USD) tokens will automatically reinvest profits into the underlying asset; so if your leveraged token position makes money, the tokens will automatically put on 3x leveraged positions with that. Conversely, (...) if you (...) buy ETHBULL, the leveraged token will automatically sell off some of its ETH as markets go down.*

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- VD in Mycelium Perpetual pools (ex Tracer DAO)

- From their [Discord](https://discord.com/channels/808906099172442122/887464543058534432/1002207147133763644):


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### Inverse strategy

- An inverse or short strategy consists in borrowing the base asset, here ETH, and selling it for USDC.
- This strategy is also affected by volatility decay.

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Leveraged short strategy

- deposit $y_0$ USDC in a lending protocol
- borrow $\delta x_0$ ETH
- sell ETH

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- Leveraged return for prices $p_0$ and $p_1$:
$$
-\dfrac{p_0 \delta x_0}{y_0} \Big( \dfrac{p_1}{p_0} - 1 \Big)
$$

- Leverage ratio: $-p_0 \delta_0/y_0$
- A strategy achieves $-1 \times$ETH by borrowing the collateral value and $-2 \times$ETH by borrowing twice the collateral value.

---

- with the same amount borrowed, two possibilities next period
-- the price has increased (loss): leverage ratio is *higher*
-- the price has decreased (profit): leverage ratio is *lower*

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To maintain a fixed leverage ratio, the fund needs to

- *buy* ETH to deleverage after a price increase
- borrow more and *sell* ETH in case of price decrease.

- buys when the price is up and sells when it is down. If the price moves back and forth, the return is affected by volatility decay.

---

### VD-free leveraged strategies

- For a protocol offering leveraged service, warning users against volatility decay and advising them to leverage only for short periods of time is self-defeating.

- How to eliminate volatility decay: do not rebalance the debt after price variations.

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- VD-free leveraged long strategies
-- deposit $x_0$ ETH
-- borrow $\bar{y}$ USDC
-- buy ETH with USDC
-- Do not rebalance

---

- Simulations:
-- ETH price is $1300
-- Deposited assets are $x_0=100$
-- Initial leverage ratio is $2$
-- $\bar{x}= 100$
-- $\bar{y}=$ 130,000 USDC

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Leveraged and unleveraged wealths


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Leverage ratio


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- The leverage ratio is decreasing with price.
- In exchange of a variable leverage ratio, investors are protected against price volatility.
- A drawback of not rebalancing the debt is that the fund may become insolvent or liquidated if the price decreases too much.

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- VD-free inverse strategies
-- deposit $y_0$ USDC
-- borrow $\bar{x}$ ETH
-- sell ETH for USDC
-- Do not rebalance

---

- Simulations:
-- ETH price is $1300
-- Deposited assets are $100$ ETH
-- debt $\bar{x}= 100$ ETH
-- Initial leverage ratio is $-1$

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Leveraged short and unleveraged wealths


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Leverage ratio


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- Is there a continuous trade-off between VD and stable leverage?
- From [Risedle](https://risedle.com/) [Discord](https://discord.com/channels/898463992991846400/900345756202983444/937314785379364885):


<!-- ETH: no rebalancing between 1.7 and 2.3 -->

- 2 $\rightarrow$ $1.7$: $-21$% price variation
- 2 $\rightarrow$ $2.5$: $+17$% price variation

---


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Cost of VD

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## Market making with leverage

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| | Market making | Leverage Long |
| -------- | -------- | -------- |
| Need to define a price range | Yes | Yes (most importantly a lower bound in leveraged longs) |
| The tighter the price range ... | the more profitable | the higher leverage |

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| | Market making | Leverage Long |
| -------- | -------- | -------- |
| The price falls below the lower bound | The strategy is paused | The position is liquidated |
| The price rises above the upper bound | The strategy is paused | Leverage is close to $1$ |

---

| | Market making | Leverage Long |
| -------- | -------- | -------- |
| Cost of repositioning the price range | Inventory risk | Volatility decay |
| Benefits of repositioning the price range | More capital concentration | Resets the leverage ratio |

---

- How to use leverage in market making?
- How to frame the MM strategy?
- Can we extend the [$\phi$ model](https://hackmd.io/@pre-vert/smm) to leveraged strategies?

---


- price range without leverage: $[1230, 3750]$
- with leverage ($1.5$ to $-0.5$): $[100, 5000]$

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### Leverage long

- At date $t$, the fund holds $x_{t-1}$ ETH
- part $x_0$ is owned by the fund
- the rest has been purchased with debt $y_{t-1}$
- The manager chooses $\delta y_t$, the new borrowed amount or repayment.

---

- Leveraged return between dates $t$ and $t+1$:
$$
\Big( \dfrac{p_{t+1}}{p_t} -1 \Big) \Big( 1+\dfrac{ y_t }{p_t x_{t-1} - y_{t-1}} \Big)
$$

$$
= \text{price variation rate} \times \big( 1+\dfrac{\text{debt}}{\text{net worth}} \big)
$$

---

### Making a profit from volatility

- Date $0$ net worth: $p_0 x_0 - y_0$
- Date $1$ net worth: $p_1 x_0 - y_0$
- Adding or repaying some debt $\delta y_1$
- Make a profit at date 2 with $p_2 = p_0$ (price loop):

$$
p_0 (x_0 + \dfrac{\delta y_1}{p_1}) - y_0 - \delta y_1 \geq p_0 x_0 - y_0
$$

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Two cases to profit from volatility

- $p_1 > p_0$, $\delta y_1 < 0$ (deleverage)
- $p_1 < p_0$, $\delta y_1 > 0$ (more leverage)

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| | Price increase | Price decrease |
| ----------------- | -------------- | -------------- |
| Constant leverage | more debt | less debt |
| VD-free leverage | same debt | same debt |
| Market making | less debt | more debt |

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ETH portfolio weight
$$
\omega_t = \dfrac{p_t x_t}{p_t x_t - y_t} = \dfrac{\text{ETH value}}{\text{net worth}}
$$

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- In the particular case of a VD-free leveraged strategy (fixed amount of debt $\bar{y}$ and borrowed ETH $\bar{x}$), the leverage ratio is *equal* to the ETH portfolio weight :
$$
\dfrac{\bar{y}}{p_t ( x_0+\bar{x}) - \bar{y} } + 1 = \omega_t
$$

---

Leverage ratio = portfolio weight


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- The VD-free leveraged strategy is like a passively held portfolio (no rebalancing)
- in which the ETH weight is *decreasing* with price
- Feels like a MM strategy, but it's not
- The portfolio weight is decreasing with price, but *not enough*.

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### Minimalist modeling

- Choose a minimum price $p_{min}$
- $\alpha >1$ is the minimum collateral ratio (lowest ratio of collateral to debt that will not trigger a liquidation)
- Maximum debt for minimum price:
$$
y_{max} = \dfrac{p_{min}x_0}{1-\alpha}
$$

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Leveraged long strategy (linear) with $p_{min}=100$


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Path independent leverage (example)


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Leveraged short strategy (linear) with $p_{max}=5000$


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