June 17, 2022

Our quant wizard breaks down how Voltz Protocol's risk engine enables large leverage safely.

**We use state-of-the-art financial engineering to bring you the Voltz Risk Engine.****The Risk Engine simulates a variety of Voltz IRS pools, covering different APY volatilities, fixed-rate markets, actor leverages, and liquidity pool concentrations.****New liquidation and insolvency risk measures have been developed: LVaR and IVaR.****Combining the above, we demonstrate that actors taking >100x leverage with Voltz are safe against liquidations and insolvencies.****In specific market scenarios, LPs can safely take leverage >1000x and traders >500x.**

Voltz Protocol1 is a noncustodial Automated Market Maker (AMM) for Interest Rate Swaps, unlocking the power to trade interest rates, with substantial leverage. On the protocol, you can exchange ‘variable rates for fixed rates’ or ‘fixed rates for variable rates’ in an *interest rate swap* (IRS).

In an IRS, we describe one side as Fixed Takers (FTs), where you swap a variable rate and end up with a fixed rate of return. This provides traders with the ability to de-risk their portfolio by ‘locking in’ a fixed interest rate. Whilst we describe the other side as Variable Takers (VTs), where you swap a fixed rate and end up with a variable rate of return. This lets more sophisticated traders increase potential profit capture on the rates of individual assets. The money market is generated using concentrated liquidity pools, where Liquidity Providers (LPs) deposit funds and collect fees. An excellent overview of all these actors, and more besides, can be found in the *Voltz Protocol General Concepts *guide.

The true power of the Voltz Protocol is unlocked when an actor trades or deposits with *leverage*: the ratio of their notional supplied to initial margin deposited, as controlled by the protocol. Under large leverages (100x +) even **small differences in fixed and variable rates can give rise to substantial profits**. The all-important question for any actor is therefore **what is the maximum afforded leverage, in order to ensure the risk of liquidations and insolvencies in my portfolio is minimized?** Here we present the new Voltz *Risk Engine*, a framework for determining such leverages in a Voltz IRS pool, simulating a suite of different market volatility scenarios, and optimizing the fee parameters of a pool to maximize performance and lock in potentially enormous profit.

We need to understand the maximum amount of leverage a given FT, VT, or LP can act with, such that the probability of their portfolio undergoing liquidation or insolvency issues is below a certain significance level, e.g. can my VT position safely take 300x leverage and keep liquidation chances below 5%? To understand how to assess these risks we need to first define what liquidation and insolvency actually are, and borrow an important concept from traditional finance: the *value-at-risk* (VaR).

Liquidation quantities how close an actor is to their liquidation margin, *M*liquidation, given their deposited margin *M*deposited. Values below zero signify liquidation, unless the margin is topped up. Similarly, insolvency is a measure of the actor’s net cashflow i.e. the sum of their deposited margin and cumulative profit-and-loss at some given time *t*. Net cashflows below zero indicate insolvent conditions, where the returns from the IRS pool can’t cover the deposited margin. Let’s put all of these ingredients together into a couple of handy mathematical definitions, and normalize so that everything is relative to the initial margin deposited. We then have the liquidation and insolvency time series along an actor’s returns curve:

In order to extract liquidation and insolvency probabilities, we need a way of estimating the *distribution* of possible liquidation and insolvency values. **The Risk Engine achieves this by simulating Voltz Protocol IRS aUSDC, aDAI, and cDAI pools in a variety of different market APY volatility conditions, bull/bear/neutral fixed rate markets, and different tick ranges to capture different liquidity pool concentrations.** For each market we study, the mean liquidation and insolvency are calculated from 100 replicates generated using circular block bootstrapping[2] of the original liquidation/insolvency series. This methodology, combined with the central limit theorem3, allows us to generate Gaussian probability distributions of the liquidation and insolvency, integrated across many different Voltz Protocol IRS markets. Given the distribution of possible liquidation and insolvency values we can now ask the very important question: what is the liquidation/insolvency value below which the bottom 𝞪 % of the distribution resides e.g. bottom 5%? **This is the all-important VaR measure. **In traditional finance, this is typically applied to the portfolio profit-and-loss distribution e.g. the returns value corresponding to the bottom 5th percentile of portfolio performance. Analogously, we can do exactly the same for our liquidation and insolvency distributions, and use the Gaussian shapes to write down immediate estimates for the **liquidation-VaR (LVaR) and insolvency-VaR (IVaR)**:

Here the 𝝁 and 𝞼 are the mean and standard deviation of the liquidation and insolvency mean Gaussians, and the *Z*-score takes different values for different 𝞪% downside probabilities e.g. 5% downside corresponds to a *Z*𝞪 of 1.96. An example simulated mean liquidation distribution is provided in Figure 2, with the corresponding LVaR indicated by the dotted black line, well approximated by taking a Gaussian fit of the simulated distribution with a Z-score of 1.96, corresponding to the 5% downside liquidations in the tail. Under our methodology, we would not select a leverage which drags the liquidations further to the left of this LVaR position, which therefore defines a maximum “liquidation leverage” for the process.

*Figure 2 (above): Simulated mean liquidation, with corresponding Gaussian fit and LVaR for a 5% downside significance level. *

Since the LVaR and IVaR represent worst case liquidation and insolvency values for a given actor and market, we can reverse engineer them to determine the amount of leverage an actor requires to be exposed to a 5% downturn for a fixed margin deposited. These values can be interpreted as reasonable maximum leverage to safely avoid liquidations and insolvencies beyond the bottom 5% of the tails. Some nice algebra and an assumption that the PnL can be approximated by its mean over the short-horizon gets us leverage estimates (for *N* notional):

The minimum of each FT, VT, and LP leverage provides a conservative cap for avoiding liquidations and insolvencies. But is this cap 1x, 10x, 100x ? We’re getting to that, but first a little more on **how we optimize Voltz protocol to keep liquidations and insolvencies low, and leverages high**.

From navigating the litepaper[1], a Voltz Protocol IRS pool depends on a number of parameters that control the upper and lower bounds of the APY rates, the liquidation margin requirements, initial margin requirements, and fee structure. These act as fee parameters in the Risk Engine, with the following definitions and notation provided in Figure 3.

*Figure 3 (above): Voltz Protocol Risk Engine free parameters, to be optimized under a particular objective function and leverage constraints. *

We seek to find an optimal set of values of the above parameters, unique to each Voltz Protocol IRS pool, which ensure that FTs, VT, and LPs can separately enter highly leveraged positions with at the same time keeping the number of liquidations and insolvency across the different market scenarios to a minimum. To capture broad market features, we simulate many different scenarios and average over them all, before using *Optuna*4 to maximize our objective function:

Here we seek to optimize using three key components:

**First term: mean leverage across all actors**, which we wish to maximize across market conditions, LPs, VTs, and FTs.**Second term: spread in possible values of the leverage**(𝞼L) between different market conditions under control. Very large spreads will give rise to unsuitably small leverages, so we seek to minimize this whilst maintaining a large mean leverage between different markets.**Third term: regularisations via liquidation and insolvency rates.**The VaRs are used as regularization terms, where mean VaR values per actor below 0 (corresponding to liquidations and insolvencies) are strongly penalized, as shown with the indicator function*I*. The two leading non-regularisation terms are rescaled to the [0,1) domain to ensure no scale bias enters the optimisation.

The market conditions we optimized over cover different simulation volatilities, rates, and LP tick ranges (for our separate 60-day aUSDC, aDAI, and cDAI pools):

- Different market volatilities that correspond to the historic token/protocol APY volatilities are scaled by factors of
**0.5, 1, 2, 3, and 5,**to capture moderate and large swings in the future APY. - Different fixed rate market conditions:
**Neutral**: assumed the fixed and variable rates are equal.**Bullish on rate**: assumes the fixed rate is above the variable rate, and as such we expect the variable rate to correct upwards.**Bearish on rate**: assumes the fixed rate is below the variable rate, and as such we expect the variable rate to correct downwards.

- Different tick ranges to capture different liquidity concentrations in the AMM:
**(0.002, 1), (1,3), (3, 10).**For a given (*x*,*y*) set of rates,*x*corresponds to the upper tick value and*y*to the lower tick value, in percentage points.

The aUSDC pool objective function is illustrated in Figure 4, optimized over 200 trials with the *Optuna* Tree-structured Parzen Estimator algorithm[5]. Note that we at present make a simplifying assumption about the fee parameters, setting 𝞬fee and 𝞴fee to 0.003 and 0 respectively, based on initial studied which will be revisited in future Voltz pools and optimisations. In Figure 5 we ran a feature importance calculation with Optuna, determining the relative importance and information gain each feature in the optimisation contributes to the over maximization of the objective functions. We can see that the aDAI optimization is driven by contributions across all parameters, particularly the parameters used in the liquidation margin calculation, and the lower bound APY multiplier relevant for the liquidation margin. This is no surprise, given that liquidations directly impact the liquidation variable, and consequently the LVaR and regularization of the objective function.

Table 1 summarizes the safe leverages different actors may use across the optimized aUSDC, aDAI, and cDAI pools. Note that in almost all cases the possible safe leverages, ensuring that liquidations and insolvencies are kept out of the bottom 5% of the tails, leverages > 100x are easily achievable, and in some extreme cases above 1000x (**or even 2000x!**) can also be confidently implemented within Voltz.

Let’s consider the implications of this.

If an actor wants to trade rates, they need to be trading with leverage, since if they’re e.g wanting to take advantage of a 100 bps movement then that needs leverage to make the returns profile attractive. When developing Voltz Protocol, separating out the AMM from the Margin Engine allowed us to solve this problem architecturally. Then within the Margin Engine itself, we had to figure out a way to maximize leverage without risk of liquidation, culminating in the Risk Engine. **This is what makes Voltz Protocol so unique - you can trade rates with 100x leverage, safely**. To add to this, even the suboptimal 30x leverage constraint on FTs in the cDAI market is conservative, and factoring it up to even 50-60x would still be reasonably risk-averse.

It is important to point out that Table 1 summarizes the mean leverages, averaged over different market volatilities, tick ranges, and rate markets. If we relax these conditions, considering market volatilities based on the historical previous 60-day APY volatility, neutral rates, and (1, 3)% tick ranges consistent with present APYs then we may generate even broader leverage constraints. Each of these conditions can be separately simulated with the Risk Engine.

**The complete Voltz Risk Engine, and further documentation, is available to clone on GitHub: ****https://github.com/Voltz-Protocol/voltz-risk-engine**

- Voltz Litepaper (2021) https://www.voltz.xyz/litepaper
- Dimitris N. Politis & Joseph P. Romano (1994) The Stationary Bootstrap, Journal of the American Statistical Association, 89:428, 1303-1313, DOI: 10.1080/01621459.1994.10476870
- Y. Filmus (2010) Two Proofs of the Central Limit Theorem https://www.cs.toronto.edu/~yuvalf/CLT.pdf
- Takuya Akiba, Shotaro Sano, Toshihiko Yanase, Takeru Ohta,and Masanori Koyama.(2019) Optuna: A Next-generation Hyperparameter Optimization Framework. In KDD.
- Yoshihiko Ozaki, Yuki Tanigaki, Shuhei Watanabe, and Masaki Onishi. (2020) Multiobjective tree-structured parzen estimator for computationally expensive optimization problems. In Proceedings of the 2020 Genetic and Evolutionary Computation Conference (GECCO '20). Association for Computing Machinery, New York, NY, USA, 533–541. https://doi.org/10.1145/3377930.3389817