Grant Proposal: The Impermanent Loss in Yield Farming

Grant ID: 86854fa0n



Yield farming is the practice of seeking rewards in the form of transaction fees by depositing tokens into a liquidity pool on a decentralized application (dApp). In contrast to traditional finance, yield farming is different from traditional liquidity provision because it occurs in the absence of a central market maker (intermediary). If a trader (the liquidity taker) enters the liquidity pool, he can buy and sell tokens by swapping them in the pool according to a known pricing rule governed by the Automated Market Maker (AMM). In exchange for providing liquidity, the yield farmer (liquidity provider) earns a share of the trading fees from the trades that occur in the liquidity pool. Liquidity is of critical importance to the stability and efficiency of financial markets. In yield farming, tokens are lent (to a protocol), and, most likely, the protocol will not return the same quantities of individual tokens to the lender, due to the so-called impermanent loss.
Impermanent loss defines the opportunity-cost dynamic between offering assets for exchange and holding the underlying assets to potentially profit from the price movement. This occurs when the price of the deposited assets changes compared to when one deposited them. The bigger this change is, the more one is exposed to impermanent loss. It is important to mention that an impermanent loss does not necessarily mean that the LP experiences a negative return on the investment: It simply means that the gains from a buy-and-hold strategy outperformed the returns obtained for the liquidity provision.


The academic literature on impermanent loss as the major risk driver in DeFi is not developed yet and especially little is known about how to hedge or minimize the impermanent loss for investors.
The main contributions we plan to achieve in this project are as follows:
i) To shed light on the concrete mechanism of impermanent loss, we propose a continuous-time stochastic model which we calibrate to the data and then use to characterize the impermanent loss dynamics in the liquidity pool. The construction of our model starts with the mathematical formulation of the AMM governing prices in the liquidity pool. We then fully derive and characterize the impermanent loss as the major source of risk for the agent who has invested in the pool. For a single pool with a single pair of tokens, the first calculations show that the impermanent loss is a function of the respective token return volatilities and the correlation between them.
ii) We then study the replication (hedging) of the impermanent loss: Therefore, we solve a portfolio problem using stochastic control theory with the goal to minimize the (one-step ahead) impermanent loss occurring to the yield farmer. The optimal portfolio strategy will then command the optimal holdings in either the underlying token pair (outside the pool) or derivatives (call or put options) written on the respective token individually.

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Describe what your project does or is aiming to do in 50 characters or less:
Our project seeks to fill the existing gap in academic literature and understanding surrounding impermanent loss while providing practical solutions for investors to hedge and minimize this risk.

If available, please enter a link to your project below:

How far along are you?:
We successfully wrote a first paper on Yield Farming which is available here:

The here proposed idea can be viewed as one of the follow-up papers.

Why did you pick this idea to work on? Do you have domain expertise in this area? How do you know people need what you’re making?:
Prof. Dr. Lorenzo Schoenleber is an Assistant Professor in Finance at the Collegio Carlo Alberto and the University of Turin. He obtained his PhD at the Frankfurt School of Finance. He is also associated with the Fintech & Digital Finance Chair at Paris Dauphine University. His area of specialization is empirical asset pricing (option-implied information) and DeFi (Yield Farming).

Prof. Dr. Andrew Papanicolaou is an associate professor in the Department of Mathematics at North Carolina State University (NCSU). His PhD is in applied mathematics from Brown University. His research interests are computational finance and stochastic systems for control and optimization. The applications of this work include financial data analysis and the challenges associated with these highly complex data sets. My background is in probability theory and nonlinear filtering.

Dr. Siddharth Naik is a quantitative portfolio manager and trader with over 7 years of experience at institutional hedge funds (systematic macro and low latency, respectively) with extensive experience in trading fx, futures, and crypto instruments. He has previously published multiple papers on non-convex optimization mechanism design and trading crypto since 2017. Previously he was a CEO at RadioLytics a startup, which aimed at making spectrum tradable.

What’s new about what you’re making? What substitutes do people resort to because it doesn’t exist yet (or they don’t know about it)?:
Our project stands out from the competition for grant funding due to several key factors. Firstly, it addresses a unique and pressing need in understanding and hedging impermanent loss within its industry. We have identified a gap that existing solutions have yet to adequately address. Secondly, our project offers an innovative approach outlined in our paper “Yield Farming for Liquidity Provision,” which has the potential to disrupt the status quo and make a significant impact. Furthermore, our highly qualified and experienced team ensures the project’s success and sustainability. Lastly, our comprehensive and well-thought-out plan, including a detailed roadmap, sets us apart. Overall, our project’s unique value proposition, innovative approach, experienced team, and well-structured plan make it a worthy investment for grant funding with long-term sustainability and positive impact potential.

What’s your vision for how your project will impact The Graph and/or web3?:
Benefits for your Foundation and Ecosystem

The stability and efficiency of financial markets heavily rely on the crucial significance of liquidity. You would sponsor a high-level academic research paper in DeFi and hence gain visibility among the academic community. Our current paper “Yield Farming for Liquidity Provision” ( got presented already at various conferences. Besides the visibility, the results which we provide in our paper can be useful for the enhancement and development of the current architecture of your dApp. We cite The Graph in the data section and promote the usage of your dApp among academic reasearchers.

In addition, we just held a one-day conference by the Collegio Carlo Alberto and the University of Turin, called ToDeFi: Torino Decentralized Finance Conference 2023 (, which promoted the highest level and up-to-date research in DeFi within the academic community. Speakers and discussants were selected among the most innovative junior researchers and established scholars. We are planning to make the conference a yearly event. Beyond the research support, we are potentially open to discuss about a deeper cooperation. In particular, the Collegio Carlo Alberto hosts a postgraduate Master’s in finance, insurance, and risk management (Mafirm), which is a small high-quality program that exists for more than 20 years. The program appears in several international rankings (e.g., 1st in Italy, 6th in Europe, and 25th worldwide in the quantitative finance masters’ ranking by The students attending our master lectures are excellent but more importantly, really motivated

How will you measure success?:
Measuring success for a grant related to an academic paper can be assessed using various criteria.
We aim for the successful completion of the academic paper and its subsequent publication in a reputable journal. The publication demonstrates that the research was conducted effectively and contributes to the academic community. In addition, the impact of the paper can be measured by the number of citations which suggests that the research has been influential and has generated interest within the academic community. Success can also be evaluated based on the opportunities for collaborations and networking that arise from the research (such as our yearly conference ToDeFi). If the grant helps establish connections with other researchers, institutions, or organizations, it can enhance the visibility and impact of the work. In addition, we are aim for research which is practically applicable having a real-world impact. If the findings of the paper are implemented in practice or contribute to solving real-world problems, it demonstrates the relevance and value of the research.

What milestones will you achieve in the coming weeks/months, with or without the grant?:
Timetable - Project Roadmap & Milestones - Expected Timeline for the Grant Completion

Dates - Project stage
06-08/2023 – Accessing, preparing, and processing spot and option data via the Tardis API

08-09/2023 – Accessing liquidity pool data using The Graph

09-12/2023 – Developing the mathematical optimization framework

01-02/2024 – Programming and numerically solving the optimization framework

02-04/2024 – Calibrating the model output to the data (tokens and options)

04-06/2024 – First paper draft

07-08/2024 – Submission to a Journal

Please select who you think will benefit from your project?:
Subgraph Developers, Delegators, Indexers, Curators

How much funding are you looking for? (USD):

Provide a breakdown of your requested funding:
To analyze Pool data which can be sourced for free via The Graph as done in our current project, we need to hire a Research Assistant, who helps us to access and analyze the relevant data. Hiring a Research Assistant for about 1.5 years accumulates to approximately 8000€. Hardware can be acquired for less than 2000€.

We want to emphasize that any amount of sponsorship or donation would be beneficial for us since we are trying to request funding from different funding sources to realize this project.

Have you applied for (or already received) funding from somewhere else?:

Why are you interested in contributing to The Graph and/or web3?:
The Graph provides free data to study Liquidity Pools and DEXs.

Are you applying as a team or individual?:

What type of grant are you pursuing?:
Project (one-off)

Please select the category your project best fits into:

Primary Community Poll

Do you fundamentally support this grant application, knowing that the final scope and grant amount may differ from the requested proposal?
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Grant Committee Vote

Do you recommend this grant to the DAO?
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I believe this grant was rejected on May 15th(Grant Proposal: The Impermanent Loss in Yield Farming (May 2023)), and this one looks the same in structure.

Were any changes made to the proposal?


Greetings, thank you for taking the time to reapply to us for a grant and in general for taking the time to write a proposal. After reviewing your information and materials, I would like to clarify a few things.

I was quite interested in reading all of it, but I didn’t quite understand how it would benefit The Graph specifically. Since these are materials that are more oriented to a rather narrow sphere of crypto and web3 in general, which in addition has only slight relation to The Graph.

Unfortunately, I could not see the value of this work for the protocol. Perhaps you can change my opinion. And I would also like to hear your suggestions, what has been changed if we compare these two grants.

Thank you! And good luck!

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Dear Colson and Doxe01,

thanks for your reply.

Indeed we recently applied with the same project but the scope of the study got narrowed and the benefits for your ecosystem got worked out more precisely.

Our goal is to analyze large amounts of liquidity pools and to write another academic article about liquidity provision and impermanent loss. As in our previous work, the data is obtained from The Graph. The Graph is cited in our paper and therefore facilitates the usage of your protocol in academia. When we present the paper at conferences or academic seminars my feeling is that obtaining on-chain data in still a significant hurdle for many (non-technical) researchers.

Last year my team already obtained a small grant to pay a research assistant to help us sourcing data from The Graph (Wave6). This project can be seen as a follow-up paper.

We where hoping to continue our relationship with The Graph Foundation and to obtain some funding to pay a master student to work with us together on this project.

Please let me know if you have any further questions.

Best Regards

Hello Lorenzo,

The Graph AdvocatesDAO primarily oversees Community Grants and RFPs that are focused on “community-building” efforts within The Graph ecosystem. This generally involves initiatives that enhance the community’s awareness, engagement, understanding, or participation in The Graph or broader web3 ecosystem. After review, it appears that your proposed research does not align directly with these objectives.

That said, your project may be of interest to the Graph Foundation, a separate entity in the Graph ecosystem that has previously funded your team’s work. We encourage you to apply for a grant from the Graph Foundation again. You can do so through this link: The Graph: Grants Application

We appreciate your understanding and wish you the best of luck with your research.

Best regards,
Graph AdvocatesDAO