gen_report.rmd

---
title: "Expected IL for Uniswap v3 positions"
author:
- 0xAlbert.eth
- folch.eth
- 0xOli
date: \today
abstract: ""
pdf_document: null
geometry: "margin=45mm"
output: 
  pdf_document:
      toc: true
      number_sections: true
      toc_depth: 2
      keep_md: false
      keep_tex: false
      template: NULL
      #pandoc_args: [
      #  --template=extras/haptic_template.tex
      #]
html_document: default
tables: yes
latex_engine: pdflatex
header-includes:
    - \usepackage{float}
    - \usepackage{pdfpages}
    - \usepackage{tabu}
    - \usepackage{lipsum}
    - \usepackage{booktabs}
    - \usepackage[justification=raggedright,labelfont=bf,singlelinecheck=false]{caption}
    - \usepackage{array}
    - \usepackage{xcolor} 
    - \usepackage{color, colortbl}
    - \usepackage{amsmath}
    - \usepackage{mathtools,mathptmx}
    - \usepackage{tabularx}
    - \usepackage{background}
    - \usepackage[english]{babel}
    - \usepackage{csquotes}                
    - \usepackage[style=alphabetic, backend=bibtex]{biblatex}
    - \bibliography{bibliography/haptic.bib}
    - \usepackage{tikz}
    - \usepackage[font=large,labelfont=bf]{caption}
    - \usetikzlibrary{shapes,positioning}
    - \usepackage{wrapfig}
    - \usepackage{eso-pic,graphicx,transparent}
    - \DeclareUnicodeCharacter{2212}{-}
    - \backgroundsetup{pages={some},contents={}, opacity={0.3}, color={gray}}
    - \usepackage{multirow}
    - \usepackage{caption}
    - \newcommand{\Tau}{\textstyle{\mathcal{T}}}
knit: (function(inputFile, encoding) {
  rmarkdown::render(inputFile, encoding = encoding, output_dir = "pdf") })
---
\captionsetup{font=small}
\captionsetup[table]{labelformat=empty}
\captionsetup[table]{labelfont=bf}
```{r results="asis", warning=FALSE, echo=FALSE}
options(knitr.kable.NA = '') 
options(scipen=999) # avoid scientific notation
options(width = 60)

path = getwd()
source(paste(path, "/config.r", sep=""))
source(paste(path, "/uil.r", sep=""))

source(paste(path, "/eil.r", sep=""))
source(paste(path, "/static.r", sep=""))

T = 168
P = S0

chunk_one <- run_calc(mu, sigma, T) 
chunk_two <- run_calc(mu, sigma,  T * 2) 
df_uil_one <- run_uil_calc(ranges, sigma, T)

```
\newpage

\section{Synopsis}

Impermament loss (IL) or divergence-loss, is a phenomena experienced by LP providers on almost all types of AMM platforms, especially when providing liquidity for non-mean reverting assets. The effects of IL have been studied extensively because of their negative impact on LP providers' returns \cite{LOE}. When modelling an asset's price with a geometric brownian motion (GBM) the expected value of IL (EIL) can be estimated numerically by using techniques such as the Monte Carlo method. To our knowledge, at least one previous fellow scholar deemed  the computation of EIL for Uniswap v3 positions analytically intractable due to the complexity of the math expressions involving the GBM \cite{GSL}, however, a recent paper provided a framework for the analytical computation of EIL for Uniswap v3 positions \cite{CV3}. Previous work \cite{DZ} introduced the "unit impermanent loss per liquidity" (UIL) and efficient static replication formulas. We implement these ideas and present the results obtained with Monte Carlo simulations. 

\subsection{Simulation parameters}
\label{simpar}

  - Number of simulations: `r n_sim` 
  - Initial price: $`r S0` 

\newpage

\section{Implementation details}

\subsection{EIL}
\label{eilint}

The integrals in the analytical computation of EIL and UIL are calculated using the "_integrate_" R function, based on QUADPACK routines dqags and dqagi by R. Piessens et al. \cite{PIE}. The code shown below is simplified for displaying purposes. Checkout the Github repository for the full version.

```{r a, eval=FALSE}

EIL <- function(f, lb, ub, P, Pa, Pb, mu, sigma, T) {

    integral <- integrate(
        f,
        # Lower integration bound
        lb, 
        # Upper integration bound
        ub, 
        subdivisions = 5000,
        abs.tol = 1e-5, 
        rel.tol = 1e-7,  
        stop.on.error = F,
        T, 
        P, 
        Pa,    
        Pb,
        mu,
        sigma
    )$value;

    coefficient <- (1 / (sigma * sqrt(2 * pi * T)))
    return(coefficient * integral)
}
```

\newpage

\subsection{UIL}
\label{uilint}

```{r x, eval=FALSE}
UIL <- function(P0, Pa, Pb, sigma, t, flag) {
    
    psi <- function(x, T, sigma, flag) {
        option_price <- bs_option(P0, x, sigma, 0, T, flag) 
        return(x^(-(3/2)) * option_price)
    }

    integral <- integrate(
        psi,
        # Lower integration bound
        lower = Pa, 
        # Upper integration bound
        upper = Pb, 
        subdivisions = 5000,
        abs.tol = 1e-5, 
        rel.tol = 1e-7,  
        stop.on.error = F,
        T = t ,
        sigma = sigma,
        flag = flag
    )$value;

    coefficient <- (-0.5)
    return(coefficient * integral)
}

```

\newpage

\subsection{Time in-the-money (TITM)}
\label{itimint}

 The two-dimensional integrals for the computation of "_time in-the-money_" ($\Tau$) are calculated using the dblquad function from the pracma library. 

```{r b, eval=FALSE}

TITM <- function(P, Pa, Pb, mu, sigma, T) {

    lower_bound <- log(Pa / P)
    upper_bound <- log(Pb / P)

    f <- function(x, y, mu, sigma) {
        t <- y
        first_term <- -(x - (mu - (sigma^2 / 2)) * t)^2
        second_term <- (2 * t * (sigma^2))
        exp(first_term / second_term) / sqrt(t)
    }

    integral <- dblquad(
        f,
        # Lower integration bound 
        log(Pa / P), 
        # Upper integration bound
        log(Pb / P), 
        0, 
        T,
        tol=1e-5,
        subdivs=5000, 
        mu=mu, 
        sigma=sigma
    )

    result <- (1 / (sigma * (sqrt(2 * pi)))) * integral[1]
    
    return(result)
}
```
 
\newpage

\section{Monte Carlo simulation results} 
\label{simres}

\subsection{EIL for 168 hours (7 days)}

```{r, echo=FALSE, warning=FALSE, results='asis'}

collapse_rows_dt <- data.frame(
    C1 = chunk_one[1:nrow(chunk_one), 1],
    C2 = chunk_one[1:nrow(chunk_one), 2],
    C3 = chunk_one[1:nrow(chunk_one), 3],
    C4 = chunk_one[1:nrow(chunk_one), 4],
    C5 = chunk_one[1:nrow(chunk_one), 5],
    C6 = chunk_one[1:nrow(chunk_one), 6],
    C7 = chunk_one[1:nrow(chunk_one), 7],
    C8 = chunk_one[1:nrow(chunk_one), 8],
    C9 = chunk_one[1:nrow(chunk_one), 9]
)

sigmaMsg <- paste("$\\\\sigma = $ ", sigma, sep="")
sigmaMsg2 <- paste("$\\\\sigma = $ ", sigma * 2, sep="")

collapse_rows_dt %>%
kbl(booktabs = T, 
    align = "c", 
    longtable = T, 
    escape = FALSE, 
    col.names = c( "$\\mu$", "r", "CE", "$\\Tau$", "EIL", "IL","$\\Tau$", "EIL", "IL"),
    caption = "Table 1: Results for setting $\\mu$ = 7.5e−5, 1.5e−4 and $\\sigma$ = 0.001 and $\\sigma$ = 0.002 for 7 days (T = 168 hours) of providing liquidity with initial price \\$1000.\\label{tab:table1}"
    ) %>%  
    kable_styling(font_size = 9) %>%
    column_spec(1, bold=F) %>%
    collapse_rows(columns = 1, latex_hline = "major", valign = "middle")  %>%
    add_header_above(c(" ", " ", " ", "Formula" = 2, "Simulation" = 1, "Formula" = 2, "Simulation" = 1), escape = FALSE)  %>%
    add_header_above(escape =FALSE, c(" ", " ", " ", setNames(3,sigmaMsg), setNames(3,sigmaMsg2)))   %>%
    kable_styling(font_size = 9, latex_options = c("hold_position", "repeat_header")) %>%
        print()

```

\subsection{EIL for 720 hours (30 days)}

```{r, echo=FALSE, warning=FALSE, results='asis'}
collapse_rows_dt <- data.frame(
    C1 = chunk_two[1:nrow(chunk_two), 1],
    C2 = chunk_two[1:nrow(chunk_two), 2],
    C3 = chunk_two[1:nrow(chunk_two), 3],
    C4 = chunk_two[1:nrow(chunk_two), 4],
    C5 = chunk_two[1:nrow(chunk_two), 5],
    C6 = chunk_two[1:nrow(chunk_two), 6],
    C7 = chunk_two[1:nrow(chunk_two), 7],
    C8 = chunk_two[1:nrow(chunk_two), 8],
    C9 = chunk_two[1:nrow(chunk_two), 9]
)

sigmaMsg <- paste("$\\\\sigma = $ ", sigma, sep="")
sigmaMsg2 <- paste("$\\\\sigma = $ ", sigma * 2, sep="")

collapse_rows_dt %>%
kbl(booktabs = T, 
    align = "c", 
    longtable = T,
    escape = FALSE, 
    col.names = c( "$\\mu$", "r", "CE", "$\\Tau$", "EIL", "IL","$\\Tau$", "EIL", "IL"),
    caption = "Table 2: Results for setting $\\mu$ = 7.5e−5, 1.5e−4 and $\\sigma$ = 0.001 and $\\sigma$ = 0.002 for 30 days (T = 720 hours) of providing liquidity with initial price \\$1000.\\label{tab:table2}"
    ) %>%  
    kable_styling(font_size = 9) %>%
    column_spec(1, bold=F) %>%
    collapse_rows(columns = 1, latex_hline = "major", valign = "middle")  %>%
    add_header_above(escape =FALSE, c(" ", " ", " ", setNames(3,sigmaMsg), setNames(3,sigmaMsg2)))   %>%
    add_header_above(escape =FALSE, c(" ", " ", setNames(3,sigmaMsg), setNames(4,sigmaMsg2)))   %>%
    kable_styling(font_size = 9, latex_options = c("hold_position", "repeat_header")) %>%
        print()
```

\subsection{Expected prices}

```{r, echo=FALSE, warning=FALSE, results='asis'}
static_expected_prices <- get_expected_prices(P, mu)
data <- static_expected_prices$data

df_expected_prices <- data.frame(data)
colnames(df_expected_prices) <- c("$\\mu$", "$\\Tau$ = 168", "$\\Tau$ = 720")
df_expected_prices <-  setNames(df_expected_prices, c("$\\mu$", "$\\Tau$ = 168", "$\\Tau$ = 720"))
row.names(df_expected_prices) <- NULL

df_expected_prices %>%
kbl( escape = F, caption = paste("Table 3: Expected price of the asset, for 7 ($\\Tau$ = 168 hours) and 30 days ($\\Tau$ = 720 hours).\\label{tab:table3}"), booktabs = T) %>%
    kable_styling(latex_options = "HOLD_position", position="left") %>%
    kable_classic(full_width = F, html_font = "Cambria") %>%
        print()
```

\subsection{Unit impermanent loss per liquidity (UIL)}

```{r, echo=FALSE, warning=FALSE, results='asis'}
df_uil_one %>%
    kbl(
        booktabs = T, 
        align = "c", 
        longtable = T,
        escape = FALSE, 
        caption = "Table 4 - Unit IL per liquidity (UIL).\\label{tab:table4} for $\\sigma$ = 0.001 and $\\sigma$ = 0.005"
    ) %>%  
    kable_styling(font_size = 9)  %>%
    collapse_rows(columns = c(1), latex_hline = "major", valign = "middle")  %>%
        add_header_above(escape = FALSE, c(" ", "Formula " = 2, "Static replication " = 2))  %>%
        kable_styling(latex_options = "HOLD_position", position="left") %>%
        kable_classic(full_width = F, html_font = "Cambria") %>%
            print()
```
\section{Remarks}
\label{remarks}

The method introduced by \cite{CV3} provides a reliable way to determine the expected value of impermanent loss under a geometric brownian motion assumption. The model is sensitive to parameters such as drift and volatility as seen in Table \ref{tab:table1} and \ref{tab:table2}. The expected value of impermanent loss computed analytically converges with the result of the Monte Carlo simulation with low error rate. \cite{DZ} characterizes analytically the option-like payoff structure of impermanent loss for concentrated liquidity positions and provides formulas for static replication of the impermanent loss with a combination of European options. Table \ref{tab:table4} reports the result of the computation obtained using the analytical approach and highlights the high accuracy of the replication formulas. 

\newpage

\printbibliography