# Ticks, Ratios, and Prices ### Ticks A tick on Mangrove represents a discrete point in the price space of a market. Rather than allowing continuous prices, Mangrove divides the space into fixed increments, each tick corresponds to a 0.01% (1 basis point) change in price. This discrete representation allows for precise, gas-efficient price calculations and easy integration with on chain logic. Formally: $$ \text{ratio} = 1.0001^{\text{tick}} $$ A higher tick value represents a higher price on the ask side and lower price on the bids side. ### Ratios In Mangrove, every offer is defined not directly by a "price", but by a ratio. The relationship between the amount offered and the amount requested: $$ \text{ratio} = \frac{\text{wants}}{\text{gives}} $$ This ratio determines how much of a token is received for a given amount of another. It's the fundamental internal measure of price used throughout the protocol and codebase, ensuring consistency across all token pairs. ### The Relation Between Ticks and Ratios Ticks and ratios are directly linked: each tick corresponds to one unique ratio, and each ratio can be expressed as a tick using a logarithmic function: $$ \text{tick} = \log\_{1.0001}(\text{ratio}) $$ This relationship allows prices to be converted between human-readable form and the compact on-chain tick representation The meaning of a ratio depends on whether the offer is on the asks or bids side of a market. * Asks side, selling the base token, receiving the quote tokens: $$ \text{price} = \frac{\text{wants}}{\text{gives}} = \text{ratio} = 1.0001^{\text{tick}} $$ * Bids side, buying the base token, giving the quote tokens: $$ \text{price} = \frac{\text{gives}}{\text{wants}} = \frac{1}{\text{ratio}} = 1.0001^{\text{-tick}} $$ This distinction ensures consistent logic across both side of the book, even though mathematical expressions are inverted. **Example:**\ In a WETH/DAI market (prices expressed in DAI per WETH): * On the **asks** side (selling WETH), price = wants/gives = 2,224 DAI/WETH. * On the **bids** side (buying WETH), price = gives/wants = 0.0004496 WETH/DAI. ### Prices, Decimals, and Raw Values All ERC-20 token amounts are stored on-chain as integers, often called **raw values**. Each token defines its own number of decimals, and Mangrove ignores these decimals internally for consistency.
TokenDecimalsRaw AmountUser Amount
USDC68765432187.654321
DAI188765432109876543210.876543210987654321
WETH18109876543210987654321109.876543210987654321
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To display human-readable prices, the SDK or UI must account for the difference in decimals between the base and quote tokens. The conversions between raw ratios & prices and their user representations are: $$ \def\arraystretch{1.5} \begin{array}{rcl} userRepresentation(raw\_ratio) & = & raw\_ratio \* 10^{(outbound\_decimals - inbound\_decimals)} \\ raw(user\_ratio) & = & user\_ratio / 10^{(outbound\_decimals - inbound\_decimals)} \\ userRepresentation(raw\_price) & = & raw\_price \* 10^{(base\_decimals - quote\_decimals)} \\ raw(user\_price) & = & user\_price / 10^{(base\_decimals - quote\_decimals)} \end{array} $$ This ensures correct display of "real-world" prices, even when token decimals differ. **Example:** | **Market** | **Token Decimals** | **Ask (quote/base)** | **Ask ratio (raw)** | **Bid (base/quote)** | **Bid ratio (raw)** | | ------------ | ---------------------------- | -------------------- | ------------------------------------------------------------ | -------------------- | ---------------------------------------------------------------- | | **DAI/USDC** |

DAI = 18
USDC = 6

| $$1$$ | $$1 \times 10^{-12} \ = 10^{6}/10^{18}$$ | $$1$$ | $$1 \times 10^{12} \ = 10^{18}/10^{6}$$ | | **WETH/DAI** |

WETH = 18
DAI = 18

| $$2{,}224$$ | $$2{,}224 \times 10^{0} \ = 2{,}224 \times 10^{18}/10^{18}$$ | $$0.0004410$$ | $$0.0004410 \times 10^{0} \ = 0.0004410 \times 10^{18}/10^{18}$$ | > **Beware decimals!** As an important example of the implications, note that while the price space contains the *raw* price 1, it may not contain the user readable price 1. This is likely if base and quote have different numbers of decimals. ### Converting Prices and Ticks Because the price space is discrete, not every price maps exactly to an integer tick. When converting a price to a tick: $$ \text{tick} =\ \begin{cases}\ \log\_{1.0001}(\text{price}) & \text{for asks} \\\ -\log\_{1.0001}(\text{price}) & \text{for bids}\ \end{cases} $$ The resulting tick is then rounded: * Takers usually round down, ensuring they never pay more or receive less than expected. * Makers usually round up, ensuring they never receive less or sell for too little. ### Tick Spacing Mangrove allows each market to define a `tickSpacing` parameter, controlling how dense the available price points are. Only ticks where: $$ \text{tick} \bmod \text{tickSpacing} = 0 $$ are valid for offer in that market. **Example:** With `tickSpacing = 5` , valid ticks are `..., -15. -10, -5, 0, 5, 10, 15,...` The smallest price increment is then: $$ 1.0001^5 - 1= 0.0005001 (≈0.05%) $$ ### Ranges Mangrove supports a wide range of ticks, allowing for extremely high or low prices: | Value | Min | Max | | ----- | ----------------------- | --------------------- | | Tick | -887,272 | 887,272 | | Ratio | $$2.9\times10^{-39}$$ | $$3.4\times 10^{38}$$ | | Price | $$2.9 \times 10^{-39}$$ | $$3.4\times 10^{38}$$ | ### The TickLib Library Mangrove provides a Solidity library called [`TickLib`](https://github.com/mangrovedao/mangrove-core/blob/2ae172805fd8b309c30b2dc877dba66245abbb3e/lib/core/TickLib.sol) which includes: * Functions to convert between ticks, ratios, and prices. * Utilities for rounding ticks appropriately for maker or taker use. * Helpers for computing inbound/outbound volumes at specific tick levels. ### Comparison to Uniswap Ticks Mangrove's ticks are inspired by Uniswap V3 ticks, but with the following notable differences: * It operates directly on ratios (not √ratios), simplifying computation. * Ratios are handled as floating-point numbers, improving precision.