> For the complete documentation index, see [llms.txt](https://docs.btcd.fi/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://docs.btcd.fi/research-and-mathematical-foundations-1.md).

# Research & Mathematical Foundations

The Bitcoin Dollar rests on a precise, falsifiable mathematical foundation: a synthetic asset pegged to a continuous 50/50 BTC/USD exposure, backed by a portfolio that is continuously hedged so its collateral ratio is insensitive to the price of Bitcoin. This page collects that foundation into a single technical whitepaper.

The paper develops the hedging calculus behind the BTCD portfolio from first principles, shows how it admits otherwise-awkward yield sources — Yield Basis ybBTC being the headline example — and validates the whole system against four years of hourly market data.

## Download the whitepaper

{% file src="/files/CxNAQYIgIFeb8Ah4QTOR" %}

### What's inside

The central claim is simple to state and the paper spends its length defending it: the BTCD architecture harvests yield efficiently, and in particular it can hold yield-bearing assets whose effective BTC exposure is *variable* or *path-dependent* — something conventional single-mandate pools cannot do without mismarking the position or carrying unhedged risk.

The machinery is a hedging calculus built on two log–log "Greeks":

* **Elasticity** $$\omega = \partial \log P / \partial \log P\_{btc}$$ — the true effective Bitcoin exposure of an asset or portfolio (the log-space analogue of delta).
* **Convexity** $$\xi = \partial^2 \log P / \partial \log P\_{btc}^2$$ — its curvature (the log-space analogue of gamma).

In this language the BTCD peg is just $$\omega\_{btcd} = \tfrac{1}{2}$$, i.e. $$P\_{btcd} \propto \sqrt{P\_{btc}}$$. Maintaining the hedge reduces to driving the diffusion of the collateral ratio to zero — a continuous delta hedge in log coordinates — and the residual drift is precisely the harvestable yield.

### Key results

* **The core 50/50 portfolio hedges Bitcoin price risk almost perfectly.** Over four years of hourly data, the rebalancing strategy earns roughly 3% annualized with a maximum drawdown on the order of a couple of basis points, even with conservative trading friction.
* **ybBTC can be held *****and***** hedged.** Marking ybBTC to its true, live elasticity — computed by look-through against the underlying Curve pool — lets the portfolio capture its Bitcoin-denominated yield while suppressing Temporary Redemption Discount (TRD) drawdowns from \~6% (naive treatment) to under 0.3% (hedged): an order-of-magnitude improvement in tail risk.
* **Pure USD and pure BTC exposure, with amplified yield.** The bdUSD and bdBTC vaults run continuously leveraged sBTCD loops to manufacture $$\omega = 0$$ and $$\omega = 1$$ exposure. bdBTC is, in effect, yield-bearing Bitcoin that a holder can exit at fair value at any time — a clean alternative to instruments that carry redemption risk.

### Who this is for

The paper is written for a technical reader — comfortable with Itô calculus, AMM mechanics, and leverage/liquidation math. If you build, allocate to, or stress-test DeFi yield strategies, it is meant to be read critically and reproduced. If you would rather start with the intuition, the companion blog post below covers many of the same ideas with fewer derivations.

### Related Reading

* Egorov, M. "Eliminating impermanent loss by leveraged liquidity." 2025. (The Yield Basis whitepaper.)
* Milionis, Moallemi, Roughgarden, Zhang. "Automated Market Making and Loss-Versus-Rebalancing." 2022. (The original LVR paper.)
* Ethena Labs documentation. (For the USDe/sUSDe parallel.)
* [How the BTCD Portfolio Captures ybBTC Yield While Hedging TRD](https://btcd.fi/blog/how-the-btcd-portfolio-captures-ybbtc)
* Additional protocol-internal research notes will be linked as they are published.