> 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.md).

# Research and 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

[**The Bitcoin Dollar: Mathematics and Research**](https://docsend.com/view/nrgs2efucm6xvf4i)

## 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.

## Chapter guide

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**1 · Introduction** — The thesis: why BTCD captures yield sources that ordinary hedge-fund-style pools structurally cannot, with ybBTC as the central case study.
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**2 · The BTCD Portfolio** — The peg and collateral ratio; the continuous-rebalancing hedge and drift conditions in the $$\omega/\xi$$ framework; the canonical USD + BTC rebalancing harvest; and ybBTC as a convex asset — Curve Cryptoswap mechanics, the origin of TRD, and the look-through computation of $$\omega\_{yb}$$.
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**3 · Composability with Leverage** — The general leveraged looping strategy, specialized to the bdUSD and bdBTC vaults.
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**4 · Backtests** — Results over four years of hourly BTC data: the non-convex portfolio, the full ecosystem with both looping vaults running simultaneously, and a portfolio holding live ybBTC through the early-2026 TRD episode.
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**5 · Future Work** — Open problems, including a complete closed-form treatment of TRD.
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## 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 the same ideas with fewer derivations.

## Related reading

[How the BTCD Portfolio Captures ybBTC Yield While Hedging TRD](https://btcd.fi/blog/how-the-btcd-portfolio-captures-ybbtc)

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This material is for informational and educational purposes only and does not constitute investment advice, a solicitation, or a recommendation to buy, sell, or hold any digital asset. DeFi protocols carry risks including, but not limited to, smart-contract vulnerabilities, liquidation risk, peg instability, and loss of funds. Past simulated performance is not indicative of future results. Do your own research and consult a qualified advisor before making any investment decisions.
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