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We give a detailed account of correlations between credit sector/quality and treasury curve factors, using the robust framework of the Barclays POINT Global Risk Model. Consistent with earlier studies, we find a strong negative correlation between sector spreads and rate shifts. However, we also observe that the correlations between spreads and Treasury twists reversed recently, which is likely attributable to the Fe…
Models which postulate lognormal dynamics for interest rates which are compounded according to market conventions, such as forward LIBOR or forward swap rates, can be constructed initially in a discrete tenor framework. Interpolating interest rates between maturities in the discrete tenor structure is equivalent to extending the model to continuous tenor. The present paper sets forth an alternative way of performing …
The financial industry has undergone a significant transition from the London Interbank Offered Rates (LIBORs) to Risk Free Rates (RFRs) such as, e.g., the Secured Overnight Financing Rate (SOFR) in the U.S. and the Cash Rate (AONIA) in Australia, as primary benchmark rates for borrowing costs. The paper examines the pricing and hedging method for financial products in a cross-currency framework with the special emph…
Overnight rates, such as the SOFR (Secured Overnight Financing Rate) in the US, are central to the current reform of interest rate benchmarks. A striking feature of overnight rates is the presence of jumps and spikes occurring at predetermined dates due to monetary policy interventions and liquidity constraints. This corresponds to stochastic discontinuities (i.e., discontinuities occurring at ex-ante known points in…
We coin the term *Protocols for Loanable Funds (PLFs)* to refer to protocols which establish distributed ledger-based markets for loanable funds. PLFs are emerging as one of the main applications within Decentralized Finance (DeFi), and use smart contract code to facilitate the intermediation of loanable funds. In doing so, these protocols allow agents to borrow and save programmatically. Within these protocols, inte…
Fixed income has received far less attention than equity portfolio optimisation since Markowitz' original work of 1952, partly as a result of the need to model rates and credit risk. We argue that the shape of the efficient frontier is mainly controlled by linear constraints, with the standard deviation relatively unimportant, and propose a two-factor model for its time evolution.
Explicitly taking into account the risk incurred when borrowing at a shorter tenor versus lending at a longer tenor ("roll-over risk"), we construct a stochastic model framework for the term structure of interest rates in which a frequency basis (i.e. a spread applied to one leg of a swap to exchange one floating interest rate for another of a different tenor in the same currency) arises endogenously. This rollover r…
Pricing extremely long-dated liabilities market consistently deals with the decline in liquidity of financial instruments on long maturities. The aim is to quantify the uncertainty of rates up to maturities of a century. We assume that the interest rates follow the affine mean-reverting Vasicek model. We model parameter uncertainty by Bayesian distributions over the parameters. The cross-sectional and time series par…
This text grew out of a historical introduction initially written for a study of interest rates in cryptocurrency markets. The difficulty of defining a term structure for a currency without a conventional bond market led naturally to a more fundamental question: under what historical conditions does a yield curve become observable at all? Credit existed long before modern money, and interest-bearing loans are documen…
Turnpike theorems state that if an investor's utility is asymptotically equivalent to a power utility, then the optimal investment strategy converges to the CRRA strategy as the investment horizon tends to infinity. This paper aims to derive the convergence rates of the turnpike theorem for optimal feedback functions in stochastic factor models. In these models, optimal feedback functions can be decomposed into two t…
Interest rates in decentralized lending protocols are set algorithmically and adjust to supply and demand for liquidity. In this study, we propose an optimal interest rate model that maximizes the expected lender wealth while incorporating penalties for liquidity risk and interest rate stabilization. This objective benefits both sides of the market: it improves yield and reduces liquidity risk for lenders, while enco…
A term structure model in which the short rate is zero is developed as a candidate for a theory of cryptocurrency interest rates. The price processes of crypto discount bonds are worked out, along with expressions for the instantaneous forward rates and the prices of interest-rate derivatives. The model admits functional degrees of freedom that can be calibrated to the initial yield curve and other market data. Our a…
It is well known that the Cox-Ingersoll-Ross (CIR) stochastic model to study the term structure of interest rates, as introduced in 1985, is inadequate for modelling the current market environment with negative short interest rates. Moreover, the diffusion term in the rate dynamics goes to zero when short rates are small; both volatility and long-run mean do not change with time; they do not fit with the skewed (fat …
We present a quantitative study of the markets and models evolution across the credit crunch crisis. In particular, we focus on the fixed income market and we analyze the most relevant empirical evidences regarding the divergences between Libor and OIS rates, the explosion of Basis Swaps spreads, and the diffusion of collateral agreements and CSA-discounting, in terms of credit and liquidity effects. We also review t…
The daily return of a stock is often restricted to an exchange-imposed band to curb extreme fluctuations. Any attempted price movement beyond this band is clipped, leaving an unobserved excess. We introduce a minimal stochastic latent-state model in which a fraction of this hidden excess is retained for the next day. This retention generates memory, even though the daily stochastic driving shocks are independent. For…
This paper studies a singular dividend control problem for a firm with heterogeneous shareholders whose discount rates follow a given distribution. The central planner aggregates expected discounted payoffs using an ambiguity aggregation function $phi$, which captures shareholder heterogeneity and ambiguity attitudes but also leads to time inconsistency. To address this issue, we seek a time-homogeneous equilibrium d…
In traditional financial markets, yield curves are widely available for countries (and, by extension, currencies), financial institutions, and large corporates. These curves are used to calibrate stochastic interest rate models, discount future cash flows, and price financial products. Yield curves, however, can be readily computed only because of the current size and structure of bond markets. In cryptocurrency mark…
Recent studies have identified long-range dependence as a key feature in the dynamics of both mortality and interest rates. Building on this insight, we develop a novel bi-variate stochastic framework based on mixed fractional Brownian motions to jointly model their long-memory behavior and instantaneous correlation. Analytical solutions are derived under the risk-neutral measure for explicitly pricing zero-coupon bo…
Batch normalization re-centers and re-scales layer inputs using mini-batch statistics, then learns a scale and shift, reducing internal covariate shift and allowing higher learning rates.
Baltic Dry Index tracks shipping rates for dry bulk commodities and offers a real-economy signal on trade flows, freight conditions, and industrial demand.
Commodity Inventory Financing — Repo-like financing of physical stocks linking curve to rates.
Gold Real Yields Correlation — Gold as non-yielding asset inversely sensitive to real rates and USD.
A CTA is a manager — often CFTC/NFA registered — that runs client money in futures and options on futures, long and short, across rates, FX, equities, and commodities.
Harvest roll-down and yield carry in bond and STIR futures — a rates-specific premia book that can fight the trend sleeve in a hiking cycle.
The MOVE Index tracks implied volatility in the U.S. Treasury market and serves as a benchmark for rates uncertainty and macro stress.
Greeks are the sensitivities of option value to spot, vol, time, and rates — the risk report of a non-linear book.
Animal spirits is Keynes’s name for the non-model confidence that makes people invest or refuse to — the residual when rates and cash flows are not enough to explain the tape.
A bear steepener is a curve move where long yields rise more than front yields (or fronts fall less) as the market prices more term premium, more deficit, or less faith in long-run restraint — and duration loses.
A bull flattener is the curve flattening as yields fall, usually because the front end rallies more than the long end into easier policy or a growth scare.
DV01 is the dollar value of one basis point: how much the position’s mark changes if the yield (or the curve point you risk on) moves by 0.01%.
TINA — There Is No Alternative — was the 2010s slogan that zero rates left no choice but equities (or credit), compressing risk premia because cash paid nothing.
Crowding out is when public borrowing or spending raises rates or absorbs real resources so private investment or net exports fall, shrinking the net fiscal impulse.
Housing Starts — Rate-sensitive construction activity with multiplier effects across materials and labor.
Nonfarm Payrolls — The headline US jobs report that routinely moves rates, FX, and equity index volatility.
Equity Risk Premium measures the excess return investors expect from equities over risk-free assets and is a core framework for evaluating relative equity valuation.
Sector Rotation Signals — Cyclical versus defensive leadership indicating growth and rates regime.
The 2000–02 dot-com crash was an equity-valuation collapse after a narrative IPO bubble — brutal for NASDAQ, milder as a banking crisis because leverage sat more in households and corporates than in dealer warehousing of the story.
The 2013 taper tantrum was a fast global rates-and-EM selloff after Bernanke hinted at slowing QE — a rehearsal of how the world’s dollar duration is one speech.
Asset Swap Spread — Spread between bond yield and floating leg, linking credit and funding markets.
Convexity Risk — Non-linear price response to yield changes, especially relevant in MBS and long bonds.
Duration Risk — Interest-rate sensitivity of bond portfolios, amplified in low-yield high-duration regimes.
An inverted curve is short rates above long rates — a market statement about expected cuts, term premium, and sometimes recession risk.
Macaulay duration is the present-value-weighted average time to receive a bond’s cash flows — duration in years, before the modified-duration hedge ratio.
TIPS are US Treasuries whose principal adjusts with CPI — a real-rate instrument, not a magic inflation hedge for every horizon.
Balance of Payments Crisis — Sudden stop in capital flows forcing adjustment through FX, rates, or austerity.
A peg is a policy that holds the exchange rate to a target or band — a promise that spends reserves and rates when the market disagrees.
FRA-OIS spread measures the difference between interbank funding expectations and overnight indexed swap rates, often used as a gauge of banking and short-term funding stress.
LIBOR-OIS spread tracks the gap between unsecured bank funding rates and overnight indexed swap rates, historically serving as a benchmark for banking-system stress.
SOFR is the Secured Overnight Financing Rate, a key benchmark for U.S. dollar funding based on overnight Treasury repo transactions.
TED Spread measures the difference between interbank lending rates and short-term U.S. government bill yields, historically used as a gauge of credit and funding stress.
Federal Funds Rate — The effective overnight policy rate anchor that transmits through the entire USD funding stack and global risk appetite.
The FOMC is the Fed body that sets the funds-rate target and the balance-sheet stance — the US rates committee.
A Financial Conditions Index aggregates variables such as rates, credit spreads, equities, and the dollar to measure how supportive or restrictive the market environment is for growth and risk assets.
Monetary policy is the central bank’s control of short rates, liquidity, and sometimes the balance sheet — the price of reserves and the path of the front end.
The 2s10s Treasury curve measures the spread between 10-year and 2-year Treasury yields and is a key indicator of growth expectations, policy path, and term structure dynamics.
The 3M10Y Treasury curve compares 10-year Treasury yields with 3-month Treasury bill yields and is closely watched as a recession and policy-cycle indicator.
Animal spirits is Keynes’s name for the non-model confidence that makes people invest or refuse to — the residual when rates and cash flows are not enough to explain the tape.
Balance of Payments Crisis — Sudden stop in capital flows forcing adjustment through FX, rates, or austerity.
Baltic Dry Index tracks shipping rates for dry bulk commodities and offers a real-economy signal on trade flows, freight conditions, and industrial demand.
Bank Term Funding Program Legacy — Crisis facility allowing par advances against securities.
Batch normalization re-centers and re-scales layer inputs using mini-batch statistics, then learns a scale and shift, reducing internal covariate shift and allowing higher learning rates.
Commodity Inventory Financing — Repo-like financing of physical stocks linking curve to rates.
A CTA is a manager — often CFTC/NFA registered — that runs client money in futures and options on futures, long and short, across rates, FX, equities, and commodities.
Crowding out is when public borrowing or spending raises rates or absorbs real resources so private investment or net exports fall, shrinking the net fiscal impulse.
Harvest roll-down and yield carry in bond and STIR futures — a rates-specific premia book that can fight the trend sleeve in a hiking cycle.
A peg is a policy that holds the exchange rate to a target or band — a promise that spends reserves and rates when the market disagrees.
The 2000–02 dot-com crash was an equity-valuation collapse after a narrative IPO bubble — brutal for NASDAQ, milder as a banking crisis because leverage sat more in households and corporates than in dealer warehousing of the story.
Eurodollar Futures Legacy — Legacy CME short-rate futures still relevant for curve history.
The FOMC is the Fed body that sets the funds-rate target and the balance-sheet stance — the US rates committee.
A Financial Conditions Index aggregates variables such as rates, credit spreads, equities, and the dollar to measure how supportive or restrictive the market environment is for growth and risk assets.
FRA-OIS spread measures the difference between interbank funding expectations and overnight indexed swap rates, often used as a gauge of banking and short-term funding stress.
Gold Real Yields Correlation — Gold as non-yielding asset inversely sensitive to real rates and USD.
An interest-rate swap exchanges fixed coupons for floating (or the reverse) on a notional — the vanilla rates derivative.
An inverted curve is short rates above long rates — a market statement about expected cuts, term premium, and sometimes recession risk.
Liability Driven Investing — Pension hedging of liabilities with long duration bonds/swaps.
LIBOR-OIS spread tracks the gap between unsecured bank funding rates and overnight indexed swap rates, historically serving as a benchmark for banking-system stress.
Monetary policy is the central bank’s control of short rates, liquidity, and sometimes the balance sheet — the price of reserves and the path of the front end.
The MOVE Index tracks implied volatility in the U.S. Treasury market and serves as a benchmark for rates uncertainty and macro stress.