Convexity Adjustment Research Articles

Interest rate convexity in a Gaussian framework

The contributions of this paper are twofold: we define and investigate the properties of a short rate model driven by a general Gaussian Volterra process and, after defining precisely a notion of convexity adjustment, derive explicit formulae for it.

SABR Model Convexity Adjustment for the Valuation of an Arithmetic Average RFR Swap

SABR Model Convexity Adjustment for the Valuation of an Arithmetic Average RFR Swap

Bond futures: Delivery Option with Term Structure Modelling

Bond futures are characterized by a set of underlying bonds; the short party has the option to deliver at expiry any of those underlying bonds. Bond futures thus embed a choice option between bonds with different maturities and coupons. The delivery mechanism also incorporates conversion factors that create an implicit strike. The option is impacted by different maturities and different moneyness for each bond. It is important to take into account the full term structure of volatility with smile. A recent paper Bang and Daboussi (2022) developed such an approach for swap ratebased products like CMS. In this paper, we extend their approach to cover futures and apply it to the specific case of bond futures. We show the impact of smile, term structure of volatility and correlations between rates on the delivery option and convexity adjustment values.

Swap Rate fallback: unreasonable effectiveness of approximations and alternatives

Observes that cash-settled swaptions with collateral discounting are impacted by the Swap Rate fallback mechanisms decided by working groups/ISDA. The legacy vanilla swaptions are becoming exotic products, as the mechanism is based on a non-linear transformation of the OIS swap rate, and generate convexity adjustments. It turns out that those two effects almost cancel each other and lead to almost vanilla products. Henrard analyses those cancelling effects and the risk management impacts. Based on those insights, Henrard proposes an adjusted fallback mechanism that reduces further the exotic features and simplifies further the risk management of the legacy book.

Dynamic term structure models for SOFR futures

AbstractThe London InterBank Offered Rate is scheduled for discontinuation, and the replacement advocated by US regulators is the Secured Overnight Financing Rate (SOFR). The only SOFR‐linked derivative with significant liquidity and trading history is the SOFR futures contract, traded at the Chicago Mercantile Exchange. We use the historical record of futures prices to construct dynamic arbitrage‐free models for the SOFR term structure. We find that a Gaussian arbitrage‐free Nelson–Siegel model describes term structure well without accounting for jumps and seasonal effects observed in SOFR. However, a shadow‐rate extension is needed to describe volatility near the zero‐boundary impacting the futures convexity adjustment and option pricing.

Derivative pricing with two collateral rates

This note analyses derivative pricing in the context of a collateral rate switch during the life of a financial product or the existence of two overnight rates. In particular we analyse the impact of forward change of collateral, the impact on OISs when the collateral rate is different from the OIS underlying, and the impact of bilateral swaptions collateral rate different from the one implied by the cleared market. In each case, we evidence new convexity adjustment impacts previously not accounted for. The order of magnitude of those impacts is also proposed for realistic values of model parameters; the individual relative impacts are not huge, but when applied on trillions of notional, as it is the case, the monetary impacts can be substantial.

SOFR In-Advance Derivative Pricing: Convexity Adjustment in the Forward Market Model

As the USD Libor interest rate benchmark is in the process of being discontinued, there is an increasing occurrence of SOFR-based loans and derivatives. Whereas Libor was most commonly set in-advance, an important difference is that SOFR is commonly set in-arrears when it is used in instruments such as overnight index swaps. Loans and interest-rate cap options used as hedges may both choose to fix the SOFR rate in-advance, which is similar to Libor by allowing the rate to be known in-advance of the interest accrual period. Here we adopt the mild assumption of bivariate normal log-forward rates for adjacent rate periods in the forward market model to derive the convexity adjustment for contracts which reference SOFR in-advance. Due to martingale conditions, when adjacent forward rates are uncorrelated there is no convexity adjustment, but when there is non-zero correlation, there is a convexity adjustment for any derivative payoff including the SOFR in-advance forward rate and the caplet/floorlet payoff which references this rate. These results demonstrate that SOFR in-advance derivatives are correlation-sensitive in their pricing and risk management, which is notably more complex than the situation for Libor.

Markov-modulated jump-diffusion models for the short rate: Pricing of zero coupon bonds and convexity adjustment

In this article, we consider a Markov-modulated model with jumps for the short rate. Using the main properties of a telegraphic process with jumps we compute the expected short rate. We obtain closed formulas for the zero coupon bond price assuming the Unbiased Expectation Hypothesis for the forward rates. Next, we obtain the coupled system of partial differential equations for the bond price using only no-arbitrage arguments. Numerical solutions are provided for some selected examples. The results obtained from both methods are compared and allow to estimate the magnitude of the convexity-adjustment.

A Fistful of Deltas

We present a model-independent method for calculating delta, vega and rho based on a comparison of the sensitivities of any derivative payoff with those of its underlying observables. In doing so, we obtain generic definitions for these greeks with an intuitive geometric interpretation. By means of examples, we show how our definitions reduce to standard formulae for familiar special cases. For pure delta hedging of quanto options, we show that the standard formula for delta should be corrected to account for the convexity adjustment in the underlying forward. Finally, by applying the technique to physical and cash-settled swaptions, we illustrate that a systematic approach for calculating delta and vega ensures that contributions from the annuity are captured in a consistent manner.

Pricing Path-Independent Payoffs with Exotic Features in the Fractional Diffusion Model

We provide several practical formulas for pricing path-independent exotic instruments (log options and log contracts, digital options, gap options, power options with or without capped payoffs …) in the context of the fractional diffusion model. This model combines a tail parameter governed by the space fractional derivative, and a subordination parameter governed by the time-fractional derivative. The pricing formulas we derive take the form of quickly convergent series of powers of the moneyness and of the convexity adjustment; they are obtained thanks to a factorized formula in the Mellin space valid for arbitrary payoffs, and by means of residue theory. We also discuss other aspects of option pricing such as volatility modeling, and provide comparisons of our results with other financial models.

Discounting Transition: Big Bang Impacts

CCPs are planning a big bang-like collateral and discounting transition for USD. In theory this transition is done with value compensation and risk exchange at fair market value. Such a transition would conduce to the absence of value and risk impact. But by definition of big bang, the transition is done in an illiquid market for which the fair theoretical value is unknown. To understand the actual impact on valuation and risk, one has to look at the practical details of the transition and how the absence of data for half of the required theoretical quantities is overcome in practice. The resulting situation prompts exotic convexity adjustments for cleared swap and unknown valuation for non-cleared products.

Calculation of the convexity adjustment to the forward rate in the Vasicek model for the forward in-arrears contracts on LIBOR rate

Calculation of the convexity adjustment to the forward rate in the Vasicek model for the forward in-arrears contracts on LIBOR rate

Rational Models for Inflation-Linked Derivatives

Related DatabasesWeb of Science You must be logged in with an active subscription to view this.Article DataHistorySubmitted: 27 December 2018Accepted: 02 July 2020Published online: 14 October 2020Keywordsinflation-linked derivatives, rational term structure models, convexity adjustment, calibration, pricing kernels, year-on-year swap, limited price indexAMS Subject Headings91B25, 91G20, 91G30, 91G99Publication DataISSN (online): 1945-497XPublisher: Society for Industrial and Applied MathematicsCODEN: sjfmbj

STIRs and OIS Futures in the Hull-White Model

This paper derives exact formulas and their simple approximations for STIRs and OIS futures convexity adjustment under the one-factor Hull-White model which can be efficiently used in curve stripping.

BACK-OF-THE-ENVELOPE SWAPTIONS IN A VERY PARSIMONIOUS MULTI-CURVE INTEREST RATE MODEL

We propose an elementary model in multi-curve setting that allows to price with simple exact closed formulas European swaptions. Swaptions can be both physical delivery and cash-settled ones. The proposed model is very parsimonious: it is a three-parameter multi-curve extension of the two-parameter J. Hull & A. White (1990) [Pricing interest-rate-derivative securities. Review of Financial Studies 3(4), 573–592] model. The model allows also to obtain simple formulas for all other plain vanilla Interest Rate derivatives and convexity adjustments. Calibration issues are discussed in detail.

Averaged Overnight Rate Futures: Convexity Adjustment

The convexity adjustment for averaged overnight rate futures, like SOFR 1m futures, is derived including the case where trading occurs during the reference period. These results are more general than previous work that relied solely on the HJM framework, and the results herein can easily incorporate and reuse previous derivations. Numerical results demonstrate that the averaged overnight rate futures convexity adjustment is close to the convexity adjustment for compounded overnight rate futures, which is due to the close proximity of daily compounding to the continuous compounding limit.

The Valuation of Fixed Term Securities Lending in a Multi-Curve Framework

We consider the risk neutral valuation of fixed term securities lending in a multi-curve framework, taking into account the forward basis of each component of the transaction relative to the discount curve, including basis between currencies. We show that a convexity adjustment arises from the collateral basis being applied to an unnatural notional proportional to the lent security price rather than to the collateral instrument(s) price(s). In practical cases the convexity adjustment is small and can be safely ignored.

Model-free stochastic collocation for an arbitrage-free implied volatility: Part I

This paper explains how to calibrate a stochastic collocation polynomial against market option prices directly. The method is first applied to the interpolation of short-maturity equity option prices in a fully arbitrage-free manner and then to the joint calibration of the constant maturity swap convexity adjustments with the interest rate swaptions smile. To conclude, we explore some limitations of the stochastic collocation technique.

Convexity Adjustments Made Easy - A Review of Convexity Adjustment Methodologies and Formulae in Interest Rate Markets

Convexity Adjustments Made Easy - A Review of Convexity Adjustment Methodologies and Formulae in Interest Rate Markets

Convexity Adjustments Made Easy - An Overview of Convexity Adjustment Methodologies in Interest Rate Markets

Convexity Adjustments Made Easy - An Overview of Convexity Adjustment Methodologies in Interest Rate Markets