A consistent stochastic model of the term structure of interest rates for multiple tenors

Abstract

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 risk consists of two components, a credit risk component due to the possibility of being downgraded and thus facing a higher credit spread when attempting to roll over short-term borrowing, and a component reflecting the (systemic) possibility of being unable to roll over short-term borrowing at the reference rate (e.g., LI-BOR) due to an absence of liquidity in the market. The modelling framework is of "reduced form" in the sense that (similar to the credit risk literature) the source of credit risk is not modelled (nor is the source of liquidity risk). However, the framework has more structure than the literature seeking to simply model a different term structure of interest rates for each tenor frequency, since relationships between rates for all tenor frequencies are established based on the modelled roll-over risk. We proceed to consider a specific case within this framework, where the dynamics of interest rate and roll-over risk are driven by a multifactor Cox/Ingersoll/Rosstype process, show how such model can be calibrated to market data, and used for relative pricing of interest rate derivatives, including bespoke tenor frequencies not liquidly traded in the market.

A stochastic control perspective on term structure models with roll-over risk

“…Therefore, even in the absence of counterparty risk, roll-over risk provides an explanation of the multi-curve phenomenon, since the spreads associated to term rates with different tenors are due to the increased funding-liquidity risk over longer time horizons. In this sense, the present work continues on the line of [FT13,GSS17,AGS20].…”

Rational Savings Account Models for Backward-Looking Interest Rate Benchmarks

“…Market makers may offer rates, which cannot be 'independently assessed' in the sense that a mark-to-market comparison to a FRA market is not going to be available. For first advancements on consistent approaches to term-and overnight interest rate benchmarks, we here refer to Alfeus et al (2018), Backwell et al (2019), and to an article by Macrina & Mahomed (in preparation) with a focus on term risk in interest rate markets. We conclude our reflections by noting that, at the time of writing, there is no global consensus regarding the transition to RFRs.…”

Consistent Valuation Across Curves Using Pricing Kernels

“…This is due to the fact that there is currently no market standard for pricing an emerging market FRA that is forecasted and discounted under different curves, with the only observable market quantity being the spot IBOR process L y t (t, t + δ) for t ≥ 0. It is also possible, as in the case of developed markets, to define a fair FRA rate process, K x y t (T i−1 , T i ) = L y t (T i−1 , T i ), however one would not be able to observe this quantity in the market (since these FRAs are not traded, in general), therefore this would be a model-implied quantity 3 .…”

“…As with the FRA, the fair IRS rate process is model-implied 2 Assuming that one insists on maintaining measurability of the payoff at the IBOR reset time T i−1 . 3 If the y-tenored IBOR corresponds to the most liquid and tradable tenor, i.e. y = y * , then one will also have access to the set of forward IBOR processes L y t (T i−1 , T i ) for 0 ≤ t ≤ T i−1 , from the standard and liquidly tradable set of single-curve emerging market FRAs, and K x y…”

“…Filipović & Trolle [16] propose a Vasicek process with stochastic long-term mean as the OIS short-rate model with explicit models for default and liquidity risk. Alfeus et al [3] adopt a novel approach of modelling "roll-over risk" explicitly in a reduced-form setting and consider multi-factor CIR-type processes for this and the OIS short-rate.…”