Short Rate Dynamics: A Fed Funds and SOFR perspective
SShort Rate Dynamics: A Fed Funds and SOFR perspective † Karol Gellert and Erik Schlögl University of Technology Sydney, Australia University of Technology Sydney, Australia — Quantitative Finance ResearchCentre.
[email protected] The African Institute for Financial Markets and Risk Management (AIFMRM),University of Cape Town, South Africa Faculty of Science, Department of Statistics, University of Johannesburg, SouthAfricaJanuary 13, 2021
Abstract
The Secured Overnight Funding Rate (SOFR) is becoming the main Risk–Free Ratebenchmark in US dollars, thus interest rate term structure models need to be updatedto reflect the key features exhibited by the dynamics of SOFR and the forward ratesimplied by SOFR futures. Historically, interest rate term structure modelling has beenbased on rates of substantially longer time to maturity than overnight, but with SOFRthe overnight rate now is the primary market observable. This means that the empiricalidiosyncrasies of the overnight rate cannot be ignored when constructing interest ratemodels in a SOFR–based world.As a rate reflecting transactions in the Treasury overnight repurchase market, thedynamics of SOFR are closely linked to the dynamics Effective Federal Funds Rate(EFFR), which is the interest rate most directly impacted by US monetary policytarget rate decisions. Therefore, these rates feature jumps at known times (FederalOpen Market Committee meeting dates), and market expectations of these jumps arereflected in prices for futures written on these rates. On the other hand, forward ratesimplied by Fed Funds and SOFR futures continue to evolve diffusively. The modelpresented in this paper reflects the key empirical features of SOFR dynamics and iscalibrated to futures prices. In particular, the model reconciles diffusive forward ratedynamics with piecewise constant paths of the target short rate. † The authors thank Leif Andersen for helpful comments on a previous draft of this paper. The usual disclaimersapply. a r X i v : . [ q -f i n . M F ] J a n Introduction
As the Secured Overnight Funding Rate (SOFR) is currently in the process of becoming thekey Risk–Free Rate (RFR) benchmark in US dollars, interest rate term structure modelsneed to be updated to reflect this. Historically, interest rate term structure modelling hasbeen based on rates of substantially longer time to maturity than overnight, either directlyas in the LIBOR Market Model, or indirectly, in the sense that even models based onthe continuously compounded short rate (i.e., with instantaneous maturity) are typicallycalibrated to term rates of longer maturities, with any regard to a market overnight rateat best an afterthought. However, with SOFR this situation is reversed: The overnightrate now is the primary market observable, and term rates (i.e., interest rates for longermaturities) will be less readily available and therefore must be inferred (for example fromderivatives prices).Thus the empirical idiosyncrasies of the overnight rate cannot be ignored when con-structing interest rate term structure models in a SOFR–based world, and more than longerterm rates, these idiosyncrasies are driven by monetary policy. In this paper we closely ex-amine the dynamics of both SOFR and the closely related and more established EffectiveFed Funds Rate (EFFR). We find (already by simple inspection) that models, in which theshort rate evolves as a diffusion, can no longer be justified by empirical data. Instead, theprimary driver of the short rate is the piecewise flat behaviour of the Federal Open MarketCommittee (FOMC) policy target rate. Concurrently, we observe that the forward ratesassociated with the policy target rate evolve in a more diffusive manner. A model whichreconciles these two features is the main contribution of this paper.Modelling the target rate may seem not quite reflective of reality, since the FOMC setsa target range rather than a specific rate. However, we present documentary and empiricalevidence that the target rate lives on via the Interest on Excess Reserves (IOER). TheIOER acting as the target rate is a deliberate strategy by the Federal Reserve, which hasproven much more effective at keeping the EFFR near the policy target.Another prominent empirical feature of SOFR dynamics, and to a lesser degree EFFRdynamics, is the occurrence of large spikes. The spikes tend to occur at predictable timeson the last day of the month and particularly end of quarter and end of year dates. Notall spikes occur on the last day of the month, such as the extreme spike in September2019. An explanation provided by the Federal Reserve for the September 2019 spike isthat it occurred on a day on which large corporate tax receipts and Treasury bond expiriescaused a sharp imbalance in demand and supply in the repo market. Both the reasonsgiven occur on dates easily obtainable in advance, therefore arguably this spike also couldbe classified as occurring on a predictable date. Using a similar approach to modelling thetarget rates, we construct a model for spikes occurring on known dates, where the forwardrates associated with those spikes evolve in a diffusive manner. See Miltersen, Sandmann and Sondermann (1997), Brace, Gatarek and Musiela (1997) and Musielaand Rutkowski (1997). Of these, Hull and White (1990) is the most prominent example. stochastic discontinu-ities , see for example Kim and Wright (2014), Keller-Ressel, Schmidt and Wardenga (2018),Fontana, Grbac, Gümble and Schmidt (2019). The nomenclature reflects the context thatthe discontinuities are treated as extensions to an existing continuous stochastic model. Ourapproach is distinctly different in that the discontinuity is the basis of our model for theshort rate, while simultaneously the forward rates for maturities beyond the next scheduledjump evolve as a continuous stochastic process.Specific to SOFR, Heitfield and Park (2019) model forward rates using a step function,assuming that rates remain constant for all dates between FOMC meetings. This is a staticapproach for the purposes of calibrating a piecewise flat term structure, similar to ourassumption in the calibration section. Most recently Andersen and Bang (2020) providea SOFR–inspired general spike model to enable the extension of derivative pricing modelsto spikes. While their paper includes spikes at known times as a special case, our focus isexclusively on short rate discontinuities at known times.Several papers focus on adapting existing models to SOFR without considering discon-tinuities. These include Mercurio (2018) who uses a deterministic SOFR–OIS spread witha short rate model for the OIS. Lyashenko and Mercurio (2019) propose an extension to theLIBOR Market Model to accommodate the in–arrears setting nature of term rates relatedto SOFR and overnight benchmark rates in general. Skov and Skovmand (2020) show thata three–factor Gaussian arbitrage–free Nelson/Siegel model is well suited for the SOFRfutures market, but they do not include the time series of SOFR itself in their estimation.The rest of the paper is organised as follows. Section 2 closely examines the empiricalbehaviour of SOFR and EFFR, which motivates the model proposed in this paper. Section3 presents the model for discontinuous short rates with continuous forward rates includingboth step and spike discontinuities. The model is presented within the Heath, Jarrow andMorton (1992) (HJM) framework, and also includes a Gaussian diffusion to account forresidual noise. Results from calibration of the model to futures market data are presentedin Section 4. Over the course of the last five years significant changes to the implementation of mone-tary policy have had a dramatic impact on the EFFR, resulting in a substantial divergencebetween its empirical behaviour and the dynamic assumptions of short rate models. Thechanges trace back to the 2008 financial crisis, prior to which monetary policy was admin-istered primarily by direct intervention in the Fed Funds market to maintain the EFFRclose to the target rate set by the FOMC. The approach relied on open market operationsby the Federal Reserve trading desk resulting in the EFFR gravitating around the targetrate with varying degrees of volatility. The first stochastic model of the short rate is attributed to Merton (1973) who employed See Hilton (2005) for an analysis of factors impacting EFFR volatility related to open market operations
959 1969 1979 1989 1999 2009 2019-10%-5%0%5%10% d a il y E FF R c h a n g e MertonVasicek CIR Hull-White Financial Crisis
Figure 1:
Empirical daily EFFR changes and the history of short rate models a single–dimension Brownian motion as the driver. At least on cursory visual inspection,the empirical data at the time, see Fig. 1, did not contradict the mathematically tractableGaussian assumption of the model. The next major development came from Vasicek (1977),adding mean reversion, a strong empirical feature of rate dynamics. Modelling mean rever-sion also aligned with the notion of open market operations by the Federal Reserve tradingdesk managing the rate around the monetary policy target. Cox, Ingersoll and Ross (1985)(CIR) modified the dynamics of the continuously compounded short rate by scaling thevolatility by the square root of the short rate, ensuring non-negativity of interest rates.The next milestone in short rate modelling was an extension of the Vasicek model withtime dependent drift by Hull and White (1990), allowing the model to be fitted to an initialterm structure of interest rates observed in the market — this was critical for use of themodel to price interest rate derivatives. Heath et al. (1992) developed the general frame-work into which all diffusion–based arbitrage–free interest rate term structure models mustfit. Open market operations are carried out by the Federal Reserve trading desk whosetrading goal is to maintain the EFFR near the target rate. This involves monitoring themarket and counteracting trades which move the EFFR away from target, in essence micro–managing market liquidity. The 2008 financial crisis included a crisis in liquidity and theability of the Federal Reserve’s trading desk to maintain the EFFR near the target ratesignificantly deteriorated. The trading desk did not have the means to counteract thedramatic drain in supply of desperately demanded capital.This was acknowledged by the Federal Reserve as one of the factors considered whenswitching to a target range, initially set between 0 and 25 basis points. The Federal Re-serve’s strategy in response to the financial crisis centred around two key policies: near zerointerest rates and quantitative easing. The phases of quantitative easing became known as See Federal Open Market Committee (2000-2020) December 2008, page 9.
4E 1/2/3 and involved selling Treasury bonds and purchases of various credit risky assets in a bid to boost liquidity and credit conditions. The Federal Reserve’s injection of liquidityresulted in an environment of elevated excess reserves. By historical standards, the rise inexcess reserves was extreme and without precedent. As can seen in Fig. 2, it increasedfrom under $2 billion in September 2008 to $1 trillion by November 2009, before reachinga high of over $2.5 trillion in October 2015.In October 2008, the Federal Reserve began paying IOER to help control the EFFRin response to increasing excess reserves. It was thought at the time that the IOER shouldact as a lower bound for the EFFR, since no institutions should want to lend below thisrate. As such, effective from October 9 the IOER was set to 75 basis points, with the EFFRtarget rate at 150 basis points. In the following days the EFFR was setting well below thetarget rate, including some days below the IOER. On the October 23, to lift rates closerto target, IOER was increased to 110 basis points, in response EFFR rates increased butwere still setting below the IOER. Other adjustments were made in November under theassumption of IOER acting as a lower bound, however with EFFR persisting to settle wellbelow the IOER it became clear the assumption was incorrect.In the FOMC immediately following the introduction of the IOER, it was noted thatinstitutions not eligible to receive it were willing to sell (lend) funds at rates below theIOER. However, it was not until December 2008, where together with the introduction ofthe target range, the IOER was set at the target range upper limit of 25 basis points inrecognition that due to unique circumstances the IOER was acting as an upper bound for theEFFR. The large surpluses in excess reserves eliminated demand for reserve loans. Insteadthe Fed Funds rate was driven by Government Sponsored Institutions who do not earninterest on reserve balances, lending their excess reserves at below the IOER to institutionswho would then earn the difference between the Fed Funds rate and the IOER. In effect,by paying the IOER in a market flooded with liquidity, the Federal Reserve became theborrower, rather than the lender of last resort.Plans for reversal of the post financial crisis expansionary policy were formally laid outat the FOMC September 2014 meeting as the Policy Normalization Principles and Plans. The aim of the normalisation strategy was to bring the EFFR back to normal levels andreduce the securities held by the Federal Reserve, thereby unwinding the excess reservesheld by banks. Prior to the financial crisis, controlling the supply of reserves via openmarket operations was a key tool in controlling the Fed Funds rate. However, the FederalReserve has adopted the view that with banks using reserves for liquidity more than priorto the crisis, it might be hard to predict demand for reserves and therefore open marketoperations would not be effective at precisely controlling the EFFR. Instead, the newnormal will constitute the Federal Reserve keeping excess reserves just large enough toremain on the flat part of the demand curve, a prerequisite condition for the use of theIOER to control the EFFR. Such as Agency Debt, Mortgage Backed Securities and Term Auction Facilities, see Binder (2010). See Federal Open Market Committee (2000-2020) October 2008, page 7. See Federal Open Market Committee (2000-2020) October 2008, page 2. See Federal Open Market Committee (2000-2020) September 2014, page 3. See Federal Open Market Committee (2000-2020) November 2018, page 3.
989 1994 1999 2004 2009 2014 201910 e x c e ss r e s e r v e b a l a n c e ( $ m ) Figure 2:
Excess reserves balance history
Thus the conditions in the Fed Funds market are dramatically different to when shortrate models were first conceived. The flood of liquidity in excess reserves, by constructionaimed at removing any supply–demand gradient, has removed most of the volatility from theshort rate of interest, with changes in the short rate being mainly driven by changes in theIOER, leading to jumps at known times (the FOMC meeting dates). Forward rates impliedby traded market instruments, however, continue to exhibit volatility, as the evolution ofmarket expectations of FOMC actions is priced into forward–looking instruments such asFed Fund futures.
In this section, the EFFR is examined by breaking it down into distinct components. Acomparison of EFFR and the Fed Funds target rate since the beginning of 2015, see Fig.3, demonstrates the low volatility in deviations of EFFR from the target rate. The targetrate therefore must be a major component of the EFFR dynamics. Another feature of FedFunds empirical data in the earlier part of the of the five years covered by Fig. 3 are end–of–month downward spikes. These spikes used to occur as a result of certain regulationsprescribing the last day of the month as a measurement day for reporting regulatory capital,resulting in a temporary imbalance in the demand–supply for excess reserve funds. It isinstructive to deconstruct the EFFR r E ( t ) rate into the two components, discontinuous atknown times, and a residual such that: r E ( t ) = r P ( t ) + ∆ r Z ( t ) + ζ ( t ) (1)The first component r P , the policy target rate is directly observable as the IOER rate. Thesecond component, ∆ r Z the end–of–month spike, can be deduced from the data. Here weplace any changes to the rate on the last trading day of the month regardless of magnitude,6
015 2016 2017 2018 2019 20200.0%0.5%1.0%1.5%2.0%2.5% r a t e l e v e l EFFRtarget rate
Figure 3:
EFFR and FOMC target rate history a sufficient approach for the qualitative analysis in this section. The third component ζ captures any residual noise in addition to the first two components. The variance of thedaily changes in each component, shown in Fig. 4, is an indicator of the relative contributionof each component. It is clear that over the 5 years of data used to produce these results,the target rate is the main factor in EFFR dynamics, followed by end–of–month spikes,with only a small contribution from the residual.The existence of mean reversion in the residual is examined by finding an approximateHurst exponent h for the time series of ζ . The Hurst exponent relates the variance of thelagged difference to the lag size as follows:Var (cid:2) ζ ( t + τ ) − ζ ( t ) (cid:3) ∼ τ h (2)A Hurst exponent value of 0.5 indicates a Brownian motion, h < . indicates presenceof mean reversion. For the residual noise time series we estimate h = 0 . , see Fig. 5,indicating mean reversion.In summary, the empirical characteristics of EFFR break down into the following com-ponents: piecewise flat target rates, followed by spikes occurring on known days and meanreverting residual noise. The correlation between the three components is close to zero,with the exception of a slightly negative correlation between the residual and the targetrates. The negative correlation is due to a small lag between target rate changes and EFFRadapting the full magnitude of the change, temporarily changing the spread in the oppositedirection to the target rate change. Shortly following the well–publicised LIBOR manipulation scandals, the Financial Stabil-ity Board and Financial Stability Oversight Council highlighted one of the key problems7 .0%0.5%1.0%1.5%2.0%2.5%-0.20%-0.10%0.00%0.10%0.20% 2016 2017 2018 2019 2020-0.20%-0.10%0.00%0.10%0.20% variance comparison
Figure 4:
EFFR breakdown in vertical order (i) target rates (ii) end–of–month spikes (iii) residual(iv) variance contribution v a r i a n c e
1e 7
EFFR residualHurst=0.5Hurst=0.32
Figure 5:
Variance for the difference of the residual noise time series for different lags, comparedto theoretical Hurst exponent values related to the reference rate to be the decline in transactions underpinning LIBOR and theassociated structural risks to the financial system. As argued in Schrimpf and Sushko See The Alternative Reference Rates Committee (2018), page 1.
015 2016 2017 2018 2019 20200.0%1.0%2.0%3.0%4.0%5.0%6.0% r a t e l e v e l SOFRtarget rate
Figure 6:
SOFR and FOMC target rate history (2019), partly to blame for the decline in interbank term lending are the inflated excessreserves discussed in the previous section. In response, the Federal Reserve convened theAlternative Reference Rates Committee (ARRC) to explore alternative reference rates. InJune 2017, the ARRC formally announced the Secured Overnight Financing Rate (SOFR)as the replacement for LIBOR. A key criterion for the choice was the large volume of trans-actions behind SOFR, translating to it being more representative of bank’s funding costsand less susceptible to manipulation. The calculation of SOFR is based on overnight repotransactions, which in 2017 averaged around $700b in daily transactions (compared toless than $1b for US dollar LIBOR).Official SOFR fixings have been calculated as far back 2014 and can be seen in compar-ison to the target rate in Fig. 6. Three features stand out, firstly SOFR appears to followa stepwise function, suggesting that similarly to EFFR the Fed Funds target rate plays animportant role in the SOFR dynamic. Another aspect is that SOFR is substantially morevolatile than EFFR. A third feature is the prominence of spikes, most of which, similarly toEFFR, occur on the last trading day of the month. The end–of–month spikes are related tothe measurement of dealers’ balance sheet exposures at month–end for regulatory purposes.This single snapshot approach incentivises the management of exposures around reportingdates, which as explained in Schrimpf and Sushko (2019) has been resulting in increases inthe SOFR rate on end–of–month dates. Therefore the main components of SOFR can becharacterised as follows, see Fig. 7: r S ( t ) = r P ( t ) + ∆ r Z ( t ) + ∆ r J ( t ) + ζ s ( t ) (3) This suggests an interesting causal link between the financial crisis, the Federal Reserve response andthe emergence of SOFR by linking the decline in LIBOR transactions to excess reserves. For details see The Alternative Reference Rates Committee (2018), page 7. .0%0.5%1.0%1.5%2.0%2.5%0.0%1.0%2.0%3.0%4.0%0.00%0.20%0.40%0.60%0.80% 2016 2017 2018 2019 2020-0.30%-0.15%0.00%0.15%0.30% variance comparison Figure 7:
SOFR breakdown in vertical order (i) target rates (iii) non-eom spike (iii) end–of–monthspikes (iv) residual (v) variance contribution
Here r S ( t ) is the SOFR observation at time t , r P ( t ) the policy target rate, ∆ r Z ( t ) theend–of–month SOFR spikes, ∆ r J ( t ) spikes not occurring on end–of–month dates and ζ s ( t ) the residual. The spikes not occurring on the last day of the trading month are the mostprominent in terms of contribution to the net variance over the period, however this is dueto only one very large spike occurring in September 2019. This particular spike occurredon a day of large corporate tax payments and Treasury bond expiries, therefore it couldbe argued that the date was predictable. The next largest contribution comes from theend–of–month spike component, followed closely by the policy target rate component.In contrast to EFFR, the contribution from the residual component is in the same orderof magnitude as the target rate component as well as the end–of–month component. Usingthe same approach as for EFFR, we see that the SOFR residual also exhibits strong meanreversion with an estimated Hurst parameter h = 0 . , see Fig. 8. In summary, the com-ponents of SOFR mostly mirror the components of EFFR, but with different contributionsto the overall variance. To reflect the empirical features outlined in the previous section, we assume a three–component model driven by independent factors and construct it within the HJM frame-work. The three components include a step component to reflect the target rate dynamics,10
20 40 60 80 100 120 140 160time lag (days)0.0000000.0000020.0000040.0000060.0000080.0000100.000012 v a r i a n c e SOFR residualHurst=0.5Hurst=0.24
Figure 8:
Variance for the difference of SOFR residual time series for different lags, compared totheoretical Hurst exponent values a spike component for spikes occurring at known times and a continuous diffusion compo-nent for the residual noise. Define a set of independent Brownian motions W comprisingof subsets of Brownian motions W P , W Z , W V related to the step, spike and continu-ous components respectively, where W P = [ W P , ..., W Pm ] , W Z = [ W Z , ..., W Zn ] and where W = [ W , ..., W m + n +1 ] = [ W P , ..., W Pm , W Z , ..., W Zn , W V ] . Under the spot risk–neutralmeasure, we have in the HJM framework: f ( t, T ) = f (0 , T ) + m + n +1 (cid:88) i =1 t (cid:90) σ i ( u, T ) T (cid:90) u σ i ( u, s ) dsdu + m + n +1 (cid:88) i =1 t (cid:90) σ i ( s, T ) dW i ( s ) (4)Define: σ i ( t, T ) = ( i ≤ m ) σ Pi ( t, T ) + ( m < i ≤ m + n ) σ Zi − m ( t, T ) + ( i = m + n + 1) σ V ( t, T ) (5)Therefore: m + n +1 (cid:88) i =1 t (cid:90) σ i ( s, T ) dW i ( s ) = m (cid:88) i =1 t (cid:90) σ Pi ( s, T ) dW Pi ( s )+ n (cid:88) i =1 t (cid:90) σ Zi ( s, T ) dW Zi ( s )+ t (cid:90) σ V ( s, T ) dW V ( s ) (6)11nd m + n +1 (cid:88) i =1 t (cid:90) σ i ( u, T ) T (cid:90) u σ i ( u, s ) dsdu = m (cid:88) i =1 t (cid:90) σ Pi ( u, T ) T (cid:90) u σ Pi ( u, s ) dsdu + n (cid:88) i =1 t (cid:90) σ Zi ( u, T ) T (cid:90) u σ Zi ( u, s ) dsdu + t (cid:90) σ V ( u, T ) T (cid:90) u σ V ( u, s ) dsdu (7)therefore we have f ( t, T ) = f P ( t, T ) + f Z ( t, T ) + f V ( t, T ) (8)where f P ( t, T ) = f P (0 , T ) + m (cid:88) i =1 t (cid:90) σ Pi ( u, T ) T (cid:90) u σ Pi ( u, s ) dsdu + m (cid:88) i =1 t (cid:90) σ Pi ( s, T ) dW Pi ( s ) (9) f Z ( t, T ) = f Z (0 , T ) + n (cid:88) i =1 t (cid:90) σ Zi ( u, T ) T (cid:90) u σ Zi ( u, s ) dsdu + n (cid:88) i =1 t (cid:90) σ Zi ( s, T ) dW Zi ( s ) (10) f V ( t, T ) = f V (0 , T ) + t (cid:90) σ V ( u, T ) T (cid:90) u σ V ( u, s ) dsdu + t (cid:90) σ V ( s, T ) dW V ( s ) (11)similarly for the short rate: r ( t ) = r P ( t ) + r Z ( t ) + r V ( t ) (12)and zero coupon bonds: B ( t, T ) = B P ( t, T ) B Z ( t, T ) B V ( t, T ) (13)We now proceed to discuss the modelling of each component in more detail. The main empirical feature of the target rate is that it is piecewise flat between the FOMCmeeting dates at which a policy change has occurred. Most of the meetings are scheduledat least one year ahead of time with the exception of emergency meetings. Since 2015 there have been 47 meetings (including 3 emergency meetings), of which 17 resulted in atarget rate change
015 2016 2017 2018 2019 20200.0%0.5%1.0%1.5%2.0%2.5%3.0% r a t e l e v e l target rate Figure 9:
Target rate and various forward rates implied by specific 30–day Fed Funds futures
Forward target rates do not trade directly, however the nature of their dynamics can bededuced from 30-day Fed Fund Futures which trade on the closely related EFFR rate. Fig.9 shows the historical target rate and various forward rates implied from specific futurescontracts. The point at which the forward rates end and meet the target rate coincideswith the expiry of the futures contracts. In contrast to the target rate, the dynamicsof target forward rates are more diffusive and do not jump at deterministic dates. Jumpsconceivably could occur on unexpected dates, reflecting sudden large changes in marketsentiment, but in this paper we focus only on the diffusive aspect of forward rates. This isthe main contribution of the present paper: Having observed that empirically the short rate(EFFR or SOFR) follows dynamics determined primarily by jumps at known times, butforward rates follow primarily diffusive dynamics, we construct a model which reconcilesthese two (naively contradictory) observations.An interpretation of the forward target rates is that they reflect the expectations ofprospective FOMC target rate changes. The diffusive dynamics of forward rates then reflectsthe nature of the changes in those expectations. From this perspective, the expectationscorresponding to each scheduled FOMC meetings are not independent of each other. Insome circumstances, for example, a change in the overall Federal Reserve monetary policystance, they will be positively correlated. In other cases, where for example the aggregatedchange to the target rate over some period of time is anticipated but the timing is lesscertain, the expectations may be negatively correlated to each other as the expected timingbut not the net outcome evolves.Therefore the target rate model is motivated by the following empirical features. Thetarget rate represented by the short rate r P ( t ) must be piecewise flat with respect to t . The Futures without an FOMC date in the reference month were chosen such that the target rate is expectedto be flat over the contract month and therefore the price of the futures reflects the expected target ratefor that month plus a spread rather than reflecting two flat periods before and after the FOMC date. T evolves diffusively with respect to t until the FOMC meetingimmediately preceding maturity T , reflecting the expectations of any FOMC policy targetrate change. We construct a model which reconciles these features, reflecting both thediscontinuous nature of the short rate and diffusively evolving forward rates. We construct the model for forward rates such that they are driven by the evolution ofexpectations associated with FOMC target rate changes, where the target rate change foreach scheduled meeting date evolves under its own dynamic. Define the forward rate f ( t, T ) dynamics under the empirical measure as follows: f P ( t, T ) = f P (0 , T ) + α ( t, T ) + n (cid:88) i =1 t (cid:90) ξ i ( s, T ) dZ i ( s ) (14)where f P (0 , T ) is the initial term structure of forward rates, α ( t, T ) a deterministic driftand dZ i ( s ) the Wiener increment corresponding to the i th FOMC date with correlation dZ i ( t ) dZ j ( t ) = ρ i,j dt . The stochastic term is defined as follows: ξ i ( t, T ) = ξ i ( t < x i ) ( T ≥ x i ) (15)where x i denotes the i th FOMC meeting date. The intuition behind this construction isthat each stochastic factor corresponds to an FOMC date and any changes to the targetrate are carried forward from that date. The indicator function ( T ≥ x i ) ensures thatthe i th factor is only applied to forwards with maturities greater or equal to x i . For anymaturities prior to the first meeting date T < x , therefore ( T ≥ x i ) = 0 , ∀ i ≥ , thusensuring no diffusion for forward rates with maturities prior to the first FOMC meetingdate. The indicator function ( t < x i ) terminates the diffusion from the i th stochasticfactor on the corresponding FOMC date. Solving the integral, see (59), yields: f P ( t, T ) = f P (0 , T ) + α ( t, T ) + n (cid:88) i =1 ξ i ( T ≥ x i ) Z i ( t ∧ x i ) (16)To demonstrate the behaviour of the model with an example, let x < T < x and t < x : f P ( t, T ) = f P (0 , T ) + α ( t, T ) + ξ Z ( t ) + ξ Z ( t ) (17)The interpretation being that both stochastic factor corresponding to FOMC dates x and x impact the forward rate up to time t . Any stochastic factors beyond x do not applysince the forward rate matures prior to x . Now let x < t < x : f P ( t, T ) = f P (0 , T ) + α ( t, T ) + ξ Z ( x ) + ξ Z ( t ) (18)In this case the first stochastic factor terminates at x , prior to t . That is the expectationsof the target rate change at time x evolve diffusively only up until this date.14 .1.2 Short Rates The forward rate dynamics are constructed to create the piecewise dynamic in the shortrate, which can be derived from (14) by setting r ( t ) = f ( t, t ) : r P ( t ) = f P ( t, t ) = f P (0 , t ) + α ( t, t ) + n (cid:88) i =1 t (cid:90) ξ i ( s, t ) dZ i ( s ) (19)solving the integral, see (60), yields: r P ( t ) = f P (0 , t ) + α ( t, t ) + n (cid:88) i =1 ξ i ( t ≥ x i ) Z i ( x i ) (20)From which it is evident that the short rate has no diffusion up until the first FOMC dateat which point it picks up all the diffusion from the forward rate accumulated up until thispoint in time. To see this, for t < x we have: r P ( t ) = f P (0 , t ) + α ( t, t ) (21)for x < t < x we have: r P ( t ) = f P (0 , t ) + α ( t, t ) + ξ Z ( x ) (22)for x < t < x : r P ( t ) = f P (0 , t ) + α ( t, t ) + ξ Z ( x ) + ξ Z ( x ) (23)In general, the accumulated diffusion for the forward rates creates discontinuities in theshort rate on FOMC dates, reflecting the empirical behaviour for the target rate and theassociated forward rates. The model can be easily transformed to independent factors which will make it consistentwith the HJM framework, thus allowing derivation of risk–neutral dynamics. Define Σ tobe the covariance matrix of the vector dZ = [ dZ , ..., dZ n ] . To transform the system toindependent factors we seek to find a transformation matrix λ , such that Σ = λλ T whichis applied using dZ = λdW P , to result in a vector of uncorrelated Wiener increments dW P = [ dW P , ..., dW Pn ] . Therefore: dZ i = n (cid:88) j =1 λ i,j dW Pj (24)15e can rewrite the forward rate dynamics with respect to uncorrelated factors: n (cid:88) i =1 t (cid:90) ξ i ( s, T ) dZ i ( s ) = n (cid:88) i =1 t (cid:90) σ i ( s < x i ) ( T ≥ x i ) dZ i ( s )= n (cid:88) i =1 t (cid:90) ξ i ( s < x i ) ( T ≥ x i ) n (cid:88) j =1 λ i,j dW Pj ( s )= n (cid:88) j =1 t (cid:90) σ Pj ( s, T ) dW Pj ( s ) (25)where σ Pj ( t, T ) = n (cid:88) i =1 ξ i λ i,j ( t < x i ) ( T ≥ x i ) (26) We can now formulate the risk neutral dynamics by using the result from HJM. Under thespot risk–neutral measure we have: f P ( t, T ) = f P (0 , T ) + n (cid:88) j =1 t (cid:90) σ Pj ( u, T ) T (cid:90) u σ Pj ( u, s ) dsdu + n (cid:88) j =1 t (cid:90) σ Pj ( s, T ) dW Pj ( s ) (27)Therefore, see (61) and (68), we get: f P ( t, T ) = f P (0 , T ) + n (cid:88) j =1 n (cid:88) q =1 n (cid:88) i =1 ξ q ξ i λ q,j λ i,j ( T ≥ x q ∨ i )( T − x i )[ t ∧ x q ∧ x i ]+ n (cid:88) j =1 n (cid:88) i =1 ξ i λ i,j ( T ≥ x i ) W Pj ( t ∧ x i ) (28) Short rate dynamics can are obtained as follows: r P ( t ) = f P (0 , t ) + n (cid:88) j =1 t (cid:90) σ Pj ( u, t ) t (cid:90) u σ Pj ( u, s ) dsdu + n (cid:88) j =1 t (cid:90) σ Pj ( s, t ) dW Pj ( s ) (29)16herefore, see (62) and (69), we get: r P ( t ) = f P (0 , t ) + n (cid:88) j =1 n (cid:88) q =1 n (cid:88) i =1 ξ q ξ i λ q,j λ i,j ( t ≥ x q ∨ i )( t − x i )[ x q ∧ x i ] (cid:124) (cid:123)(cid:122) (cid:125) deterministic term (**) + n (cid:88) j =1 n (cid:88) i =1 ξ i λ i,j ( t ≥ x i ) W Pj ( x i ) (cid:124) (cid:123)(cid:122) (cid:125) stochastic term (*) (30)The stochastic term (*) follows piecewise constant dynamics, jumping almost surely at each x i . Because at present we are only modelling the target rate, we would want the pathsof r P ( t ) to be constant between FOMC meeting dates. This implies that the deterministicterm (**) should not depend on t , i.e., the dependence on t of the triple sum must cancelagainst the dependence on t of the initial term structure f P (0 , t ) . When considering atime horizon of two years or less (as we do in the empirical section of this paper), thetriple sum in (**) is practically flat in t , so this is consistent with an initial term structure f P (0 , t ) which is approximately constant between FOMC meeting dates. Note, however,that (30) implies that if we require the paths of r P ( t ) to be constant between FOMCmeeting dates, we cannot arbitrarily choose an interpolation method for the initial termstructure. In particular, requiring piecewise constant paths of r P ( t ) precludes applying thepopular Nelson/Siegel interpolation to the initial term structure. Bond prices can be written as follows: B P ( t, T ) = exp (cid:18) − T (cid:90) t f P ( t, s ) ds (cid:19) = B P (0 , T ) B P (0 , t ) exp (cid:18) a ( t, T ) + b ( t, T ) (cid:19) (31) At this point, one might object that in reality, rates do not jump at every FOMC meeting date.However, one could argue that this is because target rates are only updated in discrete increments. Ourmodel could be extended to reflect this, but as a first approximation, we’ll accept the implication of acontinuous distribution of jump sizes for now, with jumps occurring at every FOMC meeting date. Note that the term ( t − x i ) appearing in the triple sum reflects a feature of a classical Gaussian termstructure model without mean reversion (as noted, for example, in Schlögl and Sommer (1998)), that theterm structure of forward rates endogenously steepens ever more (see also (28) above) as time passes —this can be avoided by introducing mean reversion. Skov and Skovmand (2020) show that a three–factor Gaussian arbitrage–free Nelson/Siegel model iswell suited for the SOFR futures market, but they do not include the time series of SOFR itself in theirestimation, i.e., their objective is not to match the SOFR dynamics, which have a substantial piecewise flatcomponent. a ( t, T ) = − T (cid:90) t n (cid:88) j =1 n (cid:88) q =1 n (cid:88) i =1 ξ q ξ i λ q,j λ i,j ( s ≥ x q ∨ i )( s − x i )[ t ∧ x q ∧ x i ] ds = − n (cid:88) j =1 n (cid:88) q =1 n (cid:88) i =1 ξ q ξ i λ q,j λ i,j [ t ∧ x q ∧ x i ] T (cid:90) t ( s ≥ x q ∨ i )( s − x i ) ds = − n (cid:88) j =1 n (cid:88) q =1 n (cid:88) i =1 ξ q ξ i λ q,j λ i,j [ t ∧ x q ∧ x i ][ I − I ] (32)where I = T (cid:90) ( s ≥ x q ∨ i )( s − x i ) ds = T (cid:90) x q ∨ i ( s − x i ) ds = ( T ≥ x q ∨ i )[ T ( T − x i ) − x q ∨ i ( x q ∨ i − x i )] (33) I = t (cid:90) ( s ≥ x q ∨ i )( s − x i ) ds = ( t ≥ x q ∨ i )[ t ( t − x i ) − x q ∨ i ( x q ∨ i − x i )] (34) b ( t, T ) = − T (cid:90) t n (cid:88) j =1 n (cid:88) i =1 ξ i λ i,j ( s ≥ x i ) W Pj ( t ∧ x i ) ds = − n (cid:88) j =1 n (cid:88) i =1 ξ i λ i,j W Pj ( t ∧ x i ) T (cid:90) t ( s ≥ x i ) ds = − n (cid:88) j =1 n (cid:88) i =1 ξ i λ i,j W Pj ( t ∧ x i ) ( T ≥ x i )[ T − ( t ∨ x i )] (35)Note that zero coupon bond prices are exponential affine functions of the W Pj ( t ∧ x i ) .However, unlike in classical Gauss/Markov HJM term structure models, here we cannotrepresent the entire term structure as an exponential affine function of n factors. As shown in Section 2, spikes in the short rate are a prominent feature of EFFR and partic-ularly SOFR dynamics. Similarly to the previous section, the forward rates associated withthe spikes can be deduced from the futures market, see Fig. 10, revealing similar empiricalbehaviour of forward rates evolving more diffusively rather than showing discontinuities onknown dates. In this section, we adapt the approach of the previous section to reflect theoccurrence of spikes at known dates. 18 un 2017 Aug 2017 Oct 2017 Dec 2017 Feb 2018 Apr 2018 Jun 2018 Aug 2018 Oct 2018 Dec 2018 Feb 20190.5%1.0%1.5%2.0%2.5%3.0%3.5% r a t e l e v e l SOFR
Figure 10:
SOFR and a forward rate associated with the end of month December 2018 spike
The model for forward rates is constructed such that the forward rate on each spike date z i is driven by its own independent factor. Therefore we can use the HJM result to formulatethe forward rates under the risk neutral measure: f Z ( t, T ) = f Z (0 , T ) + n (cid:88) i =1 t (cid:90) σ Zi ( u, T ) T (cid:90) u σ Zi ( u, s ) dsdu + n (cid:88) i =1 t (cid:90) σ Zi ( s, T ) dW Zi ( s ) (36)We assume that when spikes occur, they impact a fixed period h i starting from time z i . Let H i = [ z i , z i + h i ] , the volatility function is defined as follows: σ Zi ( t, T ) = σ Zi ( t < z i ) ( T ∈ H i ) (37)Therefore, see (63) and (71) : f Z ( t, T ) = f Z (0 , T ) + n (cid:88) i =1 (cid:0) σ Zi (cid:1) ( T ∈ H i )( T − z i )[ t ∧ z i ] + n (cid:88) i =1 σ Zi ( T ∈ H i ) W i ( t ∧ z i ) (38)To demonstrate the behaviour of the model with an example, let T ∈ H and t < z : f Z ( t, T ) = f Z (0 , T ) + (cid:0) σ Z (cid:1) ( T − z ) t + σ Z W ( t ) (39)The interpretation being that the forward rates only evolve when T ∈ H i up to the minimumof time t or z i , the beginning of the period H i . Usually this period of time would be equivalent to 1 day but could be more if for example it is a SOFRrate set on a Friday, therefore applying for compounding and averaging payoff calculations over the weekend .2.2 Spiked Short Rates By construction short rates follow the spiked trajectory, we have: r Z ( t ) = f Z ( t, t ) = f Z (0 , t ) + n (cid:88) i =1 t (cid:90) σ Zi ( u, t ) t (cid:90) u σ Zi ( u, s ) dsdu + n (cid:88) i =1 t (cid:90) σ Zi ( s, t ) dW Zi ( s ) (40)Therefore, see (64) and (72) : r Z ( t ) = f Z (0 , t ) + n (cid:88) i =1 (cid:0) σ Zi (cid:1) ( t ∈ H i )( t − z i ) z i + n (cid:88) i =1 σ Zi ( t ∈ H i ) W i ( z i ) (41)From this it is evident that the short rate is deterministic until the spike interval overwhich a spike applies, with a magnitude which includes the associated forward rate diffusionaccumulated up to the beginning of the interval. For example let t ∈ H : r Z ( t ) = f Z (0 , t ) + (cid:0) σ Z (cid:1) ( t − z ) z + σ Z W ( z ) (42) The bond prices can be written as follows: B Z ( t, T ) = exp (cid:18) − T (cid:90) t f Z ( t, s ) ds (cid:19) = B Z (0 , T ) B Z (0 , t ) exp (cid:18) a ( t, T ) + b ( t, T ) (cid:19) (43) a ( t, T ) = − T (cid:90) t n (cid:88) i =1 (cid:0) σ Zi (cid:1) ( s ∈ H i )( s − z i )[ t ∧ z i ] ds = − (cid:0) σ Zi (cid:1) [ t ∧ z i ] T (cid:90) t ( s ∈ H i )( s − z i ) ds = − (cid:0) σ Zi (cid:1) [ t ∧ z i ][ I − I ] (44)where I = T (cid:90) ( s ∈ H i )( s − z i ) ds = ( T ≥ z i ) (cid:20) h i ∧ ( T − z i ) (cid:21) (45)and I = t (cid:90) ( s ∈ H i )( s − z i ) ds = ( t ≥ z i ) (cid:20) h i ∧ ( t − z i ) (cid:21) (46)20nd b ( t, T ) = − T (cid:90) t n (cid:88) i =1 σ Zi W i ( t ∧ z i ) ( s ∈ H i ) ds = − n (cid:88) i =1 σ Zi W i ( t ∧ z i ) T (cid:90) t ( s ∈ H i ) ds = − n (cid:88) i =1 σ Zi W i ( t ∧ z i )([( T − z i ) ∧ ( T − t ) ∧ h i ∧ ( z i + h i − t )] ∨ (47) As shown in Section 2, an empirical feature of the noise component of both EFFR and SOFRis mean reversion. Since the initial bond term structure is most naturally contained in theinitial target rate term structure, the mean reverting Vasicek model should be sufficientto model the noise component of short rates. We present the model based on the resultsshown in Carmona (2007). The dynamics of the diffusive residual are given by: dr V ( t ) = ( θ − βr V ( t )) dt + σ V dW V ( t ) (48)The solution is given by: r V ( t ) = e − βt r V (0) + (1 − e − βt ) θβ + σ V t (cid:90) e − κ ( t − s ) dW V ( s ) (49)With forward rates: f V ( t, T ) = r V ( t ) e − β ( T − t ) + θβ (cid:18) − e − β ( T − t ) (cid:19) − θ β (cid:18) − e − β ( T − t ) (cid:19) (50)The zero coupon bond price is given by: B V ( t, T ) = a ( t, T ) e b ( t,T ) r (0) (51)with b ( t, T ) = − − e − β ( T − t ) β (52)and a ( t, T ) = 4 θβ − σ β + ( σ V ) − αβ β T + ( σ V ) − αββ e − βT − ( σ V ) β e − βT (53)21 Calibration to Futures Contracts
This section presents results calibrating the model to Fed Funds and SOFR futures data.Fed Fund futures are used to calibrate the target rate term structure, which is then usedas the basis for calibration to SOFR futures, from which we infer the term structure offorward rates related to end–of–month spikes. The time series of calibrated EFFR andSOFR forward rate vectors is used to examine how well the market anticipates FOMCpolicy target rate changes as well as end–of–month spikes. The time series of SOFR forwardrates is then used to compare the forward looking SOFR term rates to LIBOR.
Fed Funds futures contracts are based based on the arithmetic average of the EFFR,denoted r E over the specified contract month. Define m as the number of months from thecurrent trading month ( m = 0 ), τ m,i := as the date corresponding to day i in month m with n m denoting the total days in month m .Define the futures contract index for reference month m at time t as (cid:101) F m ( t ) , the valueof a single contract is $4,167 × (cid:101) F m ( t ) . The terminal value of the contract is determinedas (cid:101) F m ( τ m,n m ) = 100 − R m where R m is the arithmetic average of the daily EFFR fixingduring the contract month, settled on the first business day after the final fixing date.Defining R m := n m n m (cid:80) i =1 r E ( τ m,i ) , the terminal payoff is: (cid:101) F m ( τ m,n m ) = 100 − R m = 100 (cid:18) − n m n m (cid:88) i =1 r E ( τ m,i ) (cid:19) Using the generic futures pricing theorem, the expected value at t of the futures contractindex (cid:101) F m under the spot risk neutral measure is: F m ( t ) = E t [ (cid:101) F m ( τ m,n m )] = 100 (cid:18) − n m n m (cid:88) i =1 E t [ r E ( τ m,i )] (cid:19) (54)The current futures contract continues to trade during the observation month, thereforethe valuation needs to account for already observed values of r E : F ( t ) = 100 (cid:18) − n (cid:18) n (cid:88) i =1 ( t>τ ,i ) r E ( τ ,i ) + n (cid:88) i =1 ( t ≤ τ ,i ) E t [ r E ( τ ,i )] (cid:19)(cid:19) (55) Fed Funds futures contracts are available for each calendar month approximately 3 yearsahead of time. However the liquidity beyond 1 year deteriorates and therefore we limit See Hunt and Kennedy (2004), Theorem 12.6. f (0 , T ) is assumed to be piecewise flat betweenFOMC meeting dates. This aligns the initial term structure to the driving factors of thetarget rate model and therefore the daily changes in the calibrated f (0 , T ) vector providesan empirical estimate for the dynamics of f ( t, T ) . To simplify the calibration, it is assumedthat the impact on the drift component is negligible, particularly if the calibration is usedto obtain the empirical dynamics of the forward rate based on daily increments obtainedfrom the calibration. The spikes are a secondary component of EFFR empirical dynamicsand are ignored in the calibration. We also calibrate a constant spread between EFFRand the target rate which is equivalent to assuming zero volatility in the Gaussian residualnoise component of the model.Observable market prices exist in the form of current bid and offer and last observedprice, which reflects a trade at either the bid or the offer levels at the time of the transaction.We take the view that at any given time the true market state is at some point betweenthe bid and offer prices. Closing prices which are recorded at the end of each day’s tradingsession also reflect either the bid or the offer. Therefore the closing price could be either theoffer, inferring that the bid is one price fluctuation below the closing price, or converselyinfer that the offer is one price fluctuation above the closing price. Based on this reasoningwe embed a minimum price fluctuation size tolerance to the calibration error e m ( t ) formonth m : e m ( t ) = ( | F m ( t ) − (cid:101) F m ( t ) | − h m ) + Where the minimum fluctuation of the index for month m as h m with h = 0 . and h m = 0 . for m (cid:54) = 0 . The error bounds result in better solution stability less subjectto bid-ask fluctuations in the cross sectional and longitudinal data. The calibration isperformed using a genetic algorithm approach based on the method developed in Gellertand Schlögl (2019). To analyse the dynamics of the stepwise model forward rates, the calibration is performedon daily data in the period from January 2015 to September 2020. Additionally, we measurethe agreement between actual target rate changes and the corresponding change inferredfrom the initial term structure of calibrated forward rates. This demonstrates how wellthe futures market was able to predict target rate changes in the test period. It is also agood indicator of the ability of the model to translate futures data into a meaningful termstructure of anticipated target rate changes. Since there are 12 futures contracts and 8 FOMC meetings it may be possible to extract informationregarding expected EFFR spikes from futures data
50 100 150 200 250 300days to jump0.00.20.40.60.81.0 r - s q u a r e d eom nodesfomc nodes Figure 11:
R-squared of EFFR realised vs forward rates for different forward periods
To measure the agreement the R-squared is calculated between actual target ratechanges ∆ r P ( x i ) and corresponding initial forward rate term structure inferred changes f P (0 , x i ) − f P (0 , x i − h ) , grouped by the number of days in the forward rate term, thatis the number of days between the calibration date corresponding to t = 0 and x i . Theresults in Fig. 11 show the R-squared for increasing number of days between x i and thecalibration date. For comparison, the same calculation is shown with the same piecewiseflat assumption but with discontinuity dates naively set to coincide with futures contractmaturities.The results show a clear correspondence between actual and anticipated target ratechanges. The correspondence deteriorates as the forward term increases but still showsevidence of some anticipation for terms over 200 days. The results comprise of a mixtureof good long term anticipation of rate increases and poor anticipation of rate decreases.This can be attributed to the well communicated and regular increases in the target rateduring the normalisation phase following near zero target rates. The rapid drop in targetrates at the beginning of 2020 was not expected by the market, excluding this period wouldsubstantially improve the R-squared results. SOFR futures are available in monthly and quarterly contract period lengths. The SOFR1M futures contracts are defined to reflect the specification of the Fed Funds 30 dayfutures with SOFR replacing the EFFR as the reference rate. Therefore the pricing formulasdescribed in the previous section also apply to SOFR 1M futures.
24n contrast to the monthly contracts, the final payoff of the SOFR 3M futures contracts compounds SOFR, denoted by r s , over IMM quarterly dates, aligning the dates of thecontracts to the LIBOR–referenced quarterly Eurodollar futures. Define q as the number ofIMM quarters from the current trading quarter ( q = 0 ), τ ∗ q,i := as the date correspondingto day i in quarter q with n q denoting the total days in quarter q . The SOFR 3M futurescontract terminal payoff is: (cid:101) F s q ( τ q,n q ) = 100 − R s q where R s is based on SOFR compounded over the reference quarter: R s = 100 × n q (cid:20) n q (cid:89) i =1 (cid:26) ( τ q,i ∈ b ) (cid:18) d i r s ( τ q,i )360 (cid:19)(cid:27) − (cid:21) where b is the set of US government securities business days and d i the number of days therate r s ( τ q,i ) applies. Using the generic futures pricing theorem, the expected value at t ofthe futures contract index (cid:101) F s q under the spot risk neutral measure is: F s q ( t ) = E t [ (cid:101) F s q ( τ q,n q )] = 100 (cid:18) − n q (cid:20) n q (cid:89) i =1 (cid:26) ( τ q,i ∈ b ) (cid:18) d i E t [ r s ( τ q,i )]360 (cid:19)(cid:27) − (cid:21)(cid:19) (56)The current futures continues to trade during the observation quarter, therefore the valua-tion needs to account for already observed values of r s : F s ( t ) = E t [ (cid:101) F s q ( τ ,n )] = 100 (cid:18) − n (cid:20) n (cid:89) i =1 (cid:26) ( τ ,i ∈ b ) (cid:18) d i r ∗ s ( τ ,i )360 (cid:27) − (cid:21)(cid:19) (57)where r ∗ s ( τ ,i ) = ( t>τ ,i ) r s ( τ ,i ) + ( t ≤ τ ,i ) E t [ r s ( τ ,i )] Similarly to Fed Funds futures, SOFR 1M futures are available for each calendar month withliquidity approximately 1 year ahead, SOFR 3M futures are available between quarterlyIMM dates approximately 2 years ahead of expiry. Calibrating the target rate term structureto Fed Fund futures allows the use of SOFR futures to extract information regarding theexpected SOFR end–of–month spikes. We calibrate the spike component of the model toSOFR futures with similar assumptions as in the case of Fed Fund futures.The SOFR term structure is assumed to consist of the target rate term structure ob-tained from Fed Fund futures, an end–of–month spike specific to the SOFR rate and aSOFR specific spread. The drift component of the spike is ignored assuming it has a negli-gible effect on the inferred spike forward dynamics. The spread is assumed constant for allforwards which is equivalent to the assumption of zero volatility for the noise component.The treatment related to the bid-ask spread is applied in the same way as for Fed Fundfutures. Third Wednesday of March, June, September and December d i is equal to one plus the number of consecutive business days immediately following τ q,i
20 40 60 80 100 120 140 160days to jump0.000.050.100.150.20 r - s q u a r e d Figure 12:
R-squared of SOFR realised vs forward rates for different forward periods
Calibration is performed for all available SOFR futures data since the commencement oftrading in June 2018. The agreement between expected SOFR spikes and actual spikes ismeasured by calculating the R-squared between end–of–month changes in the SOFR rate ∆ r ( z i ) and the corresponding forward spike f Z (0 , z i ) − f Z (0 , z i − h ) . The comparison isgrouped by the number of days between the calibration date corresponding to t = 0 and z i .The results in Fig. 12 show the R-squared for increasing number of days between z i andthe calibration date.The results reveal some evidence of short term anticipation of spikes close to the spikedate. This is particularly true for the last trading day of the future because the tradingactivity in the repo market from which the day’s SOFR rate is calculated occurs simulta-neously to trading in the futures market. The contrast to the high R-squared for targetrate jumps anticipated by Fed Funds futures comes from the fact that FOMC target ratechanges are communicated well ahead of time, particularly for rate increases, while theSOFR spikes depend on liquidity conditions, which are only be anticipated in a short timeframe, if at all. However, the most negative impact on the results is not lack of anticipationof spikes, rather it is the over-anticipation of spikes particularly when spikes do not occur.This indicates the presence of a spike risk premium embedded in SOFR futures prices.It is worth noting that by focusing on the anticipation of SOFR spikes by SOFR futures,we are focusing on the incremental information contained in SOFR futures given thatFOMC target rate changes are anticipated by EFFR futures. If we were to consider SOFRfutures in isolation, these also anticipate FOMC target rate changes.26
015 2016 2017 2018 2019 20200.0%0.5%1.0%1.5%2.0%2.5%3.0%
SOFR Compounded 3MLIBOR
Figure 13:
SOFR 3m rolling compounded rate compared to LIBOR
One of the approaches considered as the replacement for the LIBOR indexation of loanterms is a rate based on retrospectively compounding of SOFR over the same term, seeFig. 13 for a historical comparison. Both rates appear to follow the same underlying trend,this is related to the target rate term structure, which as we argue in this paper underliesall interest rates. LIBOR also exhibits considerably more volatility. This is because theSOFR compounded rate is a rolling compounding calculation of already set rates, with onlyone new rate rolled in the calculation on each day. LIBOR, on the other hand, is a forwardlooking term rate and is not subject to the volatility reduction from rolling compounding.The two rates therefore are not really comparable, which highlights one aspect of substantialproblems with any proposal to replace LIBOR with a compounded SOFR. The calibration presented in the previous section enables a more analogous comparisonof LIBOR and the SOFR forward looking spot term rate. Additionally we can examine thebehaviour of SOFR 3M futures with respect to Eurodollar futures for a direct comparisonof SOFR and LIBOR forward term rates. The SOFR term rates are calculated accordingto the compounding formula used to calculate SOFR 3M futures terminal payoff, using thedaily forward rates obtained from the calibration.The calculated spot SOFR 3M term rate is shown in Fig. 14 in comparison to spotLIBOR. The rates are well correlated, approximately 50% of the LIBOR variance can beattributed to the SOFR 3M term rate. The impact of SOFR spikes dissipates over a 3month compounding period, instead the term rate is mostly driven by target rate termstructure. In turn, this shows that a significant proportion of LIBOR dynamics is driven bythe target rate term structure exposed in our modelling framework. From this perspective, Other problems include the disconnect due to credit risk between SOFR and the cost of funding ofprivate–sector banks, see Berndt, Duffie and Zhu (2020). ul 2018 Oct 2018 Jan 2019 Apr 2019 Jul 2019 Oct 2019 Jan 2020 Apr 2020 Jul 20200.0%0.5%1.0%1.5%2.0%2.5%3.0% SOFR 3M Term RateLIBOR
Figure 14:
Spot SOFR 3m Term Rate vs LIBOR one can think of LIBOR trading at a spread to the term rates implied from the targetrate term structure. One would expect this spread to be partly due to credit risk, but notentirely, since the term rate extracted from SOFR futures is not a “true” term rate in thesense that market participants could actually borrow at this rate — one would thereforeexpect this spread also to include a “funding liquidity risk” component analogous to the onefound in the LIBOR/OIS spread by Backwell, Macrina, Schlögl and Skovmand (2019).It is also interesting to compare the spot and forward LIBOR to SOFR spread. Asshown in Fig. 15, the spread in the forward rates appears more stable, especially duringthe market turmoil in February and March of 2020. This is also in contrast to the largeinstability exhibited by the repo rates during the financial crisis of 2008, see Andersen andBang (2020) for details. This is most likely due to Federal Reserve increasing operationsin the repo market as a response to the September 2019 spike, which also appears to haveeliminated end of month spikes. Having observed that empirically the short rate (EFFR or SOFR) follows dynamics de-termined primarily by jumps at known times, but forward rates follow primarily diffusivedynamics, we have constructed a model which reconciles these two (naively contradictory)observations. Such a model is needed because, with the transition away from the LIBORbenchmark, fixed income instruments referencing SOFR are becoming increasingly impor-tant. In addition, the actions of the Federal Reserve in response to the 2008 financial crisis If one takes into account a borrower’s risk of not being able to refinance roll–over borrowing at (aconstant spread to) a benchmark rate, this gives rise to additional basis spreads as observed in the market,see Alfeus, Grasselli and Schlögl (2020). See Federal Open Market Committee (2000-2020) September 2019 page 5. ul 2018 Oct 2018 Jan 2019 Apr 2019 Jul 2019 Oct 2019 Jan 2020 Apr 2020 Jul 20200.0%0.2%0.4%0.6%0.8%1.0%1.2%1.4%1.6% SOFR/LIBOR Spot 3M spreadSOFR/LIBOR SEP 2020 3M spread
Figure 15:
SOFR 3m Term Rate/LIBOR spread Spot vs Sep-2020 3M Term Rate over the last decade have removed much of the daily volatility from the EFFR, long thoughtof as the best empirical proxy for the short rate. This reduction in volatility has revealed anunderlying structure of short rates consisting of discontinuities directly related to FOMCpolicy target rate changes, which is also reflected in the empirical dynamics of SOFR. Onthe other hand, forward rates extracted from Fed Funds or SOFR–linked futures continueto evolve diffusively.The model requirement, that the target rate follows a path that is constant betweenFOMC meeting dates, has the interesting consequence that term structure interpolationcannot be chosen arbitrarily. In particular, the popular Nelson/Siegel approach to fittinga continuous term structure to market data contradicts this requirement. Note that this isnot just another manifestation of a violation of consistency in a term structure model inthe sense of Björk and Christensen (1999): Piecewise constant paths of a short rate implya no-arbitrage constraint on the shape of the initial term structure.Calibration of our model to Fed Funds futures showed the extent (in the form of R-squared) to which these futures prices anticipate Fed Funds target rate changes. Addition-ally calibrating the model to SOFR futures extracts incremental, short–term informationabout spikes in SOFR at known times. This suggests that such spikes need to be includedin the pricing and risk–management of short–term SOFR–linked instruments.In the present paper, the driving dynamics deliberately have been kept as simple aspossible. A conceptually trivial, but notationally tedious, extension would be to allow formean reversion — this would be necessary when considering a time horizon longer than thetwo years we are currently using. An extension beyond two years would also mean thatthe dates of FOMC meetings cannot be assumed to be precisely known (though they wouldstill be known approximately).Since the spikes and steps are driven by diffusive processes, the variance of spike andstep magnitude depends on the length of time until the spike/step is scheduled to occur. In29erms of cross–sectional calibration of the model (i.e., calibration of the model to marketprices observed at a single point in time), this can be controlled by an appropriate choice ofvolatility functions (at present, we do not calibrate volatilities in the empirical part of thepaper). Alternatively, one could modify the driving dynamics by including mean reversionand consider the model in its steady state.For the steps in the target rate modelled in this fashion, a possible economic interpre-tation is that there is a fundamental “shadow” rate of interest evolving diffusively. Onlythe central bank observes this shadow rate (perhaps imperfectly), and at known dates up-dates the central bank target rate to match this shadow rate. As noted in Section 3, ourmodel could be extended to reflect the fact that target rates are only updated in discreteincrements.
Appendix A
For a Brownian motion W ( t ) : t (cid:90) ( s < x ) dW ( s ) = W ( t ∧ x ) (58)Therefore we have the following solutions to various stochastic integrals appearing in thispaper: t (cid:90) ξ i ( s < x i ) ( T ≥ x i ) dZ i ( s ) = ξ i ( T ≥ x i ) Z i ( t ∧ x i ) (59) t (cid:90) ξ i ( s < x i ) ( t ≥ x i ) dZ i ( s ) = ξ i ( t ≥ x i ) Z i ( x i ) (60) t (cid:90) n (cid:88) i =1 ξ i λ i,j ( s < x i ) ( T ≥ x i ) dW Pj ( s ) = n (cid:88) i =1 ξ i λ i,j ( T ≥ x i ) W Pj ( t ∧ x i ) (61) t (cid:90) n (cid:88) i =1 ξ i λ i,j ( s < x i ) ( t ≥ x i ) dW Pj ( s ) = n (cid:88) i =1 ξ i λ i,j ( t ≥ x i ) W Pj ( x i ) (62) t (cid:90) σ Zi ( s < z i ) ( T ∈ H i ) dW Zi ( s ) = σ Zi ( T ∈ H i ) W Zi ( t ∧ z i ) (63)30 (cid:90) σ Zi ( s < z i ) ( t ∈ H i ) dW Zi ( s ) = σ Zi ( t ∈ H i ) W Zi ( z i ) (64)Solving the drift term t (cid:82) σ Pj ( u, T ) T (cid:82) u σ Pj ( u, s ) dsdu , we have: T (cid:90) u σ Pj ( u, s ) ds = T (cid:90) u n (cid:88) i =1 ξ i λ i,j ( u < x i ) ( s ≥ x i ) ds = n (cid:88) i =1 ξ i λ i,j ( u < x i ) T (cid:90) u ( s ≥ x i ) ds (65)where: T (cid:90) u ( s ≥ x i ) ds = , T < x i T − x i , u < x i , T ≥ x i T − u, u ≥ x i (66)therefore: T (cid:90) u σ Pj ( u, s ) ds = n (cid:88) i =1 ξ i λ i,j ( u < x i ) ( T ≥ x i )( T − x i ) (67)therefore: t (cid:90) σ Pj ( u, T ) T (cid:90) u σ Pj ( u, s ) dsdu = t (cid:90) n (cid:88) q =1 ξ q λ q,j ( u < x q ) ( T ≥ x q ) n (cid:88) i =1 ξ i λ i,j ( u < x i ) ( T ≥ x i )( T − x i ) du = n (cid:88) q =1 ξ q λ q,j ( T ≥ x q ) n (cid:88) i =1 ξ i λ i,j ( T ≥ x i )( T − x i ) t (cid:90) ( u < x q ) ( u < x i ) du = n (cid:88) q =1 n (cid:88) i =1 ξ q ξ i λ q,j λ i,j ( T ≥ x q ∨ i )( T − x i )[ t ∧ x q ∧ x i ] (68)Similarly: t (cid:90) σ Pj ( u, t ) t (cid:90) u σ Pj ( u, s ) dsdu = n (cid:88) q =1 n (cid:88) i =1 ξ q ξ i λ q,j λ i,j ( t ≥ x q ∨ i )( t − x i )[ x q ∧ x i ] (69)31olving t (cid:82) σ Zi ( u, T ) T (cid:82) u σ Zi ( u, s ) dsdu : T (cid:90) u σ Zi ( u, s ) ds = T (cid:90) u σ Zi ( u < z i ) ( s ∈ H i ) ds = σ Zi ( u < z i ) T (cid:90) u ( s ∈ H i ) ds = σ Zi ( u < z i ) ( T ≥ z i )[ h i ∧ ( T − z i )] (70)therefore t (cid:90) σ Zi ( u, T ) T (cid:90) u σ Zi ( u, s ) dsdu = t (cid:90) σ Zi ( u < z i ) H i ( T ) σ Zi ( u < z i ) ( T ≥ z i )[ h i ∧ ( T − z i )] du = (cid:0) σ Zi (cid:1) ( T ∈ H i )[ h i ∧ ( T − z i )] t (cid:90) ( u < z i ) du = (cid:0) σ Zi (cid:1) ( T ∈ H i )( T − z i )[ t ∧ z i ] (71)Similarly: t (cid:90) σ Zi ( u, t ) t (cid:90) u σ Zi ( u, s ) dsdu = (cid:0) σ Zi (cid:1) ( t ∈ H i )( t − z i ) z i (72) References
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