Swap Curve Research Articles

Examining swap butterfly risk premia in South Africa

ABSTRACT This paper studies a long butterfly strategy (which is immune to parallel curve shifts but exposed to non-parallel curve shifts) on the South African interest rate swap curve. Ten swap butterfly risk factors are identified using monthly South African swap data from 2001–2022. The top ranked butterfly strategies displayed strong and persistent outperformance over the bottom ranked strategies for each factor, resulting in improved risk-adjusted and absolute returns especially during positive and steep twists, and bull and bear flattening curve scenarios. Trade costs can be significant, so having a pragmatic rebalancing strategy with efficient market makers that limit trade costs to 1bps of spread make these swap butterfly risk factors an effective and successful portable alpha strategy.

Network Topology of Dynamic Credit Default Swap Curves of Energy Firms and the Role of Oil Shocks

Using network analysis on the connectedness of default factors in a credit default swap (CDS) dataset of U.S. and European energy firms, we provide the first evidence of differences in the shape and dynamics of the interconnectedness of the level, slope, and curvature, representing long-, short- and middle-term default factors, respectively. The interconnectedness of the three default factors increases during the European sovereign debt crisis (ESDC), whereas only the interconnectedness of the level factor increases during the oil price crash, and the interconnectedness of both level and slope factors spikes during COVID19. European firms contribute more to the transmission of long-term and short-term default risk from early 2011 till the beginning of the 2014–2105 oil price crash; afterwards, U.S. firms are major default transmitters despite some periods of parity with European firms. The impacts of oil demand and supply shocks on the various interconnectedness are quantile-dependent and more pronounced in the long term for the credit risk of the energy firms.

Volatility Level Dependence and Linear-Rational Term Structure Models

ABSTRACT We outline a subclass of linear-rational term structure models, based on CEV dynamics. A tractable and arbitrage-free term structure results from the linear-rational aspect of the model, independently of the term-structure volatility dynamics that follow from the CEV specification. This specification is devised to capture a flexible degree of volatility-level dependence, i.e., the degree to which yield-curve volatility depends on yield levels. We estimate the model based on a panel of South African swap rates, and extract the degree of volatility-level dependence inherent in the time series of rates, without interference from the shape of the swap curve. The CEV exponent parameters are found to be essential for matching the low degree of volatility-level dependence that tends to be observed in the high interest-rate environments of emerging markets.

Market co-movement between credit default swap curves and option volatility surfaces

We analyze the co-movement between the Credit Default Index (CDX) curve and the S&P 500 index's option volatility surface. We connect the reduced-form no-arbitrage model with the Nelson-Siegel (N-S) model on hazard rate implied from the CDX curve, and identify the levels, slopes, and curvatures from these two markets via the Unscented Kalman Filter (UKF). We find that the changes in the level, slope, and curvature in the CDX curve and those in the volatility surface are correlated due to the bridge of the S&P 500 index return. Finally, the co-movement between the CDX curve and S&P 500 index's volatility surface become stronger after the late 2000s global financial crisis.

Is convexity efficiently priced? Evidence from international swap markets

While it is widely claimed in the literature that convexity is correctly priced, we find evidence in four major swap markets that this is the case only on average and that extended periods occur when convexity-based trading strategies offer economically very significant exceptional returns. These abnormal returns can be reaped with fully no-peek-ahead strategies and after accounting for transaction costs. We find a strong link between the periods of highest profitability and conditions of reduced market liquidity. This suggests that, as observed in recent liquidity studies on US Treasuries, temporary deviations from market efficiency at the long end of the swap curve occur when pseudo-arbitrageurs do not have sufficient capital to correct the mispricings. • Convexity is not always correctly priced in the four most important swap markets. • Two strategies that exploit these inefficiencies are economically very significant. • Both strategies are most effective in conditions of poor market liquidity. • We explain the source of the economic differences between the two strategies.

A closed-form pricing formula for European options under a new stochastic volatility model with a stochastic long-term mean

Based upon the fact that a constant long-term mean could not provide a good description of the term structure of the implied volatility and variance swap curve, as suggested by Byelkina and Levin (in: Sixth world congress of the Bachelier Finance Society, Toronto, 2010) and Forde and Jacquier (Appl Math Finance 17(3):241–259, 2010), this paper presents a new stochastic volatility model, by assuming the long-term mean of the volatility in the Heston model be stochastic. An important feature of our model is that it still preserves the essential advantage of the Heston model, i.e., the analytic tractability, because a closed-form pricing formula for European options can be derived, which could not only facilitate the risk management process but also help save plenty of time in terms of model calibration. The effect of the newly introduced stochastic long-term mean is demonstrated through the numerical comparison with the Heston model. It is also shown that the current model can overall lead to more accurate option prices than the Heston model, through a carefully designed empirical study.

Implied monetary policy extracted from interest rate swaps in Chile

This article describes a methodology that uses interest rate swaps in Chilean Pesos to extract monetary policy paths, as well as their probability, with nodes on future Central Bank meetings. It is concluded that implied policy paths contain term premiums that correlate with the policy cycle. Accordingly, implied paths have consistently overestimated the future path of future interest rates, pricing hikes more often than cuts and underestimating the likelihood of rates becoming more expansionary. Moreover, the level of implied rates has been governed by the front end of the curve, instead of expectations of the economic cycle. The result is that extracting implied rates from the swap curve is an ineffective tool for forecasting, and that the swap curve has not been able to anticipate changes in policy. These findings provide insights on the asymmetrical risk profile of future monetary policy and may contribute to the design of hedging strategies

Are We Heading into a Recession? Yield Curve Inversion as a Recession Predictor

Yield Curves reflect the borrowing and lending rates over a range of maturities within a particular market and currency. Yield curves capture the term structure of interest rates and provide observers with a means of comparing short- and long-term interest rates. There are different types of yield curves, reflecting the different markets, institutions and instruments investors can choose to secure financing. Government Bond curves reflect the rate of return or yield required for governments to secure financing and likewise Swap Curves reflect the borrowing and lending rates available in Swap markets via interest rate swaps. Current market yield curves are on the verge of inverting. The returns from long-term Government Bonds are lower than the equivalent short term bonds and borrowing money long-term is cheaper than short-term borrowing. This is unusual, typically long-term financing should cost more than short-term financing, not less. In US markets inverted yield curves have been a reliable predictor of recessions. Each time the yield curve has inverted the US economy has entered a downturn within the subsequent 18 months. This has been the case with only one exception in the last 40 years. In this paper we review firstly what a yield curve is. Secondly we discuss the term structure of yields and interest rates and thirdly we outline the yield spread and explain why an inverted yield curve is a good recession predictor and indicator of heightened recessionary risk. Finally we conclude with an estimate of the likelihood of a US recession in the next 12 months based on current market information.

Dynamic credit default swap curves in a network topology

Systemically important banks are connected and their default probabilities have dynamic dependencies. An extraction of default factors from cross-sectional credit default swap (CDS) curves allows us to analyze the shape and the dynamics of default probabilities. In extending the Dynamic Nelson Siegel (DNS) model to an across firm multivariate setting, and employing the generalized variance decomposition of Diebold and Yilmaz [On the network topology of variance decompositions: Measuring the connectedness of financial firms. J. Econom., 2014, 182(1), 119–134], we are able to establish a DNS network topology. Its geometry yields a platform to analyze the interconnectedness of long-, middle- and short-term default factors in a dynamic fashion and to forecast the CDS curves. Our analysis concentrates on 10 financial institutions with CDS curves comprising of a wide range of time-to-maturities. The extracted level factor representing long-term default risk shows a higher level of total connectedness than those derived for short-term and middle-term default risk, respectively. US banks contributed more to the long-term default spillover before 2012, whereas European banks were major default transmitters during and after the European debt crisis, both in the long-term and short-term. The comparison of the network DNS model with alternatives proposed in the literature indicates that our approach yields superior forecast properties of CDS curves.

Yield curve fitting with artificial intelligence: a comparison of standard fitting methods with artificial intelligence algorithms

The yield curve is a fundamental input parameter of valuation theories in capital markets. Information about yields can be observed in a discrete form, either directly through traded yield instruments (eg, interest rate swaps) or indirectly through the prices of bonds (eg, government bonds). Capital markets usually create benchmark yield curves for specific and very liquid market instruments, or for issuers where many different quotes of individual yield information for specific maturities are observable. The standard methods to construct a continuous yield curve from discrete observable yield data quotes are the fit of a mathematical model function, interpolation or regression algorithms. This paper expands these standard methods to include artificial intelligence algorithms, which have the advantage of avoiding any assumptions with regard to the mathematical model functions of the yield curve, and which can conceptually adapt easily to any market changes. Nowadays, the most widely used risk-free yield curve in capital markets is the overnight index swap (OIS) curve, which is derived from observable OISs and is used in this paper as the benchmark curve to derive and compare different yield curve fits.

A SUBORDINATED CIR INTENSITY MODEL WITH APPLICATION TO WRONG-WAY RISK CVA

Credit valuation adjustment (CVA) pricing models need to be both flexible and tractable. The survival probability has to be known in closed form (for calibration purposes), the model should be able to fit any valid credit default swap (CDS) curve, should lead to large volatilities (in line with CDS options) and finally should be able to feature significant wrong-way risk (WWR) impact. The Cox–Ingersoll–Ross (CIR) model combined with independent positive jumps and deterministic shift (JCIR[Formula: see text]) is a very good candidate : the variance (and thus covariance with exposure, i.e. WWR) can be increased with the jumps, whereas the calibration constraint is achieved via the shift. In practice however, there is a strong limit on the model parameters that can be chosen, and thus on the resulting WWR impact. This is because only non-negative shifts are allowed for consistency reasons, whereas the upwards jumps of the JCIR[Formula: see text] need to be compensated by a downward shift. To limit this problem, we consider the two-side jump model recently introduced by Mendoza-Arriaga and Linetsky, built by time-changing CIR intensities. In a multivariate setup like CVA, time-changing the intensity partly kills the potential correlation with the exposure process and destroys WWR impact. Moreover, it can introduce a forward looking effect that can lead to arbitrage opportunities. In this paper, we use the time-changed CIR process in a way that the above issues are avoided. We show that the resulting process allows to introduce a large WWR effect compared to the JCIR[Formula: see text] model. The computation cost of the resulting Monte Carlo framework is reduced by using an adaptive control variate procedure.

Bond Yield Curve Convexity Trading

Bond Yield curve is an important indicator of the borrowing costs and lending returns, is also one of the most observed indicator by traders in fixed income trading desk among investment banks. The shape of the yield curve can be normal, flat or inverted. In most cases, bond yield curve is concavely shaped. There are theories to explain the economical meaning for a normal, flat or inverted yield curve. However, there is a lack of explanation for the concave shape of the yield curve. We in this article try to provide an explanation by constructing arbitrage portfolio under the assumption that the yield curve moves in parallel. We can show that under this assumption, zero coupon bond yield curve should be concavely shaped. We will also reach the same conclusion for swap curves. In this process, we also discover some interesting properties which were never discussed in the literature.

A Subordinated CIR Intensity Model with Application to Wrong-Way Risk CVA

Credit Valuation Adjustment (CVA) pricing models need to be both flexible and tractable. The survival probability has to be known in closed form (for calibration purposes), the model should be able to fit any valid Credit Default Swap (CDS) curve, should lead to large volatilities (in line with CDS options) and finally should be able to feature significant Wrong-Way Risk (WWR) impact. The Cox-Ingersoll-Ross model (CIR) combined with independent positive jumps and deterministic shift (JCIR++) is a very good candidate: the variance (and thus covariance with exposure, i.e. WWR) can be increased with the jumps, whereas the calibration constraint is achieved via the shift. In practice however, there is a strong limit on the model parameters that can be chosen, and thus on the resulting WWR impact. This is because only non-negative shifts are allowed for consistency reasons, whereas the upwards jumps of the JCIR++ need to be compensated by a downward shift. To limit this problem, we consider the two-side jump model recently introduced by Mendoza-Arriaga & Linetsky, built by time-changing CIR intensities. In a multivariate setup like CVA, time-changing the intensity partly kills the potential correlation with the exposure process and destroys WWR impact. Moreover, it can introduce a forward looking effect that can lead to arbitrage opportunities. In this paper, we use the time-changed CIR process in a way that the above issues are avoided. We show that the resulting process allows to introduce a large WWR effect compared to the JCIR++ model. The computation cost of the resulting Monte Carlo framework is reduced by using an adaptive control variate procedure.

Pricing the ECB's forward guidance with the EONIA swap curve

On July 4, 2013, following several other major central banks, the European Central Bank (ECB) gave for the first time forward guidance on interest rates, which affected market participants' expectations of future interest rates in the context of a Zero Lower Bound. Using an ARMAX(1,1) model in which the effect of the communication of negative macroeconomic news was disentangled from the commitment positive shock, the impact of the forward guidance on money market interest rates is estimated through the Euro Overnight Index Average swap, also called overnight index swap, at maturities between 2 months and 10 years using abnormal returns from an event study. The results and robustness checks suggest that the ECB's guidance lowered overnight index swap rates for maturities within 10 months to 3 years. These results imply the existence of a commitment effect from the ECB's communication. In the context of decreasing market liquidity because of the 3-year long-term refinancing operation repayment, market participants priced the low period of interest rates until mid-2016.

A model for interest rates with clustering effects

We propose a model for short-term rates driven by a self-exciting jump process to reproduce the clustering of shocks on the Euro overnight index average (EONIA). The key element of the model is the feedback effect between the absolute value of jumps and the intensity of their arrival process. In this setting, we obtain a closed-form solution for the characteristic function for interest rates and their integral. We introduce a class of equivalent measures under which the features of the process are preserved. We infer the prices of bonds and their dynamics under a risk-neutral measure. The question of derivatives pricing is developed under a forward measure, and a numerical algorithm is proposed to evaluate caplets and floorlets. The model is fitted to EONIA rates from 2004 to 2014 using a peaks-over-threshold procedure. From observation of swap curves over the same period, we filter the evolution of risk premiums for Brownian and jump components. Finally, we analyse the sensitivity of implied caplet volatility to parameters defining the level of self-excitation.

Pricing under rough volatility

From an analysis of the time series of realized variance using recent high-frequency data, Gatheral et al. [Volatility is rough, 2014] previously showed that the logarithm of realized variance behaves essentially as a fractional Brownian motion with Hurst exponent H of order 0.1, at any reasonable timescale. The resulting Rough Fractional Stochastic Volatility (RFSV) model is remarkably consistent with financial time series data. We now show how the RFSV model can be used to price claims on both the underlying and integrated variance. We analyse in detail a simple case of this model, the rBergomi model. In particular, we find that the rBergomi model fits the SPX volatility markedly better than conventional Markovian stochastic volatility models, and with fewer parameters. Finally, we show that actual SPX variance swap curves seem to be consistent with model forecasts, with particular dramatic examples from the weekend of the collapse of Lehman Brothers and the Flash Crash.

Quadratic variance swap models

We introduce a novel class of term structure models for variance swaps. The multivariate state process is characterized by a quadratic diffusion function. The variance swap curve is quadratic in the state variable and available in closed form, greatly facilitating empirical analysis. Various goodness-of-fit tests show that quadratic models fit variance swaps on the S&P 500 remarkably well, and outperform affine models. We solve a dynamic optimal portfolio problem in variance swaps, index option, stock index and bond. An empirical analysis uncovers robust features of the optimal investment strategy.

Can Structural Models Price Default Risk? Evidence from Bond and Credit Derivative Markets

Using a set of structural models, we evaluate the price of default protection for a sample of US corporations. In contrast to previous evidence from corporate bond data, credit default swap (CDS) premia are not systematically underestimated. In fact, one of our studied models has little difficulty on average in predicting their level. For robustness, we perform the same exercise for bond spreads by the same issuers on the same trading date. As expected, bond spreads relative to the treasury curve are systematically underestimated. This is not the case when the swap curve is used as a benchmark, suggesting that previously documented underestimation results may be sensitive to the choice of risk-free rate.

Pricing Under Rough Volatility

From an analysis of the time series of volatility using recent high frequency data, Gatheral, Jaisson and Rosenbaum previously showed that log-volatility behaves essentially as a fractional Brownian motion with Hurst exponent H of order 0.1, at any reasonable time scale. The resulting Rough Fractional Stochastic Volatility (RFSV) model is remarkably consistent with financial time series data. We now show how the RFSV model can be used to price claims on both the underlying and integrated volatility. We analyze in detail a simple case of this model, the rBergomi model. In particular, we find that the rBergomi model fits the SPX volatility markedly better than conventional Markovian stochastic volatility models, and with fewer parameters. Finally, we show that actual SPX variance swap curves seem to be consistent with model forecasts, with particular dramatic examples from the weekend of the collapse of Lehman Brothers and the Flash Crash.

Estimating the South African Overnight Indexed Swap Curve

Abstract The 2007/2008 financial crisis exposed the fragility in the global banking sector - accordingly, inter-bank defined interest rates, viz. LIBOR, EURIBOR, JIBAR, etc., that were once deemed to be default-free, or at least close proxies thereof, are now deemed to be credit risky. In response to this, global financial markets have adopted a credit and liquidity homogenous multi-curve interest rate framework. The risk-neutral valuation of financial instruments has also been fundamentally altered, following the acceptance of a new proxy for a default-free discounting curve - this being the Overnight Indexed Swap (OIS) curve. Major financial markets, like the U.S. and the Euro zone, have swiftly developed liquid OIS markets, and have therefore adopted OIS discounting. The lack of consensus on, and the inaccessibility of, a tradable overnight rate in South Africa has hindered the development of an OIS market. Nonetheless, there is still a need for a South African default-free discounting curve, as the prevalent South African defined inter-bank JIBAR rates also carry an element of credit risk. In this paper we provide a potential solution, using cointegration to estimate an OIS curve for the South African market from a JIBAR-linked swap curve.