Generic market models

Currently, there are two market models for valuation and risk management of interest rate derivatives: the LIBOR and swap market models. We introduce arbitrage-free constant maturity swap (CMS) market models and generic market models featuring forward rates that span periods other than the classical LIBOR and swap periods. We develop generic expressions for the drift terms occurring in the stochastic differential equation driving the forward rates under a single pricing measure. The generic market model is particularly apt for pricing of, e.g., Bermudan CMS swaptions and fixed-maturity Bermudan swaptions.     

The Impact of Collateralization on Swap Rates

Interest rate swap pricing theory traditionally views swaps as a portfolio of forward contracts with net swap payments discounted at LIBOR rates. In practice, the use of marking‐to‐market and collateralization questions this view as they introduce intermediate cash flows and alter credit characteristics. We provide a swap valuation theory under marking‐to‐market and costly collateral and examine the theory's empirical implications. We find evidence consistent with costly collateral using two different approaches; the first uses single‐factor models and Eurodollar futures prices, and the second uses a formal term structure model and Treasury/swap data.     

LIBOR and swap market models and measures

A self-contained theory is presented for pricing and hedging LIBOR and swap derivatives by arbitrage. Appropriate payoff homogeneity and measurability conditions are identified which guarantee that a given payoff can be attained by a self-financing trading strategy. LIBOR and swap derivatives satisfy this condition, implying they can be priced and hedged with a finite number of zero-coupon bonds, even when there is no instantaneous saving bond. Notion of locally arbitrage-free price system is introduced and equivalent criteria established. Stochastic differential equations are derived for term structures of forward libor and swap rates, and shown to have a unique positive solution when the percentage volatility function is bounded, implying existence of an arbitrage-free model with such volatility specification. The construction is explicit for the lognormal LIBOR and swap “market models”, the former following Musiela and Rutkowski (1995). Primary examples of LIBOR and swap derivatives are discussed and appropriate practical models suggested for each.     

On the Term Structure of Default Premia in the Swap and LIBOR Markets

Existing theories of the term structure of swap rates provide an analysis of the Treasury–swap spread based on either a liquidity convenience yield in the Treasury market, or default risk in the swap market. Although these models do not focus on the relation between corporate yields and swap rates (the LIBOR–swap spread), they imply that the term structure of corporate yields and swap rates should be identical. As documented previously (e.g., in Sun, Sundaresan, and Wang (1993)) this is counterfactual. Here, we propose a model of the default risk imbedded in the swap term structure that is able to explain the LIBOR–swap spread. Whereas corporate bonds carry default risk, we argue that swap contracts are free of default risk. Because swaps are indexed on "refreshed"-credit-quality LIBOR rates, the spread between corporate yields and swap rates should capture the market's expectations of the probability of deterioration in credit quality of a corporate bond issuer. We model this feature and use our model to estimate the likelihood of future deterioration in credit quality from the LIBOR–swap spread. The analysis is important because it shows that the term structure of swap rates does not reflect the borrowing cost of a standard LIBOR credit quality issuer. It also has implications for modeling the dynamics of the swap term structure.     

Decomposing swap spreads

We analyze a six-factor model for Treasury bonds, corporate bonds, and swap rates and decompose swap spreads into three components: a convenience yield from holding Treasuries, a credit risk element from the underlying LIBOR rate, and a factor specific to the swap market. The convenience yield is by far the largest component of spreads. There is a discernible contribution from credit risk as well as from a swap-specific factor with higher variability which in certain periods is related to hedging activity in the mortgage-backed security market. The model also sheds light on the relation between AA hazard rates and the spread between LIBOR rates and General Collateral repo rates and on the level of the riskless rate compared to swap and Treasury rates.     

Bounding Bermudan swaptions in a swap-rate market model

Abstract

We develop a new method for finding upper bounds for Bermudan swaptions in a swap-rate market model. By comparing with lower bounds found by exercise boundary parametrization, we find that the bounds are well within bid-offer spread. As an application, we study the dependence of Bermudan swaption prices on the number of instantaneous factors used in the model. We also establish an equivalence with LIBOR market models and show that virtually identical lower bounds for Bermudan swaptions are obtained.     

A displaced-diffusion stochastic volatility LIBOR market model: motivation,definition and implementation

Abstract

We present an extension of the LIBOR market model which allows for stochastic instantaneous volatilities of the forward rates in a displaced-diffusion setting. We show that virtually all the powerful and important approximations that apply in the deterministic setting can be successfully and naturally extended to the stochastic volatility case. In particular we show that (i) the caplet market can still be efficiently and accurately fit; (ii) that the drift approximations that allow the evolution of the forward rates over time steps as long as several years are still valid; (iii) that in the new setting the European swaption matrix implied by a given choice of volatility parameters can be efficiently approximated with a closed-form expression without having to carry out a Monte Carlo simulation for the forward rate process; and (iv) that it is still possible to calibrate the model virtually perfectly via simply matrix manipulations so that the prices of the co-terminal swaptions underlying a given Bermudan swaption will be exactly recovered, while retaining a desirable behaviour for the evolution of the term structure of volatilities.     

On the distributional distance between the lognormal LIBOR and swap market models

We consider the distributional difference in forward swap rates from the LIBOR market model (LFM) and the swap market model (LSM), the two fundamental market models for interest-rate derivatives. We explain how the Kullback–Leibler information (KLI) can be used to measure the distance of a given distribution from the lognormal (exponential) family of densities and then apply this to our models' comparison. The volatility of the projection of the LFM swap-rate distribution onto the lognormal family is compared to an industry synthetic swap volatility approximation in the LFM. Finally, we analyse how the above distance changes, in some cases, according to the parameter values and to the parameterizations themselves. We find a small distance in all cases.     

The cost of bank loans in relation to bonds swapped into a floating rate

Banks never lend at less than the interbank floating rate, LIBOR. We argue that this must be because it is insufficiently profitable for those that could lend at less than LIBOR to do so and discuss circumstances in which this would be the case. Using data from 1988–1991, we show that LIBOR varies in relation to the cost of corporate bonds swapped into a floating rate, and suggest that the relative cost of LIBOR may affect bank and bond market pricing policies. the data also indicates that changes in the compensation for credit risk demanded by the bank and bond markets are not synchronous, and that swap rates have an appreciable impact on the cost of bonds swapped into floating.     

Valuation of certain CMS spreads

In this paper, we derive an approximate lognormal process for the swap rate under the multifactor LIBOR market model using a Levy approach. Using the approximate dynamics for the swap rate, the constant maturity swap spread digital range notes with different strike rates are valued in analytic and semi-analytic form. The CMS spread digital range notes are widely traded in the marketplace, or embedded in structure notes.     

Information Transmission in the World Money Markets

Using 1,966 daily observations since the introduction of the euro, we apply cointegration and error correction tests to examine information transmission in the major world money markets as represented by the domestic CD markets and the Eurocurrency market for the US dollar, euro, Japanese yen, and British pound sterling. Our inter‐market tests show a high degree of integration and interdependency among inter‐market interest rates. Our intra‐market results show that $ LIBOR and LIBOR rates drive LIBOR and £ LIBOR. Application of Johansen's (1988) multivariate test procedure and Gonzalo and Granger's (1995) long‐memory components technique confirms and reinforces our intra‐market findings that the system of four LIBOR rates is fully integrated (i.e., three cointegrating vectors), with the single common trend driven by $ LIBOR and LIBOR. These results are consistent with the strength of the dollar and yen relative to the pound sterling and the euro during the developing world financial crisis in late 2008.     

An almost Markovian LIBOR market model calibrated to caps and swaptions

A new variant of the LIBOR market model is implemented and calibrated simultaneously to both at-the-money and out-of-the-money caps and swaptions. This model is a two-factor version of a new class of the almost Markovian LIBOR market models with properties long sought after: (i) the almost Markovian parameterization of the LIBOR market model volatility functions is unique and asymptotically exact in the limit of a short time horizon up to a few years, (ii) only minimum plausible assumptions are required to derive the implemented volatility parameterization, (iii) the calibration yields very good results, (iv) the calibration is almost immediate, (v) the implemented LIBOR market model has a related short-rate model. Numerical results for the two-factor case show that the volatility functions for the LIBOR market model can be imported into its short-rate model cousin without adjustment.     

The Lévy LIBOR model

Models driven by Lévy processes are attractive because of their greater flexibility compared to classical diffusion models. First we derive the dynamics of the LIBOR rate process in a semimartingale as well as a Lévy Heath-Jarrow-Morton setting. Then we introduce a Lévy LIBOR market model. In order to guarantee positive rates, the LIBOR rate process is constructed as an ordinary exponential. Via backward induction we get that the rates are martingales under the corresponding forward measures. An explicit formula to price caps and floors which uses bilateral Laplace transforms is derived.     

A comparison of the information in the LIBOR and CMT term structures of interest rates

We investigate the information contained in the London Interbank Offered Rate (LIBOR) and the U.S. Constant Maturity Treasury (CMT) term structure of interest rates and report three novel findings. First, we document that the information contained in term structures are significantly different from one another. Second, we provide evidence of a significant change in the nature of this difference as the financial crisis began. Third, we find that the significant changes in the information content of CMT and LIBOR are consistent with significant shocks to credit default swap rates and tenor swap rates.     

Macroeconomic releases and the interest rate term structure

We extract two systematic economic factors from a wide array of noisy and sparsely observed macroeconomic releases, and link the dynamics and market prices of the two factors to the interest rate term structure. The two factors predict 77.9–82.1% of the daily variation in LIBOR and swap rates from one month to 10 years. Shocks on inflation-related releases have large, positive impacts on interest rates of all maturities, leading to parallel shifts of the yield curve, but shocks on output-related releases have larger impacts on the short rate than on the long rate, thus generating a slope effect.     

Volatility as an Asset Class: European Evidence

Abstract

Volatility movements are known to be negatively correlated with stock index returns. Hence, investing in volatility appears to be attractive for investors seeking risk diversification. The most common instruments for investing in pure volatility are variance swaps, which now enjoy an active over-the-counter (OTC) market. This paper investigates the risk-return tradeoff of variance swaps on the Deutscher Aktienindex and Euro STOXX 50 index over the time period from 1995 to 2004. We synthetically derive variance swap rates from the smile in option prices. Using quotes from two large investment banks over two months, we validate that the synthetic values are close to OTC market prices. We find that variance swap returns exhibit an option-like profile compared to returns of the underlying index. Given this pattern, it is crucial to account for the non-normality of returns in measuring the performance of variance swap investments. As in the US, the average returns of selling variance swaps are found to be strongly positive and too large to be compatible with standard equilibrium models. The magnitude of the estimated risk premium is related to variance uncertainty and past index returns. This indicates that the variance swap rate does not seem to incorporate all past information relevant for forecasting future realized variance.     

Relationship between gold and stock markets during the global financial crisis: Evidence from nonlinear causality tests

This paper investigates the nonlinear dynamic co-movements between gold returns, stock market returns and stock market volatility during the recent global financial crisis for the UK (FTSE 100), the US (S&P 500) and Japan (Nikkei 225). Initially, the bivariate dynamic relationships between i) gold returns and stock market returns and ii) gold returns and stock market volatility are tested; both of these relationships are further investigated in the multivariate nonlinear settings by including changes in the three-month LIBOR rates. In this paper correlation integrals based on the bivariate model show significant evidence of nonlinear feedback effect among the variables during the financial crisis period for all the countries understudy. Very limited evidence of significant feedback is found during the pre-crisis period. Results from the multivariate tests including changes in the LIBOR rates provide results similar to the bivariate results. These results imply that gold may not perform well as a safe haven during the financial crisis period due to the bidirectional interdependence between gold returns and, stock returns as well as stock market volatility. However, gold may be used as a hedge against stock market returns and volatility in stable financial conditions.