Empirical Studies on Interest Rate Derivatives

On pricing of interest rate derivatives

ABSTRACTMost existing dynamic term structure models assume that interest rate derivatives are redundant securities and can be perfectly hedged using solely bonds. We find that the quadratic term structure models have serious difficulties in hedging caps and cap straddles, even though they capture bond yields well. Furthermore, at‐the‐money straddle hedging errors are highly correlated with cap‐implied volatilities and can explain a large fraction of hedging errors of all caps and straddles across moneyness and maturities. Our results strongly suggest the existence of systematic unspanned factors related to stochastic volatility in interest rate derivatives markets.

Hedging with Interest Rate and Credit Derivatives by Banks

In the past 25 years, tremendous progress has been made in modeling the dynamics of the term structure of interest rates, which play an instrumental role in determining prices and hedging portfolios of fixed-income derivative securities. This article reviews the theoretical development of the dynamic models of the default-free term structure and their applications in pricing interest rate options. Classic models, sometimes termed equilibrium models, and their multifactor extensions are outlined. These models provide clear economic intuitions connecting the term structure with economic fundamentals. They also lay a foundation for the framework of the arbitrage models that price interest rate derivatives on the basis of the market prices of bonds. This framework has been expanded and enriched by recent advances in directly modeling observable market rates through the market models and in incorporating an internally consistent correlation structure through the “infinite-dimensional” models.

Dynamic term structure models explain the yield curve variation well but perform poorly in pricing and hedging interest rate options. Most existing option pricing practices take the yield curve as given, thus having little to say about the fair valuation of the underlying interest rates. This paper proposes an m + n model structure that bridges the gap in the literature by successfully pricing both interest rates and interest rate options. The first m factors capture the yield curve variation, whereas the latter n factors capture the interest rate options movements that cannot be effectively identified from the yield curve. We propose a sequential estimation procedure that identifies the m yield curve factors from the LIBOR and swap rates in the first step and the n options factors from interest rate caps in the second step. The three yield curve factors explain over 99% of the variation in the yield curve but account for less than 50% of the implied volatility variation for the caps. Incorporating three additional options factors improves the explained variation in implied volatilities to over 99%.

Stochastic volatility of underlying assets has been shown to affect significantly the price of many financial derivatives. In particular, a fast mean-reverting factor of the stochastic volatility plays a major role in the pricing of options. This paper deals with the interest rate model dependence of the stochastic volatility impact on defaultable interest rate derivatives. We obtain an asymptotic formula of the price of defaultable bonds and bond options based on a quadratic term structure model and investigate the stochastic volatility and default risk effects and compare the results with those of the Vasicek model.

Usually, firms use interest rate derivatives to hedge interest rate risk. Similarly, some non-financial firms in Pakistan are also engaged in the use of interest rate derivatives to hedge their exposure to interest rate risk. Purpose: The purpose of this study is to identify determinants of interest rate derivatives used by non-financial firms in Pakistan and to identify any differences between non-financial firms using interest rate derivatives and non-interest rate derivatives. Methodology: Since interest rate derivatives are taken as dependent binary variables, binary digits 1 and 0 are assigned to firm and non-user interest rate derivatives. As a result, the Logit model is used to identify the determinants of interest rate derivative use. Besides, users and non-users of interest rate derivatives are two groups independent of each, and therefore the Mann Whitney U test is used to identify the difference between them. Results: In line with the literature, the non-parametric Mann Whitney U test resulted in significant differences between interest rate derivative users and non-users. While using the Logit model, there was a significant positive relationship between financial distress costs, asset growth, cash flow, firm size, and interest rate derivatives use. However, there was a negative relationship between profitability (ROA) and interest rate derivatives. Limitation: Since most financial firms use derivatives for trading rather than hedge purposes, the scope of the study is limited to non-financial firms that use derivatives for hedge purposes. Also, the study was launched in 2013-14, so the data collected and used for the period 2006-12 will be used. Implication: The findings suggest that large, financially distressed firms can hedge their interest rate risk and benefit from the optimal utilization of interest rate derivatives. Originality: The study was the first attempt to explore interest rate derivatives in non-financial firms in Pakistan and pave the way for further exploration of the subject matter. Besides, the original data were extracted from the annual reports of the sampled firms.

Interest Rate Derivatives: Features, Valuation and Benefits

Around the time the Bretton Woods exchange rate system was breaking down in global markets, interest rates in many local markets became much less predictable, leaving businesses and investors exposed to far more interest rate risk than they were familiar with. The financial community began looking for ways to hedge and speculate this risk. To address this need to hedge interest rate risk, the futures exchanges in Chicago and London (and other financial centers) began offering interest rate contracts in addition to their historically traded commodity futures. In 1981, IBM and the World Bank entered into the first interest rate swap, soon followed by a foreign exchange swap. As the Black-Scholes model and other options pricing models became generally accepted, options on interest rates became widely traded in deep, liquid markets. Decades later, the interest rate derivatives markets, both exchange traded and over the counter (OTC), have grown into a huge global marketplace (notional amounts in billions of U.S. dollars equivalent). (See Exhibit 10.1.) These so-called plain vanilla interest rate derivatives totaled almost U.S. $355 trillion at the end of 2006, and there are many other interest rate derivatives that the Bank for International Settlements (BIS) does not report statistics on. At first glance, the BIS total appears to be roughly 10 times global gross domestic product. However, some OTC derivatives are double counted (e.g., each side of the same risk may be on the books of two banks; in practice this is difficult to net). A significant percentage of the total (exchange traded plus OTC) is a hedge against other interest rate risk but does not get netted in the statistics just cited. Most important, the numbers shown reflect notional amounts (i.e., the amounts used to as a benchmark for determining the dollar value of interest rate payments). The market value of these instruments is much smaller. Whole books have been written on each type of interest rate derivative, but the next section provides a sampling of the common types of interest rate derivatives you are likely to see on a broker-dealer fixed income trading floor.

There are many examples of interest rate derivatives that are actively traded in over-the counter markets and in organized exchanges. Caps, floors, Treasury bond options, Treasury bond futures options, Eurodollar futures options, and swaptions are just some examples of this important class of derivatives in financial markets. They are classified as “interest rate derivatives” because their stochastic movements are directly related to the interest rate movements in a way that is analogous to the stock option price that moves in step with the underlying stock price.

Explicit Solutions of Quadratic FBSDEs Arising from Quadratic Term Structure Models

In this paper, we study the pricing of contracts in fixed income markets under volatility uncertainty in the sense of Knightian uncertainty or model uncertainty. The starting point is an arbitrage-free bond market under volatility uncertainty. The uncertainty about the volatility is modeled by a G-Brownian motion, which drives the forward rate dynamics. The absence of arbitrage is ensured by a drift condition. Such a setting leads to a sublinear pricing measure for additional contracts, which yields either a single price or a range of prices and provides a connection to hedging prices. Similar to the forward measure approach, we define the forward sublinear expectation to simplify the pricing of cashflows. Under the forward sublinear expectation, we obtain a robust version of the expectations hypothesis, and we show how to price options on forward prices. In addition, we develop pricing methods for contracts consisting of a stream of cashflows, since the nonlinearity of the pricing measure implies that we cannot price a stream of cashflows by pricing each cashflow separately. With these tools, we derive robust pricing formulas for all major interest rate derivatives. The pricing formulas provide a link to the pricing formulas of traditional models without volatility uncertainty and show that volatility uncertainty naturally leads to unspanned stochastic volatility.

Recent events have highlighted the potential importance of nonlinear efiects of flscal variables (notably debt and deflcits) on interest rates: While in times when government solvency is not a concern the standard crowding-out efiects are of moderate magnitude, in times when default risk becomes an issue the interest rate efiects can become very large. This paper provides new evidence on the magnitude of these efiects. For the case when default risk is not a concern, it uses an arbitrage-free term structure model to estimate the dynamic efiects of flscal policy shocks on interest rates along the entire maturity spectrum. For the case when default risk becomes a concern (thereby violating a central assumption of the term structure model), I present evidence based on EMU government bond spread regressions on time-varying efiects of national flscal policies on spreads as well as the time-varying sensitivity of yield spreads to international risk aversion as a function of the state of flscal policy. JEL classiflcation: E6, H6.

ABSTRACTWe study sources of potentially serious errors of popular numerical realizations of the Fourier method in affine models and explain that, in many cases, a calibration procedure based on such a realization will be able to find a “correct parameter set” only in a rather small region of the parameter space, with a blind spot: an interval of strikes depending on the model and time to maturity, where accurate calculations are extremely time-consuming. We explain how to construct more accurate and faster pricing and calibration procedures. An important ingredient of our method is the study of the analytic continuation of the solution of the associated system of generalized Riccati equations, and contour deformation techniques. As a byproduct, we show that the straightforward application of the Runge–Kutta method may lead to sizable errors, and suggest certain remedies. In the paper, the method is applied to a wide class of stochastic volatility models with stochastic interest rate and interest rate models of An(n) class. The methodology of the paper can be applied to other models (e.g., quadratic term structure models, Wishart dynamics, 3/2-model).

In this paper, quadratic term structure models (QTSMs) are analyzed and characterized in a general Markovian setting. The primary motivation for this work is to find a useful extension of the traditional QTSM, which is based on an Ornstein–Uhlenbeck (OU) state process, while maintaining the analytical tractability of the model. To accomplish this, the class of quadratic processes, consisting of those Markov state processes that yield QTSM, is introduced. The main result states that OU processes are the only conservative quadratic processes. In general, however, a quadratic potential can be added to allow QTSMs to model default risk. It is further shown that the exponent functions that are inherent in the definition of the quadratic property can be determined by a system of Riccati equations with a unique admissible parameter set. The implications of these results for modeling the term structure of risk‐free and defaultable rates are discussed.