Interest Rate Contingent Claims Research Articles

Factor Investing in Currency Markets: Does it Make Sense?

The concept of factor investing emerged at the end of the 2000s and has completely changed the landscape of equity investing. Today, institutional investors structure their strategic asset allocation around five risk factors: size, value, low beta, momentum and quality. This approach has been extended to multi-asset portfolios and is known as the alternative risk premia model. This framework recognizes that the construction of diversified portfolios cannot only be reduced to the allocation policy between asset classes, such as stocks and bonds. Indeed, diversification is multifaceted and must also consider alternative risk factors. More recently, factor investing has gained popularity in the fixed income universe, even though the use of risk factors is an old topic for modeling the yield curve and pricing interest rate contingent claims. Factor investing is now implemented for managing portfolios of corporate bonds or emerging bonds. In this paper, we focus on currency markets. The dynamics of foreign exchange rates are generally explained by several theoretical economic models that are commonly presented as competing approaches. In our opinion, they are more complementary and they can be the backbone of a Fama-French-Carhart risk factor model for currencies. In particular, we show that these risk factors may explain a significant part of time-series and cross-section returns in foreign exchange markets. Therefore, this result helps us to better understand the management of forex portfolios. To illustrate this point, we provide some applications concerning basket hedging, overlay management and the construction of alpha strategies.

(The Estimation of Chinese Term Structure of Interest Rate and Treasury Bonds Pricing)

Chinese Abstract: 基于上交所国债数据，沿袭Bliss（1997）的研究方法，并加以改进和扩展，本文考察了五种静态利率期限结构模型在样本内拟合和样本外预测上的表现，与Bliss（1997）的结果不同，我们发现三次平滑样条方法更适合用来估计中国上交所国债的利率期限结构。在此基础上，通过对各模型的拟合误差和预测误差的分析，我们发现，利率期限结构并不是决定国债定价的唯一系统性因素，上交所国债的定价还需考虑国债的其他特征因素，投资者在交易国债时会将息票金额的大小和债券价格的溢价程度考虑到定价中来,这一点与Bliss（1997）的结论相同。作为本文结论的一个应用以及对Bliss（1997）方法的扩展，我们发现息票金额低的投资组合和溢价因子高的投资组合分别高于息票金额高的投资组合和溢价因子低的投资组合，本文的结论对国债投资有一定的指导意义。English Abstract: The term structure of interest rate is a concept central to economic and financial theory and the pricing of interest rate contingent claims. An estimate of the term structure of interest rate is the necessary starting point for applying them to pricing and hedging interest rate derivatives. Since September 06, 2013 when treasury bond futures was launched by China Financial Futures Exchange, it had become more important to learn about the relation between the estimate of the term structure and the treasury bond pricing which is crucial to the pricing of treasury bond futures. To study their relationship, using Treasury bonds data from Shanghai Stock Exchange and methods proposed by Bliss (1997), this paper first compares five distinct models for estimating the term structure in China. Combining in-sample performance and out-of-sample performance, we find that the smoothed cubic spline method does best overall, which is different from Bliss (1997) who studied the USA treasury data.However, is the term structure of interest rate the only factor that determines the pricing of treasury bonds? To exploit this question, we examine the behavior of fitted-price errors of individual bonds from period to the next and across different models. Specifically, we have done three following studies:In the first place, we test the transition matrix for fitted-price errors classified as positive, zero, or negative. If the errors are indeed noise, the transition probabilities should reflect the frequency of positive, zero, and negative errors in the overall population. In this test, we find that the fitted-price errors for specific bonds are correlated through time.In the second place, we test the coincidence matrices of positive, zero, and negative errors across estimation models, that is, we examine the probability that a positive (zero, or negative) fitted-price error produced by one model will also have a positive (zero, or negative) fitted-price error under the other model. In this test, we find that the fitted-price errors are related across models.In the third place, we regress the fitted-price errors against variables that may be relevant to the bond pricing process. In this test, we identify coupon rates and Price Premium might be relevant to the pricing of Treasury bonds.All the three tests have the same results with Bliss (1997), meaning that the term structure of interest rate is not the only pricing element in the price of treasury bond market in China.Finally, as an application of our finding and an extension to the Bliss (1997), we exploit its implication for bond portfolio investment. We find that small-coupon portfolio and high-price-premium portfolio have better return than large-coupon portfolio and low-price-premium portfolio respectively.

Closed-Form Expansion, Conditional Expectation, and Option Valuation

Enlightened by the theory of Watanabe [Watanabe S (1987) Analysis of Wiener functionals (Malliavin calculus) and its applications to heat kernels. Ann. Probab. 15:1–39] for analyzing generalized random variables and its further development in Yoshida [Yoshida N (1992a) Asymptotic expansions for statistics related to small diffusions. J. Japan Statist. Soc. 22:139–159], Takahashi [Takahashi A (1995) Essays on the valuation problems of contingent claims. Ph.D. thesis, Haas School of Business, University of California, Berkeley, Takahashi A (1999) An asymptotic expansion approach to pricing contingent claims. Asia-Pacific Financial Markets 6:115–151] as well as Kunitomo and Takahashi [Kunitomo N, Takahashi A (2001) The asymptotic expansion approach to the valuation of interest rate contingent claims. Math. Finance 11(1):117–151, Kunitomo N, Takahashi A (2003) On validity of the asymptotic expansion approach in contingent claim analysis. Ann. Appl. Probab. 13(3):914–952] etc., we focus on a wide range of multivariate diffusion models and propose a general probabilistic method of small-time asymptotic expansions for approximating option price in simple closed-form up to an arbitrary order. To explicitly construct correction terms, we introduce an efficient algorithm and novel closed-form formulas for calculating conditional expectation of multiplication of iterated stochastic integrals, which are potentially useful in a wider range of topics in applied probability and stochastic modeling for operations research. The performance of our method is illustrated through various models nested in constant elasticity of variance type processes. With an application in pricing options on VIX under GARCH diffusion and its multifactor generalization to the Gatheral double lognormal stochastic volatility models, we demonstrate the versatility of our method in dealing with analytically intractable non-Lévy and non-affine models. The robustness of the method is theoretically supported by justifying uniform convergence of the expansion over the whole set of parameters.

Valuation of Two-Factor Interest Rate Contingent Claims Using Green's Theorem

Over the years a number of two-factor interest rate models have been proposed that have formed the basis for the valuation of interest rate contingent claims. This valuation equation often takes the form of a partial differential equation that is solved using the finite difference approach. In the case of two-factor models this has resulted in solving two second-order partial derivatives leading to boundary errors, as well as numerous first-order derivatives. In this article we demonstrate that using Green's theorem, second-order derivatives can be reduced to first-order derivatives that can be easily discretized; consequently, two-factor partial differential equations are easier to discretize than one-factor partial differential equations. We illustrate our approach by applying it to value contingent claims based on the two-factor CIR model. We provide numerical examples that illustrate that our approach shows excellent agreement with analytical prices and the popular Crank–Nicolson method.

A binomial approximation for two-state Markovian HJM models

This article develops a lattice algorithm for pricing interest rate derivatives under the Heath et al. (Econometrica 60:77–105, 1992) paradigm when the volatility structure of forward rates obeys the Ritchken and Sankarasubramanian (Math Financ 5:55–72) condition. In such a framework, the entire term structure of the interest rate may be represented using a two-dimensional Markov process, where one state variable is the spot rate and the other is an accrued variance statistic. Unlike in the usual approach based on the Nelson-Ramaswamy (Rev Financ Stud 3:393–430) transformation, we directly discretize the heteroskedastic spot rate process by a recombining binomial tree. Further, we reduce the computational cost of the pricing problem by associating with each node of the lattice a fixed number of accrued variance values computed on a subset of paths reaching that node. A backward induction scheme coupled with linear interpolation is used to evaluate interest rate contingent claims.

Maximizing the Probability of a Perfect Hedge in the Case of Stochastic Interest Rate

SYNOPTIC ABSTRACTIn the financial market including n risky assets with prices following the full-observed geometric-Brownian Motion model and a bank account with stochastic interest rate, we study the problem of maximizing the probability of an investor's wealth at terminal time T meeting or exceeding the value of a contingent claim C which expires at T. In this incomplete market, using an adapted duality methodology familiar in the utility maximization literatures, we characterize the optimal probability and the corresponding portfolio strategies, in the most general framework of interest rate models and contingent claims. Then we calculate in detail the case of Vasicek model for interest rate and some specific contingent claim, via the classical theorem of Time-Changing for Martingales and a generalized Cameron-Martin formula developed in the literatures.

Implied Bond and Derivative Prices Based on Non-Linear Stochastic Interest Rate Models

In this paper we expand the Box Method of Sorwar et al. (2007) to value both default free bonds and interest rate contingent claims based on one factor non-linear interest rate models. Further we propose a one-factor non-linear interest rate model that incorporates features suggested by recent research. An example shows the extended Box Method works well in practice.

The Pricing of Interest Rate Contingent Claims

The most important result in this working paper is the construction of a multidimensional Gaussian interest-rate term structure model, where, based on a construction of equivalent martingale measures and a suitable selection of numerators, it is shown that it is possible to derive analytical expressions for a wide range of derived instruments.The instruments for which analytical expressions are derived in this connection are forward contracts, futures contracts, options on zero-coupon bonds, options on interest rates (including options on the slope and curvature of the term structure of interest rates), caps and floors, options on both forward contracts and futures contracts, options on CIBOR futures, including options on FRAs, the pricing of floating-rate bonds, swaps, swaptions and, finally, options on coupon bonds.As regards the price expression for options on coupon bonds, a generalization is made of the Karoui, Myneni and Viswanathan (1993) model. The analytical expression derived here is namely a real analytical expression as opposed to Karoui, Myneni and Viswanathan which must be considered to be a semi-analytical expression.Using numerical tests we even managed to show that this new “true” closed form formula for the pricing of options on coupon bonds even seems to be valid in a general multi-factor Markovian HJM framework.

Fast Calibration of Interest Rate Claims in the Quadratic Gaussian Model : 2 the Swaptions

In the second part of a series of articles on the pricing of interest rate contingent claims in the multifactor Quadratic Gaussian model, we concentrate on the pricing of swaptions. Assuming the zero-coupon bond volatility to be a deterministic function of some Markov processes, we derive the true volatility of the coupon-bond as a weighted sum of some zero-coupon bond volatility with different maturities. Bounding the stochastic weights such that the misspecified volatility dominates the true one, we obtain bounds and hedges to the true price which are solved with approximate solutions of the Black type to the prices of call option and binary option when volatility, rates and dividends are function of the Markov processes.

Empirical performance of multifactor term structure models for pricing and hedging Eurodollar futures options

This article compares two one-factor, two two-factor, two three-factor models in the HJM class and Black's [Black, F. (1976). The pricing of commodity contracts. Journal of Financial Economics, 3, 167–179.] implied volatility function in terms of their pricing and hedging performance for Eurodollar futures options across strikes and maturities from 1 Jan 2000 to 31 Dec 2002. We find that three-factor models perform the best for 1-day and 1-week prediction, as well as for 5-day and 20-day hedging. The moneyness bias and the maturity bias appear for all models, but the three-factor models produce lower bias. Three-factor models also outperform other models in hedging, in particular for away-from-the-money and long-dated options. Making Black's volatility a square root or exponential function performs similar to one-factor HJM models in pricing, but not in hedging. Correctly specified and calibrated multifactor models are thus important and cannot be replaced by one-factor models in pricing or hedging interest rate contingent claims.

Managing Interest Rate Volatility Risk

Interest rate contingent claims such as swaptions, caps and floors, callable bonds, mortgage-backed securities and many balance sheet items are subject to vega risk. Vega risk is defined as the interest rate volatility risk of an interest rate contingent claim, the uncertain change in volatility affecting the embedded option value. Since these contingent claims9 values are affected by the interest rate volatilities and these volatilities are stochastic, any hedging strategies, or more generally risk management, of these interest rate contingent claims should take the vega risk into account. However, based on market conventions, the interest rate volatilities are represented by a set of approximately 120 implied volatilities of the at-the-money swaptions over a range of tenors and expiration dates, called the volatility surface. And this volatility surface is stochastic. Controlling for these vega risks poses a significant practical challenge to managing these large number of risk sources. Thus far, there is a lack of solutions. This article introduces the key rate vega measures to manage the vega risks. Furthermore, the article shows how the key rate vega measures can depict the option risks embedded in an interest rate contingent claim and how they can be used for hedging a mortgage interest-only strip using key rate vega measures and key rate duration measures together, as an illustration of broader applications of the key rate vega measures.

A Note on Monetary Policy with Interest-Rate Contingent Claims as Indicators

In this paper, we consider the use of interest rate contingent claims as indicators for the monetary policy. We analyze two approaches: one based on the term structure of zero bonds and another based on interest-rate option derivatives. We show how traditional tools based on the Black framework could be biased to build indicators for monetary policy. In fact, the second method could not be viewed as an alternative approach, but as a complementary approach of the term structure approach. The conduct of monetary policy by central banks could be achieved by using some targets and instruments. By doing that, they need to build indicators. In most case, these indicators are based on interest rates expectations. Two main methods are available to compute these expactations. The Þrst one is to consider the term structure of zero bonds (or interest rates). The second one is more complex and is based on derivatives. Contingent claims could be useful to extract expectations because these assets look forward. In this short note, we present briesy these two approaches and we illustrate the fact that these two approaches are not competitive but complementary. Moreover, the particularity of interest rate contingent claims imply that we have to imagine new speciÞc tools for the second approach. 2 Monetary policy with term structure of zero bonds as indicators The term structure of zero bonds can be used to determine the yield curve and the term structure of forward interest rates. These forward rates can be used as indicators, which contain some informations on interest rates market expectations (Svensson [1994]). So, they can be used to forecast interest rates and we can forecast foreign exchange rates (under the assumption of uncovered interest rates parity) or insation (under the assumption of Fisher relationship). They have even been used to estimate the credibility of EMU.

Construction of Zero-Coupon Yield Curve From Coupon Bond Yield Using Australian Data

This paper briefly surveys the various approaches to modelling the zero coupon yield curve is the starting point for much finance research. The method adopted here for the Australian Treasury bond data is based upon polynomial spline fitting, but with the constraint that the long end of the term structure is stable. This approach has also been successfully applied to the Danish bond market (Tanggaard and Jakobsen (1988)). The forward rate curve then becomes the important input data for the modelling of the term structure of interest rates and pricing of interest rate contingent claims using the Heath-Jarrow-Morton (1992) model.

Pricing and hedging interest rate options: Evidence from cap–floor markets

We examine the pricing and hedging performance of interest rate option pricing models using daily data on US dollar cap and floor prices across both strike rates and maturities. Our results show that fitting the skew of the underlying interest rate probability distribution provides accurate pricing results within a one-factor framework. However, for hedging performance , introducing a second stochastic factor is more important than fitting the skew of the underlying distribution. This constitutes evidence against claims in the literature that correctly specified and calibrated one-factor models could replace multi-factor models for consistent pricing and hedging of interest rate contingent claims.

A characterization of hedging portfolios for interest rate contingent claims

We consider the problem of hedging a European interest rate contingent claim with a portfolio of zero-coupon bonds and show that an HJM type Markovian model driven by an infinite number of sources of randomness does not have some of the shortcomings found in the classical finite-factor models. Indeed, under natural conditions on the model, we find that there exists a unique hedging strategy, and that this strategy has the desirable property that at all times it consists of bonds with maturities that are less than or equal to the longest maturity of the bonds underlying the claim.

A Closed-Form Multifactor Binomial Interest Rate Model

A number of binomial interest rate models have found broad application in valuing interest rate-contingent claims. While researchers are seeking to extend the one-factor model to multifactor models, so far all multifactor models are non-recombining interest rate models, which are not as accurate in valuing securities calibrated to market prices. The multifactor closed-form binomial interest rate model proposed here is simple to implement and can capture a broad range of interest rate movements that are arbitrage-free. Empirical evidence supports the robustness of the model, which can be calibrated to 70 at-the-money swaptions with under 1.3% average error.

Pricing and hedging interest rate options: Evidence from cap–floor markets

We examine the pricing and hedging performance of interest rate option pricing models using daily data on US dollar cap and floor prices across both strike rates and maturities. Our results show that fitting the skew of the underlying interest rate probability distribution provides accurate pricing results within a one-factor framework. However, for hedging performance, introducing a second stochastic factor is more important than fitting the skew of the underlying distribution. This constitutes evidence against claims in the literature that correctly specified and calibrated one-factor models could replace multi-factor models for consistent pricing and hedging of interest rate contingent claims.

Valuation of Common Two-Factor Interest Rate Contingent Claims

Numerous two-factor interest rate models have been proposed that have attempted to overcome the limitations of one-factor interest rate models. These two-factor interest rate models lead to complex valuation equations for interest rate contingent claims. These valuation equations can be expressed in the form of partial differential equations that can only be solved numerically. Sorwar and Barone-Adesi (2002) have recently extended the Box Method to solve second order partial differential equations. They find that the Box Method outperforms the existing finite difference scheme. In this paper we use their approach to value interest rate contingent claims based on a wide range of two-factor interest rate models. Our approach is to treat the first factor as the short term interest rate and the second factor as the long term interest rate. In this paper we use the Markov Chain Monte Carlo approach to estimate parameters based on two-factor, Vasicek, CIR and CKLS models. We find wide variation in parameter values across different models. We also find that these parameters have a major impact on contingent claim values. In particular we find a wide variation in contingent claim price across different models.

Pricing and Hedging Interest Rate Options: Evidence from Cap-Floor Markets

This paper examines the pricing and hedging performance of interest rate option pricing models in the U.S. dollar interest rate cap and floor markets. We evaluate alternative one-factor and two-factor term structure models on daily data from March-December 1998, consisting of actual cap and floor prices across both strike rates and maturities. Results show that fitting the skew of the underlying interest rate distribution provides accurate pricing results within a one-factor framework. However, for hedging performance, introducing a second stochastic factor is more important than fitting the skew of the underlying distribution. Modeling the second factor allows a better representation of the dynamic evolution of the term structure by incorporating expected twists in the yield curve. Thus,the interest rate dynamics embedded in two-factor models appears to be closer to the one driving the actual economic environment, leading to more accurate hedges. This constitutes evidence against claims in the literature that correctly specified and calibrated one-factor models could replace multi-factor models for consistent pricing and hedging of interest rate contingent claims.

Algorithms behind Term Structure Models of Interest Rates: I. Valuation and Hedging of Interest Rates Derivatives with the Ho-Lee Model

In this article we implement the well known Ho-Lee Model of the term structure of interest rates and describe the algorithm behind this model. After a brief discussion of interest rates and bonds we construct a binomial tree and show how to replicate any fixed income type security. This allows us to value any interest rate contingent claim by means of the replicating portfolio. We also discuss the problem of negative interest rates arising in this model and show how to calibrate the model to an observed set of bond prices.