Economics: Current and Future Developments

This chapter introduces basic concepts of managing risk from interest rates. We summarize the fundamental concepts of interest rate theory, such as discount factors, bond pricing, forward rate, swap rate, and term structures of interest rates. Additionally, historical data on interest rates of markets are shown, and these data suggest to us that market uncertainty is an important factor in financial risk.

Next, we introduce several risk measures used to evaluate interest rate risk: sensitivity, convexity, and value at risk (VaR). And, we briefly explain three methods to measure the VaR: covariance VaR, historical VaR, and Monte Carlo VaR. Additionally, nested simulation is explained for risk assessment in a derivative portfolio. After considering the validity of VaR as a risk measure, we address coherent risk measures.

This chapter briefly summarizes basic concepts of stochastic calculus, using intuitive examples. First, the fundamentals of probability spaces are introduced by working with a simple example of a stochastic process. Next, stochastic processes are introduced in connection with a natural filtration and a martingale. Then, we introduce a stochastic integral and Ito`s formula, which is an important tool for solving stochastic differential equations. Finally, we address some fundamental examples of stochastic differential equations, which simply model the price process of a financial asset.

Although these subjects are applied in practice to interest rate modeling, the definitions are given for the one-dimensional case for the sake of simplicity. We complement this with some basic results for multi-dimensional cases in Section 2.7, at the end of this chapter.

This chapter summarizes arbitrage theory in the framework of martingale theory. First, we introduce an arbitrage-free market and arbitrage price for the general asset market, where the key concepts are the state price deflator and a martingale. Next, a numéraire and a numéraire measure are introduced to generalize arbitrage theory. Accordingly, we will see that the arbitrage price does not vary with the choice of numéraire.

Next, we work with a bond market where the bond prices are represented by Ito processes. For this, the market price of risk is introduced to ensure the arbitrage-free condition in the market. The market price of risk widely plays an important role in traditional interest-rate models, as an example, which will appear in the basic theory of the HJM model in Chapter 4. The estimation of the market price of risk is the most important subject of this book and is studied after Chapter 6.

For management of interest rate risk, we create an interest rate scenario by using an arbitrage-free model of the bond market, which describes the evolution of the forward rate. With this understanding, this chapter addresses the forward rate model introduced by Heath et al. (1992) (hereinafter, HJM). Additionally, we introduce the short rate model introduced by Hull and White (1990), which we treat as a special case in the HJM model.

In interest rate models, the option price is typically valuated under the riskneutral measure, and so these models have been developed as models specified under the risk-neutral measure.

On the one hand, when we apply a model to risk management, we must use a model specified under the real-world measure. We consider this further by valuating the VaR of a simple example. On the other hand, to construct an interest rate model under the real-world measure, it is necessary to estimate the market price of risk. We briefly summarize some approaches to estimation of that price in the short rate models.

This chapter introduces the LIBOR market model, which is the standard model for derivatives pricing. Because the topic of this book is risk management, we do not deal with the details of pricing. Instead, this chapter introduces the model, focusing on the implications of the real-world model.

First, we give a definition of the LIBOR market model, following Jamshidian (1997). Next, we define the LIBOR market model under the real-world measure (hereinafter, LMRW), and show, following the method of Yasuoka (2013b), that the model exists. Additionally we find the models under the spot LIBOR measure and under a forward measure that are implied by the LMRW.

Finally, we verify the numerical differences of the LIBOR process according to choice of measure. The study on the real-world model will be developed in Chapter 9.

This chapter theoretically investigates a real-world model within the Gaussian HJM model. In order to construct the real-world model, it is vital to estimate the market price of risk. For this purpose, we assume that the market price of risk is constant during each observation period. Representing the forward rate process in a principal component space, we introduce a formula for the market price of risk as the maximum likelihood estimate.

Next, we investigate the numerical properties of the market price of risk, after which we give an interpretation of that price with respect to the historical trend of the forward rates. Furthermore, we show that the interest rate simulation admits historical drift and volatility. Finally, we present a numerical procedure for real-world modeling. These results are essentially those from Yasuoka (2015). Of particular note, however, is that applying maximum likelihood estimation to finding the market price of risk is newly written for this book, in Section 6.2. Additionally, a numerical procedure is introduced in Section 6.9 for implementing the real-world model.

This chapter presents some remarks about real-world modeling. First, we examine the numerical differences between real-world simulations and riskneutral simulations, comparing the drift terms in each type of model.

Next, we investigate the reason that the market price of risk is negative by using a simplified model, which we refer to as a flat yield model. This subject is motivated by the following research question: Why does long-period observation tend to imply a negative value for the market price of risk? We obtain an answer to this question by introducing simplified models (specifically, the flat yield model and the positive slope model).

Through this book, we have estimated the market price of risk under the assumption that it is constant in the sample period. Addressing this, we examine the validity of the constancy assumption for risk management by using a simplified model. It is worth nothing here that Section 7.1 is basically the contents of Yasuoka (2015), and Sections 7.2 and 7.3 are newly written for this book.

This chapter studies construction of a Hull-White-type real-world model, using the results of Section 6 to do so.

First, we briefly summarize some approaches to volatility estimation in the short rate model. Next, we present two methods for calibrating the Hull-White model. One is to analyze the short rate dynamics. The other is to analyze the forward rate dynamics, working within the HJM framework. Additionally, we remark on some practical aspects of volatility estimation with respect to the mean reversion rate.

Accordingly, we present a method for constructing a Hull-White model under the real-world measure. The chief benefit of this is that the real-world Hull-White model is simple to compute; Section 8.6 summarizes the numerical procedures necessary to construct the real-world model. Further, some numerical examples will be presented in Chapter 10.

This section of the book develops the theory of simulation in the LMRW. Although the theory for this is developed similarly to that for the Gaussian HJM model, the results here are somewhat more complicated than those. In particular, the drift term in the LMRW has an additional feature that makes it different from that in the Gaussian model. Moreover, the methods for reducing dimensionality and constructing the drift term for use in simulation are different from those for the Gaussian model. Readers are recommended to review the corresponding results in Chapter 6 to more deeply understand the properties of the LMRW.

Most of the arguments in this chapter are based on Yasuoka (2013a); Section 9.2 is newly written to describe maximum likelihood estimation for the market price of risk.

Some numerical examples will be shown in Chapter 10, using the same historical data as in the example for the Gaussian HJM model.

This chapter presents numerical examples of real-world modeling using actual market data. First, we show examples within the Gaussian HJM model, working with the Japanese LIBOR swap market data. We calculate the market price of risk from the data, referring to the interpretation of the market price of risk given in Chapter 6. After that, we show numerical examples in the LMRW in parallel to the above. Since the simulation model in the LMRW is more complicated than that of the Gaussian HJM model, we consider four different cases in the LMRW to clearly illustrate the properties of the real-world model. In these, Sections 10.1 and 10.2 present detailed examinations of the examples given in Yasuoka (2015) and Yasuoka (2012, 2013a), respectively.

Additionally, we present an actual example that admits a positive market price of risk. For this, we employ the Hull-White model, working with data on U.S. Treasury yields. Finally, working with long-period observations of U.S. Treasury yields, we calculate the market prices of risk in the Hull-White model. With this, we verify that long-period observation tends to cause a negative market price of risk. These examples, in Sections 10.3 and 10.4, are original to this book.