Interest rates volatility

CoUin-Dufresne P, Goldstein R, Christopher J (2004) Can Interest Rate Volatility be Extracted from the Cross Section of Bond Yields An Investigation of Unspanned Stochastic Volatility, Working prater. 

Longstaff FA, Schwartz ES (1992) Interest Rate Volatility an the Term Structure A Two-Factor General Equilibrium Model. Journal of Finance 47 1259-1282. 

Longstaff, F., Schwartz, E., 1992. Interest rate volatility and the term structure a two-factor general equUibrium model. J. Finance 47, 1259-1282. 

H. Gifford Fong and Oldrich A. Vasicek, Interest Rate Volatility and a Stochastic Factor, Gifford Fong Associates, working paper (1992). 

Francis A. Longstaff and Eduardo Schwartz, Interest Rate Volatility and the Term Structure A Two-Factor General Equilibrium Model, Journal of Finance 47 (1992), pp. 1259-1282 and Fletcher A. Longstaff and Eduardo Schwartz, A Two-Factor Interest Rate Model and Contingent Claim Valuation, Journal of Fixed Income 3 (1992), pp. 16-23. 

The relationship is reduced as maturity is increased because of the low liquidity of futures markets beyond two to five years, increased futures execution risk, and increased interest rate volatility. The convexity problem when using futures contracts (nonconvex instruments) to hedge interest rate swap positions (convex instruments) is more pronounced for long-term transactions, resulting in reduced hedge efficiency. The convexity issue is addressed in detail later in this chapter. 

Although published officially in 1985, the Cox-Ingersoll-Ross model was described in academic circles in 1977, or perhaps even earlier, which would make it the first interest rate model. Like Vasiceks it is a one-factor model that defines interest rate movements in terms of the dynamics of the short rate. It differs, however, in incorporating an additional feature, which relates the variation of the short rate to the level of interest rates. This feature precludes negative interest rates. It also reflects the fact that interest rate volatility rises when rates are high and correspondingly decreases when rates are low. The Cox-lngersoll-Ross model is expressed by equation (4.11). 

Li H, Zhao F (2006) Unspanned Stochastic Volatility Evidence from Hedging Interest Rate Derivatives. Journal of Finance 61 341-378. 

The price processes of shares and bonds, as well as interest rate processes, are stochastic processes. That is, they exhibit a random change over time. For the purposes of modelling, the change in asset prices is divided into two components. These are the drift of the process, which is a deterministic element, also called the mean, and the random component known as the noise, also called the volatility of the process. 

Implementing an interest-rate model requires the input of the term structure yields and volatility parameters, which are used in the prcx ess of calibrating... 

The Vasicek, Cox-Ingersoll-Ross, Hull-White and other models incorporate mean reversion. As the time to maturity increases and as it approaches infinity, the forward rates converge to a point at the long-run mean reversion level of the current short-rate. This is the limiting level of the forward rate and is a function of the volatility of the current short-rate. As the time to maturity approaches zero, the short-term forward rate converges to the same level as the instantaneous short-rate. In the Merton and Vasicek models, the mean of the short-rate over the maturity period T is assumed to be constant. The same constant for the mean, or the drift of the interest rate, is described in the Ho-Lee model, but not the extended Vasicek or Hull-White model. 

A landmark development in interest-rate modelling has been the specification of the dynamics of the complete term stracture. In this case, the volatility of the term structure is given by a specified functiOTi, which may be a function of time, term to maturity or zero-coupon rates. A simple approach is described in the Ho-Lee model, in which the volatility of the term structure is a parallel shift in the yield curve, the extent of which is independent of the current time and the level of current interest rates. The Ho-Lee model is not widely used, although it was the basis for the HJM model, which is widely used. The HJM model describes a process whereby the whole yield curve evolves simultaneously, in accordance with a set of volatility term structures. The model is usually described as being one that describes the evolution of the forward rate however, it can also be expressed in terms of the spot rate or of bond prices (see, e.g., James and Webber (1997), Chapter 8). For a more detailed description of the HJM framework refer to Baxter and Rennie (1996), Hull (1997), Rebonato (1998), Bjork (1996) and James and Webber (1997). Baxter and Reimie is very accessible, while Neftci (1996) is an excellent introduction to the mathematical background. 

In Equation (4.3), the drift and volatility coefficients are functions of time t and T. For all forward rates7) in the period [0, T, the only source of uncertainty is the Brownian motion. In practice, this would mean that all forward rates would be perfectly positively correlated, irrespective of their terms to maturity. However, if we introduce the feature that there is more than one source of uncertainty in the evolution of interest rates, it would result in less than perfect correlation of interest rates, which is what is described by the HIM model. 

Model consistency As we have noted elsewhere, using models requires their constant calibration and re-calibration over time. For instance, an arbitrage model makes a number of assumptions about the interest rate drift rate and volatility, and in some cases, the mean reversion of the dynamics of the rate process. Of course, these values will fluctuate constantly over time so that the estimate of these model parameters used one day will not remain the same over time. So, the model will be inconsistent over time and must be re-calibrated to the market Equilibrium models use parameters that are estimated from historical data, and so there is no unused daily change. Model parameters remain stable. Over time therefore these models remain consistent, at least with themselves. However, given the points we have noted above, market participants usually prefer to use arbitrage models and re-calibrate them frequently. 

Andersen, T., Lund, J., 1997. Estimating continuous-time stochastic volatility models of the shortterm interest rate. J. Econometrics 77, 343-377. 

From market observation we know that index-linked bonds can experience considerable volatility in prices, similar to conventional bonds, and therefore, there is an element of volatility in the real yield return of these bonds. Traditional economic theory states that the level of real interest rates is cmistant however, in practice they do vary over time. In addition, there are liquidity and supply and demand factors that affect the market prices of index-linked bonds. In this chapter, we present analytical techniques that can be applied to index-linked bonds, the duration and volatility of index-linked bonds and the concept of the real interest rate term structure. Moreover, we show the valuation of inflation-linked bonds with different cash flow structures and embedded options. 

The embedded option component in convertible bonds makes the valuation sensitive from three main parameters share price, volatility and interest rate. These parameters affect the value of a convertible bond for both situations ... 

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