Abstract
The main inputs of portfolio selection models are the expected values and covariances of the assets under consideration. In equity portfolio selection, the expected values and covariances are oftentimes estimated by analyzing the historical time series of the stocks. Because of bond characteristics and properties of bond portfolio selection models that will be discussed in greater detail in Chapter 4, such an approach is generally ruled out for fixed income instruments. In order to determine the bond portfolio selection parameters consistently, a theoretical model for the evolution of bond prices over time is needed.
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References
See Wilhelm (1992), p. 213.
See Wilhelm (1992), p. 209.
See Wilhelm (1992), p. 209.
Choudhry (2004), p. 178.
The model dates back to the seminal analysis of Macaulay (1938).
See Cairns (2004), p. 18.
See Baz/Chacko (2004), p. 108.
Hull/ White (1996), pp. 261–262.
See Hull/White (1996), p. 228.
See Hull/White (1996), p. 229.
See Hull/White (1996), p. 229.
A stochastic process has the Markov property if the conditional distribution of its future values depends only on the current value and not on the past. See Cvitanic/Zapatero (2004), p. 65.
See Hull/White (1996), p. 229.
Hull/ White (1996), p. 229.
It has been common to assume that only one factor drives the term structure. According to Martellini/Priaulet/Priaulet (2003), p. 388, there is a general consensus among researchers that this factor should be the short rate.
See Table 3.1 on page 70 of Martellini/Priaulet/Priaulet (2003) for empirical results of interest-rate co-movements.
See Martellini/Priaulet/Priaulet (2003), p. 388.
See Martellini/Priaulet/Priaulet (2003), p. 388.
Svoboda (2004), p. vii.
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Heath/ Jarrow/ Morton (1992), p. 80.
See Heath/Jarrow/Morton (1992), p. 80. For notational convenience this dependence is not indicated in Equation (3.2).
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See Heath/Jarrow/Morton (1992), p. 79.
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For a brief exposition of the two approaches see Cairns (2004), p. 55 and Cairns (2004), p. 60.
For a book-length treatment of asset pricing that contains a chapter on the stochatic discount factor methodology see Cochrane (2005). Wilhelm (2005) derives Gaussian interest rate models from the specification of the stochastic discount factor. The Ho-Lee interest rate model was derived by stochastic discounting already in Wilhelm (1999). For stochastic discounting in a discrete time setting see Wilhelm (1996).
If the market is free of arbitrage and complete, there exists a unique stochastic discount factor, see Harrison/Kreps (1979) and Harrison/Pliska (1981).
Based on the one-factor formulation in Baz/Chacko (2004), p. 51.
The market price of interest rate risk gives the expected return over the riskless rate per unit of volatility. It must be equal for all traded assets, see Branger/Schlag (2004), p. 115.
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This follows from the definition of the stochastic discount factor, see Duffie (1996), p. 103. A stochastic process X is a martingale if Et(Xs) = Xt, see Duffie (1996), p. 22.
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Vasicek (1977).
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Svoboda (2004), p. 10.
Svoboda (2004), p. 10.
Vasicek (1977), p. 186.
Svoboda (2004), p. 9.
This can be derived as well from Equation (3.20). See also Vasicek (1977), p. 185.
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Munk (2004b), p. 157.
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Munk (2004b), p. 159.
It is \( N\left( { - \frac{{E_t [R(T)]}} {{\sqrt {\operatorname{var} _t (rT))} }}} \right) \) , where N(x) is the standard-normal cumulative probability function, see Munk (2004b), p. 163.
Chan et al. (1992), p. 1218.
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Vasicek (1977), pp. 186–187.
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This formulation is equivalent to the original one with correlated Brownian motions. See e.g. Brigo/Mercurio (2001), p. 134
For a general integration by parts formula see Oksendal (1992), p. 46.
Brigo/ Mercurio (2001), p. 150.
The sum of the two stochastic integrals is normally distributed. If x is normally distributed, then \( E[e^x ] = e^E [x] + \frac{1} {2}\operatorname{var} (x) \) , see Rinne (1997), p. 365. In this case, E[x] = 0.
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See Golub/Tilman (2000), p. 89.
See Hull/White (1996), p. 334.
See Hull/White (1996), p. 369.
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(2008). Term Structure Modeling in Continuous Time. In: Bond Portfolio Optimization. Lecture Notes in Economics and Mathematical Systems, vol 605. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-76593-6_3
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