Abstract
In order to implement the previously proposed pricing procedure, we need in addition to the payoff transformations derived in Chapter 5, the particular characteristic functions. The goal of this chapter is to provide these necessary functions for the case of an underlying one-factor interest-rate model and to examine the behavior of the particular density functions and prices of selected contingent claims, according to Table 4.1, influenced by jump components. Thus, we focus our efforts exclusively on the exponential-affine term-structure models generated by the one-factor version of equation (2.23). Since a one-factor model implicates the incorporation of one Brownian motion, this statement does not entail the restriction of including one sole jump component. Therefore, we apply different jump components in our examples. The general version of the one-factor instantaneous interest rate is then given by
and the factor x t is defined by the one-dimensional stochastic differential equation
where j ∈ IRN, μℚ(x t ) and σ(x t ) are the one-factor counterparts of the original parameters J, μsℚ(x t ) and Σ(x t ) used in equation (2.23). All parameters are postulated under the risk-neutral probability measure ℚ. Therefore, the solution of the general characteristic function ψ(x t , z, w0, w1, g0, g1, τ) for these models is given by the simplified versions of the ODEs (2.40) and (2.41), which are
and
with terminal conditions a(z, 0) = 0 and b(z, 0) = ιzg1.
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References
The process used in Vasicek (1977) and the process discussed in this section coincide for the case of rt = xt, thus setting the discount parameters to w0 = 0 and w1 = 1.
In Das and Foresi (1996), the parameter ψ denotes the probability that the sign of the jump is positive.
See Arapis and Gao (2006), Figure 3. The authors apply alternatively a nonparametric estimator for the short-rate probability density of three-month Treasury bill rates and seven-day Eurodollar deposit rates.
See Ahn and Thompson (1988), p. 168.
See Feller (1951), p. 173.
However, Ahn and Thompson (1988) implemented a constant, negatively sized jump component in a CIR short-rate model. Accordingly, they have to choose carefully the fixed jump amplitude to ensure that interest rates remain positive over the trading interval τ.
See, for example, Arapis and Gao (2006).
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(2008). Jump-Enhanced One-Factor Interest-Rate Models. In: Pricing Interest-Rate Derivatives. Lecture Notes in Economics and Mathematical Systems, vol 607. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-77066-4_8
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DOI: https://doi.org/10.1007/978-3-540-77066-4_8
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