CHAPTER 7Rates‐Equity Hybrid Modelling

7.1 STATEMENT OF PROBLEM

Having derived the general form of multi‐factor pricing kernels in the preceding chapter, we look now to apply this knowledge first to a relatively simple problem, considering the joint distribution of equity and rates with a view to calculating a pricing kernel for rates‐equity hybrid derivatives. For rates we assume the short‐rate model of Hull and White [1990], as described in Chapter 4, using the notation there introduced. We suppose the equity process upper S Subscript t to be given by

(7.1)StartFraction d upper S Subscript t Baseline Over upper S Subscript t Baseline EndFraction equals left-parenthesis r Subscript dt Baseline minus q left-parenthesis t right-parenthesis right-parenthesis dt plus sigma Subscript z Baseline left-parenthesis t right-parenthesis d upper W Subscript t Superscript z Baseline comma

with r Subscript d t given by (4.2), q left-parenthesis t right-parenthesis the dividend rate and upper W Subscript t Superscript z a Brownian motion. Further, take this Brownian motion to be correlated with that driving the Hull–White rates model, with

(7.2)corr left-parenthesis upper W Subscript t Baseline comma upper W Subscript t Superscript z Baseline right-parenthesis equals rho Subscript d z Baseline period

As previously, define an auxiliary process z Subscript t implicitly by (3.45). It follows ...

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