The valuation of interest rate derivative securities by Jeroen F. J. De Munnik

By Jeroen F. J. De Munnik

The elevated volatility of rates of interest in the course of contemporary years and the corresponding advent of numerous rate of interest by-product securities like bond innovations, futures and embedded thoughts in mortgages, rigidity the necessity for a finished monetary idea to figure out values of mounted source of revenue tools and by-product securities regularly.

This ebook offers: a close assessment and category of different methods to price rate of interest established securities; a comparability of the numerical techniques to worth advanced securities; and an empirical exam for the Dutch mounted source of revenue marketplace of a few famous rate of interest versions which demonstrates contemporary advancements to explain rate of interest hobbies with regards to contingent declare valuation.

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36, p. ■ The Black and Scholes (1973) formula will now be derived by closely following the lines suggested by the theorems above. This example serves as a nice illustration of the strong implications of the exclusion of arbitrage opportunities and the valuation of The valuation of interest rate derivative securities 28 contingent claims and illustrates, more generally, the procedure to be followed in the next chapters when interest rate contingent claims will be discussed. The continuous-time economy in which Black and Scholes derived a closed-form solution for a European call option on a non-dividend paying stock is mainly characterized by a Brownian motion {W(t), 0 ≤ t ≤ T} defined on a filtered probability described at the beginning of this section.

8 Because 9 Because 10 Brennan (1979) showed that under the assumption of a bivariate lognormal distribution of the price of the underlying asset and aggregate wealth, a sufficient condition to obtain the Black-Scholes formula is the utility function to exhibit constant proportional risk aversion. Duan (1990) extended this valuation technique in the case of Garch(p, q) distributed lognormal returns. 11 See Gihman and Skorohod (1972, p. 40). 12 For the details of Girsanov’s Theorem, see Elliott (1982).

As maturity decreases, the volatility decreases and becomes zero in the end. However, the drift term of the Schaefer and Schwartz bond price process is assumed to be constant, ignoring thereby the above-mentioned price effect, by which bond prices are forced to equal their face value at the final maturity date. In addition, the short-term rate of interest is assumed to be constant, restricting the empirical applicability of their model to short-term options on long-term bonds. The model of Ball and Torous (1983) incorporates the “drift-to-face-value” effect by using the Brownian Bridge to model bond prices.

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