Bond Pricing and Portfolio Analysis by Olivier de La Grandville

By Olivier de La Grandville

"Protecting traders ultimately" within the identify displays the author's curiosity within the proposal of immunization - an instructive dialogue, yet no longer anything particular, whether MIT Press disagrees. should you clutch this fabric fast sufficient, and choose to seek advice different books on derivatives math, you'll locate little cause to maintain this one; Martellini et al. will be a superb ebook to check earlier than figuring out.

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He will then sell it for V0 …b B0 ˆ BT ‰1 ‡ i…0Y T†ŠÀT † and invest the proceeds at rate i…0Y T† for period T. At T, he will receive V0 ‰1 ‡ i…0Y T†Š T . Now (3) can also be written V0 ‰1 ‡ i…0Y T† T Š b BT …4† 29 An Arbitrage-Enforced Valuation of Bonds So, with his earnings V0 ‰1 ‡ i…0Y T†Š T at time T, he can buy the bond for its par value BT , give it back to the person he borrowed it from, and make a pro®t V0 ‰1 ‡ i…0Y T† T Š À BT . 2 recapitulates these transactions and their associated cash ¯ows.

We can write: . value of straight bond: min …V Y BT † C . value of warrant: min …0Y lV À BT † C This last value is the value of a call on the part of the ®rm with an exercise price (per convertible) of BT al. Most convertible bonds have more complicated features than the one depicted above. In particular, conversion can be performed before maturity. Furthermore, often the ®rm can call back the convertible, and the holder can as well put it to the issuer under speci®ed conditions. Thus a convertible has a number of imbedded options: a warrant, a written (sold) call feature, and a put feature.

Its sum1 is 1Àa 1Àa , and therefore we have: 4 5 4 5 1 T 1 À …1‡i † c BT c 1 BT ˆ 1À …13† ‡ ‡ Bˆ T T 1 1 ‡ i 1 À 1‡i i …1 ‡ i† …1 ‡ i† …1 ‡ i† T Let us ®rst check that it has the right value at maturity (when the remaining maturity T becomes zero). For T ˆ 0, 1 À 1a…1 ‡ i† T is equal to zero, and BT a…1 ‡ i†ÀT becomes BT . So indeed the value of the bond is at its par, as it should be. Now look at what happens when there is no reimbursement (or, equivalently, when maturity is in®nity). The bond, in this case, is called a perpetual, or a consol; when T 3 y, B tends toward cai.

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