# How would I value a perpetual bond with an embedded option?

I am trying to work out how to value the following transactions. It should be straight forward, since it breaks down into a series of well known instruments, yet I am not sure how to evaluate it:

1. Receive Cash payment amount of \$X 2. Subsequent pay out \$Y per calendar month into perpetuity
3. Have the option to "close out" the implied perpetuity, by paying the original received $X amount, any time after 1 year. I would like to know how to value such an instrument, which consists of effectively: • a perpetuity • an embedded option If I was to make such an instrument available to someone, how much would I sell it for? As an interesting aside, this is clearly a debt instrument, and would be recorded in the other parties 'liability column'. What value would be recorded in the books? Clearly, not the original$X ...?

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There are "perpetual" bonds and preferred shares that are traded in the corporate credit markets that exactly match your conditions above. They are recorded in the 10-K at notional value $X$. The "close-out" feature is an embedded call.

You should assume your favorite stochastic interest rate (and/or credit) model and run a PDE solver, tree, or other grid scheme from some time $T$ far in the future (we used to let $T$=70 years). Basically, you want $T$ far enough away that, after discounting, $X$ paid at time $T$ is negligible. Most of the time your credit model has a pretty high probability of default within 100 years.

Many of these instruments have their embedded calls starting only past some date in the future, and in that case traders often tend to lend great consideration to the PV of payments only up to the first call date.

The main risks to your valuation (aside from model or calibration error in the credit and interest rate models) are

• Suspension of preferred coupon/dividend payments
• Mergers and takeovers
• Capital structure changes
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I agree with this answer intuitively - although I must admit that I don't exactly understand your answer. My understanding of your answer is that the instrument should be valued as a perpetual bond (with an embedded option) and recorded at Notional value $X (is my understanding correct?). If yes, then it seems that the instrument I described is really, a callable "perpetual" bond?. It is still not clear to me how to price the instrument - could you please be more explicit?. Its been a while since I did any FM/pricing stuff so please bear with me being slow .. – Homunculus Reticulli Dec 1 '11 at 10:34 The BDT model is, despite its age, still one of the most common models employed for pricing callable bonds. In this case you would need to modify it with a jump to default branch on the trees corresponding to your credit risk. The accounting step of recording$X\$ on the books is a convenience, as actual instrument value is more dependent on what someone else is willing to pay and hence depends on their perception of how risky your credit is. – Brian B Dec 1 '11 at 14:37