# Convertible perpetuity valuation

How would one value the following security: 1. it is a convertible perpetuity 2. the issuer has (at any time) the option to withhold the payment of interest 3. the issuer and the holder have at any time the option to convert the security into common stock. All I know is that one need to value three different securities: a perpetuity, a call and a put, but given the indefinite time frame and the issuer option to withhold interest payments I have no idea how to proceed further. Why?For one thing the BS formula tells me that both put and call with extreme time to maturity (I put in 5000 years) have a 0 value when risk free rate is 4% and dividend yield is 2%. Any thoughs?

first of all, perpetual puts and calls are of the american type (obviously, otherwise you have to wait for perpetuity to be over before you can exercise which can take a while). you can’t value them with BS at all, which is for european-type of exercise. for a dividend-paying stock, an optimal exercise boundary exist and there is a non-trivial formula (due to Robert Merton) for the value of a perpetual puts/calls which is very different from Black-Scholes. look it up in a textbook. however, what you are describing doesn’t fully make sense. if both the issuer and the holder have the right to force conversion at any time, at the same conversion rate, then either one or the other will do it today. then the value of your security is trivially equal to the stock price times the number of conversion shares.

MS, thanks for the insight, I never thought that different pricing method for perpetual options might exist (can’t remember reading about them in the textbooks either). Regarding the conversion - you make a lot of sense. The conversion price however was not said to be the same for the put and the call, so I guess I’ll need to dig further into the “wonderfull” world of perpetual option pricing methods.

i see - if the conversion prices are different, that makes things a lot more *interesting*. here is a link for pricing formulas for perpetual options for your reference: http://finance.bi.no/~bernt/gcc_prog/recipes/recipes/node9.html#formula:perpetual_call often this type of instrument will be priced in a lattice model because the ability of both the holder and issuer to force conversion are not independent of each other, so it affects the optimal exercise boundary. when you decouple the convertible into pure fixed income bond + two stand-alone options, you might be changing some of the facts although it is a good start for an approximation

The formulas on the stated web page give very doubtful results. Found another user friendly page though (http://www.coggit.com/coggit.php), I just wonder what the formulas behind the calculations look like and whether the output values resemble true values at all…

I trade and value convertibles, convertible preferreds, mandatories, etc. Most registered perpetual convertibles are callable by the company (say 5-7 years from issue) at par, are noncumulative, and a strike with a 15-30% premium. So to value this, main assumptions are volatility, underlying dividend (the preferred may be protected), credit assumption for the preferred (typical rule of thumb is +200 bp from where debt trades), and underlying borrow cost (today (given) but future borrow cost may rise). In some cases, an IB will detach the warrants associated with the preferred (called a unit when combined) and trade the unit and straight preferred separately. If this is a registered security post the ticker/cusip and I will see if I can value, otherwise you will need to post some specifics for me to help get a valuation.

The bank that I work for is considering to issue such a security. The problem is how to set both strike prices (for put and call) as well the coupon rate. And to make things even sexier, the bank doesn’t have listed shares so market price of common stock is a mystery as well.

the formulas on that website are legit - but don’t trust any web templates, or random websites, pick any derivative pricing textbook and you’ll see them in there (Haug for example, he has an extensive collection of formulas). it’s not nw stuff, they have been worked out a long time ago, and the underlying assumptions are consistent with black-scholes (i.e. stock price follows geometric brownian motion, risk-neutral pricing, etc.) if the results don’t make sense, either your intuition is misleading you or, more likely, your choice of decoupling the convertible into pure bond + pure equity options is not that great here.