How to look at embedded CB option?

[volguy]

I'd like to separate the call/put from a convertible bond and look at it in vol terms.

How does one go about this?

 

[Joy Pathak]

Can you explain the question a bit more?

You want to look at a Convertible bond without the embedded properties of options it holds?

You want to look at it in volatility terms?

Do you want to decompose a convertible bond into it's basic components again? The volatility of CB's are usually affected by equity and interest rate risk.

 

[volguy]

Do you want to decompose a convertible bond into it's basic components again? The volatility of CB's are usually affected by equity and interest rate risk.

Thanks Joy. Yes, I would like to decompose a convertible bond into its basic components -- The bond and the option, I assume?

Reason is, I would like to isolate the embedded call/put and compare it to similar OTC calls/puts that are trading.

Is there a standard model with which to do this? Sorry, such a basic question, but trying to understand.

 

[Sarang]

Let's say that the fair value of convertible bond is the sum of fair price of option at time t, and that of a straight bond at time t.

i.e. CB(t) = F(t) + C(t)

Now the fair values of both these components can be computed using theoretical models - should be pretty straight.

The fair value of a straight bond with a face value N, a continuously compounded coupon rate c, and a credit spread s can be calculated using the discounting formula -

F(t) = N * exp[-(r(t,T) + s(t) - c)(T - t)] + N (k - 1) * exp[-r(t,T) + s(t))(T-t)]

k is the final repayment ratio.

For the fair price of the call option, the black-scholes framework results in the fair value being computed as -

C(t) = S(t)N(d1) exp[-d(s) (T-t)] - X * exp[-r(T,t)(T-t)] N(d2)

where standard notations apply.


But before you run off to compare these 'split-up' values with their stripped-tradable-counterparts, let me warn you of the severe drawbacks,

- The Black-Scholes model is a crude approximation and street-values will never reflect the computed ones, so the computation is pointless - even if you bring implied vol into the picture

- Unlike call options, where the strike is known in advance, convertible bonds contain an option component with a stochastic strike price. This is because the value of the bond to be delivered in exchange for the shares is usually not known in advance in most cases.

I had tried something similar for my employer a few years ago, and we used asian bonds to run this exercise. But the vol dependency was so crazy that we gave up the experiment - as it looked sexier only on paper.

 

[myampol]

What you have described is really a "bond with detachable warrants." While not common in the US, such structures are often issued in certain foreign markets.

Here is an (old) paper by a friend of ours on the above topic:

http://www.math.nyu.edu/research/carrp/papers/pdf/bdw.pdf

For convertible bonds, it's a bit more complicated than this.

The key reason is that Convertible Bonds generally contain not one option, but two:

A) You, as the bondholder, have the option to exchange the bond for (usually) a specified number of shares.

So, if a $1,000 bond has a "conversion ratio" of 50 and, thus, a "conversion price" of $20 (meaning that it converts into 50 shares of stock, at a strike price of $20 per share), then if the stock is trading at, say, $30 per share, then you could exercise your option to convert the bond into shares worth $1,500 (this figure is known as the "parity value" of the bond.) If you were convert the bond and if you were to simultaneously execute a separate trade to sell 50 shares of the stock, then you would lock in a $500 net payoff.

Note, however, that when converting, you forfeit any interest accrual on the bond, so exercising your conversion option when the bond has 5 1/2 months of accrued interest may not be the brightest idea.

Also, the market price of the bond is likely a bit above the parity value. This excess (known as the "conversion premium") would be lost when you convert the bond. (Of course, if the option is deep-in-the-money, the conversion premium is likely to be small.) When you do convert the bond, you lose this premium.

B) I, as the bond issuer, usually grant myself the option to "call" the bond, for cash.

This means that if the stock is trading well above the bond's conversion price, but you have not yet converted your bond into shares (a good reason would be if the coupon rate which I pay you on this bond exceeds the rate of dividends -- if any -- that I would pay you on the converted shares) I can "force" you to convert it.

The way this works is as follows:

I (the company that issued the bond) make an announcement that the company is going to redeem all such bonds at their face value ($1,000) plus whatever interest has accrued as of a specific date (usually 30 days from now.)

In this case, you had better hurry up and convert your bonds -- because while they are worth $1,500 now, if you hold on to them and fail to exercise your conversion option (suppose, for example, that you are taking a long cruise, and cannot be reached) then you would lose all this extra money. Of course when you do convert the bond, I give you shares -- but (generally) I do not give you any of the accrued interest.

But, things get even more complicated.

Suppose that my company's stock is currently trading at $15 per share, and I sell convertible bonds on the above terms, paying interest of, say, 6 percent. Suppose further that my company does indeed do very well over the next several months -- the share price doubles.

Would you want to purchase a convertible bond from me if I had the right to call it back right away? Suppose that 4 months and 4 weeks later, I were to announce that I shall redeem all such bonds at par in 30 days -- that's 5 months and 4 weeks following the issuance date. It's also a couple of days before the first coupon payment would have been paid. By forcing you to convert the bond, I have "screwed" you out of 6 months of interest. The ability for me to do such unto you is thus referred to as a "screw clause."

Once upon a time, convertible issuers used to be able to do just that.

Because such action tends to make investors unhappy, more recent issues usually contain a "make-whole" provision, which states that if I were to call the bond before a certain date, I would have to make an additional payment to you (often, a function of the unpaid coupons during a specified initial period).

Some bonds may have a "hard call" provision, which would forbid the issuer from calling the bond for a specified period of time after issuance, regardless of the share price.

Another provision, known as a "soft call", says that the issuer cannot call the bond unless the stock has traded above a certain price for a minimum period of time (say, 20 days out of 30). Here is a paper which discusses valuation of bonds with this restriction:

http://www.realtimerisksystems.com/pdf/paper1.pdf

A good summary of various convertible bond features, some common and others rare, can be found here:

Convertible bond - Wikipedia, the free encyclopedia

Happy pricing,

m.y.

 

[Yike Lu]

Whatever the case, the value of the CB is just the sum of its constituent parts (i.e., the bond itself plus whatever options are embedded).

Bond pricing proceeds by present value.

The option on the issuer's stock is just an equity option, which have standard methods for valuation, Black-Scholes being one of them.

The option of the issuer is a short position to the bondholder on an interest rate derivative. Pick your stochastic interest rate model and value away.

If the options get fancier... use fancier option pricing techniques.

 

[Sarang]

The option on the issuer's stock is just an equity option, which have standard methods for valuation, Black-Scholes being one of them.

The option of the issuer is a short position to the bondholder on an interest rate derivative. Pick your stochastic interest rate model and value away.

If the options get fancier... use fancier option pricing techniques.

You seem to believe that the models 'predict' the right price.
Oh, dear.

 

[Yike Lu]

I certainly don't think so. That's why I'm careful to say "value" and not price.

I'm not here to discuss model risk. Whenever one picks a model, one should be aware of the inherent strengths and weaknesses of that model. However, to go into these strengths and weaknesses for all models would get quite lengthy (and in many cases reach beyond the scope of my knowledge). That is why I don't recommend any specific model for valuation. I only mention BS since it is in some sense a standard first approximation.

And don't put words in my mouth - or implied mouth.

 

[Sarang]

I certainly don't think so. That's why I'm careful to say "value" and not price.

I'm not here to discuss model risk. Whenever one picks a model, one should be aware of the inherent strengths and weaknesses of that model. However, to go into these strengths and weaknesses for all models would get quite lengthy (and in many cases reach beyond the scope of my knowledge). That is why I don't recommend any specific model for valuation. I only mention BS since it is in some sense a standard first approximation.

Appreciate that you understand the weaknesses as well.
I've seen so many arrogant SOBs who put 'faith' in BS (pun intended), that I look at everyone sceptically.

And don't put words in my mouth - or implied mouth.

Haha ... Funny one there ...

 

[Yike Lu]

When you realize they believe in the predictive power of models - that's when you take their mo... err... charge for liquidity!