Exchangeable Convertible Bond Model

A convertible bond issuer pays periodic coupons to the convertible bond holder. The bond holder can convert the bond into the underlying stock within the period(s) of time specified by the conversion

Exchangeable Convertible Bond Analytics

A convertible bond issuer pays periodic coupons to the convertible bond holder. The bond holder can convert the bond into the underlying stock within the period(s) of time specified by the conversion schedule. The bond issuer can call the bond and the holder can put it according to the call and put provisions.

The Exchangeable feature assumes that the convertible bond and the underlying stock are issued by different parties. There are two possible cases with respect to stock conversion:

· covered: the bond issuer holds the underlying stock for the life time of the

convertible;

· vulnerable: the bond issuer does not hold the stock prior to the stock conversion.

We assume a convertible bond that is specified by

· (t1, t2, …, tM) - the set of coupon dates, tM is also the bond maturity date;

· (c1, c2, …, cM) - the set of coupons, each coupon payable on the respective coupon date;

· P - the bond principal, payable at maturity date, tM.

· Tx - the exercise date on which the bond can be converted into the

underlying stock, which pays dividends.

We consider discrete dividends that are specified by absolute amounts paid at pre-determined times. In particular we assume that an absolute amount, , is paid at a time, , for , where (here is the convertible’s maturity). We model the stock price based on a Hull-White approach. Specifically, let

• denote a capital gains process,

• represent the stock price,

• denote the bond issuer’s default time.

Let

(1)

for , where is a deterministic risk-free rate (see https://finpricing.com/lib/IrCurveIntroduction.html). We then assume that

, (2)

for , where

• is a deterministic hazard-rate,

(here we set ).

Furthermore the process satisfies, under the risk-neutral probability measure, an SDE of the form

(3)

where

• is a constant volatility parameter,

• is a standard Brownian motion,

• . (4)

Thus,

(5)

Let

· Tj be the dividend date immediately after the option maturity date,

i.e. Tj = min(T1, T2, …, TN) such that Tj ³ Tx, where (T1, …, TN) is the set of dividend payment dates;

· tm is the coupon date immediately after the option maturity date,

i.e. tm = min(t1, t2, …, tM) such that tm ³ Tx.

Then

(6)

Using arguments similar to those used in ref [1] one can show that the Tx-time price of the straight bond, , is

(7)

and its present value is

(8)

If the conversion ratio is g, the present value of the conversion option is

Using the assumption that the time to default process is independent from the Brownian motion process, Wt, that drives the capital gain process (see ref[1]), and the fact that the probability of default before time Tx, , we get further

(9)

where

(10)

Finally, one gets

(11)

where

, (12)

, (13)

, (14)

and is the standard normal cumulative distribution function.

The convertible bond price is

(15)

where V is given by eq. (9) and by eq. (8).