Convertible Bond Pricing

A convertible bond gives the holder the best of both worlds, in principle: it behaves like an ordinary bond, paying regular coupons and returning principal at maturity, right up until the issuer's stock rises high enough that swapping the bond for a fixed number of shares becomes more valuable than holding it as debt — at which point the holder can convert and participate in the stock's upside instead. It's a bond with a call option sewn into it, and once you see it that way, the pricing is just addition: value the bond piece, value the option piece, add them together.

The payoff decomposition

$$\text{CB Value} \approx \text{Straight Bond Value} + \text{Conversion Option Value}.$$

In words: the straight bond value is what the convertible would be worth if it could never be converted — just the present value of its coupons and principal, discounted at a rate reflecting the issuer's credit risk. The conversion option value is the value of a call option on the issuer's stock, struck at the "conversion price" (the effective price per share you're paying by giving up the bond), with a strike set so that $\text{conversion ratio} \times \text{conversion price} = \text{bond face value}$. At maturity, a holder converts only if the stock's value via conversion exceeds the bond's cash value — exactly the same "exercise only if it's worth more" logic as an ordinary call.

Worked example 1 — the two pieces separately

A $1,000 face convertible bond has a straight-bond value (just the coupons and principal, no conversion feature) of $920, reflecting the issuer's credit spread. The conversion ratio is 20 shares per bond, so the conversion price is $1000/20=$ $50. The stock currently trades at $45, so a call struck at $50 on 20 shares, given the stock's volatility and time to maturity, is worth $60. Total estimated convertible value: $920+60=$ $980 — trading below face value, because the option piece, while real, isn't yet worth enough to fully offset the bond's own credit-driven discount.

Worked example 2 — what happens as the stock rallies

Same bond, same 20-share conversion ratio and $50 conversion price, but the stock rallies hard to $80. Converting now delivers $20\times80=$ $1,600 worth of stock, far more than the bond's $1,000 face value or its $920 straight-bond value — the option piece is now deep in the money and dominates the bond's value entirely. The convertible's price will trade very close to $1,600, i.e. 20 shares at $80 each (minus a small amount for the coupons the holder gives up by converting before maturity), moving almost one-for-one with the stock from here — the bond floor is no longer relevant because the option piece is worth so much more than it.

The embedded conversion option is an ordinary long call, struck at the conversion price — drag the strike above and watch how, below it, the convertible's total value is dominated by the bond floor (roughly flat), while above it the option payoff takes over and the total value tracks the stock almost directly.

What this means in practice

Convertible bonds let issuers borrow at a lower coupon than a plain bond would require, because investors accept less current income in exchange for the equity upside; in return, existing shareholders face dilution if bonds get converted. On the buy side, "convertible arbitrage" funds specifically try to isolate the cheap option embedded in a convertible — buying the bond and hedging out the straight-bond and credit risk with other instruments, to be left holding something close to a cheap call on the stock (see convertible arbitrage).