Convertible Bonds: Half Bond, Half Option

Imagine a bond that can transform into stock. That is a convertible bond. Chapter 33 of Wilmott’s book dives into one of the most fascinating instruments in finance, a hybrid security that sometimes acts like debt and sometimes acts like equity. It sounds simple on the surface, but underneath it is a deeply complex contract involving American option features, stochastic interest rates, path dependence, dilution, and credit risk.

What Is a Convertible Bond?

A convertible bond (CB) pays coupons and returns principal at maturity, just like a regular bond. But it has one extra feature: at any time before maturity, the holder can exchange the bond for a specified number of shares of the issuing company’s stock. This exchange is called conversion.

The value of a CB is bounded below by two things:

1. Its conversion value, the amount you get if you convert immediately (number of shares times the current stock price)
2. Its straight value, the value as a regular corporate bond with coupons and principal

When the stock price is low, the conversion option is worthless and the CB trades like a bond. When the stock price is high, the CB trades like the stock. In between, you get the best of both worlds: upside participation with downside protection. At least in theory. In practice, credit risk complicates things.

Why Do Companies Issue Convertible Bonds?

Companies need capital. They can issue equity (diluting shareholders) or debt (interest payments). A convertible bond is a middle ground. The issuer pays a lower coupon because the conversion option adds value for the buyer. If the company succeeds, investors convert and the company issues new shares instead of repaying principal.

The typical CB issuer is a smaller or more speculative company: tech startups, biotech firms, companies with high cash burn rates. From the buyer’s perspective, convertibles offer upside exposure with a safety net. The coupon is usually higher than the dividend yield, and in bankruptcy, bond holders rank above equity holders.

The Market’s Measures

Practitioners track a few key metrics. The market conversion price tells you how much you effectively pay per share if you convert right now. The market conversion premium ratio measures the extra cost over the stock price. The favorable income differential compares coupon income versus dividend income from holding the equivalent shares. And the premium payback period estimates how long it takes for the extra coupon income to cover the premium you paid.

Pricing the Simple Way

For a basic CB with no call or put features and no dividends, you can decompose it into two pieces:

1. A straight bond (present value of all coupons plus principal)
2. A call option on the stock with strike equal to principal divided by conversion ratio

Price each piece separately using standard tools (discounting for the bond, Black-Scholes for the option) and add them together.

Wilmott’s worked example: a five-year CB with $1000 principal, 2% coupon, conversion ratio of 20, stock at $47. The straight bond is $866.18, the option adds $259.40, total $1125.58. The delta is 0.74, so this CB acts more like a stock than a bond. Simple, but it falls apart when you add features that real convertibles have.

The Full Framework

For a proper pricing model, Wilmott starts with the Black-Scholes setup. The CB value depends on the stock price and time. You build a hedged portfolio with one CB and short some amount of the stock. Eliminate the risk, set the return equal to the risk-free rate. The result is a Black-Scholes inequality (not an equation, because of the American-style features).

The boundary conditions are straightforward. When the stock price is high, the CB behaves like the stock. When the stock price is zero, the CB is just the present value of all remaining coupons and principal. Across coupon dates, there is a jump in value equal to the coupon payment.

The early conversion feature makes this similar to an American option. There is a free boundary: above a certain stock price, it is optimal to convert. Below that level, you hold the bond. Interestingly, with no dividends on the stock and no coupons on the bond, early conversion is never optimal, just like with American calls on non-dividend-paying stocks.

When the stock pays dividends, early conversion can become optimal because you are missing out on dividends by holding the bond. When the bond pays coupons, things go the other way, the coupon income gives you a reason to hold rather than convert.

Call and Put Features

Real convertibles almost always have a call feature. This lets the issuing company buy back the bond at a specified price. If the stock rallies and the CB becomes very valuable, the company can force conversion by threatening to call. This caps the bond’s upside and reduces its value to the holder.

Mathematically, the call feature adds an upper constraint: V cannot exceed the call price. Combined with the conversion constraint (V must be at least the conversion value), you are solving a problem where the price is squeezed between two bounds.

Some CBs also have a put feature, letting the holder sell the bond back to the company for a specified price. This adds a lower constraint and increases the bond’s value. The put protects against both falling stock prices and rising rates.

Two-Factor Modeling

Convertible bonds typically live for five years or more. Over that time horizon, assuming constant interest rates is unreasonable. So Wilmott introduces a two-factor model where both the stock price and the interest rate are random.

Now the CB value depends on three variables: stock price, interest rate, and time. You need to hedge with both the stock (to eliminate equity risk) and a zero-coupon bond (to eliminate interest rate risk). The resulting pricing equation has cross-partial derivatives reflecting the correlation between stock and interest rate movements.

This equation has two nice special cases. If interest rates are constant, it collapses to regular Black-Scholes. If there is no stock dependence, it reduces to the basic bond pricing equation.

For the Vasicek interest rate model, there is a clever similarity reduction that simplifies the computation. But unfortunately, call and put features break the similarity, so you cannot use this shortcut for most real convertibles.

Path Dependence

Some convertibles have features that make them path dependent. For example, the conversion ratio might be reset based on where the stock price was at a specific past date. This adds a new state variable (the stock price at the reset date), turning a two-dimensional problem into a three-dimensional one. Adding stochastic interest rates would make it four-dimensional. At that point, computation becomes extremely expensive.

Dilution

Here is something that distinguishes convertibles from regular options. When a CB is converted, the company issues new shares. This dilutes the value of existing shares. If N shares exist before conversion and n new shares are created, the share price after conversion is not S but (NS - V) / (N + n). The factor N/(n + N) is the dilution factor.

For most convertibles, n is small relative to N, so dilution is minor. But for smaller companies issuing large amounts of convertible debt, it can be significant.

Credit Risk

This is the elephant in the room. Convertible bonds are corporate debt. Companies can default. If the stock price drops to zero, the company is probably bankrupt, and the bond is worth nothing too.

Without credit risk, a CB trades like a bond when the stock is low. With credit risk, the CB value drops toward zero as the stock price approaches zero. The shape of the price curve changes dramatically. We will see proper credit risk models later in the book, but for now, just know that ignoring default risk in CB pricing is a serious mistake.

The Hedge Fund Angle

Hedge funds hold about 70% of all convertible bonds. They delta-hedge by shorting the underlying stock, which creates a stabilizing effect that dampens volatility. At one point, CB arbitrage was the third-best hedge fund strategy. But so many funds piled in that the opportunities evaporated. A classic case of a strategy being arbitraged away by too many participants.

Key Takeaways

Convertible bonds are one of the most complex instruments in finance. They combine the early exercise features of American options (in three forms: conversion, call, and put) with long time horizons that demand stochastic interest rate models. They can be path dependent. They involve dilution. They carry credit risk. And yet they are hugely popular because they offer something genuinely useful: a way for companies to raise capital cheaply and for investors to get equity-like upside with bond-like protection.

The pricing challenge reflects this complexity. Simple decomposition into bond plus option works for textbook cases. Real convertibles need the full machinery of two-factor models, free boundary problems, and credit risk adjustments.

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