Convertible Bond

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Convertible Bond Definition and Valuation

FinPricing provides valuation models for the following convertible bonds:

  • Convertible Bond
  • Callable Convertible Bond
  • Putable Convertible Bond
  • Total Return Basket Swap
  • Cancelable Reverse Convertible Bond
  • Check FinPricing valuation models

All the equity models in FinPricing take volatility skew/smile and dividend into account.


1. Convertible Bond Introduction

A convertible bond has an embedded call option that gives bondholders the right to convert their bonds into equity at a given time for a predetermined number of shares in the issuing company. Whereas a reverse convertible bond has an embedded put option that gives the issuer the right to convert the bond’s principal into shares of equity at a set date.

Convertible bonds typically have lower yields than the yields on similar fixed rate bonds without the convertible option. Reverse convertible bonds usually have shorter terms to maturity and higher yields than most other fixed rate bonds.

Most convertible bonds are subordinated debt of the issuer. In the event of bankruptcy, the claims of other bondholders take priority over convertible bondholders, who themselves have priority over owners of the preferred and common stock.

Issuers have several reasons to use convertible financing. By issuing convertibles they can lower their cost of debt funding compared to straight debt alone. Lower-credit companies who may not be able to access the straight debt market can often still issue convertible debt. Companies who anticipate equity appreciation can use convertibles to defer equity financing to a time when growth has been achieved.

Investors find several features of convertibles appealing. They offer greater satiability of income than common stock. They provide a yield that is often higher than the dividend yield of common stock. Finally, because they are often theoretically underpriced, they may provide a cheap source of common stock volatility.

2. Valuation

Convertible bonds are hybrid securities that have both debt and equity features. The valuation of convertible or reverse convertible bonds can be quite complex because of its dual nature as a normal fixed rate bond and as an equity call option or equity put option. There is no closed-form solution for convertibles. Convertible prices can only be solved by numerical methods, such as, Monte Carlo simulation, tree/lattice approaches, or partial differential equation (PDE) solutions.

Three sources of randomness exist in a convertible bond: the stock price, the interest rate, and the credit spread. Interest rate is assumed to be constant as the effect of a stochastic interest rate on convertible bond prices is so small that it can be neglected. Accurately modeling the equity process appears crucial. Since convertible bonds are issued mainly by start-up or small companies (while more established firms rely on other means of financing), credit risk plays an important role in the valuation. FinPricing uses PDE to price convertible and reverse convertible bonds, and use Monte Carlo simulation to value convertibles with exotic path-dependent trigger provisions.

The value of the convertible at each node is divided into two components: a component of bond and a component of stock. The PDE of the equity component G is given by

Equity component PDE of convertible bond valuation in FinPricing

The PDE of the bond component B is

Bond component PDE of convertible bond valuation in FinPricing

The final conditions at maturity T can be generalized as

Convertible bond PDE condition in FinPricing

The upside constraints at time are

Equity convertible bond PDE constraints in FinPricing
3. Implementation

The valuation can be done via backward induction. The procedure is as follows.

For i = penultimateTime to currentTime

  • Determine accrual interest and call/put prices.
  • Determine boundary nodes
  • use the PSOR (Projected Successive over Relaxation) method to obtain the continuation value of the fixed rate bond component B and the continuation value of the equity component G, applying the constraints

EndFor

The value at node[0][y] is the convertible bond price where the equity price at node[0][y] is equal to the current market stock price.

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6. Related Topics