FinPricing offers:

Four user interfaces:

  • Data API.
  • Excel Add-ins.
  • Model Analytic API.
  • GUI APP.

Convertible Bond Definition and Valuation

FinPricing covers the following convertible bond models:

1. Convertible Bond Introduction

A convertibnle bond is a financial instrument that pays periodic coupon payments to the holder and can also convert itselt into the issuer’s stock. The convertible bond may also include call and put provisions, which respectively allow the issuer to buy back the convertible bond and the owner to put the convertible bond for respective preset amounts.

Convertible bond is not only a coupon paying bond but also can be converted at the discretion of the holder within the periods of time specified by the conversion schedule. Typically, the issuer has the option to buy the bond back at a predetermined strike price(s) during the callable period(s). Also, there are provisions that allow the holder to return the bond to the issuer in exchange for a predetermined cash price during certain period(s).

When stock price rises high, the bondholder can ride on the high stock prices and exercise the convertible option. When stock price deteriorates, the bondholder can still receive stable coupon payment if no default occurs. In general, a convertible bond consists of two components: bond and equity option.

Issuers have several reasons to use convertible financing. By issuing convertibles they can lower their cost of funding compared to regular bond. Lower-credit companies who may not be able to access the straight debt market can often still issue convertible debt.

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. Also investors who anticipate equity appreciation can use convertibles to defer equity financing until when equity starts to grow.

The purchaser of a convertible bond can choose to hold the bond, put the bond, or convert the bond into stock; furthermore, if the convertible bond issuer calls the bond, the holder can choose to take the better of the call strike, put strike, or stock value.

2. Valuation

Convertible bonds are hybrid instruments that have both debt and equity features. The valuation of convertible bond can be complex because of its dual nature as a normal fixed rate bond plus equity options.

Convertible bond is not as liquid as regular bond or equity. People notice that using the market closing prices of convertible bonds leads to large variations in the daily P&L. This can be explained by the fact that the closing prices of Convertible Bonds and their underlying stocks are not synchronous, i.e., normally the stock is traded more often than convertible bond.

Since convertible bonds are mostly issued by a small company, credit risk can not be ignored. Valuation of convertible bond should handle credit risk properly. In case of default, the stock price drops by a fixed percentage and the convertible bond holder recovers a fixed fraction of the face value of the bond.

Following general market pricing practice, pricing of convertible bonds uses lattice for the stock price with the interest rate curves static just like pricing a conventional bond. This is the so-called one-factor (stock price) model. The call/put feature of the bond therefore has price effect mainly contributed by the stock price movement rather than the interest rate movement.

Convertible bond has two components: a component of bond and a component of stock. We assume that the stock’s price follows geometric Brownian motion with drift. For tractability, however, the stock’s price SDE is modified so that its drift does not depend on the stochastic short-term interest rate.

We assume that the stock’s price process follows geometric Brownian motion with drift, of the form

Hull-White PDE
  • σ is the stock’s price volatility and W is standard Brownian motion.

Here the Brownian motions respectively driving the short-term interest rate and stock’s price processes have constant instantaneous correlation coefficient.

We build respective partial differential equation (PDE) for the approximate stock’s price process above.

Convertible bond has an embedded option that allows the holder to convertible the bond into equity and benift from increase in the underlying stock price. The option will be exercised if the stock price increase substantially. The PDE of the equity component is given by

Equity component PDE of convertible bond valuation in FinPricing

Prior to conversion, the instrument has the characteristics of a bond. The PDE of the bond component B is

Bond component PDE of convertible bond valuation in FinPricing

Convertible valuation model needs to account for the possibility that the issuer may suffer a credit event, such as default. Credit event may lead to a significat drop in the price of the underlying stock. The risk of default comlicates the valuation of convertible bond, as the issuer will always be able to provide the required shares but may find it difficult to deliver cash. The two components of a convertible bond are thus sensitive to different levels of credit risk. The final conditions at maturity T can be generalized as

Convertible bond PDE condition in FinPricing

The price of the underlying stock is assumed to drop immediately after a credit event. Convertible bond valuation is performed using PDE solver. A mesh is constructed with time and stock price as axes. API tridiagonal matrix is then solved at each time node by moving backward through the messh. These calculation are fully implicit in nature. The upside constraints at time are

Equity convertible bond PDE constraints in FinPricing
3. Related Topics