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This paper deals with the pricing of convertible bond with call provision based on the traditional B-S formula. By applying the principle of no arbitrage, the partial differential equation for the bond is established with identified boundary conditions, which solution results in the closed form of the pricing formula.

Convertible bonds are complicated and broadly used financial instruments combining the characteristics of stocks and bonds. In recent years, convertible bond has become a new investment product for the investors. The possibility to convert the bond into a predetermined number of stocks offers the chance to participate in rising stock prices with limited loss, given that the issuer does not default on its bond obligation. Convertible bonds often contain other embedded options such as call and put provisions. These options can be specified in various different ways, which make the products more complex. Especially, conversion and call opportunities may occur in case of certain restricted time periods with given stock price conditions, which results in the changing of the call price with time. The study of the convertible bond has a long history, firstly appearing in 1843; however, the pricing theory is relatively backward. With the work of Black and Scholes (1973) [

The purpose of this paper is to study the pricing of convertible bonds with call provision. By using the method for option pricing, the closed form of the price for convertible bond is presented.

The paper is organized as follows. In Section 2, we introduce the content of convertible bond with call provision briefly, and establish the pricing model for the convertible bonds with call provision. In Section 3, we solve the partial differential equation established for the convertible bonds, and give the closed form solution. Section 4 is a conclusion.

Convertible bond has the characteristics of the implied stock option, the convertible bond holder once decided to execute the options, it becomes the shareholders of the company, and the right has no difference to the original company shareholder. The main impact for the holders to convert or not is the underlying asset price, the bond holders often choose to continuously hold or immediately convert into shares of the company based on the stock prices, when stock prices continued to slump or located in a special given regime, the bonds holder always held in hand to the maturity time or directly sell it to other investors; when stock prices continue to rise or the conversion is profitable, the holders usually choose convert the bonds into shares, and the amount of the trading profits depends of the specific stock price in the market.

However, there is possible that its stock price in the market continues to rise, its converted value far exceeds the profit obtained by holding to the maturity, which, to some extent, has serious impact on the interests of the previously-existing shareholders of the company. Therefore, the benefit issuers can reduce their cost of the issuance through establishing the call provision, to avoid the loss due to stock price soaring and market interest rates. The call provisioncan accelerate the conversion process and relieve the company’s financial pressure. Redemption usually occurs in case that the stock market price is far higher than the conversion price. When the company announces the redemption, bond holders usually immediately opted conversion to avoid loss.

Here, we employ a standard assumption that the stock price movement S_{t} meet geometric Brownian motion,

where μ is expect return rate (constant), σ is the volatility of the stock price, dW_{t} is standard Brownian motion. Since the value of convertible bonds is related to the stock price and time, we use _{B}), the issuer announced the redemption of bonds. At this point, investors immediately implement the contract to convert the convertible bonds into stocks in order to obtain a higher interest. If one chooses to continue to hold the bond, he will get the bond value. When the stock price S reaches barrier value S_{B}, the convertible bond will be executed. Let D a solution area as follows

Then the final revenue function of the convertible bonds value is

where S_{T} is stock price at maturity date, K is transforming shares price stipulated in the contract, S_{B} is stock price redemptive threshold stipulated by the issuer. Here

Clearly,

In the given region

To sum up, the pricing of convertible bonds with call provision is a specific boundary value problem for the Black-Scholes equation, which has similar properties of the up-and-out options to some extent. Compared with the standard options, it has more boundary conditions.

The pricing process of convertible bonds is, in the area

Make the transformation

Then

where

Define

with

Based on (1.9), W is suitable for definite solution problems on the area

By using the mirror methods, defining

It is clearly

which is a odd function. The limitations on

The solution of Cauchy problem (1.14) can be expressed as the Poisson equation

Then

From (1.9), inserting

where

then

and

then

This, together (1.7), yields

with

In this paper, based on the analysis of the execution conditions of convertible bonds with call provision, by solving a certain boundary value problem of the Black-Scholes equation, the closed form of the pricing formula is obtained.

In reality, the holders of convertible bonds tend to be risk-averse; owners generally hold the convertible bonds until the maturity date to obtain the bond interest as income, but when stock prices rise to a certain level (as defined for redemption), in order to obtain more profits, the holder will convert the bonds to the related stock, and sell the stocks in the secondary market. This paper just provides a theoretical evaluation of the bond for the investors. In this sense, the studied model in the paper is interesting and realistic.

Bin Zhang,Dianli Zhao, (2016) The Pricing of Convertible Bonds with a Call Provision. Journal of Applied Mathematics and Physics,04,1124-1130. doi: 10.4236/jamp.2016.46117