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This article investigates the pricing of the warrant bonds with default risk under a jump diffusion process. We assume that the stock price follows a jump diffusion model while the interest rate and the default intensity have the feature of mean reversion. By the risk neutral pricing theorem, we obtain an explicit pricing formula of the warrant bond. Furthermore, numerical analysis is provided to illustrate the sensitivities of the proposed pricing model.

In recent years, warrant bond is one of the major investment instruments in financial market. The warrant bonds are made to keep the features of both convertible bonds and warrants. The holder may convert the bond into a predetermined number of stock or continue to hold the bond to maturity depending on the market. Differently from the convertible bond, the essential characteristic of the warrant bond is that the bond and the option are separable. That is to say, when the bond is converted into stock, the value of the bond still exists.

The seminal work of Brennan and Schwartz [

The aforementioned papers have made significant contributions to the study of pricing convertible bonds and warrant bonds. Since the 2008 financial crisis, the credit risk has been one of the most important sources of risks that should be taken into account. Bond holders also face credit risk as bonds issuer may default before the bond is delivered. Among a vast amount of literature on credit risk, two main approaches are used to model credit risk: structural model and reduced form model. The structural model is originated by Black and Scholes [

This article investigates the pricing of warrant bonds with credit risk. From the characteristic of the warrant bond, we find that its value can be divided into the value of a bond and the value of a call option. In order to price the warrant bond, we should utilize the theory of option pricing. It is known that certain vital features of financial time series cannot be depicted by the classical Black-Scholes models. Therefore, Merton [

The rest of the paper is organized as follows. In Section

Let

We assume that the stock price

In addition, the money market account

In this article, we use the reduced form model proposed in Jarrow and Turnbull [

Furthermore, the filtration

We adopt the assumption of Jarrow and Yu [

A warrant bond (see Payne et al. [

Here,

In addition to the intensity of default, another important quantity in the credit risk studies is the recovery rate. As in Jarrow and Yu [

In this section we investigate the pricing of the warrant bonds with credit risk. By the risk neutral valuation formula, under the equivalent martingale measure

In terms of the default intensity, we obtain the following expression:

We substitute formula (

For simplifying the notations, denote

In the following, we calculate

According to Jaimungal and Wang [

Moreover,

In the presence of stochastic interest rate, we will define the forward-neutral measure

Let

From (

By Bayes rule,

According to Lemma

By the law of iterated conditional expectation, we obtain that

Let

Next, we introduce Lemma

Define a measure

Analogously to the proof of Lemma

From Lemma

In the following, we give the main result in Theorem

The price of the warrant bond with credit risk under the jump diffusion model at time

In order to calculate

By Girsanov theorem,

In addition, by Lemma

For the calculation of

Then, a direct application of Girsanov’s theorem implies that

Combining (

In the following, we present a few remarks below to discuss some special results.

When

If the stock price is modeled without compound Poisson jump, the result of (

In this section, we shall perform the numerical analysis of the results obtained in Theorem

In Figure

The warrant bond price with different recovery rate.

Figure

The warrant bond price with different

Figure

The warrant bond price with different

As assumed in (

The warrant bond price with different

Finally, in Figure

The warrant bond price with different

The primary purpose of this paper is to value the warrant bond with credit risk under the jump diffusion model. We assume that the stock price follows a jump diffusion model while the market interest rate and the default intensity are described by mean reversion models. The technique of measure transformation is applied to provide an efficient way to evaluate the warrant bond prices. Finally, from the numerical analysis, we obtain the effects of the recovery rate

The data used to support the findings of this study are available from the corresponding author upon request.

There are no conflicts of interest related to this paper.

The authors gratefully acknowledge the support from the Zhejiang Provincial Natural Science Foundation of China (LY17G010003), Open Project of Jiangsu Key Laboratory of Financial Engineering (NSK2015-12), Natural Science Foundation of the Higher Education Institutions of Jiangsu Province (14KJB110014), and National Natural Science Foundation of China (11671115, 11526112).