# Pricing Warrant Bonds with Credit Risk under a Jump Diffusion Process

Authors: Xiaonan Su; Wei Wang; Wensheng Wang
Journal: Discrete Dynamics in Nature and Society
 (2018)

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## Abstract

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.


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## Body

## 1. Introduction

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 [1] and Ingersoll [2] popularized the studies on pricing convertible bond. Liao and Huang [3] considered the pricing of convertible bond with credit risk under the geometric Brownian motion model. Zhou and Wang [4] assumed that the interest rate follows the geometric Brownian motion and obtained the valuation of convertible bond by the method of measure transformation. Laura and Ioannis [5] defined the firm’s optimal call policy and proposed the pricing framework for convertible bond based on a structural default model. There has been a considerable interest in investigating the valuation of warrant bond since the study of Payne et al. [6]. Zhu [7] extended Payne et al. [6] to a stochastic interest rate frame and considered the pricing of warrant bond. It is well known that traditional asset price models fail to handle discrete movements (such as random environment, market trends, interest rates, business cycles, etc.). To reflect the reality, Wang and Zhao [8] used a regime switching model to describe the price dynamics of asset and investigated the pricing of warrant bond. Chen [9] assumed that the stochastic interest rate and the underlying stock follow fractional Brownian motion, respectively, and deduced the pricing formula of warrant bond. Hu et al. [10] built a structure model under portfolio constraints, discussed the pricing of warrant bond and investment portfolios under prohibition of short-selling and borrowing, and obtained an arbitrage-free price interval.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 [11]. Furthermore, Merton [12] assumed that the default is specified as the firm’s asset value reaches a specific threshold boundary. Major investigations about the structural model are to characterize the evolution of the firm’s value and capital structure. Related papers include Merton [12], Johnson and Stulz [13], Klein and Inglis [14], Ammann [15], and Wang and Wang [16]. In contrast, the reduced form model which considers that the default is controlled by an exogenous intensity process is more flexible and tractable in the real market. Since the pioneering work by Jarrow and Turnbull [17], more advanced settings and methods have been proposed on the reduced form model, such as Jarrow and Yu [18], Su and Wang [19], Liang et al. [20], and Wang et al. [21].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 [12] and many scholars introduced the jump diffusion process to describe the price dynamics of assets and improved the pricing model of Black-Scholes. Comparing with these studies about warrant bonds, the differences between theirs and ours are evident. First, based on Merton [12], we assume that the stock price follows a jump diffusion model in order to capture its large or sudden changes. Second, we use a reduced form model to describe the default risk. Finally, we provide numerical experiments to illustrate the effect of some parameters on the price of the warrant bond.The rest of the paper is organized as follows. In Section2, we give some basic assumptions of the model. In Section 3, we derive the pricing of the warrant bonds. In Section 4, we present some numerical analysis of the result obtained.


## 2. Modeling Framework

### 2.1. The Underlying Market

LetT>0 be a finite time horizon and (Ω,F,{Ft}t≥0,Q) be a filtered probability space satisfying the usual conditions. Let Q represent an equivalent martingale measure under which the discounted asset price processes are martingales. We assume that WtS,Wtr,Wtλ are standard Brownian motions under Q, and Nt is a Poisson process with constant arrival rate ν(ν>0). Assume that the covariance matrix of the Brownian motions W=(WtS,Wtr,Wtλ) is (1)1ρ12ρ13ρ211ρ23ρ31ρ321t,where ρij=ρji, and -1<ρij<1 for i≠j.We assume that the stock priceS=St follows a jump diffusion process. The dynamic of the stock price process St is specified as(2)dStSt-=rtdt+σ1dWtS-νβdt+d∑i=1NtXi,where rt is the instantaneous interest rate and σ1>0 is the volatility of St. If the jump happens, the jump size is controlled by independent identical distributed random variables Xi(Xi>-1,i=1,2,…). Here, Xi>-1 is to make sure that the stock price is nonnegative. Furthermore, we denote f(y) as the probability density of ln⁡1+Xi and β=EQ[Xi], where EQ[·] denotes the mathematical expectation under the probability measure Q. Throughout this paper, we suppose that Ntt∈[0,T], Xii=1,2,… and Wt∈[0,T] are mutually independent.In addition, the money market accountB=Bt and the market interest rate r=rt are governed by(3)dBt=rtBtdt,B0=1,drt=kθ-rtdt+σ2dWtr,where k>0, θ>0, and σ2>0 represent the speed of reversion, the long term mean level, and the volatility of rt, respectively.In this article, we use the reduced form model proposed in Jarrow and Turnbull [17] to model the default risk. Let τ denote the default time of the warrant bonds issuer with default intensity process λt. We model the default intensity λt having the feature of mean reversion(4)dλt=ab-λtdt+σ3dWtλ,where a>0, b>0, and σ3>0 represent the speed of reversion, the long term mean level, and the volatility of λt, respectively.Furthermore, the filtrationFt is generated by Ft=FtS∨Ftr∨Ftλ∨Ht, where FtS=σ(Ss,s≤t), Ftr=σ(rs,s≤t), Ftλ=σ(λs,s≤t), and Ht=σ(I(τ≤s),s≤t). Define a new filtration Gt=FTS∨FTr∨FTλ∨Ht, and G0=FTS∨FTr∨FTλ.We adopt the assumption of Jarrow and Yu [18]; the conditional and unconditional distributions of τ are given by (5)Qτ>t∣G0=exp⁡-∫0tλsds,Qτ>t=EQexp⁡-∫0tλsds,t∈0,T.


### 2.2. Warrant Bonds

A warrant bond (see Payne et al. [6]) offers the investor the option to convert it into a predetermined amount of stock or continue to hold the bond to maturity. When the bond is converted into stock, the value of the bond still exists. We assume that the holder chooses to convert the bond into stock only at expiration time T. Thus, the value of the warrant bond can be divided into two parts, the value of a bond and the value of a European call option. The assumption about the conversion time t∗ may be more realistic if we assume that t∗∈[0,T]. Wang and Bian [22], Yang et al. [23], and Laura and Ioannis [5] considered the pricing of convertible bonds when the holder converts the bond into stocks before maturity. The major differences between their papers and this one are the following: first, Wang and Bian [22] assumed that the stock price is driven by a Poisson process and the interest rate is constant. Second, in Yang et al. [23] the interest rate and default intensity were assumed to be constants. Finally, Laura and Ioannis [5] described the default risk based on a structural default model. As mentioned above, we make assumptions about the stock price, the interest rate, and the default intensity as described by (2), (3), and (4). In fact, the result may not have explicit solution for the price of the warrant bond if the conversion time is chosen at any time before T under our pricing frame and we shall explore such extension in future works. Then, the cash flows of the warrant bond at T can be expressed as follows:(6)ΨT=Pb,ST<Cv,Pb+αγST-Cv,ST≥Cv.Here,Pb=DeiT denotes the value of a bond with the coupon rate i, and the face value D. Cv is the agreed conversion price and α is the number of warrants that a bond can receive. γ is the exercise proportion; that is, one bond can be converted into γ shares of stocks.In addition to the intensity of default, another important quantity in the credit risk studies is the recovery rate. As in Jarrow and Yu [18], we assume that the recovery rate is a constant ω(0≤ω<1). When the warrant bonds issuer defaults, the value is given by ω times the payoff of the default-free bond at maturity. The valuation of the warrant bond with credit risk at time T is given by (7)VT=Iτ>TΨT+Iτ≤TωΨT.


## 2.1. The Underlying Market

LetT>0 be a finite time horizon and (Ω,F,{Ft}t≥0,Q) be a filtered probability space satisfying the usual conditions. Let Q represent an equivalent martingale measure under which the discounted asset price processes are martingales. We assume that WtS,Wtr,Wtλ are standard Brownian motions under Q, and Nt is a Poisson process with constant arrival rate ν(ν>0). Assume that the covariance matrix of the Brownian motions W=(WtS,Wtr,Wtλ) is (1)1ρ12ρ13ρ211ρ23ρ31ρ321t,where ρij=ρji, and -1<ρij<1 for i≠j.We assume that the stock priceS=St follows a jump diffusion process. The dynamic of the stock price process St is specified as(2)dStSt-=rtdt+σ1dWtS-νβdt+d∑i=1NtXi,where rt is the instantaneous interest rate and σ1>0 is the volatility of St. If the jump happens, the jump size is controlled by independent identical distributed random variables Xi(Xi>-1,i=1,2,…). Here, Xi>-1 is to make sure that the stock price is nonnegative. Furthermore, we denote f(y) as the probability density of ln⁡1+Xi and β=EQ[Xi], where EQ[·] denotes the mathematical expectation under the probability measure Q. Throughout this paper, we suppose that Ntt∈[0,T], Xii=1,2,… and Wt∈[0,T] are mutually independent.In addition, the money market accountB=Bt and the market interest rate r=rt are governed by(3)dBt=rtBtdt,B0=1,drt=kθ-rtdt+σ2dWtr,where k>0, θ>0, and σ2>0 represent the speed of reversion, the long term mean level, and the volatility of rt, respectively.In this article, we use the reduced form model proposed in Jarrow and Turnbull [17] to model the default risk. Let τ denote the default time of the warrant bonds issuer with default intensity process λt. We model the default intensity λt having the feature of mean reversion(4)dλt=ab-λtdt+σ3dWtλ,where a>0, b>0, and σ3>0 represent the speed of reversion, the long term mean level, and the volatility of λt, respectively.Furthermore, the filtrationFt is generated by Ft=FtS∨Ftr∨Ftλ∨Ht, where FtS=σ(Ss,s≤t), Ftr=σ(rs,s≤t), Ftλ=σ(λs,s≤t), and Ht=σ(I(τ≤s),s≤t). Define a new filtration Gt=FTS∨FTr∨FTλ∨Ht, and G0=FTS∨FTr∨FTλ.We adopt the assumption of Jarrow and Yu [18]; the conditional and unconditional distributions of τ are given by (5)Qτ>t∣G0=exp⁡-∫0tλsds,Qτ>t=EQexp⁡-∫0tλsds,t∈0,T.


## 2.2. Warrant Bonds

A warrant bond (see Payne et al. [6]) offers the investor the option to convert it into a predetermined amount of stock or continue to hold the bond to maturity. When the bond is converted into stock, the value of the bond still exists. We assume that the holder chooses to convert the bond into stock only at expiration time T. Thus, the value of the warrant bond can be divided into two parts, the value of a bond and the value of a European call option. The assumption about the conversion time t∗ may be more realistic if we assume that t∗∈[0,T]. Wang and Bian [22], Yang et al. [23], and Laura and Ioannis [5] considered the pricing of convertible bonds when the holder converts the bond into stocks before maturity. The major differences between their papers and this one are the following: first, Wang and Bian [22] assumed that the stock price is driven by a Poisson process and the interest rate is constant. Second, in Yang et al. [23] the interest rate and default intensity were assumed to be constants. Finally, Laura and Ioannis [5] described the default risk based on a structural default model. As mentioned above, we make assumptions about the stock price, the interest rate, and the default intensity as described by (2), (3), and (4). In fact, the result may not have explicit solution for the price of the warrant bond if the conversion time is chosen at any time before T under our pricing frame and we shall explore such extension in future works. Then, the cash flows of the warrant bond at T can be expressed as follows:(6)ΨT=Pb,ST<Cv,Pb+αγST-Cv,ST≥Cv.Here,Pb=DeiT denotes the value of a bond with the coupon rate i, and the face value D. Cv is the agreed conversion price and α is the number of warrants that a bond can receive. γ is the exercise proportion; that is, one bond can be converted into γ shares of stocks.In addition to the intensity of default, another important quantity in the credit risk studies is the recovery rate. As in Jarrow and Yu [18], we assume that the recovery rate is a constant ω(0≤ω<1). When the warrant bonds issuer defaults, the value is given by ω times the payoff of the default-free bond at maturity. The valuation of the warrant bond with credit risk at time T is given by (7)VT=Iτ>TΨT+Iτ≤TωΨT.


## 3. Pricing the Warrant Bonds with Credit Risk

In this section we investigate the pricing of the warrant bonds with credit risk. By the risk neutral valuation formula, under the equivalent martingale measureQ, the valuation at time t of the warrant bond is given by (8)Vt,T=EQe-∫tTruduIτ>TΨT+Iτ≤TωΨT∣Ft.In terms of the default intensity, we obtain the following expression:(9)Vt,T=ωEQe-∫tTruduΨT∣Ft+1-ωEQe-∫tTruduIτ>TΨT∣Ft=ωEQe-∫tTruduΨT∣Ft+1-ωIτ>tEQe-∫tTru+λuduΨT∣Ft.The details about the above equation are in Su and Wang [19].We substitute formula (6) into (9) and obtain (10)Vt,T=ωEQe-∫tTruduPb∣Ft+EQe-∫tTruduαγSTIST≥Cv∣Ft-EQe-∫tTruduαγCvIST≥Cv∣Ft+1-ωIτ>tEQe-∫tTru+λuduPb∣Ft+EQe-∫tTru+λuduαγSTIST≥Cv∣Ft-EQe-∫tTru+λuduαγCvIST≥Cv∣Ft.For simplifying the notations, denote(11)I1=EQe-∫tTruduPb∣Ft;I2=EQe-∫tTruduαγSTIST≥Cv∣Ft;(12)I3=EQe-∫tTruduαγCvIST≥Cv∣Ft;I4=EQe-∫tTru+λuduPb∣Ft;(13)I5=EQe-∫tTru+λuduαγSTIST≥Cv∣Ft;(14)I6=EQe-∫tTru+λuduαγCvIST≥Cv∣Ft.Then V(t,T) can be rewritten as(15)Vt,T=ωI1+I2-I3+1-ωIτ>tI4+I5-I6.


### 3.1. The Useful Lemmas

In the following, we calculateI1,I2,I3,I4,I5,I6, respectively. In order to use the method of measure transformation to obtain the price of the warrant bonds, we first present two lemmas to introduce two new measures QT and Qλ. Let P(t,T) denote the price of the zero coupon bond at time t, with maturity T. From (11), we have(16)I1=PbPt,T.According to Jaimungal and Wang [24], we get the zero coupon with the affine structure as follows: (17)Pt,T=exp⁡-rtσt,T,k+At,T,where (18)σt,T,k=1-e-kT-tk,At,T=θ-σ222k2σt,T,k-T-t-σ224kσ2t,T,k.Moreover,P(t,T) satisfies(19)dPt,T=rtPt,Tdt-σ2σt,T,kPt,TdWtr.In the presence of stochastic interest rate, we will define the forward-neutral measureQT equivalent to the risk neutral measure Q by Lemma 1.Lemma 1.
Letη1T denote the Radon-Nikodým derivative(20)η1T=dQTdQ=PT,TP0,TBT,and, then, (21)W~tr=Wtr+∫0tσ2σu,T,kdu,W~tS=WtS+∫0tρ12σ2σu,T,kdu,W~tλ=Wtλ+∫0tρ23σ2σu,T,kduare the standard Brownian motions under measure QT. The covariance matrix of (W~tS,W~tr,W~tλ) is the same as (WtS,Wtr,Wtλ). Moreover, the intensity of Nt and the distribution of Xi under QT are the same as those under Q.Proof.
From (19) and (20), the Radon-Nikodým derivative η1T is given by (22)η1T=dQTdQ=exp⁡-∫0Tσ2σu,T,kdWur-12∫0Tσ22σ2u,T,kdu.By virtue of Girsanov’s theorem, we immediately get the result of Lemma 1.By Bayes rule,I3 can be calculated under QT:(23)I3=EQe-∫tTruduαγCvIST≥Cv∣Ft=αγCvPt,TEQTIST≥Cv∣Ft.According to Lemma1 and the Itô lemma, we can rewrite ST under QT as (24)ST=Stexp⁡θT-t+rt-θσt,T,k-σ122T-t-νβT-t+∑i=Nt+1NTln⁡1+Xi+σ1W~TS-W~tS+∫tTσ2σu,T,kdW~ur-∫tTσ22σ2u,T,kdu-∫tTρ12σ1σ2σu,T,kdu.By the law of iterated conditional expectation, we obtain that(25)I3=αγCvPt,TEQTEQTIST≥Cv∣Ft∨σ∑i=Nt+1NTln⁡1+Xi∣Ft=αγCvPt,T∑n=1∞e-νT-tνnT-tnn!∫-∞∞Nd3t,T,yfnydy+e-νT-tNd3t,T,0,where N(·) denotes the cumulative distribution function for a standard normal random variable, fn(y) is the n-th convolution of the density function f(y) of ln(1+Xi), and d3(t,T,y) is given by formula (39) in Theorem 3. Further,(26)Λt,T=θ-σ122-νβT-t+rt-θσt,T,k-∫tTσ22σ2u,T,kdu-∫tTρ12σ1σ2σu,T,kdu.Let(27)Xt,T=EQe-∫tTru+λudu∣Ft.From (12), we get(28)I4=PbXt,T.By (3) and (4), we have(29)∫tTru+λudu=θ+brt-θσt,T,k+λt-bσt,T,a+∫tTσ2σu,T,kdWur+∫tTσ3σu,T,adWuλ.Direct calculation yields(30)Xt,T=exp⁡-θ+bT-t-rr-θσt,T,k-λt-bσt,T,a+12∫tTσ22σ2u,T,kdu+12∫tTσ32σ2u,T,kdu+∫tTρ23σ2σ3σu,T,kσu,T,adu.Next, we introduce Lemma2.Lemma 2.
Define a measureQλ by the Radon-Nikodým derivative (31)η2T=dQλdQ=e-∫0Tru+λuduEQe-∫0Tru+λudu,and then (32)W¯tr=Wtr+∫0tM1udu,W¯tλ=Wtλ+∫0tM2udu,W¯tS=WtS+ρ12∫0tM1udu+ρ13∫0tM2uduare standard Qλ Brownian motions, where M1(u)=σ2σ(u,T,k)+ρ23σ3σ(u,T,a), and M2(u)=σ3σ(u,T,a)+ρ23σ2σ(u,T,k). The covariance matrix of (W¯tS,W¯tr,W¯tλ) is the same as (WtS,Wtr,Wtλ). Moreover, the intensity of Nt and the distribution of Xi under Qλ are the same as those under Q.Proof.
Analogously to the proof of Lemma1, we can get the Radon-Nikodým derivative (33)η2T=exp⁡-∫0Tσ2σu,T,kdWur-∫0Tσ3σu,T,adWuλ-12∫0Tσ22σ2u,T,kdu-12∫0Tσ32σ2u,T,adu-∫0Tρ23σ2σ3σu,T,kσu,T,adu.By virtue of Girsanov’s theorem, we can complete the proof.From Lemma2, Itô lemma, and (2), ST can be written as (34)ST=Stexp⁡Mt,T+σ1W¯TS-W¯tS+∫tTσ2σu,T,kdW¯ur+∑i=Nt+1NTln⁡1+Xi,where(35)Mt,T=θT-t+rt-θσt,T,k-12σ12T-t-νβT-t-∫tTσ2σu,T,kM1udu-∫tTρ12σ1M1udu-∫tTρ13σ1M2udu.Thus, by Bayes rule and the law of iterated conditional expectation, we get(36)I6=αγCvXt,TEQλEQλIST≥Cv∣Ft∨σ∑i=Nt+1NTln⁡1+Xi∣Ft=αγCvXt,T∑n=1∞e-νT-tνnT-tnn!∫-∞∞Nd6t,T,yfnydy+e-νT-tNd6t,T,0,where d6(t,T,y) can be obtained by formula (41) in Theorem 3.


### 3.2. Main Results

In the following, we give the main result in Theorem3.Theorem 3.
The price of the warrant bond with credit risk under the jump diffusion model at timet is(37)Vt,T=ωPbPt,T-αγCvPt,T∑n=1∞e-νT-tνnT-tnn!∫-∞∞Nd3t,T,yfnydy+e-νT-tNd3t,T,0+αγSte-ν^T-tNd2t,T,0+∑n=1∞e-ν^T-tν^nT-tnn!∫-∞∞Nd2t,T,yf^nydy+1-ωIτ>tPbXt,T+αγXt,TY~t,Te-ν∗T-tNd5t,T,0+∑n=1∞e-ν∗T-tν∗nT-tnn!∫-∞∞Nd5t,T,yf∗nydy-αγCvXt,T∑n=1∞e-νT-tνnT-tnn!∫-∞∞Nd6t,T,yfnydy+e-νT-tNd6t,T,0,where N(·) is the cumulative distribution function of a standard normal distribution, ν^=ν∗=ν(β+1), and fn(y),f^n(y),f∗n(y) denote the n-th convolution of f(y),f^(y),f∗(y), respectively. The definition of Λ(t,T), X(t,T), M(t,T), and Γ(t,T) can be referred to in (26), (30), (35), and (44). Further,(38)d2t,T,y=ln⁡St/Cv+Γt,T+y1/2∫tTσ22σ2u,T,kdu+1/2σ12T-t+∫tTρ12σ1σ2σu,T,kdu,(39)d3t,T,y=ln⁡St/Cv+Λt,T+y1/2∫tTσ22σ2u,T,kdu+1/2σ12T-t+∫tTρ12σ1σ2σu,T,kdu,(40)d5t,T,y=ln⁡St/Cv+Mt,T+∫tTσ2σu,T,kM3udu+∫tTσ1M4udu+y1/2∫tTσ22σ2u,T,kdu+1/2σ12T-t+∫tTρ12σ1σ2σu,T,kdu,(41)d6t,T,y=ln⁡St/Cv+Mt,T+y1/2∫tTσ22σ2u,T,kdu+1/2σ12T-t+∫tTρ12σ1σ2σu,T,kdu.Proof.
In order to calculateI2, we first define a measure QS by the Radon-Nikodým derivative (42)dQSdQ=ST/BTEQST/BT.
By Girsanov theorem,W^tS=Wtr-σ1t, W^tr=Wtλ-ρ12σ1t, and W^tλ=WtS-ρ13σ1t are standard QS Brownian motions. Under QS, the intensity of Nt is ν(β+1), and the density function of ln(1+Xi) is f^(y)=eyf(y)/1+β. According to Bayes rules and the law of iterated conditional expectation, I2 can be calculated under QS:(43)I2=αγStEQSIST≥Cv∣Ft=αγSt∑n=1∞e-ν^T-tν^nT-tnn!∫-∞∞Nd2t,T,yf^nydy+e-ν^T-tNd2t,T,0,where (44)d2t,T,y=ln⁡St/Cv+Γt,T+y1/2∫tTσ22σ2u,T,kdu+1/2σ12T-t+∫tTρ12σ1σ2σu,T,kdu,Γt,T=θ+σ122-νβT-t+rt-θσt,T,k+∫tTρ12σ1σ2σu,T,kdu.
In addition, by Lemma2 and Bayes rules, we get (45)I5=αγXt,TEQλSTIST≥Cv∣Ft.
For the calculation ofI5, we perform a measure change to Qλ by the Radon-Nikodým derivative (46)dQ∗dQλ=STEQλST.
Then, a direct application of Girsanov’s theorem implies thatW¯^tr=W¯tr-∫tT‍M3(u)du, W¯^tS=W¯tλ-∫tT‍M4(u)du, and W¯^tλ=W¯tS-∫tT‍ρ23M3(u)du-∫tT‍ρ23M4(u)du are standard Q∗ Brownian motions, where M3(u)=σ2σ(u,T,k)+ρ12σ1, and M4(u)=σ1+ρ12σ2(u,T,k). The intensity of Nt is ν∗=ν(β+1), and the density function of ln(1+Xi) is f∗(y)=eyf(y)/1+β. Here, we can calculate I5 in the following way:(47)I5=αγXt,TEQλST∣FtEQ∗IST≥Cv∣Ft=αγXt,TY~t,TEQ∗IST≥Cv∣Ft=αγXt,TY~t,T∑n=1∞e-ν∗T-tν∗nT-tnn!∫-∞∞Nd5t,T,yf∗nydy+e-ν∗T-tNd5t,T,0,where (48)Y~t,T=EQλST∣Ft=Stexp⁡Mt,T+νβT-t+12∫tTσ22σ2u,T,kdu+12σ12T-t+ρ12∫tTσ1σ2σu,T,kdu.
Combining (15), (16), (25), (28), (36), (43), and (47), we can obtain the result of the theorem.In the following, we present a few remarks below to discuss some special results.Remark 4.
Whenω=1, (37) reduces to the formula for the price of the warrant bond under a jump diffusion without credit risk. This result is consistent with Wang and Zhao [8]. In this case, (37) is simplified to(49)Vt,T=PbPt,T-αγCvPt,T∑n=1∞e-νT-tνnT-tnn!∫-∞∞Nd3t,T,yfnydy+e-νT-tNd3t,T,0+αγSte-ν^T-tNd2t,T,0+∑n=1∞e-ν^T-tν^nT-tnn!∫-∞∞Nd2t,T,yf^nydy.Remark 5.
If the stock price is modeled without compound Poisson jump, the result of (37) is given by the following formula which is similar to that of Zhu [7]:(50)Vt,T=PbPt,T-αγCvPt,TNd3t,T+αγStNd2t,T,where (51)d2t,T=ln⁡St/Cv+Γt,T1/2∫tTσ22σ2u,T,kdu+1/2σ12T-t+∫tTρ12σ1σ2σu,T,kdu,d3t,T=ln⁡St/Cv+Λt,T1/2∫tTσ22σ2u,T,kdu+1/2σ12T-t+∫tTρ12σ1σ2σu,T,kdu,Γt,T=θ+σ122-νβT-t+rt-θσt,T,k+∫tTρ12σ1σ2σu,T,kdu,Λt,T=θ-σ122-νβT-t+rt-θσt,T,k-∫tTσ22σ2u,T,kdu-∫tTρ12σ1σ2σu,T,kdu.


## 3.1. The Useful Lemmas

In the following, we calculateI1,I2,I3,I4,I5,I6, respectively. In order to use the method of measure transformation to obtain the price of the warrant bonds, we first present two lemmas to introduce two new measures QT and Qλ. Let P(t,T) denote the price of the zero coupon bond at time t, with maturity T. From (11), we have(16)I1=PbPt,T.According to Jaimungal and Wang [24], we get the zero coupon with the affine structure as follows: (17)Pt,T=exp⁡-rtσt,T,k+At,T,where (18)σt,T,k=1-e-kT-tk,At,T=θ-σ222k2σt,T,k-T-t-σ224kσ2t,T,k.Moreover,P(t,T) satisfies(19)dPt,T=rtPt,Tdt-σ2σt,T,kPt,TdWtr.In the presence of stochastic interest rate, we will define the forward-neutral measureQT equivalent to the risk neutral measure Q by Lemma 1.Lemma 1.
Letη1T denote the Radon-Nikodým derivative(20)η1T=dQTdQ=PT,TP0,TBT,and, then, (21)W~tr=Wtr+∫0tσ2σu,T,kdu,W~tS=WtS+∫0tρ12σ2σu,T,kdu,W~tλ=Wtλ+∫0tρ23σ2σu,T,kduare the standard Brownian motions under measure QT. The covariance matrix of (W~tS,W~tr,W~tλ) is the same as (WtS,Wtr,Wtλ). Moreover, the intensity of Nt and the distribution of Xi under QT are the same as those under Q.Proof.
From (19) and (20), the Radon-Nikodým derivative η1T is given by (22)η1T=dQTdQ=exp⁡-∫0Tσ2σu,T,kdWur-12∫0Tσ22σ2u,T,kdu.By virtue of Girsanov’s theorem, we immediately get the result of Lemma 1.By Bayes rule,I3 can be calculated under QT:(23)I3=EQe-∫tTruduαγCvIST≥Cv∣Ft=αγCvPt,TEQTIST≥Cv∣Ft.According to Lemma1 and the Itô lemma, we can rewrite ST under QT as (24)ST=Stexp⁡θT-t+rt-θσt,T,k-σ122T-t-νβT-t+∑i=Nt+1NTln⁡1+Xi+σ1W~TS-W~tS+∫tTσ2σu,T,kdW~ur-∫tTσ22σ2u,T,kdu-∫tTρ12σ1σ2σu,T,kdu.By the law of iterated conditional expectation, we obtain that(25)I3=αγCvPt,TEQTEQTIST≥Cv∣Ft∨σ∑i=Nt+1NTln⁡1+Xi∣Ft=αγCvPt,T∑n=1∞e-νT-tνnT-tnn!∫-∞∞Nd3t,T,yfnydy+e-νT-tNd3t,T,0,where N(·) denotes the cumulative distribution function for a standard normal random variable, fn(y) is the n-th convolution of the density function f(y) of ln(1+Xi), and d3(t,T,y) is given by formula (39) in Theorem 3. Further,(26)Λt,T=θ-σ122-νβT-t+rt-θσt,T,k-∫tTσ22σ2u,T,kdu-∫tTρ12σ1σ2σu,T,kdu.Let(27)Xt,T=EQe-∫tTru+λudu∣Ft.From (12), we get(28)I4=PbXt,T.By (3) and (4), we have(29)∫tTru+λudu=θ+brt-θσt,T,k+λt-bσt,T,a+∫tTσ2σu,T,kdWur+∫tTσ3σu,T,adWuλ.Direct calculation yields(30)Xt,T=exp⁡-θ+bT-t-rr-θσt,T,k-λt-bσt,T,a+12∫tTσ22σ2u,T,kdu+12∫tTσ32σ2u,T,kdu+∫tTρ23σ2σ3σu,T,kσu,T,adu.Next, we introduce Lemma2.Lemma 2.
Define a measureQλ by the Radon-Nikodým derivative (31)η2T=dQλdQ=e-∫0Tru+λuduEQe-∫0Tru+λudu,and then (32)W¯tr=Wtr+∫0tM1udu,W¯tλ=Wtλ+∫0tM2udu,W¯tS=WtS+ρ12∫0tM1udu+ρ13∫0tM2uduare standard Qλ Brownian motions, where M1(u)=σ2σ(u,T,k)+ρ23σ3σ(u,T,a), and M2(u)=σ3σ(u,T,a)+ρ23σ2σ(u,T,k). The covariance matrix of (W¯tS,W¯tr,W¯tλ) is the same as (WtS,Wtr,Wtλ). Moreover, the intensity of Nt and the distribution of Xi under Qλ are the same as those under Q.Proof.
Analogously to the proof of Lemma1, we can get the Radon-Nikodým derivative (33)η2T=exp⁡-∫0Tσ2σu,T,kdWur-∫0Tσ3σu,T,adWuλ-12∫0Tσ22σ2u,T,kdu-12∫0Tσ32σ2u,T,adu-∫0Tρ23σ2σ3σu,T,kσu,T,adu.By virtue of Girsanov’s theorem, we can complete the proof.From Lemma2, Itô lemma, and (2), ST can be written as (34)ST=Stexp⁡Mt,T+σ1W¯TS-W¯tS+∫tTσ2σu,T,kdW¯ur+∑i=Nt+1NTln⁡1+Xi,where(35)Mt,T=θT-t+rt-θσt,T,k-12σ12T-t-νβT-t-∫tTσ2σu,T,kM1udu-∫tTρ12σ1M1udu-∫tTρ13σ1M2udu.Thus, by Bayes rule and the law of iterated conditional expectation, we get(36)I6=αγCvXt,TEQλEQλIST≥Cv∣Ft∨σ∑i=Nt+1NTln⁡1+Xi∣Ft=αγCvXt,T∑n=1∞e-νT-tνnT-tnn!∫-∞∞Nd6t,T,yfnydy+e-νT-tNd6t,T,0,where d6(t,T,y) can be obtained by formula (41) in Theorem 3.


## 3.2. Main Results

In the following, we give the main result in Theorem3.Theorem 3.
The price of the warrant bond with credit risk under the jump diffusion model at timet is(37)Vt,T=ωPbPt,T-αγCvPt,T∑n=1∞e-νT-tνnT-tnn!∫-∞∞Nd3t,T,yfnydy+e-νT-tNd3t,T,0+αγSte-ν^T-tNd2t,T,0+∑n=1∞e-ν^T-tν^nT-tnn!∫-∞∞Nd2t,T,yf^nydy+1-ωIτ>tPbXt,T+αγXt,TY~t,Te-ν∗T-tNd5t,T,0+∑n=1∞e-ν∗T-tν∗nT-tnn!∫-∞∞Nd5t,T,yf∗nydy-αγCvXt,T∑n=1∞e-νT-tνnT-tnn!∫-∞∞Nd6t,T,yfnydy+e-νT-tNd6t,T,0,where N(·) is the cumulative distribution function of a standard normal distribution, ν^=ν∗=ν(β+1), and fn(y),f^n(y),f∗n(y) denote the n-th convolution of f(y),f^(y),f∗(y), respectively. The definition of Λ(t,T), X(t,T), M(t,T), and Γ(t,T) can be referred to in (26), (30), (35), and (44). Further,(38)d2t,T,y=ln⁡St/Cv+Γt,T+y1/2∫tTσ22σ2u,T,kdu+1/2σ12T-t+∫tTρ12σ1σ2σu,T,kdu,(39)d3t,T,y=ln⁡St/Cv+Λt,T+y1/2∫tTσ22σ2u,T,kdu+1/2σ12T-t+∫tTρ12σ1σ2σu,T,kdu,(40)d5t,T,y=ln⁡St/Cv+Mt,T+∫tTσ2σu,T,kM3udu+∫tTσ1M4udu+y1/2∫tTσ22σ2u,T,kdu+1/2σ12T-t+∫tTρ12σ1σ2σu,T,kdu,(41)d6t,T,y=ln⁡St/Cv+Mt,T+y1/2∫tTσ22σ2u,T,kdu+1/2σ12T-t+∫tTρ12σ1σ2σu,T,kdu.Proof.
In order to calculateI2, we first define a measure QS by the Radon-Nikodým derivative (42)dQSdQ=ST/BTEQST/BT.
By Girsanov theorem,W^tS=Wtr-σ1t, W^tr=Wtλ-ρ12σ1t, and W^tλ=WtS-ρ13σ1t are standard QS Brownian motions. Under QS, the intensity of Nt is ν(β+1), and the density function of ln(1+Xi) is f^(y)=eyf(y)/1+β. According to Bayes rules and the law of iterated conditional expectation, I2 can be calculated under QS:(43)I2=αγStEQSIST≥Cv∣Ft=αγSt∑n=1∞e-ν^T-tν^nT-tnn!∫-∞∞Nd2t,T,yf^nydy+e-ν^T-tNd2t,T,0,where (44)d2t,T,y=ln⁡St/Cv+Γt,T+y1/2∫tTσ22σ2u,T,kdu+1/2σ12T-t+∫tTρ12σ1σ2σu,T,kdu,Γt,T=θ+σ122-νβT-t+rt-θσt,T,k+∫tTρ12σ1σ2σu,T,kdu.
In addition, by Lemma2 and Bayes rules, we get (45)I5=αγXt,TEQλSTIST≥Cv∣Ft.
For the calculation ofI5, we perform a measure change to Qλ by the Radon-Nikodým derivative (46)dQ∗dQλ=STEQλST.
Then, a direct application of Girsanov’s theorem implies thatW¯^tr=W¯tr-∫tT‍M3(u)du, W¯^tS=W¯tλ-∫tT‍M4(u)du, and W¯^tλ=W¯tS-∫tT‍ρ23M3(u)du-∫tT‍ρ23M4(u)du are standard Q∗ Brownian motions, where M3(u)=σ2σ(u,T,k)+ρ12σ1, and M4(u)=σ1+ρ12σ2(u,T,k). The intensity of Nt is ν∗=ν(β+1), and the density function of ln(1+Xi) is f∗(y)=eyf(y)/1+β. Here, we can calculate I5 in the following way:(47)I5=αγXt,TEQλST∣FtEQ∗IST≥Cv∣Ft=αγXt,TY~t,TEQ∗IST≥Cv∣Ft=αγXt,TY~t,T∑n=1∞e-ν∗T-tν∗nT-tnn!∫-∞∞Nd5t,T,yf∗nydy+e-ν∗T-tNd5t,T,0,where (48)Y~t,T=EQλST∣Ft=Stexp⁡Mt,T+νβT-t+12∫tTσ22σ2u,T,kdu+12σ12T-t+ρ12∫tTσ1σ2σu,T,kdu.
Combining (15), (16), (25), (28), (36), (43), and (47), we can obtain the result of the theorem.In the following, we present a few remarks below to discuss some special results.Remark 4.
Whenω=1, (37) reduces to the formula for the price of the warrant bond under a jump diffusion without credit risk. This result is consistent with Wang and Zhao [8]. In this case, (37) is simplified to(49)Vt,T=PbPt,T-αγCvPt,T∑n=1∞e-νT-tνnT-tnn!∫-∞∞Nd3t,T,yfnydy+e-νT-tNd3t,T,0+αγSte-ν^T-tNd2t,T,0+∑n=1∞e-ν^T-tν^nT-tnn!∫-∞∞Nd2t,T,yf^nydy.Remark 5.
If the stock price is modeled without compound Poisson jump, the result of (37) is given by the following formula which is similar to that of Zhu [7]:(50)Vt,T=PbPt,T-αγCvPt,TNd3t,T+αγStNd2t,T,where (51)d2t,T=ln⁡St/Cv+Γt,T1/2∫tTσ22σ2u,T,kdu+1/2σ12T-t+∫tTρ12σ1σ2σu,T,kdu,d3t,T=ln⁡St/Cv+Λt,T1/2∫tTσ22σ2u,T,kdu+1/2σ12T-t+∫tTρ12σ1σ2σu,T,kdu,Γt,T=θ+σ122-νβT-t+rt-θσt,T,k+∫tTρ12σ1σ2σu,T,kdu,Λt,T=θ-σ122-νβT-t+rt-θσt,T,k-∫tTσ22σ2u,T,kdu-∫tTρ12σ1σ2σu,T,kdu.


## 4. Numerical Experiments

In this section, we shall perform the numerical analysis of the results obtained in Theorem3. We assume that the parameters are as follows if there is no special instruction: a=0.25, b=0.1, k=0.1, θ=0.05, σ1=0.2, σ2=0.2, σ3=0.25, ν=2, α=2, ω=0.8, i=0.05, M=100, r0=0.03, S0=100, γ=1, ρ12=0.2, ρ13=0.1, ρ23=0.3, t=0, and T=2. Furthermore, we assume that ln(1+Xi) satisfies the standard normal distribution for obtaining the numerical results of the price of the warrant bond.In Figure1, for each ω=0.4,0.6,0.8, we consider the impact of conversion price Cv on the warrant bond price. As mentioned above, the value of the warrant bond includes the value of a European call option, and the conversion price amounts to the exercised price of the option. So, the warrant bond price decreases as Cv increases. It is also found in Figure 1 that the price of the warrant bond increases with the value of the recovery rate, i.e., ω. In fact, the greater ω means that the holder of the warrant bond will obtain more payoff once a credit event occurs. Hence, it is not surprising that the value of the warrant bond increases as the recovery rate ω increases.Figure 1
The warrant bond price with different recovery rate.Figure2 indicates that the initial stock price S0 has a significant effect on the price of the warrant bond. As the values of S0 increase, the values of the warrant bond increase as well. In fact, the greater the stock price is, the more likely the convertible bond will be converted. Hence the holder of the warrant bond can get more benefit from the higher stock price.Figure 2
The warrant bond price with differentS0.Figure3 provides the impact of the exercise proportion γ on the warrant bond. As we can see, the price of the warrant bond increases as γ increases. In fact, the larger the exercise proportion γ, the more the stocks that can be converted into and the more the profit the holder may get which leads to the higher price of the warrant bonds.Figure 3
The warrant bond price with differentγ.As assumed in (4), the default intensity λ has the property of mean reversion with the long term mean level b. Larger b leads to the more chances of default which implies that the valuation of the warrant bond may be lower. It is shown in Figure 4 that the price of the warrant bond decreases as b increases.Figure 4
The warrant bond price with differentb.Finally, in Figure5 we compare the warrant bond price with different θ which is the long term mean level of the interest rate. The value of a warrant bond includes the value of a bond and the value of an option. The higher interest rate makes the value of the bond lower but makes the value of the option price higher. Combining the two facets, the higher the interest rate, the lower the price of the warrant bond.Figure 5
The warrant bond price with differentθ.


## 5. Conclusion

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 rateω, the agreed conversion price Cv, the initial price of stock, the exercise proportion γ, and the long term mean level of interest rate and default intensity b and θ on the warrant bond price.



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*Source: 4601395-2018-07-08.xml*