Pricing Options with Credit Risk in Markovian Regime-Switching Markets

Abstract

This paper investigates the valuation of European option with credit risk in a reduced form model when the stock price is driven by the so-called Markov-modulated jump-diffusion process, in which the arrival rate of rare events and the volatility rate of stock are controlled by a continuous-time Markov chain. We also assume that the interest rate and the default intensity follow the Vasicek models whose parameters are governed by the same Markov chain. We study the pricing of European option and present numerical illustrations.

1. Introduction

Pricing options with credit risk is an important topic in finance from both theoretical and practical perspectives. Credit risk refers to an investor’s risk that a borrower will default on making payments as promised. There are basically two kinds of models to describe the default: structural models and reduced form models. The structural approach was firstly introduced by Merton [1] who investigated European option pricing for modeling single corporate default. The approach is further extended by recent literature: see Ammann [2] and Klein and Inglis [3]. Another tractable approach is called reduced form model, which models the intensity of arrival of default events directly. The reduced form models can be seen in Artzner and Freddy [4], Duffie and Singleton [5], Duffie and Gârleanu [6], and Leung and Kwok [7] and are developed extensively by Su and Wang [8].

Recently, Markovian regime-switching models have attracted attention among researchers and practitioners in economics and mathematics. Elliott et al. [9] introduce a self-calibrating model for short-term interest rate by assuming that the short rate is governed by a finite state space Markov process. Elliott et al. [10] and Elliott and Osakwe [11] use Markov-modulated market parameters to capture the time inhomogeneity generated by the financial market. Elliott et al. [12] perform the valuation of option under a generalized Markov-modulated jump-diffusion model. Siu et al. [13] consider the pricing currency options under two-factor Markov-modulated stochastic volatility models. Bo et al. [14] derive the valuation of currency option when the spot foreign exchange rates follow Markov-modulated jump-diffusion model.

In this paper, we investigate the valuation of European option with credit risk in a reduced form model in Markovian regime-switching markets. We assume that the recovery rate is constant; that is, when the writer of the option defaults, a specified constant fraction times the payoff will be paid at maturity. In order to incorporate both rare events and time-inhomogeneity in finance market, we model the stock price by the so-called Markov-modulated jump-diffusion process, in which rare events are described as a compound Poisson process and the arrival rate of Poisson process and the volatility rate of stock are governed by a continuous-time Markov chain. The states of Markov chain can be interpreted as the states of the market. The transitions of the states of the market may describe changes of economy, finance, business cycles, and other conditions. In addition, we assume that the interest rate and the default intensity both follow the Vasicek models and the parameters of models are correlated with the same Markov chain. By the method of changing measures, we obtain the closed form formula for the valuation of the European option.

The rest of this paper is organized as follows. Section 2 presents the model description. In Section 3, by applying Girsanov’s measure changing theorem, we derive the formula of the pricing of European-style call option. We provide numerical analysis in Section 4. The concluding remarks are contained in Section 5.

2. The Model Description

Let (Ω,ℱ,𝑃) be a complete probability space, where 𝑃 is a neutral-risk probability measure. Define 𝜉={𝜉_𝑡,𝑡≥0} on (Ω,ℱ,𝑃) as a continuous-time, finite state Markov chain with 𝑛-state space 𝐸. We interpret the state of 𝜉 as the states of the economy as follows (Elliott et al. [10] and Elliott and Osakwe [11]). Without loss of generality, we take the state space of 𝜉 to be a finite set of unite vectors {𝑒₁,𝑒₂,…,𝑒_𝑛}  with 𝑒_𝑖=(0,…,1,…,0)∈𝑅^𝑛. And 𝜉 has the following semimartingale representation: 𝑑𝜉_𝑡=𝑄𝜉_𝑡𝑑𝑡+𝑑𝑀_𝑡, (1) where 𝑄=(𝑞_𝑖𝑗)_{𝑖,𝑗=1,2,…,𝑛} is 𝑄-matrix of 𝜉 and 𝑀={𝑀_𝑡}_{0≤𝑡≤𝑇} is an 𝑅^𝑛-valued martingale with respect to the filtration generated by {𝜉_𝑡,0≤𝑡≤𝑇} under 𝑃. Suppose that the stock price 𝑆_𝑡 and interest rate 𝑟_𝑡 satisfy the following stochastic differential equations (SDE): 𝑑𝑆_𝑡𝑆_𝑡−=𝑟_𝑡−𝑘𝜈_𝑡𝑑𝑡+𝜎_1𝑡𝑑𝑊_1𝑡+𝑒^𝑌𝑡−−1𝑑𝑁(𝑡),𝑑𝑟_𝑡=𝑎_𝑡−𝑏_𝑡𝑟_𝑡𝑑𝑡+𝜎_2𝑡𝑑𝑊_2𝑡, (2) where 𝑊_1𝑡, 𝑊_2𝑡 are standard Brownian motions and 𝜎_1𝑡, 𝜎_2𝑡 are the stochastic volatility of the stock and the interest rate, respectively. {𝑁_𝑡}_{0≤𝑡≤𝑇} is Poisson process with the stochastic jump intensity {𝜈_𝑡}_{0≤𝑡≤𝑇}, and the jump amplitude is controlled by {𝑌_𝑡}. 𝑌_𝑠 and 𝑌_𝑡 for 𝑠≠𝑡 independently identify distribution, and write 𝑘=𝐸𝑒^𝑌𝑖−1. Moreover, 𝑌_𝑡, 𝑁_𝑡 are assumed to be mutually independent. 𝜎_1𝑡, 𝜎_2𝑡, 𝑣_𝑡, 𝑎_𝑡, 𝑏_𝑡 are controlled by 𝜉_𝑡, that is, 𝜎_1𝑡=⟨𝜎₁,𝜉_𝑡⟩,𝜎₁=𝜎₁₁,𝜎₁₂,…,𝜎_1𝑛∈(0,∞)^𝑛,𝜎_2𝑡=⟨𝜎₂,𝜉_𝑡⟩,𝜎₂=𝜎₂₁,𝜎₂₂,…,𝜎_2𝑛∈(0,∞)^𝑛,𝜈_𝑡=⟨𝜈,𝜉_𝑡⟩,𝜈=𝜈₁,𝜈₂,…,𝜈_𝑛∈(0,∞)^𝑛,𝑎_𝑡=⟨𝑎,𝜉_𝑡⟩,𝑎=𝑎₁,𝑎₂,…,𝑎_𝑛∈(0,∞)^𝑛,𝑏_𝑡=⟨𝑏,𝜉_𝑡⟩,𝑏=𝑏₁,𝑏₂,…,𝑏_𝑛∈(0,∞)^𝑛, (3) where ⟨⋅,⋅⟩ denotes the inner product in 𝑅^𝑛. Let 𝜏 denote the default time of the writer of the option with default intensity process 𝜆_𝑡, and 𝜆_𝑡 is given by 𝑑𝜆_𝑡=𝛼_𝑡−𝛽_𝑡𝜆_𝑡𝑑𝑡+𝜎_3𝑡𝑑𝑊_3𝑡, (4) where 𝑊_3𝑡 is standard Brownian motion and 𝜎_3𝑡 is the stochastic volatility of default intensity. 𝜎_3𝑡, 𝛼_𝑡,  and 𝛽_𝑡 are also controlled by 𝜉_𝑡 and satisfy 𝜎_3𝑡=⟨𝜎₃,𝜉_𝑡⟩,𝜎₃=𝜎₃₁,𝜎₃₂,…,𝜎_3𝑛∈(0,∞)^𝑛,𝛼_𝑡=⟨𝛼,𝜉_𝑡⟩,𝛼=𝛼₁,𝛼₂,…,𝛼_𝑛∈(0,∞)^𝑛,𝛽_𝑡=⟨𝛽,𝜉_𝑡⟩,𝛽=𝛽₁,𝛽₂,…,𝛽_𝑛∈(0,∞)^𝑛. (5) Moreover, we assume that 𝑁_𝑡,𝑌_𝑡 are independent of 𝑊_1𝑡, 𝑊_2𝑡, and 𝑊_3𝑡 and the covariance matrix of the Brownian motion (𝑊_1𝑡,𝑊_2𝑡,𝑊_3𝑡) is 1𝜌₁₂𝜌₁₃𝜌₁₂1𝜌₂₃𝜌₁₃𝜌₂₃1𝑡. (6) The filtration 𝓕_𝑡 is generated by 𝓕_𝑡=𝓕^𝑆_𝑡∨𝓕^𝑟_𝑡∨𝓕^𝜆_𝑡∨𝓕^𝜉_𝑇∨𝓗_𝑡, where 𝓕^𝑆_𝑡=𝜎(𝑆_𝑠,0≤𝑠≤𝑡), 𝓕^𝑟_𝑡=𝜎(𝑟_𝑠,0≤𝑠≤𝑡), 𝓕^𝜆_𝑡=𝜎(𝜆_𝑠,0≤𝑠≤𝑡), 𝓕^𝜉_𝑡=𝜎(𝜉_𝑠,0≤𝑠≤𝑡), and 𝓗_𝑡=𝜎(1_{(𝜏≤𝑠)},𝑠≤𝑡).

3. Pricing Options with Credit Risk

We consider the case of a European call option with credit risk. Assume that the recovery rate is a constant 𝑤. When the seller of option defaults, the payoff is given by 𝑤 times the payoff of the default-free option at maturity. Therefore, by the risk neutral pricing theorem, the valuation of the European call option at time 𝑡, with strike price 𝐾 and maturity 𝑇, is given by𝐶()𝑒𝑡,𝑇=𝐸^{−∫𝑇𝑡𝑟𝑠𝑑𝑠}𝑤𝑆_𝑇−𝐾⁺1_{(𝜏≤𝑇)}+𝑆_𝑇−𝐾⁺1_(𝜏>𝑇)∣𝓕_𝑡=𝐸𝑒^{−∫𝑇𝑡𝑟𝑠𝑑𝑠}𝑤𝑆_𝑇−𝐾⁺+(1−𝑤)𝑆_𝑇−𝐾⁺1_(𝜏>𝑇)∣𝓕_𝑡. (7)Following Lando [15], 𝐸𝑒^{−∫𝑇𝑡𝑟𝑠𝑑𝑠}𝑆_𝑇−𝐾⁺1_(𝜏>𝑇)∣𝓕_𝑡=1_(𝜏>𝑡)𝐸𝑒^{−∫𝑇𝑡(𝑟𝑠+𝜆𝑠)𝑑𝑠}𝑆_𝑇−𝐾⁺∣𝓕_𝑡. (8) We can obtain the following expression: 𝐶(𝑡,𝑇)=𝑤𝐸𝑒^{−∫𝑇𝑡𝑟𝑠𝑑𝑠}𝑆_𝑇−𝐾⁺∣𝓕_𝑡+(1−𝑤)1_(𝜏>𝑡)𝐸𝑒^{−∫𝑇𝑡(𝑟𝑠+𝜆𝑠)𝑑𝑠}𝑆_𝑇−𝐾⁺∣𝓕_𝑡=𝐼₁+𝐼₂. (9) Next, we calculate 𝐼₁ and 𝐼₂, respectively. For 𝑠>𝑡, we have 𝑟_𝑠=𝑒^{−∫𝑠𝑡𝑏𝑢𝑑𝑢}𝑟_𝑡+^𝑠_𝑡𝑒^{−∫𝑠𝑣𝑏𝑢𝑑𝑢}𝑎_𝑣𝑑𝑣+^𝑠_𝑡𝑒^{−∫𝑠𝑣𝑏𝑢𝑑𝑢}𝜎_2𝑣𝑑𝑊_2𝑣. (10) Integrated from 𝑡 to 𝑇 in both sides of (10), ^𝑇_𝑡𝑟_𝑠𝑑𝑠=^𝑇_𝑡𝑒^{−∫𝑠𝑡𝑏𝑢𝑑𝑢}𝑑𝑠𝑟_𝑡+^𝑇_𝑡𝑎_𝑣𝑑𝑣^𝑇_𝑣𝑒^{−∫𝑠𝑣𝑏𝑢𝑑𝑢}+𝑑𝑠_𝑇𝑡𝜎_2𝑣𝑑𝑊_2𝑣^𝑇_𝑣𝑒^{−∫𝑠𝑣𝑏𝑢𝑑𝑢}𝑑𝑠. (11) Let 𝑀(𝑥,𝑦,𝑇)=∫^𝑇_𝑦𝑒^{−∫𝑠𝑦𝑥𝑢𝑑𝑢}𝑑𝑠; then ^𝑇_𝑡𝑟_𝑠𝑑𝑠=𝑀(𝑏,𝑡,𝑇)𝑟_𝑡+^𝑇_𝑡𝑎_𝑣𝑀()+𝑏,𝑣,𝑇𝑑𝑣_𝑇𝑡𝜎_2𝑣𝑀(𝑏,𝑣,𝑇)𝑑𝑊_2𝑣. (12) Similarly, ^𝑇_𝑡𝜆_𝑠𝑑𝑠=𝑀𝛽,𝑡,𝑇𝜆_𝑡+^𝑇_𝑡𝛼_𝑣𝑀+𝛽,𝑣,𝑇𝑑𝑣_𝑇𝑡𝜎_3𝑣𝑀𝛽,𝑣,𝑇𝑑𝑊_3𝑣. (13) From (12) and (13), we have that 𝑍(𝑡,𝑇)∶=𝐸exp−^𝑇_𝑡𝑟_𝑠+𝜆_𝑠𝑑𝑠∣𝓕_𝑡=exp−𝑀(𝑏,𝑡,𝑇)𝑟_𝑡−𝑀𝛽,𝑡,𝑇𝜆_𝑡−^𝑇_𝑡𝑎_𝑢𝑀()−𝑏,𝑢,𝑇𝑑𝑢_𝑇𝑡𝛼_𝑢𝑀1𝛽,𝑢,𝑇𝑑𝑢×exp2^𝑇_𝑡𝜎²_2𝑢𝑀²()+1𝑏,𝑢,𝑇𝑑𝑢2^𝑇_𝑡𝜎²_3𝑢𝑀²𝜌𝛽,𝑢,𝑇𝑑𝑢×exp₂₃^𝑇_𝑡𝜎_2𝑢𝜎_3𝑢𝑀(𝑏,𝑢,𝑇)𝑀𝛽,𝑢,𝑇𝑑𝑢. (14) Thus, we can define the probability measure 𝑄 by 𝑑𝑄𝑑𝑃_𝓕𝑡=exp−∫^𝑇_𝑡𝑟_𝑠+𝜆_𝑠𝑑𝑠𝐸exp−∫^𝑇_𝑡𝑟_𝑠+𝜆_𝑠𝑑𝑠∣𝓕_𝑡−1=exp2^𝑇_𝑡𝜎²_2𝑢𝑀²()−1𝑏,𝑢,𝑇𝑑𝑢2^𝑇_𝑡𝜎²_3𝑢𝑀²𝛽,𝑢,𝑇𝑑𝑢×exp−𝜌₂₃^𝑇_𝑡𝜎_2𝑢𝜎_3𝑢𝑀()𝑀−𝑏,𝑢,𝑇𝛽,𝑢,𝑇𝑑𝑢×exp^𝑇_𝑡𝜎_2𝑣𝑀(𝑏,𝑣,𝑇)𝑑𝑊_2𝑣−^𝑇_𝑡𝜎_3𝑣𝑀𝛽,𝑣,𝑇𝑑𝑊_3𝑣. (15) By Girsanov’s theorem, 𝑑𝑊^𝑄_2𝑡=𝑑𝑊_2𝑡+𝜎_2𝑡𝑀(𝑏,𝑡,𝑇)𝑑𝑡+𝜌₂₃𝜎_3𝑡𝑀𝛽,𝑡,𝑇𝑑𝑡,𝑑𝑊^𝑄_3𝑡=𝑑𝑊_3𝑡+𝜎_3𝑡𝑀𝛽,𝑡,𝑇𝑑𝑡+𝜌₂₃𝜎_2𝑡𝑀(𝑏,𝑡,𝑇)𝑑𝑡,𝑑𝑊^𝑄_1𝑡=𝑑𝑊_1𝑡+𝜌₁₂𝑀₁(𝑡)𝑑𝑡+𝜌₁₃𝑀₂(𝑡)𝑑𝑡, (16) where 𝑀₁(𝑡)=𝜎_2𝑡𝑀(𝑏,𝑡,𝑇)+𝜌₂₃𝜎_3𝑡𝑀,𝑀𝛽,𝑡,𝑇₂(𝑡)=𝜎_3𝑡𝑀𝛽,𝑡,𝑇+𝜌₂₃𝜎_2𝑡𝑀(𝑏,𝑡,𝑇). (17)𝑊^𝑄_1𝑡, 𝑊^𝑄_2𝑡, and 𝑊^𝑄_3𝑡 are standard Brownian motions under probability measure 𝑄, and (𝑊^𝑄_1𝑡,𝑊^𝑄_2𝑡,𝑊^𝑄_3𝑡) has the same covariance matrix as (𝑊_1𝑡,𝑊_2𝑡,𝑊_3𝑡). Therefore, 𝐸𝑒^{−∫𝑇𝑡(𝑟𝑠+𝜆𝑠)𝑑𝑠}𝑆_𝑇−𝐾⁺∣𝓕_𝑡=𝑍(𝑡,𝑇)𝐸^𝑄𝑆_𝑇−𝐾⁺∣𝓕_𝑡. (18) In addition, 𝐸^𝑄𝑆_𝑇−𝐾⁺∣𝓕_𝑡=𝐸^𝑄𝑆_𝑇1_{(𝑆𝑇≥𝐾)}∣𝓕_𝑡−𝐾𝐸^𝑄1_{(𝑆𝑇≥𝐾)}∣𝓕_𝑡. (19) By the solution of SDE (2), 𝑆_𝑇=𝑆_𝑡exp^𝑇_𝑡𝑟_𝑠−𝑘𝜈_𝑠−12𝜎²_1𝑠+𝑑𝑠_𝑇𝑡𝜎_1𝑠𝑑𝑊_1𝑠+^𝑇_𝑡𝑌_𝑠−𝑑𝑁_𝑠. (20) Under 𝑄, 𝑆_𝑇=𝑆_𝑡expΛ(𝑡,𝑇)+^𝑇_𝑡𝜎_1𝑠𝑑𝑊^𝑄_1𝑠+^𝑇_𝑡𝜎_2𝑠𝑀(𝑏,𝑠,𝑇)𝑑𝑊^𝑄_2𝑠+^𝑇_𝑡𝑌_𝑠−𝑑𝑁_𝑠, (21) where Λ(𝑡,𝑇)=−^𝑇_𝑡𝑘𝜈_𝑠+12𝜎²_1𝑠𝑑𝑠+𝑀(𝑏,𝑡,𝑇)𝑟_𝑡+^𝑇_𝑡𝑎_𝑠𝑀(𝑏,𝑠,𝑇)𝑑𝑠−^𝑇_𝑡𝜎²_2𝑠𝑀²(𝑏,𝑠,𝑇)𝑑𝑠−𝜌₁₂^𝑇_𝑡𝜎_1𝑠𝑀₁(𝑠)𝑑𝑠−𝜌₁₃^𝑇_𝑡𝜎_1𝑠𝑀₂(𝑠)𝑑𝑠−𝜌₂₃^𝑇_𝑡𝜎_2𝑠𝜎_3𝑠𝑀(𝑏,𝑠,𝑇)𝑀𝛽,𝑠,𝑇𝑑𝑠. (22) So, 𝑃^𝑄𝑆_𝑇≥𝐾∣𝓕_𝑡=𝑃^𝑄𝑇_𝑡𝜎_1𝑠𝑑𝑊^𝑄_1𝑠+^𝑇_𝑡𝜎_2𝑠𝑀(𝑏,𝑠,𝑇)𝑑𝑊^𝑄_2𝑠≥ln𝐾/𝑆_𝑡−Λ(𝑡,𝑇)−^𝑇_𝑡𝑌_𝑠𝑑𝑁_𝑠=𝑃^𝑄∫^𝑇_𝑡𝜎_1𝑠𝑑𝑊^𝑄_1𝑠+∫^𝑇_𝑡𝜎_2𝑠𝑀(𝑏,𝑠,𝑇)𝑑𝑊^𝑄_2𝑠√Δ≤ln𝑆_𝑡/𝐾+Λ(𝑡,𝑇)+∫^𝑇_𝑡𝑌_𝑠𝑑𝑁_𝑠√Δ=^∞_𝑛=0∫^𝑇_𝑡𝜈_𝑠𝑑𝑠^𝑛𝑒𝑛!^{−∫𝑇𝑡𝜈𝑠𝑑𝑠}𝐸𝑁𝑑₁, (23) where Δ=^𝑇_𝑡𝜎²_1𝑠𝑑𝑠+^𝑇_𝑡𝜎²_2𝑠𝑀²(𝑏,𝑠,𝑇)𝑑𝑠+2𝜌₁₂^𝑇_𝑡𝜎_1𝑠𝜎_2𝑠𝑀()𝑑𝑏,𝑠,𝑇𝑑𝑠,₁=ln𝑆_𝑡/𝐾+Λ(𝑡,𝑇)+∑^𝑛_𝑗=1𝑌_𝑗√Δ. (24)𝐸(⋅) is the expectation of 𝑌_𝑗 under 𝑃, and 𝑁(⋅) is distribution function of standard normal distribution. On the other hand, let 𝓛(𝑡,𝑇)∶=𝐸^𝑄𝑆_𝑇∣𝓕_𝑡=𝑆_𝑡𝑒^{Λ(𝑡,𝑇)}1exp2^𝑇_𝑡𝜎²_1𝑠1𝑑𝑠+2^𝑇_𝑡𝜎²_2𝑠𝑀²()𝜌𝑏,𝑠,𝑇𝑑𝑠×exp₁₂^𝑇_𝑡𝜎_1𝑠𝜎_2𝑠𝑀(𝑏,𝑠,𝑇)𝑑𝑠+𝑘^𝑇_𝑡𝜈_𝑠𝑑𝑠. (25) We can define the probability measure 𝑅 as follows: 𝑑𝑅𝑑𝑄_𝓕𝑡=𝑆_𝑇𝐸^𝑄𝑆_𝑇∣𝓕_𝑡=exp^𝑇_𝑡𝜎_1𝑠𝑑𝑊^𝑄_1𝑠+^𝑇_𝑡𝜎_2𝑠𝑀(𝑏,𝑠,𝑇)𝑑𝑊^𝑄_2𝑠−12^𝑇_𝑡𝜎²_1𝑠−1𝑑𝑠×exp2^𝑇_𝑡𝜎²_2𝑠𝑀²(𝑏,𝑠,𝑇)𝑑𝑠−𝜌₁₂^𝑇_𝑡𝜎_1𝑠𝜎_2𝑠𝑀(𝑏,𝑠,𝑇)𝑑𝑠×exp^𝑇_𝑡𝑌_𝑠𝑑𝑁_𝑠−𝑘^𝑇_𝑡𝜈_𝑠𝑑𝑠. (26) By Girsanov’s theorem, we have that 𝑑𝑊^𝑅_1𝑡=𝑑𝑊^𝑄_1𝑡−𝜎_1𝑡𝑑𝑡−𝜌₁₂𝜎_2𝑡𝑀(𝑏,𝑡,𝑇)𝑑𝑡,𝑑𝑊^𝑅_2𝑡=𝑑𝑊^𝑄_2𝑡−𝜎_2𝑡𝑀(𝑏,𝑡,𝑇)𝑑𝑡−𝜌₁₂𝜎_1𝑡𝑑𝑡, (27) where, under 𝑅, 𝑊^𝑅_1𝑡, 𝑊^𝑅_2𝑡 are standard Brownian motions and their correlation coefficient is 𝜌₁₂, 𝑁_𝑡 is Poisson process with intensity (𝑘+1)𝜈_𝑡, and the density function of 𝑌₁ is 𝑒^𝑦𝑓(𝑦)/(𝑘+1), where 𝑓(⋅) is the density function of 𝑌₁ under 𝑃. Since, under 𝑅, 𝑆_𝑇=𝑆_𝑡expΛ(𝑡,𝑇)+Δ+^𝑇_𝑡𝜎_1𝑠𝑑𝑊^𝑅_1𝑠+^𝑇_𝑡𝜎_2𝑠𝑀(𝑏,𝑠,𝑇)𝑑𝑊^𝑅_2𝑠+^𝑇_𝑡𝑌_𝑠−𝑑𝑁_𝑠, (28) then 𝐸^𝑄𝑆_𝑇1_{(𝑆𝑇≥𝐾)}∣𝓕_𝑡=𝓛(𝑡,𝑇)𝑃^𝑅𝑆_𝑇≥𝐾∣𝓕_𝑡=𝓛(𝑡,𝑇)𝑃^𝑅𝑇_𝑡𝜎_1𝑠𝑑𝑊^𝑅_1𝑠+^𝑇_𝑡𝜎_2𝑠𝑀(𝑏,𝑠,𝑇)𝑑𝑊^𝑅_2𝑠𝐾≥ln𝑆_𝑡−Λ(𝑡,𝑇)−Δ−^𝑇_𝑡𝑌_𝑠𝑑𝑁_𝑠=𝓛(𝑡,𝑇)𝑃^𝑅∫^𝑇_𝑡𝜎_1𝑠𝑑𝑊^𝑅_1𝑠+∫^𝑇_𝑡𝜎_2𝑠𝑀(𝑏,𝑠,𝑇)𝑑𝑊^𝑅_2𝑠√Δ≤ln𝑆_𝑡/𝐾+Λ(𝑡,𝑇)+Δ+∫^𝑇_𝑡𝑌_𝑠𝑑𝑁_𝑠√Δ=𝓛(𝑡,𝑇)^∞_𝑛=0∫^𝑇_𝑡𝜈_𝑠𝑑𝑠^𝑛𝑒𝑛!^{−∫𝑇𝑡𝜈𝑠𝑑𝑠}𝐸^𝑅𝑁𝑑₂, (29) where 𝑑₂=ln𝑆_𝑡/𝐾+Λ(𝑡,𝑇)+Δ+∑^𝑛_𝑗=1𝑌_𝑗√Δ=𝑑₁+√Δ (30) and 𝐸^𝑅(⋅) is the expectation of 𝑌_𝑗 under 𝑅. By (23) and (29), we have the following lemma.

Lemma 1. Consider the Following: 𝐸𝑒^{−∫𝑇𝑡(𝑟𝑠+𝜆𝑠)𝑑𝑠}𝑆_𝑇−𝐾⁺∣𝓕_𝑡()×=𝑍𝑡,𝑇_∞𝑛=0∫^𝑇_𝑡𝜈_𝑠𝑑𝑠^𝑛𝑛!×𝑒^{−∫𝑇𝑡𝜈𝑠𝑑𝑠}𝓛(𝑡,𝑇)𝐸^𝑅𝑁𝑑₂−𝐾𝐸𝑁𝑑₁. (31) In the following; in order to calculate 𝐼₁, we write 𝐸𝑒^{−∫𝑇𝑡𝑟𝑠𝑑𝑠}𝑆_𝑇−𝐾⁺∣𝓕_𝑡=𝐸𝑒^{−∫𝑇𝑡𝑟𝑠𝑑𝑠}𝑆_𝑇1_{(𝑆𝑇≥𝐾)}∣𝓕_𝑡−𝐾𝐸𝑒^{−∫𝑇𝑡𝑟𝑠𝑑𝑠}1_{(𝑆𝑇≥𝐾)}∣𝓕_𝑡=𝐼₃−𝐼₄. (32) Denote by 𝑃(𝑡,𝑇) the value at time 𝑡 of a 𝑇 maturity zero coupon bond whose face value is 1. Then 𝑃(𝑡,𝑇)=𝐸𝑒^{−∫𝑇𝑡𝑟𝑠𝑑𝑠}∣𝓕_𝑡. (33) From (12), 𝑃(𝑡,𝑇)=exp−𝑀(𝑏,𝑡,𝑇)𝑟_𝑡−^𝑇_𝑡𝑎_𝑣𝑀()+1𝑏,𝑣,𝑇𝑑𝑣2^𝑇_𝑡𝜎²_2𝑣𝑀²(),()𝑏,𝑣,𝑇𝑑𝑣𝑑𝑃𝑡,𝑇𝑃(𝑡,𝑇)=𝑟_𝑡𝑑𝑡−𝑀(𝑏,𝑡,𝑇)𝜎_2𝑡𝑑𝑊_2𝑡. (34)(35) Define the Radon-Nikodym derivative given by 𝑑𝑄^𝑇=𝑒𝑑𝑃𝐸𝑒^{−∫𝑇𝑡𝑟𝑠𝑑𝑠}∣^{−∫𝑇𝑡𝑟𝑠𝑑𝑠}𝓕_𝑡=exp−^𝑇_𝑡𝑀(𝑏,𝑣,𝑇)𝜎_2𝑣𝑑𝑊_2𝑣−12^𝑇_𝑡𝜎²_2𝑣𝑀²(𝑏,𝑣,𝑇)𝑑𝑣, (36) and by Girsanov’s theorem, under 𝑄^𝑇, 𝑑𝑊^𝑄𝑇_2𝑡=𝑑𝑊_2𝑡+𝜎_2𝑡𝑀(𝑏,𝑡,𝑇)𝑑𝑡𝑑𝑊^𝑄𝑇_1𝑡=𝑑𝑊_1𝑡+𝜌₁₂𝜎_2𝑡𝑀(𝑏,𝑡,𝑇)𝑑𝑡. (37) Thus 𝑆_𝑇 can be rewritten as 𝑆_𝑇=𝑆_𝑡exp𝓐(𝑡,𝑇)+^𝑇_𝑡𝑀(𝑏,𝑣,𝑇)𝜎_2𝑣𝑑𝑊^𝑄𝑇_2𝑣+^𝑇_𝑡𝜎_1𝑣𝑑𝑊^𝑄𝑇_1𝑣+^𝑇_𝑡𝑌_𝑠−𝑑𝑁_𝑠, (38) where 𝓐(𝑡,𝑇)=−^𝑇_𝑡𝑘𝜈_𝑠+12𝜎²_1𝑠𝑑𝑠+𝑀(𝑏,𝑡,𝑇)𝑟_𝑡+^𝑇_𝑡𝑎_𝑠𝑀(𝑏,𝑠,𝑇)𝑑𝑠−^𝑇_𝑡𝜎²_2𝑠𝑀²(𝑏,𝑠,𝑇)𝑑𝑠−𝜌₁₂^𝑇_𝑡𝜎_1𝑠𝜎_2𝑠𝑀(𝑏,𝑠,𝑇)𝑑𝑠. (39) Then 𝑃^𝑄𝑇𝑆_𝑇≥𝐾∣𝓕_𝑡=𝑃^𝑄𝑇𝑇_𝑡𝜎_1𝑠𝑑𝑊^𝑄𝑇_1𝑠+^𝑇_𝑡𝜎_2𝑠𝑀(𝑏,𝑠,𝑇)𝑑𝑊^𝑄𝑇_2𝑠𝐾≥ln𝑆_𝑡−𝓐(𝑡,𝑇)−^𝑇_𝑡𝑌_𝑠−𝑑𝑁_𝑠=𝑃^𝑄𝑇∫^𝑇_𝑡𝜎_1𝑠𝑑𝑊^𝑄𝑇_1𝑠+∫^𝑇_𝑡𝜎_2𝑠𝑀(𝑏,𝑠,𝑇)𝑑𝑊^𝑄𝑇_2𝑠√Δ≤ln𝑆_𝑡/𝐾+𝓐(𝑡,𝑇)+∫^𝑇_𝑡𝑌_𝑠−𝑑𝑁_𝑠√Δ=^∞_𝑛=0∫^𝑇_𝑡𝜈_𝑠𝑑𝑠^𝑛𝑒𝑛!^{−∫𝑇𝑡𝜈𝑠𝑑𝑠}𝐸𝑁𝑑₃, (40) where 𝐸(⋅) is the expectation of 𝑌_𝑗 under 𝑃, 𝑑₃=ln𝑆_𝑡/𝐾+𝓐(𝑡,𝑇)+∑^𝑛_𝑗=1𝑌_𝑗√Δ. (41) Therefore, 𝐼₄=𝐾𝑃(𝑡,𝑇)𝐸^𝑄𝑇1_{(𝑆𝑇≥𝐾)}∣𝓕_𝑡=𝐾𝑃(𝑡,𝑇)^∞_𝑛=0∫^𝑇_𝑡𝜈_𝑠𝑑𝑠^𝑛𝑒𝑛!^{−∫𝑇𝑡𝜈𝑠𝑑𝑠}𝐸𝑁𝑑₃. (42) Finally, let 𝐴(𝑡,𝑇)∶=𝐸𝑒^{−∫𝑇𝑡𝑟𝑠𝑑𝑠}𝑆_𝑇∣𝓕_𝑡=𝑆_𝑡exp−^𝑇_𝑡𝑘𝜈_𝑠+12𝜎²_1𝑠+1𝑑𝑠2^𝑇_𝑡𝜎²_1𝑠𝑑𝑠+𝑘^𝑇_𝑡𝜈_𝑠𝑑𝑠. (43) We define 𝑑𝑄^𝑆=𝑒𝑑𝑃^{−∫𝑇𝑡𝑟𝑠𝑑𝑠}𝑆_𝑇𝐸𝑒^{−∫𝑇𝑡𝑟𝑠𝑑𝑠}𝑆_𝑇∣𝓕_𝑡=exp^𝑇_𝑡𝜎_1𝑣𝑑𝑊_1𝑣−12^𝑇_𝑡𝜎²_2𝑣𝑑𝑣+^𝑇_𝑡𝑌_𝑠−𝑑𝑁(𝑠)−𝑘^𝑇_𝑡𝜈_𝑠𝑑𝑠, (44) and by Girsanov’s theorem, under 𝑄^𝑆, 𝑑𝑊^𝑄𝑆_1𝑡=𝑑𝑊_1𝑡−𝜎_1𝑡𝑑𝑡𝑑𝑊^𝑄𝑆_2𝑡=𝑑𝑊_2𝑡−𝜌₁₂𝜎_1𝑡𝑑𝑡, (45) and, under 𝑄^𝑆, 𝑁_𝑡 is Poisson process with intensity (𝑘+1)𝜈_𝑡, and the density function of 𝑌₁ is 𝑒^𝑦𝑓(𝑦)/(𝑘+1), where 𝑓(⋅) is the density function of 𝑌₁ under 𝑃. Since, under 𝑄^𝑆, 𝑆_𝑇=𝑆_𝑡exp𝓐(𝑡,𝑇)+Δ+^𝑇_𝑡𝑀(𝑏,𝑣,𝑇)𝜎_2𝑣𝑑𝑊^𝑄𝑆_2𝑣+^𝑇_𝑡𝜎_1𝑣𝑑𝑊^𝑄𝑆_1𝑣+^𝑇_𝑡𝑌_𝑠−𝑑𝑁_𝑠, (46) then 𝑃^𝑄𝑆𝑆_𝑇≥𝐾∣𝓕_𝑡=𝑃^𝑄𝑆𝑇_𝑡𝜎_1𝑠𝑑𝑊^𝑄𝑆_1𝑠+^𝑇_𝑡𝜎_2𝑠𝑀(𝑏,𝑠,𝑇)𝑑𝑊^𝑄𝑆_2𝑠𝐾≥ln𝑆_𝑡−𝓐(𝑡,𝑇)−Δ−^𝑇_𝑡𝑌_𝑠−𝑑𝑁_𝑠=𝑃^𝑄𝑇∫^𝑇_𝑡𝜎_1𝑠𝑑𝑊^𝑄𝑆_1𝑠+∫^𝑇_𝑡𝜎_2𝑠𝑀(𝑏,𝑠,𝑇)𝑑𝑊^𝑄𝑆_2𝑠√Δ≤ln𝑆_𝑡/𝐾+𝓐(𝑡,𝑇)+Δ+∫^𝑇_𝑡𝑌_𝑠−𝑑𝑁_𝑠√Δ=^∞_𝑛=0∫^𝑇_𝑡𝜈_𝑠𝑑𝑠^𝑛𝑒𝑛!^{−∫𝑇𝑡𝜈𝑠𝑑𝑠}𝐸^𝑆𝑁𝑑₄, (47) where 𝐸^𝑆(⋅) is the expectation of 𝑌_𝑗 under 𝑄^𝑆, 𝑑₄=ln𝑆_𝑡/𝐾+𝓐(𝑡,𝑇)+Δ+∑^𝑛_𝑗=1𝑌_𝑗√Δ=𝑑₃+√Δ. (48) From (42) and (33), we can conclude the following lemma.

Lemma 2. Consider the following: 𝐸𝑒^{−∫𝑇𝑡𝑟𝑠𝑑𝑠}𝑆_𝑇−𝐾⁺∣𝓕_𝑡=^∞_𝑛=0∫^𝑇_𝑡𝜈_𝑠𝑑𝑠^𝑛𝑛!×𝑒^{−∫𝑇𝑡𝜈𝑠𝑑𝑠}𝐴(𝑡,𝑇)𝐸^𝑆𝑁𝑑₄−𝐾𝑃(𝑡,𝑇)𝐸𝑁𝑑₃. (49)
Combined with the previous lemmas, the price of European call option at time 𝑡 is given by the following theorem.

Theorem 3. The price of European call option with credit risk at time 𝑡 is 𝐶()=𝑡,𝑇_∞𝑛=0∫^𝑇_𝑡𝜈_𝑠𝑑𝑠^𝑛𝑛!×𝑒^{−∫𝑇𝑡𝜈𝑠𝑑𝑠}𝑤𝐴(𝑡,𝑇)𝐸^𝑆𝑁𝑑₄−𝐾𝑃(𝑡,𝑇)𝐸𝑁𝑑₃+(1−𝑤)1_(𝜏>𝑡)𝑍()×𝓛()𝐸𝑡,𝑇𝑡,𝑇^𝑅𝑁𝑑₂−𝐾𝐸𝑁𝑑₁. (50)

4. Numerical Analysis

In this section, we will employ Monte Carlo simulation and perform a numerical analysis for European-style call option with credit risk under the Markov-modulated jump-diffusion model. We consider that the Markov chain 𝜉 has two states, which means that macroeconomic shifts between the two states: 𝑒₁  (“boom” or “good”) and 𝑒₂ (“recession” or “bad”). We assume that the current economy is in boom and that the transition probability matrix of the two-state Markov chain 𝜉 is given by 𝑝₁₁𝑝₁₂𝑝₂₁𝑝₂₂=0.70.30.20.8. (51) Assume that 𝑌_𝑡 is normally distributed with mean 𝜇_𝐽 and standard deviation 𝜎_𝐽. We will adopt the specimen values for the model parameters as Table 1. We consider a range of spot-to-strike ratios 𝑆₀/𝐾 from 0.8 to 1.2 and assume that one year has 252 trading days. We perform 10 000 simulations for computing the option price.

Table 1 

The parameter values.

For each 𝜔=0.4, 0.6, 0.8, 1, we consider the impact of the spot-to-strike ratio on the option price. From Figure 1, we observe that the option price increases as the spot-to-strike ratio increases. We can also see that the greater the 𝜔, the greater the option price. When 𝜔=1, it follows that there is not default risk. It is a result of the fact that the payoff at the maturity will increase as the recovery rate increases. Figure 2 depicts the plot of the option price against the spot-to-strike ratio for each maturity 𝑇=0.5, 1, 1.5. From these plots, we find that the longer the maturities, the greater the option price.

Figure 1 

The option price with different recovery rate for 𝑇=1, 𝜌12=0.7, 𝜌13=0.5, and 𝜌23=0.6.

Figure 2 

The option price with different maturity 𝑇 for 𝜔=0.4, 𝜌12=0.7, 𝜌13=0.5, and 𝜌23=0.6.

In the following, we compare the option price with different correlation coefficients for fixed maturity 𝑇=1 and recovery rate 𝜔=0.4. Figure 3 illustrates that option price increases as correlation coefficient 𝜌₁₂ increases. From Figure 4, we can also see that option price decreases as correlation coefficient 𝜌₁₃ increases.

Figure 3 

The option price with different correlation coefficients 𝜌12 for 𝑇=1, 𝜔=0.4, 𝜌13=0.5, and 𝜌23=0.6.

Figure 4 

The option price with different correlation coefficients 𝜌13 for 𝑇=1, 𝜔=0.4, 𝜌12=0.7, and 𝜌23=0.6.

In Figure 5, we present how the option prices vary with the changes of the annual jump intensity 𝜈. Figure 5 displays a large change in the option price due to the variation of the jump intensity. The option price increases as jump intensity increases.

Figure 5 

The option price with different 𝜈 for 𝜔=0.4, 𝜌12=0.7, 𝜌13=0.5, and 𝜌23=0.6.

Finally, we investigate the difference of the option price in Markov-modulated model and non-Markov-modulated models. From Figure 6, we can observe that the call option price in good economy is lower than that in bad economy. The reason is that when economy is bad, the volatility of the risky asset is great so that the option price is higher. In our model, we assume that the current economy is good, so Markov-modulated model is close to the model for a good economy, while the two plots and the model for a bad economy are relatively far apart. In Markov-modulated model, we take into consideration the changes of the state of the economy, so Markov-modulated model is more in accord with the reality for the pricing of the defaultable options.

Figure 6 

The option price in Markov-modulated model and non-Markov-modulated models.

5. Conclusion

The pricing of European option with credit risk in a reduced form model was studied, while the stock price was driven by Markov-modulated jump-diffusion models. The interest rate and the default intensity followed the Vasicek models, and the parameters were also controlled by the same Markov chain. Compared with most of the credit risk models, the main advantage is that we incorporated Markov-modulated rates into the models. We applied Girsanov’s theorem to obtain the equivalent martingale measure and derived the closed form formula for the valuation of the European option. Finally, from the numerical illustrations, we obtain the effects of the recovery rate, the maturity, the correlation coefficients, and the jump intensity of stock on the option price.

Acknowledgments

The authors would like to thank the referee for many helpful and valuable comments and suggestions. Jinzhi Li would like to acknowledge the National Commission of Ethnic Affair (no. 12ZYZ003) and the Ministry of Education of Humanities and Social Sciences Project (no. 11YJC790102).