1.Introduction JinzhiLi andShixiaMa PricingOptionswithCreditRiskinMarkovianRegime-SwitchingMarkets ResearchArticle

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Volume 2013, Article ID 621371,9pages http://dx.doi.org/10.1155/2013/621371

Research Article

Pricing Options with Credit Risk in Markovian Regime-Switching Markets

Jinzhi Li

¹

and Shixia Ma

²

1College of Sciences, Minzu University of China, Beijing 100081, China

2College of Sciences, Hebei University of Technology, Tianjin 300401, China

Correspondence should be addressed to Jinzhi Li; [email protected] Received 16 February 2013; Revised 16 May 2013; Accepted 20 May 2013 Academic Editor: Shan Zhao

Copyright © 2013 J. Li and S. Ma. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 Gˆarleanu [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,

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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 inSection 4. The concluding remarks are contained inSection 5.

2. The Model Description

Let (Ω,F, 𝑃) be a complete probability space, where 𝑃 is a neutral-risk probability measure. Define𝜉 = {𝜉_𝑡, 𝑡 ≥ 0}

on(Ω,F, 𝑃)as a continuous-time, finite state Markov chain with𝑛-state space𝐸. We interpret the state of𝜉as 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 of𝜉to 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) where𝑄 = (𝑞_𝑖𝑗)𝑖,𝑗=1,2,...,𝑛is𝑄-matrix of𝜉and𝑀 = {𝑀_𝑡}_{0≤𝑡≤𝑇} is an 𝑅^𝑛-valued martingale with respect to the filtration generated by{𝜉_𝑡, 0 ≤ 𝑡 ≤ 𝑇}under𝑃. Suppose that the stock price𝑆_𝑡 and interest rate𝑟_𝑡 satisfy the following stochastic differential equations (SDE):

𝑑𝑆_𝑡

𝑆_𝑡− = (𝑟_𝑡− 𝑘]_𝑡) 𝑑𝑡 + 𝜎_1𝑡𝑑𝑊_1𝑡+ (𝑒^𝑌^𝑡− − 1) 𝑑𝑁 (𝑡) , 𝑑𝑟_𝑡= (𝑎_𝑡− 𝑏_𝑡𝑟_𝑡) 𝑑𝑡 + 𝜎_2𝑡𝑑𝑊_2𝑡,

(2)

where𝑊_1𝑡,𝑊_2𝑡are standard Brownian motions and𝜎_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{]_𝑡}_{0≤𝑡≤𝑇}, and the jump amplitude is controlled by{𝑌_𝑡}.𝑌_𝑠and𝑌_𝑡for𝑠 ̸= 𝑡independently identify distribution, and write𝑘 = 𝐸𝑒^𝑌^𝑖 − 1. Moreover,𝑌_𝑡,𝑁_𝑡are assumed to be mutually independent.𝜎_1𝑡, 𝜎_2𝑡,V_𝑡, 𝑎_𝑡, 𝑏_𝑡are 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,

]_𝑡= ⟨], 𝜉_𝑡⟩, ]= (]₁,]₂, . . . ,]_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 in𝑅^𝑛. Let𝜏denote the default time of the writer of the option with default intensity process𝜆_𝑡, and𝜆_𝑡is given by

𝑑𝜆_𝑡= (𝛼_𝑡− 𝛽_𝑡𝜆_𝑡) 𝑑𝑡 + 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,and𝛽_𝑡are also controlled by𝜉_𝑡and 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 that𝑁_𝑡, 𝑌_𝑡are independent of𝑊_1𝑡,𝑊_2𝑡, and𝑊_3𝑡and the covariance matrix of the Brownian motion (𝑊_1𝑡, 𝑊_2𝑡, 𝑊_3𝑡)is

(1 𝜌₁₂ 𝜌₁₃ 𝜌₁₂ 1 𝜌₂₃

𝜌₁₃ 𝜌₂₃ 1 ) 𝑡. (6)

The filtrationF_𝑡is generated byF_𝑡=F^𝑆_𝑡∨F^𝑟_𝑡∨F^𝜆_𝑡∨F^𝜉_𝑇∨ H_𝑡, whereF^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), andH_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 constant𝑤. When the seller of option defaults, the payoff is given by𝑤times the payoff of the default-free option at maturity. Therefore, by the risk neutral pricing theorem, the valuation of the European call option at time𝑡, with strike price𝐾and maturity𝑇, is given by

𝐶 (𝑡, 𝑇)

= 𝐸 [𝑒^{− ∫}^𝑡^𝑇^𝑟^𝑠^𝑑𝑠(𝑤(𝑆_𝑇− 𝐾)⁺1_{(𝜏≤𝑇)}+ (𝑆_𝑇− 𝐾)⁺1_(𝜏>𝑇)) |F_𝑡]

= 𝐸 [𝑒^{− ∫}^𝑡^𝑇^𝑟^𝑠^𝑑𝑠(𝑤(𝑆_𝑇−𝐾)⁺+(1−𝑤) (𝑆_𝑇−𝐾)⁺1_(𝜏>𝑇)) |F_𝑡] . (7) Following Lando [15],

𝐸 [𝑒^{− ∫}^𝑡^𝑇^𝑟^𝑠^𝑑𝑠(𝑆_𝑇− 𝐾)⁺1_(𝜏>𝑇)|F_𝑡]

= 1_(𝜏>𝑡)𝐸 [𝑒^{− ∫}^𝑡^𝑇^(𝑟^𝑠^+𝜆^𝑠^)𝑑𝑠(𝑆_𝑇− 𝐾)⁺|F_𝑡] . (8)

We can obtain the following expression:

𝐶 (𝑡, 𝑇) = 𝑤𝐸 [𝑒^{− ∫}^𝑡^𝑇^𝑟^𝑠^𝑑𝑠(𝑆_𝑇− 𝐾)⁺ |F_𝑡]

+ (1 − 𝑤) 1_(𝜏>𝑡)𝐸 [𝑒^{− ∫}^𝑡^𝑇^(𝑟^𝑠^+𝜆^𝑠^)𝑑𝑠(𝑆_𝑇− 𝐾)⁺ |F_𝑡]

= 𝐼₁+ 𝐼₂.

(9) Next, we calculate𝐼₁and𝐼₂, respectively. For𝑠 > 𝑡, we have

𝑟_𝑠= 𝑒^{− ∫}^𝑡^𝑠^𝑏^𝑢^𝑑𝑢𝑟_𝑡+ ∫^𝑠

𝑡 𝑒^{− ∫}^V^𝑠^𝑏^𝑢^𝑑𝑢𝑎_V𝑑V+ ∫^𝑠

𝑡 𝑒^{− ∫}^V^𝑠^𝑏^𝑢^𝑑𝑢𝜎_2V𝑑𝑊_2V. (10)

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Integrated from𝑡to𝑇in both sides of (10),

∫^𝑇

𝑡 𝑟_𝑠𝑑𝑠 = ∫^𝑇

𝑡 𝑒^{− ∫}^𝑡^𝑠^𝑏^𝑢^𝑑𝑢𝑑𝑠𝑟_𝑡+ ∫^𝑇

𝑡 𝑎_V𝑑V∫^𝑇

V 𝑒^{− ∫}^V^𝑠^𝑏^𝑢^𝑑𝑢𝑑𝑠 + ∫^𝑇

𝑡 𝜎_2V𝑑𝑊_2V∫^𝑇

V 𝑒^{− ∫}^V^𝑠^𝑏^𝑢^𝑑𝑢𝑑𝑠.

(11)

Let𝑀(𝑥, 𝑦, 𝑇) = ∫_𝑦^𝑇𝑒^{− ∫}^𝑦^𝑠^𝑥^𝑢^𝑑𝑢𝑑𝑠; then

∫^𝑇

𝑡 𝑟_𝑠𝑑𝑠 = 𝑀 (𝑏, 𝑡, 𝑇) 𝑟_𝑡+ ∫^𝑇

𝑡 𝑎_V𝑀 (𝑏,V, 𝑇) 𝑑V + ∫^𝑇

𝑡 𝜎_2V𝑀 (𝑏,V, 𝑇) 𝑑𝑊_2V.

(12)

Similarly,

∫^𝑇

𝑡 𝜆_𝑠𝑑𝑠 = 𝑀 (𝛽, 𝑡, 𝑇) 𝜆_𝑡+ ∫^𝑇

𝑡 𝛼_V𝑀 (𝛽,V, 𝑇) 𝑑V + ∫^𝑇

𝑡 𝜎_3V𝑀 (𝛽,V, 𝑇) 𝑑𝑊_3V.

(13)

From (12) and (13), we have that

𝑍 (𝑡, 𝑇) := 𝐸 [exp{− ∫^𝑇

𝑡 (𝑟_𝑠+ 𝜆_𝑠) 𝑑𝑠} |F_𝑡]

= exp{−𝑀 (𝑏, 𝑡, 𝑇) 𝑟_𝑡− 𝑀 (𝛽, 𝑡, 𝑇) 𝜆_𝑡

− ∫^𝑇

𝑡 𝑎_𝑢𝑀 (𝑏, 𝑢, 𝑇) 𝑑𝑢

− ∫^𝑇

𝑡 𝛼_𝑢𝑀 (𝛽, 𝑢, 𝑇) 𝑑𝑢}

×exp{1 2∫^𝑇

𝑡 𝜎_2𝑢² 𝑀²(𝑏, 𝑢, 𝑇) 𝑑𝑢 +1

2∫^𝑇

𝑡 𝜎_3𝑢² 𝑀²(𝛽, 𝑢, 𝑇) 𝑑𝑢}

×exp{𝜌₂₃∫^𝑇

𝑡 𝜎_2𝑢𝜎_3𝑢𝑀 (𝑏, 𝑢, 𝑇) 𝑀 (𝛽, 𝑢, 𝑇) 𝑑𝑢} . (14)

Thus, we can define the probability measure𝑄by 𝑑𝑄

𝑑𝑃󵄨󵄨󵄨󵄨

󵄨󵄨󵄨󵄨F𝑡

= exp{− ∫_𝑡^𝑇(𝑟_𝑠+ 𝜆_𝑠) 𝑑𝑠}

𝐸 [exp{− ∫_𝑡^𝑇(𝑟_𝑠+ 𝜆_𝑠) 𝑑𝑠} |F_𝑡]

= exp{−1 2∫^𝑇

𝑡 𝜎_2𝑢² 𝑀²(𝑏, 𝑢, 𝑇) 𝑑𝑢

−1 2∫^𝑇

𝑡 𝜎_3𝑢² 𝑀²(𝛽, 𝑢, 𝑇) 𝑑𝑢}

×exp{−𝜌₂₃∫^𝑇

𝑡 𝜎_2𝑢𝜎_3𝑢𝑀 (𝑏, 𝑢, 𝑇) 𝑀 (𝛽, 𝑢, 𝑇) 𝑑𝑢}

×exp{− ∫^𝑇

𝑡 𝜎_2V𝑀 (𝑏,V, 𝑇) 𝑑𝑊_2V

− ∫^𝑇

𝑡 𝜎_3V𝑀 (𝛽,V, 𝑇) 𝑑𝑊_3V} .

(15) By Girsanov’s theorem,

𝑑𝑊_2𝑡^𝑄= 𝑑𝑊_2𝑡+ 𝜎_2𝑡𝑀 (𝑏, 𝑡, 𝑇) 𝑑𝑡 + 𝜌₂₃𝜎_3𝑡𝑀 (𝛽, 𝑡, 𝑇) 𝑑𝑡, 𝑑𝑊_3𝑡^𝑄= 𝑑𝑊_3𝑡+ 𝜎_3𝑡𝑀 (𝛽, 𝑡, 𝑇) 𝑑𝑡 + 𝜌₂₃𝜎_2𝑡𝑀 (𝑏, 𝑡, 𝑇) 𝑑𝑡, 𝑑𝑊_1𝑡^𝑄= 𝑑𝑊_1𝑡+ 𝜌₁₂𝑀₁(𝑡) 𝑑𝑡 + 𝜌₁₃𝑀₂(𝑡) 𝑑𝑡,

(16) where

𝑀₁(𝑡) = 𝜎_2𝑡𝑀 (𝑏, 𝑡, 𝑇) + 𝜌₂₃𝜎_3𝑡𝑀 (𝛽, 𝑡, 𝑇) ,

𝑀₂(𝑡) = 𝜎_3𝑡𝑀 (𝛽, 𝑡, 𝑇) + 𝜌₂₃𝜎_2𝑡𝑀 (𝑏, 𝑡, 𝑇) . (17) 𝑊_1𝑡^𝑄, 𝑊_2𝑡^𝑄, and 𝑊_3𝑡^𝑄 are standard Brownian motions under probability measure 𝑄, and (𝑊_1𝑡^𝑄, 𝑊_2𝑡^𝑄, 𝑊_3𝑡^𝑄) has the same covariance matrix as(𝑊_1𝑡, 𝑊_2𝑡, 𝑊_3𝑡). Therefore,

𝐸 [𝑒^{− ∫}^𝑡^𝑇^(𝑟^𝑠^+𝜆^𝑠^)𝑑𝑠(𝑆_𝑇− 𝐾)⁺|F_𝑡]

= 𝑍 (𝑡, 𝑇) 𝐸^𝑄((𝑆_𝑇− 𝐾)⁺ |F_𝑡) .

(18)

In addition,

𝐸^𝑄((𝑆_𝑇− 𝐾)⁺|F_𝑡)

= 𝐸^𝑄(𝑆_𝑇1_(𝑆_𝑇_≥𝐾)|F_𝑡) − 𝐾𝐸^𝑄(1_(𝑆_𝑇_≥𝐾)|F_𝑡) . (19) By the solution of SDE (2),

𝑆_𝑇= 𝑆_𝑡exp{∫^𝑇

𝑡 (𝑟_𝑠− 𝑘]_𝑠−1 2𝜎²_1𝑠) 𝑑𝑠 + ∫^𝑇

𝑡 𝜎_1𝑠𝑑𝑊_1𝑠+ ∫^𝑇

𝑡 𝑌_𝑠−𝑑𝑁_𝑠} .

(20)

(4)

Under𝑄,

𝑆_𝑇= 𝑆_𝑡exp{Λ (𝑡, 𝑇) + ∫^𝑇

𝑡 𝜎_1𝑠𝑑𝑊_1𝑠^𝑄 + ∫^𝑇

𝑡 𝜎_2𝑠𝑀 (𝑏, 𝑠, 𝑇) 𝑑𝑊_2𝑠^𝑄+ ∫^𝑇

𝑡 𝑌_𝑠−𝑑𝑁_𝑠} , (21)

where

Λ (𝑡, 𝑇) = − ∫^𝑇

𝑡 (𝑘]_𝑠+1

2𝜎²_1𝑠) 𝑑𝑠 + 𝑀 (𝑏, 𝑡, 𝑇) 𝑟_𝑡 + ∫^𝑇

𝑡 𝑎_𝑠𝑀 (𝑏, 𝑠, 𝑇) 𝑑𝑠 − ∫^𝑇

𝑡 𝜎_2𝑠²𝑀²(𝑏, 𝑠, 𝑇) 𝑑𝑠

− 𝜌₁₂∫^𝑇

𝑡 𝜎_1𝑠𝑀₁(𝑠) 𝑑𝑠 − 𝜌₁₃∫^𝑇

𝑡 𝜎_1𝑠𝑀₂(𝑠) 𝑑𝑠

− 𝜌₂₃∫^𝑇

𝑡 𝜎_2𝑠𝜎_3𝑠𝑀 (𝑏, 𝑠, 𝑇) 𝑀 (𝛽, 𝑠, 𝑇) 𝑑𝑠.

(22)

So,

𝑃^𝑄(𝑆_𝑇≥ 𝐾 |F_𝑡)

= 𝑃^𝑄(∫^𝑇

𝑡 𝜎_1𝑠𝑑𝑊_1𝑠^𝑄+ ∫^𝑇

𝑡 𝜎_2𝑠𝑀 (𝑏, 𝑠, 𝑇) 𝑑𝑊_2𝑠^𝑄

≥ln(𝐾/𝑆_𝑡) − Λ (𝑡, 𝑇) − ∫^𝑇

𝑡 𝑌_𝑠𝑑𝑁_𝑠)

= 𝑃^𝑄(∫_𝑡^𝑇𝜎_1𝑠𝑑𝑊_1𝑠^𝑄+ ∫_𝑡^𝑇𝜎_2𝑠𝑀 (𝑏, 𝑠, 𝑇) 𝑑𝑊_2𝑠^𝑄

√Δ

≤ ln(𝑆_𝑡/𝐾) + Λ (𝑡, 𝑇) + ∫_𝑡^𝑇𝑌_𝑠𝑑𝑁_𝑠

√Δ )

=∑^∞

𝑛=0

(∫_𝑡^𝑇]_𝑠𝑑𝑠)^𝑛

𝑛! 𝑒^{− ∫}^𝑡^𝑇^]^𝑠^𝑑𝑠𝐸 (𝑁 (𝑑₁)) ,

(23)

where

Δ = ∫^𝑇

𝑡 𝜎_1𝑠²𝑑𝑠 + ∫^𝑇

𝑡 𝜎_2𝑠²𝑀²(𝑏, 𝑠, 𝑇) 𝑑𝑠 + 2𝜌₁₂∫^𝑇

𝑡 𝜎_1𝑠𝜎_2𝑠𝑀 (𝑏, 𝑠, 𝑇) 𝑑𝑠, 𝑑₁= ln(𝑆_𝑡/𝐾) + Λ (𝑡, 𝑇) + ∑^𝑛_𝑗=1𝑌_𝑗

√Δ .

(24)

𝐸(⋅)is the expectation of𝑌_𝑗under𝑃, and𝑁(⋅)is distribution function of standard normal distribution. On the other hand, let

L(𝑡, 𝑇)

:= 𝐸^𝑄(𝑆_𝑇|F_𝑡)

= 𝑆_𝑡𝑒^{Λ(𝑡,𝑇)}exp{1 2∫^𝑇

𝑡 𝜎_1𝑠² 𝑑𝑠 +1 2∫^𝑇

𝑡 𝜎_2𝑠²𝑀²(𝑏, 𝑠, 𝑇) 𝑑𝑠}

×exp{𝜌₁₂∫^𝑇

𝑡 𝜎_1𝑠𝜎_2𝑠𝑀 (𝑏, 𝑠, 𝑇) 𝑑𝑠 + 𝑘 ∫^𝑇

𝑡 ]_𝑠𝑑𝑠} . (25)

We can define the probability measure𝑅as follows:

𝑑𝑅 𝑑𝑄󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨F𝑡

= 𝑆_𝑇

𝐸^𝑄(𝑆_𝑇|F_𝑡)

= exp{∫^𝑇

𝑡 𝜎_1𝑠𝑑𝑊_1𝑠^𝑄+ ∫^𝑇

𝑡 𝜎_2𝑠𝑀 (𝑏, 𝑠, 𝑇) 𝑑𝑊_2𝑠^𝑄

−1 2∫^𝑇

𝑡 𝜎_1𝑠² 𝑑𝑠}

×exp{−1 2∫^𝑇

−𝜌₁₂∫^𝑇

𝑡 𝜎_1𝑠𝜎_2𝑠𝑀 (𝑏, 𝑠, 𝑇) 𝑑𝑠}

×exp{∫^𝑇

𝑡 𝑌_𝑠𝑑𝑁_𝑠− 𝑘 ∫^𝑇

𝑡 ]_𝑠𝑑𝑠} .

(26)

By Girsanov’s theorem, we have that

𝑑𝑊_1𝑡^𝑅= 𝑑𝑊_1𝑡^𝑄− 𝜎_1𝑡𝑑𝑡 − 𝜌₁₂𝜎_2𝑡𝑀 (𝑏, 𝑡, 𝑇) 𝑑𝑡,

𝑑𝑊_2𝑡^𝑅= 𝑑𝑊_2𝑡^𝑄− 𝜎_2𝑡𝑀 (𝑏, 𝑡, 𝑇) 𝑑𝑡 − 𝜌₁₂𝜎_1𝑡𝑑𝑡, (27)

where, under 𝑅, 𝑊_1𝑡^𝑅, 𝑊_2𝑡^𝑅 are standard Brownian motions and their correlation coefficient is𝜌₁₂,𝑁_𝑡is Poisson process with intensity (𝑘 + 1)]_𝑡, and the density function of 𝑌₁ is 𝑒^𝑦𝑓(𝑦)/(𝑘 + 1), where𝑓(⋅)is the density function of𝑌₁under 𝑃. Since, under𝑅,

𝑆_𝑇= 𝑆_𝑡exp{Λ (𝑡, 𝑇) + Δ + ∫^𝑇

𝑡 𝜎_1𝑠𝑑𝑊_1𝑠^𝑅 + ∫^𝑇

𝑡 𝜎_2𝑠𝑀 (𝑏, 𝑠, 𝑇) 𝑑𝑊_2𝑠^𝑅+ ∫^𝑇

𝑡 𝑌_𝑠−𝑑𝑁_𝑠} , (28)

(5)

then

𝐸^𝑄(𝑆_𝑇1_(𝑆_𝑇_≥𝐾)|F_𝑡)

=L(𝑡, 𝑇) 𝑃^𝑅(𝑆_𝑇≥ 𝐾 |F_𝑡)

=L(𝑡, 𝑇) 𝑃^𝑅(∫^𝑇

𝑡 𝜎_1𝑠𝑑𝑊_1𝑠^𝑅+ ∫^𝑇

𝑡 𝜎_2𝑠𝑀 (𝑏, 𝑠, 𝑇) 𝑑𝑊_2𝑠^𝑅

≥ln(𝐾

𝑆_𝑡) − Λ (𝑡, 𝑇) − Δ − ∫^𝑇

𝑡 𝑌_𝑠𝑑𝑁_𝑠)

=L(𝑡, 𝑇) 𝑃^𝑅(∫_𝑡^𝑇𝜎_1𝑠𝑑𝑊_1𝑠^𝑅+ ∫_𝑡^𝑇𝜎_2𝑠𝑀 (𝑏, 𝑠, 𝑇) 𝑑𝑊_2𝑠^𝑅

√Δ

≤ ln(𝑆_𝑡/𝐾) + Λ (𝑡, 𝑇) + Δ + ∫_𝑡^𝑇𝑌_𝑠𝑑𝑁_𝑠

√Δ )

=L(𝑡, 𝑇)∑^∞

𝑛=0

𝑛! 𝑒^{− ∫}^𝑡^𝑇^]^𝑠^𝑑𝑠𝐸^𝑅(𝑁 (𝑑₂)) ,

(29) where

𝑑₂= ln(𝑆_𝑡/𝐾) + Λ (𝑡, 𝑇) + Δ + ∑^𝑛_𝑗=1𝑌_𝑗

√Δ

= 𝑑₁+ √Δ

(30)

and𝐸^𝑅(⋅)is the expectation of𝑌_𝑗under𝑅. By (23) and (29), we have the following lemma.

Lemma 1. Consider the Following:

𝐸 [𝑒^{− ∫}^𝑡^𝑇^(𝑟^𝑠^+𝜆^𝑠^)𝑑𝑠(𝑆_𝑇− 𝐾)⁺ |F_𝑡]

= 𝑍 (𝑡, 𝑇)

×∑^∞

𝑛=0

(∫_𝑡^𝑇]_𝑠𝑑𝑠)^𝑛 𝑛!

× 𝑒^{− ∫}^𝑡^𝑇^]^𝑠^𝑑𝑠[L(𝑡, 𝑇) 𝐸^𝑅(𝑁 (𝑑₂)) − 𝐾𝐸 (𝑁 (𝑑₁))] . (31) In the following; in order to calculate𝐼₁, we write

𝐸 [𝑒^{− ∫}^𝑡^𝑇^𝑟^𝑠^𝑑𝑠(𝑆_𝑇− 𝐾)⁺|F_𝑡]

= 𝐸 [𝑒^{− ∫}^𝑡^𝑇^𝑟^𝑠^𝑑𝑠𝑆_𝑇1_(𝑆_𝑇_≥𝐾)|F_𝑡]

− 𝐾𝐸 [𝑒^{− ∫}^𝑡^𝑇^𝑟^𝑠^𝑑𝑠1_(𝑆_𝑇_≥𝐾)|F_𝑡]

= 𝐼₃− 𝐼₄.

(32)

Denote by 𝑃(𝑡, 𝑇) the value at time𝑡 of a 𝑇maturity zero coupon bond whose face value is 1. Then

𝑃 (𝑡, 𝑇) = 𝐸 [𝑒^{− ∫}^𝑡^𝑇^𝑟^𝑠^𝑑𝑠|F_𝑡] . (33)

From(12),

𝑃 (𝑡, 𝑇) = exp{−𝑀 (𝑏, 𝑡, 𝑇) 𝑟_𝑡− ∫^𝑇

𝑡 𝑎_V𝑀 (𝑏,V, 𝑇) 𝑑V +1

2∫^𝑇

𝑡 𝜎²_2V𝑀²(𝑏,V, 𝑇) 𝑑V} ,

(34)

𝑑𝑃 (𝑡, 𝑇)

𝑃 (𝑡, 𝑇) = 𝑟_𝑡𝑑𝑡 − 𝑀 (𝑏, 𝑡, 𝑇) 𝜎_2𝑡𝑑𝑊_2𝑡. (35) Define the Radon-Nikodym derivative given by

𝑑𝑄^𝑇

𝑑𝑃 = 𝑒^{− ∫}^𝑡^𝑇^𝑟^𝑠^𝑑𝑠 𝐸 [𝑒^{− ∫}^𝑡^𝑇^𝑟^𝑠^𝑑𝑠|F_𝑡]

= exp{− ∫^𝑇

𝑡 𝑀 (𝑏,V, 𝑇) 𝜎_2V𝑑𝑊_2V

−1 2∫^𝑇

𝑡 𝜎_2V²𝑀²(𝑏,V, 𝑇) 𝑑V} ,

(36)

and by Girsanov’s theorem, under𝑄^𝑇,

𝑑𝑊_2𝑡^𝑄^𝑇= 𝑑𝑊_2𝑡+ 𝜎_2𝑡𝑀 (𝑏, 𝑡, 𝑇) 𝑑𝑡 𝑑𝑊_1𝑡^𝑄^𝑇 = 𝑑𝑊_1𝑡+ 𝜌₁₂𝜎_2𝑡𝑀 (𝑏, 𝑡, 𝑇) 𝑑𝑡.

(37)

Thus𝑆_𝑇can be rewritten as

𝑆_𝑇= 𝑆_𝑡exp{A(𝑡, 𝑇) + ∫^𝑇

𝑡 𝑀 (𝑏,V, 𝑇) 𝜎_2V𝑑𝑊_2V^𝑄^𝑇 + ∫^𝑇

𝑡 𝜎_1V𝑑𝑊_1V^𝑄^𝑇+ ∫^𝑇

𝑡 𝑌_𝑠−𝑑𝑁_𝑠} ,

(38)

where

A(𝑡, 𝑇) = − ∫^𝑇

𝑡 (𝑘]_𝑠+1

2𝜎_1𝑠²) 𝑑𝑠 + 𝑀 (𝑏, 𝑡, 𝑇) 𝑟_𝑡 + ∫^𝑇

𝑡 𝑎_𝑠𝑀 (𝑏, 𝑠, 𝑇) 𝑑𝑠 − ∫^𝑇

− 𝜌₁₂∫^𝑇

𝑡 𝜎_1𝑠𝜎_2𝑠𝑀 (𝑏, 𝑠, 𝑇) 𝑑𝑠.

(39)

(6)

Then

𝑃^𝑄^𝑇(𝑆_𝑇≥ 𝐾 |F_𝑡)

= 𝑃^𝑄^𝑇(∫^𝑇

𝑡 𝜎_1𝑠𝑑𝑊_1𝑠^𝑄^𝑇+ ∫^𝑇

𝑡 𝜎_2𝑠𝑀 (𝑏, 𝑠, 𝑇) 𝑑𝑊_2𝑠^𝑄^𝑇

≥ln(𝐾

𝑆_𝑡) −A(𝑡, 𝑇) − ∫^𝑇

𝑡 𝑌_𝑠−𝑑𝑁_𝑠)

= 𝑃^𝑄^𝑇(∫_𝑡^𝑇𝜎_1𝑠𝑑𝑊_1𝑠^𝑄^𝑇+ ∫_𝑡^𝑇𝜎_2𝑠𝑀 (𝑏, 𝑠, 𝑇) 𝑑𝑊_2𝑠^𝑄^𝑇

√Δ

≤ln(𝑆_𝑡/𝐾) +A(𝑡, 𝑇) + ∫_𝑡^𝑇𝑌_𝑠−𝑑𝑁_𝑠

√Δ )

=∑^∞

𝑛=0

𝑛! 𝑒^{− ∫}^𝑡^𝑇^]^𝑠^𝑑𝑠𝐸 (𝑁 (𝑑₃)) ,

(40)

where𝐸(⋅)is the expectation of𝑌_𝑗under𝑃, 𝑑₃=ln(𝑆_𝑡/𝐾) +A(𝑡, 𝑇) + ∑^𝑛_𝑗=1𝑌_𝑗

√Δ . (41)

Therefore,

𝐼₄= 𝐾𝑃 (𝑡, 𝑇) 𝐸^𝑄^𝑇(1_(𝑆_𝑇_≥𝐾)|F_𝑡)

= 𝐾𝑃 (𝑡, 𝑇)∑^∞

𝑛=0

𝑛! 𝑒^{− ∫}^𝑡^𝑇^]^𝑠^𝑑𝑠𝐸 (𝑁 (𝑑₃)) . (42)

Finally, let

𝐴 (𝑡, 𝑇) := 𝐸 (𝑒^{− ∫}^𝑡^𝑇^𝑟^𝑠^𝑑𝑠𝑆_𝑇|F_𝑡)

= 𝑆_𝑡exp{− ∫^𝑇

𝑡 (𝑘]_𝑠+1 2𝜎_1𝑠²) 𝑑𝑠 +1

2∫^𝑇

𝑡 𝜎²_1𝑠𝑑𝑠 + 𝑘 ∫^𝑇

𝑡 ]_𝑠𝑑𝑠} .

(43)

We define 𝑑𝑄^𝑆

𝑑𝑃 = 𝑒^{− ∫}^𝑡^𝑇^𝑟^𝑠^𝑑𝑠𝑆_𝑇 𝐸 [𝑒^{− ∫}^𝑡^𝑇^𝑟^𝑠^𝑑𝑠𝑆_𝑇|F_𝑡]

= exp{∫^𝑇

𝑡 𝜎_1V𝑑𝑊_1V−1 2∫^𝑇

𝑡 𝜎_2V² 𝑑V + ∫^𝑇

𝑡 𝑌_𝑠−𝑑𝑁 (𝑠) − 𝑘 ∫^𝑇

𝑡 ]_𝑠𝑑𝑠} ,

(44)

and by Girsanov’s theorem, under𝑄^𝑆, 𝑑𝑊_1𝑡^𝑄^𝑆= 𝑑𝑊_1𝑡− 𝜎_1𝑡𝑑𝑡 𝑑𝑊_2𝑡^𝑄^𝑆 = 𝑑𝑊_2𝑡− 𝜌₁₂𝜎_1𝑡𝑑𝑡,

(45)

and, under𝑄^𝑆,𝑁_𝑡is Poisson process with intensity(𝑘 + 1)]_𝑡, and the density function of𝑌₁is𝑒^𝑦𝑓(𝑦)/(𝑘 + 1), where𝑓(⋅)is the density function of𝑌₁under𝑃. Since, under𝑄^𝑆,

𝑆_𝑇= 𝑆_𝑡exp{A(𝑡, 𝑇) + Δ + ∫^𝑇

𝑡 𝑀 (𝑏,V, 𝑇) 𝜎2V𝑑𝑊_2V^𝑄^𝑆 + ∫^𝑇

𝑡 𝜎_1V𝑑𝑊_1V^𝑄^𝑆+ ∫^𝑇

𝑡 𝑌_𝑠−𝑑𝑁_𝑠} ,

(46)

then

𝑃^𝑄^𝑆(𝑆_𝑇≥ 𝐾 |F_𝑡)

= 𝑃^𝑄^𝑆(∫^𝑇

𝑡 𝜎_1𝑠𝑑𝑊_1𝑠^𝑄^𝑆+ ∫^𝑇

𝑡 𝜎_2𝑠𝑀 (𝑏, 𝑠, 𝑇) 𝑑𝑊_2𝑠^𝑄^𝑆

≥ln(𝐾

𝑆_𝑡) −A(𝑡, 𝑇) − Δ − ∫^𝑇

𝑡 𝑌_𝑠−𝑑𝑁_𝑠)

= 𝑃^𝑄^𝑇(∫_𝑡^𝑇𝜎_1𝑠𝑑𝑊_1𝑠^𝑄^𝑆+ ∫_𝑡^𝑇𝜎_2𝑠𝑀 (𝑏, 𝑠, 𝑇) 𝑑𝑊_2𝑠^𝑄^𝑆

√Δ

≤ln(𝑆_𝑡/𝐾) +A(𝑡, 𝑇) + Δ + ∫_𝑡^𝑇𝑌_𝑠−𝑑𝑁_𝑠

√Δ )

=∑^∞

𝑛=0

𝑛! 𝑒^{− ∫}^𝑡^𝑇^]^𝑠^𝑑𝑠𝐸^𝑆(𝑁 (𝑑₄)) ,

(47) where𝐸^𝑆(⋅)is the expectation of𝑌_𝑗under𝑄^𝑆,

𝑑₄=ln(𝑆_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:

𝐸 [𝑒^{− ∫}^𝑡^𝑇^𝑟^𝑠^𝑑𝑠(𝑆_𝑇− 𝐾)⁺ |F_𝑡]

=∑^∞

𝑛=0

× 𝑒^{− ∫}^𝑡^𝑇^]^𝑠^𝑑𝑠[𝐴 (𝑡, 𝑇) 𝐸^𝑆(𝑁 (𝑑₄))−𝐾𝑃 (𝑡, 𝑇) 𝐸 (𝑁 (𝑑₃))] . (49) Combined with the previous lemmas, the price of European call option at time𝑡is given by the following theorem.

(7)

Table 1: The parameter values.

Parameter name Value in state𝑒₁ Value in state𝑒₂

Volatility of𝑆 𝜎₁₁= 0.2 𝜎₁₂= 0.4

Jump intensity ]₁= 15 ]₂= 30

Speed of reversion of𝑟 𝑏₁= 2 𝑏₂= 1

Long-term average of𝑟 𝑎₁/𝑏₁= 0.04 𝑎₂/𝑏₂= 0.02

Volatility of𝑟 𝜎₂₁= 0.15 𝜎₂₂= 0.3

Speed of reversion of𝜆 𝛽₁= 1.5 𝛽₂= 2 Long-term average of𝜆 𝛼₁/𝛽₁= 0.01 𝛼₂𝛽₂= 0.02

Volatility of𝜆 𝜎₃₁= 0.25 𝜎₃₂= 0.45

Initial stock price 𝑆₀= 100 Initial interest rate 𝑟₀= 0.04 Initial default intensity 𝜆₀= 0.5

Mean jump size 𝜇_𝐽= 0

Standard deviation of jump size 𝜎_𝐽= 0.1

Theorem 3. The price of European call option with credit risk at time𝑡is

𝐶 (𝑡, 𝑇)

=∑^∞

𝑛=0

× 𝑒^{− ∫}^𝑡^𝑇^]^𝑠^𝑑𝑠[𝑤 (𝐴 (𝑡, 𝑇) 𝐸^𝑆(𝑁 (𝑑₄))

− 𝐾𝑃 (𝑡, 𝑇) 𝐸 (𝑁 (𝑑₃))) + (1 − 𝑤) 1_(𝜏>𝑡)𝑍 (𝑡, 𝑇)

× (L(𝑡, 𝑇) 𝐸^𝑅(𝑁 (𝑑₂))

− 𝐾𝐸 (𝑁 (𝑑₁)))] .

(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 chain𝜉has two states, which means that macroeconomic shifts between the two states:𝑒₁ (“boom” or “good”) and𝑒₂(“recession” or “bad”).

We assume that the current economy is in boom and that the transition probability matrix of the two-state Markov chain𝜉 is given by

(𝑝¹¹ 𝑝₁₂

𝑝₂₁ 𝑝₂₂) = (0.7 0.30.2 0.8) . (51) Assume that𝑌_𝑡 is normally distributed with mean 𝜇_𝐽 and standard deviation 𝜎_𝐽. We will adopt the specimen values for the model parameters asTable 1. We consider a range of spot-to-strike ratios𝑆₀/𝐾from 0.8 to 1.2 and assume that one year has 252 trading days. We perform 10 000 simulations for computing the option price.

10 12 14 16 18 20 22 24 26 28

0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 1.25 𝑆₀/𝐾

𝜔 = 0.4

𝜔 = 0.6 𝜔 = 0.8

𝜔 = 1

Option price

Figure 1: The option price with different recovery rate for𝑇 = 1, 𝜌₁₂= 0.7,𝜌₁₃= 0.5, and𝜌₂₃= 0.6.

Option price

𝑇 = 0.5 𝑇 = 1 𝑇 = 1.5

50.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 1.25 10

15 20 25 30

𝑆0/𝐾

Figure 2: The option price with different maturity𝑇for𝜔 = 0.4, 𝜌₁₂= 0.7,𝜌₁₃= 0.5, and𝜌₂₃= 0.6.

For each𝜔 = 0.4, 0.6, 0.8, 1, we consider the impact of the spot-to-strike ratio on the option price. FromFigure 1, we observe that the option price increases as the spot-to-strike ratio increases. We can also see that the greater the𝜔, the greater the option price. When𝜔 = 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 2depicts the plot of the option price against the spot- to-strike ratio for each maturity𝑇 = 0.5, 1, 1.5. From these plots, we find that the longer the maturities, the greater the option price.

In the following, we compare the option price with different correlation coefficients for fixed maturity𝑇 = 1

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𝜌₁₂= 0.7 𝜌₁₂= −0.7 𝜌₁₂= −0.9 8

10 12 14 16 18 20 22 24

0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 1.25 𝑆₀/𝐾

Option price

Figure 3: The option price with different correlation coefficients𝜌₁₂ for𝑇 = 1,𝜔 = 0.4,𝜌₁₃= 0.5, and𝜌₂₃= 0.6.

10 15 20 25

Option price

𝜌13= 1 𝜌13= 0.5 𝜌13= −0.5

0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 1.25 𝑆0/𝐾

Figure 4: The option price with different correlation coefficients𝜌₁₃ for𝑇 = 1,𝜔 = 0.4,𝜌₁₂= 0.7, and𝜌₂₃= 0.6.

and recovery rate𝜔 = 0.4. Figure 3illustrates that option price increases as correlation coefficient𝜌₁₂increases. From Figure 4, we can also see that option price decreases as correlation coefficient𝜌₁₃increases.

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

10 15 20 25

₁= 10, ₂= 20 ₁= 15, ₂= 30 ₁= 20, ₂= 40

Option price

0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 1.25 𝑆₀/𝐾

Figure 5: The option price with different]for𝜔 = 0.4,𝜌₁₂ = 0.7, 𝜌₁₃= 0.5, and𝜌₂₃= 0.6.

Markov-modulated model

Non-Markov-modulated model for a good economy Non-Markov-modulated model for a bad economy 5

10 15 25 30 35

20

Option price

0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 1.25 𝑆₀/𝐾

Figure 6: The option price in Markov-modulated model and non- Markov-modulated models.

Finally, we investigate the difference of the option price in Markov-modulated model and non-Markov-modulated models. FromFigure 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

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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.

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).

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