Pricing Options with Credit Risk in Markovian Regime-Switching Markets

This paper investigates the valuation of European option with credit risk in a reduced formmodel 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 sameMarkov chain.We study the pricing of European option and present numerical illustrations.


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,

The Model Description
Let (Ω, F, ) be a complete probability space, where  is a neutral-risk probability measure.Define  = {  ,  ≥ 0} on (Ω, F, ) as a continuous-time, finite state Markov chain with -state space .We interpret the state of  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  to be a finite set of unite vectors { 1 ,  2 , . . .,   } with   = (0, . . ., 1, . . ., 0) ∈   .And  has the following semimartingale representation: where  = (  ) ,=1,2,..., is -matrix of  and  = {  } 0≤≤ is an   -valued martingale with respect to the filtration generated by {  , 0 ≤  ≤ } under .Suppose that the stock price   and interest rate   satisfy the following stochastic differential equations (SDE): where  1 where ⟨⋅, ⋅⟩ denotes the inner product in   .Let  denote the default time of the writer of the option with default intensity process   , and   is given by where

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 .When the seller of option defaults, the payoff is given by  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 , with strike price  and maturity , is given by Following Lando [15], We can obtain the following expression: Next, we calculate  1 and  2 , respectively.For  > , we have Integrated from  to  in both sides of (10), Let Similarly, From ( 12) and ( 13), we have that Thus, we can define the probability measure  by By Girsanov's theorem, In addition, By the solution of SDE (2), Under , where where ]  } . ( We can define the probability measure  as follows: By Girsanov's theorem, we have that where, under ,   1 ,   2 are standard Brownian motions and their correlation coefficient is  12 ,   is Poisson process with intensity ( + 1)]  , and the density function of  1 is   ()/( + 1), where (⋅) is the density function of  1 under .Since, under , then where and   (⋅) is the expectation of   under .By (23) and (29), we have the following lemma.

Lemma 1. Consider the Following:
In the following; in order to calculate  1 , we write Denote by (, ) the value at time  of a  maturity zero coupon bond whose face value is 1.Then From (12), Define the Radon-Nikodym derivative given by and by Girsanov's theorem, under   , Thus   can be rewritten as where where (⋅) is the expectation of   under , Therefore, Finally, let We define and by Girsanov's theorem, under   , and, under   ,   is Poisson process with intensity ( + 1)]  , and the density function of  1 is   ()/( + 1), where (⋅) is the density function of  1 under .Since, under   , then where   (⋅) is the expectation of   under   , From (42) and (33), we can conclude the following lemma.

Lemma 2. Consider the following:
Combined with the previous lemmas, the price of European call option at time  is given by the following theorem.(50)

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  has two states, which means that macroeconomic shifts between the two states:  1 ("boom" or "good") and  2 ("recession" or "bad").We assume that the current economy is in boom and that the transition probability matrix of the two-state Markov chain  is given by Assume that   is normally distributed with mean   and standard deviation   .We will adopt the specimen values for the model parameters as Table 1.We consider a range of spot-to-strike ratios  0 / 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.For each  = 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 , the greater the option price.When  = 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 spotto-strike ratio for each maturity  = 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 maturity  = 1 and recovery rate  = 0.4.Figure 3 illustrates that option price increases as correlation coefficient  12 increases.From Figure 4, we can also see that option price decreases as correlation coefficient  13 increases.
In Figure 5, we present how the option prices vary with the changes of the annual jump intensity ]. 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.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 Markovmodulated 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.

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.

Table 1 :
The parameter values.Parameter name Value in state  1 Value in state  2