# Pricing Options with Credit Risk in Markovian Regime-Switching Markets

Authors: Jinzhi Li; Shixia Ma
Journal: Journal of Applied Mathematics
 (2013)

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## 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.


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## Body

## 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. Section2 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={ξt,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,…,en}  with ei=(0,…,1,…,0)∈Rn. And ξ has the following semimartingale representation:
(1)dξt=Qξtdt+dMt,
where Q=(qij)i,j=1,2,…,n is Q-matrix of ξ and M={Mt}0≤t≤T is an Rn-valued martingale with respect to the filtration generated by {ξt,0≤t≤T} under P. Suppose that the stock price St and interest rate rt satisfy the following stochastic differential equations (SDE):
(2)dStSt-=(rt-kνt)dt+σ1tdW1t+(eYt--1)dN(t),drt=(at-btrt)dt+σ2tdW2t,
where W1t, W2t are standard Brownian motions and σ1t, σ2t are the stochastic volatility of the stock and the interest rate, respectively. {Nt}0≤t≤T is Poisson process with the stochastic jump intensity {νt}0≤t≤T, and the jump amplitude is controlled by {Yt}. Ys and Yt for s≠t independently identify distribution, and write k=EeYi-1. Moreover, Yt, Nt are assumed to be mutually independent. σ1t, σ2t, vt, at, bt are controlled by ξt, that is,
(3)σ1t=〈σ1,ξt〉,σ1=(σ11,σ12,…,σ1n)∈(0,∞)n,σ2t=〈σ2,ξt〉,σ2=(σ21,σ22,…,σ2n)∈(0,∞)n,νt=〈ν,ξt〉,ν=(ν1,ν2,…,νn)∈(0,∞)n,at=〈a,ξt〉,a=(a1,a2,…,an)∈(0,∞)n,bt=〈b,ξt〉,b=(b1,b2,…,bn)∈(0,∞)n,
where 〈·,·〉 denotes the inner product in Rn. Let τ denote the default time of the writer of the option with default intensity process λt, and λt is given by
(4)dλt=(αt-βtλt)dt+σ3tdW3t,
where W3t is standard Brownian motion and σ3t is the stochastic volatility of default intensity. σ3t, αt,  and βt are also controlled by ξt and satisfy
(5)σ3t=〈σ3,ξt〉,σ3=(σ31,σ32,…,σ3n)∈(0,∞)n,αt=〈α,ξt〉,α=(α1,α2,…,αn)∈(0,∞)n,βt=〈β,ξt〉,β=(β1,β2,…,βn)∈(0,∞)n.
Moreover, we assume that Nt,Yt are independent of W1t, W2t, and W3t and the covariance matrix of the Brownian motion (W1t,W2t,W3t) is
(6)(1ρ12ρ13ρ121ρ23ρ13ρ231)t.
The filtration Ft is generated by Ft=FtS∨Ftr∨Ftλ∨FTξ∨Ht, where FtS=σ(Ss,0≤s≤t), Ftr=σ(rs,0≤s≤t), Ftλ=σ(λs,0≤s≤t), Ftξ=σ(ξs,0≤s≤t), and Ht=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 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 by(7)C(t,T)=E[e-∫tT‍rsds(w(ST-K)+1(τ≤T)+(ST-K)+1(τ>T))∣Ft]=E[e-∫tT‍rsds(w(ST-K)++(1-w)(ST-K)+1(τ>T))∣Ft].Following Lando [15],
(8)E[e-∫tT‍rsds(ST-K)+1(τ>T)∣Ft]=1(τ>t)E[e-∫tT‍(rs+λs)ds(ST-K)+∣Ft].
We can obtain the following expression:
(9)C(t,T)=wE[e-∫tT‍rsds(ST-K)+∣Ft]+(1-w)1(τ>t)E[e-∫tT‍(rs+λs)ds(ST-K)+∣Ft]=I1+I2.
Next, we calculate I1 and I2, respectively. For s>t, we have
(10)rs=e-∫ts‍budurt+∫ts‍e-∫vs‍buduavdv+∫ts‍e-∫vs‍buduσ2vdW2v.
Integrated from t to T in both sides of (10),
(11)∫tT‍rsds=∫tT‍e-∫ts‍bududsrt+∫tT‍avdv∫vT‍e-∫vs‍bududs+∫tT‍σ2vdW2v∫vT‍e-∫vs‍bududs.
Let M(x,y,T)=∫yT‍e-∫ys‍xududs; then
(12)∫tT‍rsds=M(b,t,T)rt+∫tT‍avM(b,v,T)dv+∫tT‍σ2vM(b,v,T)dW2v.
Similarly,
(13)∫tT‍λsds=M(β,t,T)λt+∫tT‍αvM(β,v,T)dv+∫tT‍σ3vM(β,v,T)dW3v.
From (12) and (13), we have that
(14)Z(t,T):=E[exp{-∫tT‍(rs+λs)ds}∣Ft]=exp{-M(b,t,T)rt-M(β,t,T)λt∫tT‍-∫tT‍auM(b,u,T)du-∫tT‍αuM(β,u,T)du}×exp{12∫tT‍σ2u2M2(b,u,T)du+12∫tT‍σ3u2M2(β,u,T)du}×exp{ρ23∫tT‍σ2uσ3uM(b,u,T)M(β,u,T)du}.
Thus, we can define the probability measure Q by
(15)dQdP|Ft=exp{-∫tT‍(rs+λs)ds}E[exp{-∫tT‍(rs+λs)ds}∣Ft]=exp{-12∫tT‍σ2u2M2(b,u,T)du-12∫tT‍σ3u2M2(β,u,T)du}×exp{-ρ23∫tT‍σ2uσ3uM(b,u,T)M(β,u,T)du}×exp{-∫tT‍σ2vM(b,v,T)dW2v-∫tT‍σ3vM(β,v,T)dW3v}.
By Girsanov’s theorem,
(16)dW2tQ=dW2t+σ2tM(b,t,T)dt+ρ23σ3tM(β,t,T)dt,dW3tQ=dW3t+σ3tM(β,t,T)dt+ρ23σ2tM(b,t,T)dt,dW1tQ=dW1t+ρ12M1(t)dt+ρ13M2(t)dt,
where
(17)M1(t)=σ2tM(b,t,T)+ρ23σ3tM(β,t,T),M2(t)=σ3tM(β,t,T)+ρ23σ2tM(b,t,T).W1tQ, W2tQ, and W3tQ are standard Brownian motions under probability measure Q, and (W1tQ,W2tQ,W3tQ) has the same covariance matrix as (W1t,W2t,W3t). Therefore,
(18)E[e-∫tT‍(rs+λs)ds(ST-K)+∣Ft]=Z(t,T)EQ((ST-K)+∣Ft).
In addition,
(19)EQ((ST-K)+∣Ft)=EQ(ST1(ST≥K)∣Ft)-KEQ(1(ST≥K)∣Ft).
By the solution of SDE (2),
(20)ST=Stexp{∫tT‍(rs-kνs-12σ1s2)ds+∫tT‍σ1sdW1s+∫tT‍Ys-dNs}.
Under Q,
(21)ST=Stexp{Λ(t,T)+∫tT‍σ1sdW1sQ+∫tT‍σ2sM(b,s,T)dW2sQ+∫tT‍Ys-dNs},
where
(22)Λ(t,T)=-∫tT‍(kνs+12σ1s2)ds+M(b,t,T)rt+∫tT‍asM(b,s,T)ds-∫tT‍σ2s2M2(b,s,T)ds-ρ12∫tT‍σ1sM1(s)ds-ρ13∫tT‍σ1sM2(s)ds-ρ23∫tT‍σ2sσ3sM(b,s,T)M(β,s,T)ds.
So,
(23)PQ(ST≥K∣Ft)=PQ(∫tT‍σ1sdW1sQ+∫tT‍σ2sM(b,s,T)dW2sQ≥ln(K/St)-Λ(t,T)-∫tT‍YsdNs)=PQ(∫tT‍σ1sdW1sQ+∫tT‍σ2sM(b,s,T)dW2sQΔ≤ln(St/K)+Λ(t,T)+∫tT‍YsdNsΔ)=∑n=0∞‍(∫tT‍νsds)nn!e-∫tT‍νsdsE(N(d1)),
where
(24)Δ=∫tT‍σ1s2ds+∫tT‍σ2s2M2(b,s,T)ds+2ρ12∫tT‍σ1sσ2sM(b,s,T)ds,d1=ln(St/K)+Λ(t,T)+∑j=1nYjΔ.E(·) is the expectation of Yj under P, and N(·) is distribution function of standard normal distribution. On the other hand, let
(25)L(t,T):=EQ(ST∣Ft)=SteΛ(t,T)exp{12∫tT‍σ1s2ds+12∫tT‍σ2s2M2(b,s,T)ds}×exp{ρ12∫tT‍σ1sσ2sM(b,s,T)ds+k∫tT‍νsds}.
We can define the probability measure R as follows:
(26)dRdQ|Ft=STEQ(ST∣Ft)=exp{∫tT‍σ1sdW1sQ+∫tT‍σ2sM(b,s,T)dW2sQ-12∫tT‍σ1s2ds}×exp{-12∫tT‍σ2s2M2(b,s,T)ds-ρ12∫tT‍σ1sσ2sM(b,s,T)ds}×exp{∫tT‍YsdNs-k∫tT‍νsds}.
By Girsanov’s theorem, we have that
(27)dW1tR=dW1tQ-σ1tdt-ρ12σ2tM(b,t,T)dt,dW2tR=dW2tQ-σ2tM(b,t,T)dt-ρ12σ1tdt,
where, under R, W1tR, W2tR are standard Brownian motions and their correlation coefficient is ρ12, Nt is Poisson process with intensity (k+1)νt, and the density function of Y1 is eyf(y)/(k+1), where f(·) is the density function of Y1 under P. Since, under R,
(28)ST=Stexp{Λ(t,T)+Δ+∫tT‍σ1sdW1sR+∫tT‍σ2sM(b,s,T)dW2sR+∫tT‍Ys-dNs},
then
(29)EQ(ST1(ST≥K)∣Ft)=L(t,T)PR(ST≥K∣Ft)=L(t,T)PR(∫tT‍σ1sdW1sR+∫tT‍σ2sM(b,s,T)dW2sR≥ln(KSt)-Λ(t,T)-Δ-∫tT‍YsdNs)=L(t,T)PR(∫tT‍σ1sdW1sR+∫tT‍σ2sM(b,s,T)dW2sRΔ≤ln(St/K)+Λ(t,T)+Δ+∫tT‍YsdNsΔ)=L(t,T)∑n=0∞‍(∫tT‍νsds)nn!e-∫tT‍νsdsER(N(d2)),
where
(30)d2=ln(St/K)+Λ(t,T)+Δ+∑j=1nYjΔ=d1+Δ
and ER(·) is the expectation of Yj under R. By (23) and (29), we have the following lemma.Lemma 1.
Consider the Following:(31)E[e-∫tT‍(rs+λs)ds(ST-K)+∣Ft]=Z(t,T)×∑n=0∞‍(∫tT‍νsds)nn!×e-∫tT‍νsds[L(t,T)ER(N(d2))-KE(N(d1))].
In the following; in order to calculate I1, we write
(32)E[e-∫tT‍rsds(ST-K)+∣Ft]=E[e-∫tT‍rsdsST1(ST≥K)∣Ft]-KE[e-∫tT‍rsds1(ST≥K)∣Ft]=I3-I4.
Denote by P(t,T) the value at time t of a T maturity zero coupon bond whose face value is 1. Then
(33)P(t,T)=E[e-∫tT‍rsds∣Ft].
From (12),
(34)P(t,T)=exp{-M(b,t,T)rt-∫tT‍avM(b,v,T)dv+12∫tT‍σ2v2M2(b,v,T)dv},(35)dP(t,T)P(t,T)=rtdt-M(b,t,T)σ2tdW2t.
Define the Radon-Nikodym derivative given by
(36)dQTdP=e-∫tT‍rsdsE[e-∫tT‍rsds∣Ft]=exp{-∫tT‍M(b,v,T)σ2vdW2v-12∫tT‍σ2v2M2(b,v,T)dv},
and by Girsanov’s theorem, under QT,
(37)dW2tQT=dW2t+σ2tM(b,t,T)dtdW1tQT=dW1t+ρ12σ2tM(b,t,T)dt.
Thus ST can be rewritten as
(38)ST=Stexp{A(t,T)+∫tT‍M(b,v,T)σ2vdW2vQT+∫tT‍σ1vdW1vQT+∫tT‍Ys-dNs},
where
(39)A(t,T)=-∫tT‍(kνs+12σ1s2)ds+M(b,t,T)rt+∫tT‍asM(b,s,T)ds-∫tT‍σ2s2M2(b,s,T)ds-ρ12∫tT‍σ1sσ2sM(b,s,T)ds.
Then
(40)PQT(ST≥K∣Ft)=PQT(∫tT‍σ1sdW1sQT+∫tT‍σ2sM(b,s,T)dW2sQT≥ln(KSt)-A(t,T)-∫tT‍Ys-dNs)=PQT(∫tT‍σ1sdW1sQT+∫tT‍σ2sM(b,s,T)dW2sQTΔ≤ln(St/K)+A(t,T)+∫tT‍Ys-dNsΔ)=∑n=0∞‍(∫tT‍νsds)nn!e-∫tT‍νsdsE(N(d3)),
where E(·) is the expectation of Yj under P,
(41)d3=ln(St/K)+A(t,T)+∑j=1nYjΔ.
Therefore,
(42)I4=KP(t,T)EQT(1(ST≥K)∣Ft)=KP(t,T)∑n=0∞‍(∫tT‍νsds)nn!e-∫tT‍νsdsE(N(d3)).
Finally, let
(43)A(t,T):=E(e-∫tT‍rsdsST∣Ft)=Stexp{-∫tT‍(kνs+12σ1s2)ds+12∫tT‍σ1s2ds+k∫tT‍νsds}.
We define
(44)dQSdP=e-∫tT‍rsdsSTE[e-∫tT‍rsdsST∣Ft]=exp{∫tT‍σ1vdW1v-12∫tT‍σ2v2dv+∫tT‍Ys-dN(s)-k∫tT‍νsds},
and by Girsanov’s theorem, under QS,
(45)dW1tQS=dW1t-σ1tdtdW2tQS=dW2t-ρ12σ1tdt,
and, under QS, Nt is Poisson process with intensity (k+1)νt, and the density function of Y1 is eyf(y)/(k+1), where f(·) is the density function of Y1 under P. Since, under QS,
(46)ST=Stexp{A(t,T)+Δ+∫tT‍M(b,v,T)σ2vdW2vQS+∫tT‍σ1vdW1vQS+∫tT‍Ys-dNs},
then
(47)PQS(ST≥K∣Ft)=PQS(∫tT‍σ1sdW1sQS+∫tT‍σ2sM(b,s,T)dW2sQS≥ln(KSt)-A(t,T)-Δ-∫tT‍Ys-dNs)=PQT(∫tT‍σ1sdW1sQS+∫tT‍σ2sM(b,s,T)dW2sQSΔ≤ln(St/K)+A(t,T)+Δ+∫tT‍Ys-dNsΔ)=∑n=0∞‍(∫tT‍νsds)nn!e-∫tT‍νsdsES(N(d4)),
where ES(·) is the expectation of Yj under QS,
(48)d4=ln(St/K)+A(t,T)+Δ+∑j=1nYjΔ=d3+Δ.
From (42) and (33), we can conclude the following lemma.Lemma 2.
Consider the following:(49)E[e-∫tT‍rsds(ST-K)+∣Ft]=∑n=0∞‍(∫tT‍νsds)nn!×e-∫tT‍νsds[A(t,T)ES(N(d4))-KP(t,T)E(N(d3))].
Combined with the previous lemmas, the price of European call option at timet is given by the following theorem.Theorem 3.
The price of European call option with credit risk at timet is
(50)C(t,T)=∑n=0∞‍(∫tT‍νsds)nn!×e-∫tT‍νsds[w(A(t,T)ES(N(d4))ES-KP(t,T)E(N(d3)))+(1-w)1(τ>t)Z(t,T)×(L(t,T)ER(N(d2))ES-KE(N(d1)))].


## 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 chainξ 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
(51)(p11p12p21p22)=(0.70.30.20.8).
Assume that Yt is normally distributed with mean μJ and standard deviation σJ. 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.
Parameter name
Value in statee1
Value in statee2
Volatility ofS
σ
11
=
0.2
σ
12
=
0.4
Jump intensity
ν
1
=
15
ν
2
=
30
Speed of reversion ofr
b
1
=
2
b
2
=
1
Long-term average ofr
a
1
/
b
1
=
0.04
a
2
/
b
2
=
0.02
Volatility ofr
σ
21
=
0.15
σ
22
=
0.3
Speed of reversion ofλ
β
1
=
1.5
β
2
=
2
Long-term average ofλ
α
1
/
β
1
=
0.01
α
2
β
2
=
0.02
Volatility ofλ
σ
31
=
0.25
σ
32
=
0.45
Initial stock price
S
0
=
100
Initial interest rate
r
0
=
0.04
Initial default intensity
λ
0
=
0.5
Mean jump size
μ
J
=
0
Standard deviation of jump size
σ
J
=
0.1For eachω=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 forT=1, ρ12=0.7, ρ13=0.5, and ρ23=0.6.Figure 2
The option price with different maturityT 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 maturityT=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 coefficientsρ12 for T=1, ω=0.4, ρ13=0.5, and ρ23=0.6.Figure 4
The option price with different correlation coefficientsρ13 for T=1, ω=0.4, ρ12=0.7, and ρ23=0.6.In Figure5, 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 differentν 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 Figure6, 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.



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