# The Pricing of Vulnerable Options in a Fractional Brownian Motion Environment

Authors: Chao Wang; Shengwu Zhou; Jingyuan Yang
Journal: Discrete Dynamics in Nature and Society
 (2015)

---

## Abstract

Under the assumption of the stock price, interest rate, and default intensity obeying the stochastic differential equation driven by fractional Brownian motion, the jump-diffusion model is established for the financial market in fractional Brownian motion setting. With the changes of measures, the traditional pricing method is simplified and the general pricing formula is obtained for the European vulnerable option with stochastic interest rate. At the same time, the explicit expression for it comes into being.


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## Body

## 1. Introduction

Vulnerable option is a kind of option with credit risk that refers to a risk, a borrower that will default on any type of debt by failing to make required payments. Johnson and Stulz [1] firstly substituted default risk into option pricing and advanced a new definition called vulnerable option. Klein [2] obtained the pricing formula for vulnerable option with martingale method. Ammann [3] developed Klein’s model on the basis of structural approach. He finally obtained the explicit expression for vulnerable option under the assuming of interest rate and default intensity obeying the stochastic differential equation. What is more, other academics such as Chang and Hung [4] also discussed this problem, while all the discussions stated above are in the environment of geometric Brownian motion. Because of the inadequacies of geometric Brownian motion in describing the self-similarity and long-term dependence of stock prices, fractional Brownian motion is widely used into asset pricing. Hu and Øksendal [5] developed the structural approach in the condition that the stock prices followed a fractional Brownian motion and they proved that the correspondence to fractional Black-Scholes market had no arbitrage. For more literature on fractional Brownian motion, we can refer to Øksendal [6]. But there is another problem that fractional Black-Scholes market does not have equivalent martingale measure according to Sottinen and Valkeila [7]. Necula [8] applied quasi-martingale method to the risk neutral measure. Huang et al. [9] obtained the explicit expression for the European option price under the assuming of fractional Black-Scholes market. Su and Wang [10] and Li and Ma [11] derived the closed form formula for the price of the vulnerable European option by the method of changing measures.In this paper, we will use quasi-martingale method to change measures, so we can derive the general pricing formula for the European vulnerable option under the assuming of the stock price obeying the jump-diffusion model, the interest rate and default intensity obeying Vasicek model which are driven by fractional Brownian motion.


## 2. Market Environment

Let the uncertainty in the economy be described by the filtered probability spaceΩ,F,P,Ft0≤t≤T. rt is the short-term interest rate which is consistently positive and Ft-measurable in this space. Assume that Q is a risk neutral martingale measure under which the discounted asset price processes are martingales.Suppose the stock price is given by(1)dSt=rtStdt+σ1StdBH1t+StdJt,where σ1 is positive constant and BHt,t≥0 is a fractional Brownian motion whose Hurst parameter is H1/2≤H<1 in the space Ω,F,P,Ft0≤t≤T. Jt,t≥0 is a composite Poisson process, Jt=∑i=1NtYi, and Yi represents the ith jump range of Jt (using a convention that Y0=0 if there are no jumps). Consider EYi=θ. Yi is a sequence of independent identically distributed random variables with the finite expected value. Yi and BHt are mutually independent and Yi and Nt are mutually independent. Nt,t≥0 is a Poisson process whose intensity is λt. λt,t≥0 is a nonnegative adapted stochastic process which is integrable on any finite time interval. Nt, λt, and BHt are mutually independent.Suppose thatYi+1 follows log-normal distribution lnYi+1~NμJ,σJ2 so that ln⁡θ+1=Eln⁡Yi+1=μJ+(1/2)σJ2; then we have(2)lnYi+1~Nlnθ+1-12σJ2,σJ2.Suppose that the interest rate and default intensity follow Vasicek model under the risk neutral measure(3)drt=αβ-rtdt+σ2dBH2t(4)dλt=ab-λtdt+σ3dBH3t,where α,β,a,b,σ2, and σ3 are all positive constants. The covariance matrix of BH1t,BH2t,BH3t is (5)1ρ12ρ13ρ121ρ23ρ13ρ231t.In order to prove the theorem, we introduce two lemmas firstly.Lemma 1 (see [8]).
We denote byEt~· the quasi-conditional expectation with respect to the risk neutral measure. Then the price at every t∈0,T of a bounded FT-measurable claim F is given by(6)Ft=e-rT-tEt~F.Lemma 2 (see [12]).
IfX,Y~Nμ,Σ, where μ=μXμY, Σ=σX2,ρσXσYρσXσY,σY2, then(7)EeXIX>b=expμX+σX22N-b+μY+covX,YσY,where N· is a standard normal distribution.


## 3. Pricing Options

In this section, we intend to discuss pricing vulnerable options in a Fractional Brownian Motion Environment.We define that the default time isτ and ω represents the recovery rate due to the bankruptcy or reorganization, where ω is a constant. When the writer of the option defaults, the payoff is given by ω times the payoff of the default-free option at maturity. The price at every t∈0,T of an European vulnerable call option with strike price K and maturity T is given by(8)
C
t
,
T
=
E
e
-
∫
t
T
r
u
d
u
ω
S
T
-
K
+
I
τ
≤
T
+
S
T
-
K
+
I
τ
>
T
F
t
.Note thatIτ≤T+Iτ>T=1; we have(9)Ct,T=ωEe-∫tTruduST-K+Ft+1-ωEe-∫tTruduST-K+Iτ>TFt.SinceFt0≤t≤T is a filtration, then Ft⊆FT. Suppose there is no default at present time, by the law of iterated conditional expectations; therefore, (10)Ee-∫tTruduST-K+Iτ>TFt=EEe-∫tTruduST-K+Iτ>TFtFT.Obviouslye-∫tTruduST-K+Iτ>T is bonded and absolutely integrable, so we can interchange the two expectations byFubini’s theorem(11)Ee-∫tTruduST-K+Iτ>TFt=Ee-∫tTruduST-K+EIτ>TFTFt.Since the full path ofλt is known at time T, we have [13](12)EIτ>TFT=Pτ>T=e-∫tTλudu.Then (9) can be written as(13)Ct,T=ωEe-∫tTruduST-K+Ft+1-ωEe-∫tTru+λuduST-K+Ft.For convenience, letCt,T=I+II, where I=ωEe-∫tTruduST-K+Ft and II=1-ωEe-∫tTru+λuduST-K+Ft; I and II are given as follows.Theorem 3.
Consider(14)I=ω∑n=0∞λtT-tne-λtT-tn!·1+θnStNd1n+KPt,TNd2n,where(15)
d
2
n
=
-
ln
⁡
K
/
S
t
+
Δ
t
,
T
+
Π
t
,
T
+
ln
⁡
1
+
θ
n
-
n
σ
J
2
/
2
n
σ
J
2
+
2
H
∫
t
T
σ
∗
u
,
T
2
u
2
H
-
1
d
u
d
1
n
=
d
2
n
+
n
σ
J
2
+
2
H
∫
t
T
σ
∗
u
,
T
2
u
2
H
-
1
d
u
σ
∗
u
,
T
2
=
σ
2
2
M
u
,
T
,
α
2
+
σ
1
2
+
2
ρ
12
σ
1
σ
2
M
u
,
T
,
α
P
t
,
T
=
e
x
p
-
β
T
-
t
-
r
t
-
β
M
t
,
T
,
α
+
H
σ
2
2
∫
t
T
M
u
,
T
,
α
2
u
2
H
-
1
d
u
Δ
t
,
T
=
β
T
-
t
+
r
t
-
β
M
t
,
T
,
α
-
1
2
σ
1
2
T
2
H
-
t
2
H
Π
t
,
T
=
-
2
H
∫
t
T
σ
2
2
M
u
,
T
,
α
2
+
ρ
12
σ
1
σ
2
M
u
,
T
,
α
·
u
2
H
-
1
d
u
M
t
,
T
,
α
=
1
α
1
-
e
-
α
T
-
t
.Proof.
By means of Lemma1 we can see that the discounted asset price process whose numeraire is Ft is a quasi-martingale. So we can define different equivalent quasi-martingale measures for different numeraires.
Suppose there is a bank accountBt=exp∫0trudu and a coupon with maturity T. The price at every t∈0,T of the coupon is Pt,T. Then the forward quasi-martingale measure QT equivalent to Q by theRadon-Nikodym derivative is given as (16)dQTdQ=PT,TP0,TBT=1P0,TBT.Since(17)Pt,T=Ee-∫tTruduFt,hence(18)dQTdQ=e-∫0TruduEe-∫0TruduFt.
Using (3) we can have the expression for rt [14](19)rt=e-αt-srs+β1-e-αt-s+σ2∫ste-αt-udBH2u.We integrate two sides from t,T exclusively:(20)∫tTrsds=βT-t+rt-β∫tTe-αs-tds+∫tTσ2∫tse-αs-udBH2uds=βT-t+1αrt-β1-e-αT-t+σ2∫tT∫uTe-αs-udsdBH2u.UsingFubini’s theorem [15], we exchange the last item’s integral sequence and obtain(21)∫tTrsds=βT-t+1αrt-β1-e-αT-t+σ2∫tT∫tse-αs-udBH2uds=βT-t+1αrt-β1-e-αT-t+1ασ2∫tT1-e-αT-udBH2u.Let Mt,T,α=(1/α)1-e-αT-t, using multidimensional fractionalItô’s lemma [16](22)
P
t
,
T
=
E
e
-
∫
t
T
r
u
d
u
F
t
=
E
e
x
p
-
β
T
-
t
-
r
t
-
β
M
t
,
T
,
α
+
σ
2
∫
t
T
-
M
u
,
T
,
α
d
B
H
2
u
=
e
x
p
-
β
T
-
t
-
r
t
-
β
M
t
,
T
,
α
+
H
σ
2
2
∫
t
T
M
u
,
T
,
α
2
u
2
H
-
1
d
u
.So theRadon-Nikodym derivative is(23)dQTdQ=e-∫0TruduEe-∫0TruduFt=e-σ2∫0TMu,T,αdBH2u-Hσ22∫0TMu,T,α2u2H-1du.
Using fractionalGirsanov’s theorem [14](24)BH2Tu=BH2u+2Hσ2∫0tMu,T,αu2H-1duBH1Tu=BH1u+2Hρ12σ2∫0tMu,T,αu2H-1du.BH1Tu and BH2Tu are fractional Brownian motions under measure QT.
So, we can calculateI=ωETPt,TST-K+Ft under measure QT.
Since(25)ETPt,TST-K+Ft=ETPt,TSTIST≥KFt-KEfPt,TIST≥KFt.we define I1=ETPt,TSTIST≥KFt and I2=KETPt,TIST≥KFt for convenience. We will calculate I2 firstly.
Using (1) we have the expression for ST(26)
S
T
=
S
t
e
x
p
∫
t
T
r
u
d
u
-
1
2
σ
1
2
T
2
H
-
t
2
H
+
∫
t
T
σ
1
d
B
H
1
u
+
∑
i
=
0
N
t
l
n
Y
i
+
1
.We substitute ∫tTrudu into it; then(27)ST=SteβT-t+rt-βMt,T,α+σ2∫tTMu,T,αdBH2u-1/2σ12T2H-t2H+∫tTσ1dBH1u+∑i=0NtlnYi+1.Using fractionalGirsanov’s theorem, we have the expression for ST under measure QT(28)
S
T
=
S
t
e
x
p
Δ
t
,
T
+
σ
2
∫
t
T
M
u
,
T
,
α
d
B
H
2
u
+
∫
t
T
σ
1
d
B
H
1
u
+
∑
i
=
0
N
t
l
n
Y
i
+
1
=
S
t
e
x
p
Δ
t
,
T
+
σ
2
∫
t
T
M
u
,
T
,
α
d
B
H
2
T
u
+
∫
t
T
σ
1
d
B
H
1
T
u
+
∑
i
=
0
N
t
l
n
Y
i
+
1
+
Π
t
,
T
,where(29)Πt,T=-2H∫tTσ22Mu,T,α2+ρ12σ1σ2Mu,T,α·u2H-1du.
SinceI2=KEPt,TIST≥KFt=KPt,TETIST≥KFt, we substitute ST into IST≥K; then(30)IST≥K=Iσ2∫tTMu,T,αdBH2Tu+∫tTσ1dBH1Tu+∑i=0NtlnYi+1≥ln⁡K/St-Δt,T-Πt,T,since (31)σ2∫tTMu,T,αdBH2Tu~N0,2Hσ22∫tTMu,T,α2u2H-1du∫tTσ1dBH1Tu~N0,σ12T2H-t2H.When Nt=n,(32)∑i=0NtlnYi+1~Nln1+θn-12nσJ2,nσJ2.
According to the nature of normal distribution, whenNt=n, we have(33)ETIST≥KFt=∑n=0∞λtT-tne-λtT-tn!Nd2n,where (34)d2n=-ln⁡K/St+Δt,T+Πt,T+ln1+θn-nσJ2/2nσJ2+2H∫tTσ∗u,T2u2H-1du(35)σ∗u,T2=σ22Mu,T,α2+σ12+2ρ12σ1σ2Mu,T,α.So(36)I2=KPt,T∑n=0∞λtT-tne-λtT-tn!Nd2n.
Then we will calculateI1.
SinceI1=EPt,TSTIST≥KFt=Pt,TETSTIST≥KFt, we substitute ST into IST≥K; then(37)
E
T
S
T
I
S
T
≥
K
F
t
=
S
t
e
Δ
t
,
T
+
Π
t
,
T
E
T
e
σ
2
∫
t
T
M
u
,
T
,
α
d
B
H
2
T
u
+
∫
t
T
σ
1
d
B
H
1
T
u
+
∑
i
=
0
N
t
l
n
Y
i
+
1
·
I
σ
2
∫
t
T
M
u
,
T
,
α
d
B
H
2
T
u
+
∫
t
T
σ
1
d
B
H
1
T
u
+
∑
i
=
0
N
t
l
n
Y
i
+
1
≥
l
n
⁡
K
/
S
t
-
Δ
t
,
T
-
Π
t
,
T
.
Using Lemma2, when Nt=n,(38)ETSTIST≥KFt=SteΔt,T+Πt,T+ln1+θn+Hσ22∫tTMu,T,α2u2H-1du+1/2σ12T2H-t2H+2Hρ12σ1σ2∫tTMu,T,αu2H-1du·∑n=0∞λtT-tne-λtT-tn!Nd1n,where (39)d1n=d2n+nσJ2+2H∫tTσ∗u,T2u2H-1duso (40)I1=Pt,TETSTIST≥KFt=SteΔt,T+Πt,T+ln1+θn+Hσ22∫tTMu,T,α2u2H-1du+1/2σ12T2H-t2H+2Hρ12σ1σ2∫tTMu,T,αu2H-1du·e-βT-t-rt-βMt,T,α+Hσ22∫tTMu,T,α2u2H-1du=1+θnSt∑n=0∞λtT-tne-λtT-tn!Nd1n.Hence, when Nt=n,(41)I=ωI1+I2=ω∑n=0∞λtT-tne-λtT-tn!·1+θnStNd1n+KPt,TNd2n.Theorem 4.
Consider(42)II=1-ω∑n=0∞λtT-tne-λtT-tn!Xt,T·Yt,TNd3n-KNd4n,where (43)
d
4
n
=
-
l
n
⁡
K
/
S
t
+
Δ
t
,
T
+
Λ
t
,
T
+
l
n
1
+
θ
n
-
n
σ
J
2
/
2
n
σ
J
2
+
2
H
∫
t
T
σ
∗
u
,
T
2
u
2
H
-
1
d
u
,
d
3
n
=
d
4
n
+
n
σ
J
2
+
2
H
∫
t
T
σ
∗
u
,
T
2
u
2
H
-
1
d
u
,
X
t
,
T
=
exp
⁡
-
β
+
b
T
-
t
-
r
t
-
β
M
t
,
T
,
α
-
r
t
-
b
M
t
,
T
,
a
+
H
∫
t
T
σ
2
2
M
2
u
,
T
,
α
+
σ
3
2
M
2
u
,
T
,
a
+
2
ρ
23
σ
2
σ
3
M
u
,
T
,
α
M
u
,
T
,
a
u
2
H
-
1
d
u
,
Y
t
,
T
=
S
t
e
x
p
Δ
t
,
T
+
Λ
t
,
T
+
l
n
1
+
θ
n
+
H
∫
t
T
σ
∗
u
,
T
2
u
2
H
-
1
d
u
,
Λ
t
,
T
=
-
2
H
∫
t
T
σ
2
M
u
,
T
,
α
M
1
u
+
ρ
12
σ
1
M
1
u
+
ρ
13
σ
1
M
2
u
u
2
H
-
1
d
u
,
M
1
u
=
σ
2
M
u
,
T
,
α
+
ρ
23
σ
3
M
u
,
T
,
a
,
M
2
u
=
σ
3
M
u
,
T
,
a
+
ρ
23
σ
2
M
u
,
T
,
α
,
M
t
,
T
,
a
=
1
a
1
-
e
-
a
T
-
t
.Proof.
Using Lemma1, the quasi-martingale measure Qλ equivalent to Q by theRadon-Nikodym derivative is given as (44)dQλdQ=e-∫tTru+λuduEe-∫tTru+λudu.Using (4) we can have the expression for λt(45)λt=e-at-srs+b1-e-at-s+σ3∫ste-at-udBH3u.Then we have(46)∫tTλsds=bT-t+1aλt-b1-e-aT-t+1aσ3∫tT1-e-aT-udBH3u.Let Mt,T,a=(1/a)1-e-aT-t; then(47)-∫tTru+λudu=-β+bT-t-rt-βMt,T,α-λt-bMt,T,a-σ2∫tTMu,T,αdBH2u-σ3∫tTMu,T,adBH3u.Using multidimensional fractionalItô lemma, let Xt,T=Ee-∫tTru+λuduFt; then(48)
X
t
,
T
=
exp
⁡
-
β
+
b
T
-
t
-
r
t
-
β
M
t
,
T
,
α
-
r
t
-
b
M
t
,
T
,
a
+
H
∫
t
T
σ
2
2
·
M
2
u
,
T
,
α
+
σ
3
2
M
2
u
,
T
,
a
+
2
ρ
23
σ
2
σ
3
M
u
,
T
,
α
M
u
,
T
,
a
u
2
H
-
1
d
u
.So(49)
d
Q
λ
d
Q
=
e
-
∫
t
T
r
u
+
λ
u
d
u
E
e
-
∫
t
T
r
u
+
λ
u
d
u
=
exp
⁡
-
σ
2
∫
t
T
M
u
,
T
,
α
d
B
H
2
u
-
σ
3
∫
t
T
M
u
,
T
,
a
d
B
H
3
u
-
H
∫
t
T
σ
2
2
M
2
u
,
T
,
α
+
σ
3
2
M
2
u
,
T
,
a
+
2
ρ
23
σ
2
σ
3
M
u
,
T
,
α
M
u
,
T
,
a
u
2
H
-
1
d
u
.
Using fractionalGirsanov’s theorem(50)BH2λu=BH2u+2Hσ2∫tTMu,T,αu2H-1du+2Hρ23σ3·∫tTMu,T,au2H-1du,BH3λu=BH3u+2Hσ3∫tTMu,T,au2H-1du+2Hρ23·σ2∫tTMu,T,αu2H-1du,BH1λu=BH1u+2Hρ12∫tTM1uu2H-1du+2Hρ13∫tTM2uu2H-1du,where (51)M1u=σ2Mu,T,α+ρ23σ3Mu,T,a,M2u=σ3Mu,T,a+ρ23σ2Mu,T,αUnder the probability measure Qλ, the processes BH1λu, BH2λu and BH3λu are fractional Brownian motions. The covariance matrix of BH1λ,BH2λ,BH3λ is the same as BH1,BH2,BH3.
Then we can calculateII=1-ωXt,TEλST-K+Ft under Qλ.
Since(52)EλST-K+Ft=EλSTIST≥KFt-KEλIST≥KFt,
we defineI3=EλSTIST≥KFt and I4=KEλIST≥KFt for convenience. We will calculate I4 firstly.
Using fractionalGirsanov’s theorem, we have the expression for ST under measure Qλ(53)ST=SteΔt,T+σ2∫tTMu,T,αdBH2λu+∫tTσ1dBH1λu+∑i=0NtlnYi+1+Πt,T,where (54)
Δ
t
,
T
=
β
T
-
t
+
r
t
-
β
M
t
,
T
,
α
-
1
2
·
σ
1
2
T
2
H
-
t
2
H
Λ
t
,
T
=
-
2
H
∫
t
T
σ
2
M
u
,
T
,
α
M
1
u
+
ρ
12
σ
1
M
1
u
+
ρ
13
σ
1
M
2
u
u
2
H
-
1
d
u
.We substitute ST into IST≥K; when Nt=n, then(55)EλIST≥KFt=∑n=0∞λtT-tne-λtT-tn!Nd4n,where(56)d4n=-ln⁡K/St+Δt,T+Λt,T+ln1+θn-nσJ2/2nσJ2+2H∫tTσ∗u,T2u2H-1du.So(57)I4=KEλIST≥KFt=K∑n=0∞λtT-tne-λtT-tn!Nd4n.
Then we will calculateI3.
SinceI3=EλSTIST≥KFt, we substitute ST into IST≥K(58)
E
λ
S
T
I
S
T
≥
K
F
t
=
S
t
e
Δ
t
,
T
+
Λ
t
,
T
E
λ
e
σ
2
∫
t
T
M
u
,
T
,
α
d
B
H
2
λ
u
+
∫
t
T
σ
1
d
B
H
1
λ
u
+
∑
i
=
0
N
t
l
n
Y
i
+
1
·
I
σ
2
∫
t
T
M
u
,
T
,
α
d
B
H
2
λ
u
+
∫
t
T
σ
1
d
B
H
1
λ
u
+
∑
i
=
0
N
t
l
n
Y
i
+
1
≥
l
n
⁡
K
/
S
t
-
Δ
t
,
T
-
Λ
t
,
T
.Using Lemma 2, when Nt=n, we have(59)EλSTIST≥KFt=SteΔt,T+Λt,Teln1+θn-1/2nσJ2+Hσ22∫tTMu,T,α2u2H-1du+1/2σ12T2H-t2H+2Hρ12σ1σ2∫tTMu,T,αu2H-1du+1/2nσJ2·∑n=0∞λtT-tne-λtT-tn!Nd3n.Let (60)Yt,T=SteΔt,T+Λt,T+ln1+θn+Hσ22∫tTMu,T,α2u2H-1du+1/2σ12T2H-t2H+2Hρ12σ1σ2∫tTMu,T,αu2H-1du,and then(61)EλSTIST≥KFt=Yt,T∑n=0∞λtT-tne-λtT-tn!Nd3n,where (62)d3n=d4n+nσJ2+2H∫tTσ∗u,T2u2H-1du.So (63)I3=EλSTIST≥KFt=Yt,T∑n=0∞λtT-tne-λtT-tn!Nd3n.Hence (64)II=1-ωXt,TI3-I4=1-ω·∑n=0∞λtT-tne-λtT-tn!Xt,T·Yt,TNd3n-KNd4n.Theorem 5.
The price at timet of the European vulnerable call option is(65)
C
t
,
T
=
∑
n
=
0
∞
λ
t
T
-
t
n
e
-
λ
t
T
-
t
n
!
·
ω
1
+
θ
n
S
t
N
d
1
n
+
K
P
t
,
T
N
d
2
n
+
1
-
ω
X
t
,
T
·
Y
t
,
T
N
d
3
n
-
K
N
d
4
n
.Note 1.
When there is no jump process,H=1/2, and recovery rate is 1, (65) can be simplified as(66)Ct,T=StNd1+KPt,TNd2.


## 4. Numerical Experiments

In this section, we mainly discuss the influence of different parameters on option prices. Figures are about the relationship between option prices and strike prices with different parameters.The values of different parameters are as follows:(67)α=0.3;β=0.05;a=0.2;b=0.02;σ1=0.2;σ2=0.15;σ3=0.25;σJ=0.1;r0=0.02;λ0=0.5;S0=100;T=1.For all figures, the horizontal axis shows strike price and the vertical axis shows option value.Figure1 is about the influence of Hurst parameter H on option value. As we can see from the figure, the influence will become larger with the increasing of strike prices within limits.Figure 1
The influence ofHurst parameter H.Figure2 is about the influence of recovery rate ω on option value. The default risk will decline with the increasing of recovery rate. So the option prices will fall down.Figure 2
The influence of recovery rateω.Figures3, 4, and 5 are about the influences of different covariance. According to these figures, we can see that different covariance has different influences on option value.Figure 3
The influence of default covarianceρ23.Figure 4
The influence of covarianceρ12.Figure 5
The influence of covarianceρ13.


## 5. Conclusion

The method of changing measures is widely used for pricing options. In this paper, we develop this method and prove its feasibility in pricing options under the assumption of fractional Brownian motion. What is more, we also take jump process into consideration and obtain the general pricing formula for the European vulnerable option. Finally, we verify its accuracy through the numerical experiments.



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*Source: 579213-2015-07-09.xml*