We reduce the problem of pricing continuously monitored defaultable securities (namely, barrier type options, corporate debts) under a stochastic interest rate framework to calculations of boundary crossing probabilities (BCP) for Brownian Motion (BM) with stochastic boundaries. For the case when the interest rate is governed by linear stochastic equation (Vasicek model) we suggest a numerical algorithm for calculation of BCP based on a piece-wise linear approximation for the stochastic boundaries. We also provide an estimation for a rate of convergence of the suggested approximation as a function of number of nodes and illustrate the results by numerical examples.

Alex Novikov

Nina Kordzakhia

English

Research Papers in Economics

 
 
QUANTITATIVE FINANCE RESEARCH CENTRE 
 
 
Research Paper 193  February 2007 
 
 
 
 
Pricing of Defaultable Securities under 
Stochastic Interest 
 
Nino Kordzakhia and Alexander Novikov 
 
 
ISSN 1441-8010                  www.qfrc.uts.edu.au 
  
QUANTITATIVE FINANCE 
RESEARCH CENTRE
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���������������QMF 2006 Sydney,
Pricing of Defaultable Securities under Stochastic Interest
Nino Kordzakhia
Macquarie University; NSW 2109, Australia
Email: nino.kordzakhia@mq.edu.au
Alexander Novikov
University of Technology, Sydney; NSW 2007, Australia
Email: alex.novikov@uts.edu.au
Abstract. We reduce the problem of pricing continuously monitored defaultable securities
(namely, barrier type options, corporate debts) under a stochastic interest rate framework to calcu-
lations of boundary crossing probabilities (BCP) for Brownian Motion (BM) with stochastic bound-
aries. For the case when the interest rate is governed by linear stochastic equation (Vasicek model)
we suggest a numerical algorithm for calculation of BCP based on a piece-wise linear approxima-
tion for the stochastic boundaries. We also provide an estimation for a rate of convergence of the
suggested approximation as a function of number of nodes and illustrate the results by numerical
examples.
1. Introduction. Practitioners often acknowledge an existence of a common problem with pricing
schemes of exotic (e.g. barrier, lookback) options for the contracts with long maturities. The prices
of these instruments signiﬁcantly depend on an interest rate term-structure. For the case of deter-
ministic interest rates some results in this direction have been obtained by Roberts/Shortland (1997),
Novikov/Frishling/Kordzakhia (1999,2003) (see also other references in these papers). For deterministic
interest rates the pricing problem of barrier options is reduced to calculations of BCP for BM with deter-
ministic boundaries. In Section 2 we present a modiﬁcation of the algorithm from [9] and [10] to handle
a general setup of stochastic interest rates. In Sections 3 and 4 we describe the numerical algorithm in
details and provide an estimation for a rate of convergence of the suggested approximation as a function
of number of nodes. In Section 5 the results are illustrated by numerical examples.
Further we use the standard notation St for a price of an underlying asset and rt for a default-free short
interest rate. We assume that St is a diﬀusion process of the form
St = S0eYt
with the log-return process Yt governed by the equation
dYt = µ(t)dt + σdWt,
where Wt is a standard BM with respect to a ’real-world’ measure P and a ﬁltration Ft, σ is a constant
volatility, µ(t) is a historical trend.
Let fT be a payoﬀ of an option at maturity T. In this paper we concentrate on two fundamental examples
of payoﬀ functions (there are many other examples that may be treated in a similar way).
Case 1. The payoﬀ of Up-and-Out European Barrier Call option is
fT = (ST − K)+I{τ > T},
where T is a maturity time, a default time τ = inf{t : St ≥ G+(t)}, G+(t) is a continuous deterministic
barrier, and I{τ > T} is an indicator function.
Case 2. The payoﬀ at maturity T of a defaultable zero-coupon bond St is
fT = 1 − wI{τ ≤ T}, 0 < w = const < 1,
where τ = inf{t : Yt ≤ bt}, the default threshold bt is deﬁned as follows
bt = b0 + λ
Z t
0
(Ys − v − bs)ds,Pricing of Defaultables
with some constants b0, v and λ > 0. An economic rationale behind the model is well justiﬁed in
Collin-Dufresne/Goldstein (2001).
Put for convenience
bt = Yt − lt.
Then
τ = inf{t : lt ≤ 0}, dlt = µ(t)dt + λ(v − lt)dt + σdWt. (1)
Under the assumption λ = 0 this model has been studied by Longstaﬀ - Schwartz (1997).
2. Pricing formulae. Under a free-arbitrage assumption a fair price of an option with a payoﬀ
fT is
Cf = E
∗

e
−
R T
0
rsdsfT

, (2)
where E
∗
is a symbol of expectation with respect to the equivalent risk-neutral (martingale) measure
P∗ such that the process
e
−
R t
0
rsdsSt
is (P∗,Ft)−martingale. It implies (see e.g. Liptser/Shiryaev (2003)) that
Yt = σW∗
t +
Z t
0
rsds −
σ2
2
t, (3)
where W∗
t is a standard BM with respect P∗.
In general, the fair price (2) can be approximated by Monte Carlo methods via discretization of a time
parameter. However, such algorithms typically have a signiﬁcant bias; this aspect is discussed in Section
5 for the case of discretely monitored options.
The price of a risk-free zero-coupon bond with maturity T is
P(0,T) = E∗(e
−
R T
0
rsds)
Deﬁne the forward measure PF as follows:
PF(A) = E∗(I(A)e
−
R T
0
rsds)/P(0,T), A ∈ FT.
Then
Cf = E
∗

e
−
R T
0
rsdsfT

= P(0,T)EF (fT), (4)
where EF is a symbol of expectation with respect to PF.
For aﬃne models (e.g. Vasicek, CIR models) the function P(0,T) can be calculated analytically. There-
fore, a pricing problem is reduced to evaluation of EF (fT). Further we assume that the risk-free short
rate rt is a Ornstein-Uhlenbeck (or, Vasicek) process governed by a linear stochastic equation
drt = ar(¯ r − rt)dt + σrdW
(r)
t , (5)
where a parameter ar > 0, W
(r)
t is another standard BM with respect to the risk-neutral measure P∗
and given ﬁltration Ft such that E∗(W
(r)
t W∗
t ) = 0. A general case of correlated St and rt we plan to
discuss in a separate paper.
The solution of (5) has the following representation
rt = a(t) + σrξt, a(t) = ¯ r + (r0 − ¯ r)e−art,
where
dξt = −arξtdt + dW
(r)
t ,ξ0 = 0Kordzakhia, Novikov
and, therefore,
ξt = e−art
Z t
0
earsdW(r)
s . (6)
By direct calculations
P(0,T) = E
∗
(e
−
R T
0
rsds) = e−A(T)+D
2(T)/2
with
A(T) =
Z T
0
a(s)ds = ¯ rT + (r0 − ¯ r)(1 − e−arT)/ar,
D(T) = σ2
rV ar(
Z T
0
ξsds) =
σ2
r
2a3
r
 
2Tar − e−2Tar − 4a2
r + 4a2
re−Tar + 1

.
Also, by the Girsanov theorem one could show that under the forward measure PF the BM W
(r)
t has a
drift term
q(t,T) = −
σr
ar
Z t
0
(1 − e−ar(T−u))du (7)
(see e.g. Brigo/Mercurio (2006), p. 886).
Case 1. Using (3) we obtain
τ = inf{t : St > G+(t)} = inf{t : σW∗
t >
σ2
2
t + log(G+(t)/S0) −
Z t
0
rsds}.
The fair price Cf can be written as follows
Cf = E∗[e
−
R T
0
rsds(ST − K)I{τ > T,ST > K}].
In view of (4) and (7 we have
= S0[E∗

eσW
∗
T− σ2
2 TI{τ > T,ST > K}

− KP(0,T)PF{τ > T,ST > K}] =
S0q1 − P(0,T)Kq2 (8)
with
q1 = ˜ P{σW∗
t <
σ2
2
t + log(G+(t)/S0) −
Z t
0
rsds,σW∗
T >
σ2
2
T +
Z T
0
rsds + log(K/S0)},
where the measure ˜ P is deﬁned as follows
˜ P(A) = E∗(I{A}eσW
∗
T− σ2
2 T) , A ∈ FT.
Note that, the BM W∗
t has a drift σt under the measure ˜ P.
The second probability in (8) has a similar form:
q2 = PF{σW∗
t <
σ2
2
t + log(G+(t)/S0) −
Z t
0
rsds,σW∗
T >
σ2
2
T +
Z T
0
rsds + log(K/S0)},
(recall that under PF the BM Wr
t has the drift q(t,T) deﬁned in (7)).
Case 2. We have
Cf = E
∗
e
−
R T
0
rsds(1 − w + wI{τ > T}) = P(0,T)[1 − w + PF{lt ≥ 0,t ≤ T}],
where due to (3) and (1) we have
dlt = rtdt + λ(v − lt)dt + σdW∗
t ,l0 = −b0.Pricing of Defaultables
With a simple algebra one can show that the problem of computation of PF{lt ≥ 0,t ≤ T} can be
reduced to ﬁnding of probabilities
P{ηt < g(t) − e−λt
Z t
0
eλsξsds, t ≤ T},
where g(t) is a deterministic function, ξs is deﬁned in (6) and ηt is a Ornstein-Uhlenbeck process deﬁned
by the equation
dηt = −ληtdt + σdWt,η0 = 0
and therefore
ηt = σe−λt
Z t
0
eλsdWs.
Applying the change of time
u =
Z t
0
e2λsds
we obtain Z t(u)
0
eλsdWs = ˜ Wu, t(u) =
log(2λu + 1)
2λ
,
where ˜ Wu is another standard BM. Hence, we can reduce the pricing problem to computation of the
probability
P{σWu <
√
2λu + 1g(t(u)) −
Z t(u)
0
eλsξsds, u ≤
Z T
0
e2λsds}.
3. Approximations to BCP. For Cases 1 and 2 discussed above we have reduced the option
pricing problem to ﬁnding BCP of the form
P(b,h) := P{WT > b;Wt < h(t),t ≤ T},
where b is a random variable, h(t) is a stochastic boundary such that
h(t) = g(t) − zRt,Rt =
Z t
0
ξsds,
g(t) is smooth deterministic function, z = const, ξt has the form (6) with a standard BM Wt.
Further we study Case 1 in somewhat more details. Note that for this case λ = 0 and for probabilities
qi = P(bi,hi),i = 1,2 deﬁned in (8) we have
σbi = σr
Z T
0
ξsds + Hi,
σhi(t) = gi(t) − σrR
(i)
t ,
where
g1(t) = −
σ2
2
t + log(G+(t)/S0) −
Z t
0
a(s)ds,
H1 = −
σ2
2
T +
Z T
0
a(s)ds + log(K/S0)},Rt =
Z t
0
ξsds
ξt has the form (6) with a standard BM W
(r)
t ;
g2(t) =
σ2
2
t + log(G+(t)/S0) −
Z t
0
a(s)ds,Kordzakhia, Novikov
H2 =
σ2
2
T +
Z T
0
a(s)ds + log(K/S0)}.
We suggest to use approximating probabilities P(ˆ b,ˆ h) with piece-wise linear stochastic boundaries ˆ h(t)
instead of h(t) = g(t) − zRt such that for some partition {ti},0 = t0 < t1 < ... < tm = T,
ˆ h(tj) = h(tj) = g(tj) − zRtj,j = 0,...,m. (9)
Further we assume that z = 1.
Note that the vector R = (Rt1,...,Rtm) and the process W∗
t are independent due to the assumption on
independency of St and rt.
The approximating probabilities P(ˆ b,ˆ h) can be calculated using the following formula
P(ˆ b,ˆ h) = E[I{WT > ˆ b}
m−1 Y
j=0
(1 − e
−
2(ˆ h(tj)−Wtj )+(ˆ h(tj+1)−Wtj+1)+
tj+1−tj )]. (10)
This formula was derived by Wang/P¨ otzelberger (1997) for the special case when b = −∞ and boundaries
ˆ h(t) are deterministic piece-wise linear functions. In a more general context, including double barriers,
although restricted to deterministic boundaries, a similar result was derived in [9]. To prove formula
(10) for stochastic boundaries one could follow the proof in [9] or, alternatively, Proposition 1 formulated
below may be helpful.
Let us consider
ηj(u) := Wtj+u − Wtj − u
Wtj+1 − Wtj
∆tj
, 0 ≤ u ≤ ∆tj = tj+1 − tj, j = 0,...,m − 1
and
W = {Wt1,...,Wtm}.
Proposition 1. For any u and v
Cov(ηi(u),ηj(v)|W) = 0,i 6= j,
Cov(ηi(u),ηi(v)|W) = min(u,v) − uv/∆ti.
The proof of this result is based on standard properties of Gaussian random variables.
As a simple consequence of Proposition 1 one can see that the random processes {ηj(u),j = 0,...,m−1}
are jointly independent Brownian bridges conditioned to W. Recall another well-known fact that for a
linear (nonrandom) function h(t)
P{Wt < h(t),ti ≤ t ≤ ti+1|(Wti,Wti+1)} = 1 − e
−
2(h(ti)−Wti)+(h(ti+1)−Wti+1)+
ti+1−ti .
Now using conditioning with respect to a vector (W,R) with R = (Rt1,...,Rtm) one can obtain (10) for
the general case under discussion.
Note that to compute (10) we need to simulate 2m-Gaussian r.v.’s, Wtj and Rtj, j = 1,...,m or, alter-
natively, we could use 2m numerical integrations with respect to space variables.
4. Accuracy of the approximation. Our estimation of accuracy of the suggested approxima-
tion will be based on the following result proved in Borovkov/Novikov (2005).
Notation. Lip(κ) is the class of Lipschitz functions h(t) on [0,T] :
|h(t + h) − h(t)| ≤ κh,0 = t < t + h = T,
where κ is a ﬁnite nonrandom constant.
Proposition 2 (Borovkov-Novikov (2005)). Let stochastic boundaries h(t) ∈ Lip(κ).Then for any
ε > 0
P(b,h + ε) − P(b,h) ≤ (5κ/2 + 1/
√
T)ε.Pricing of Defaultables
The proof of this results is given in [3], Lemma 1. The latter was formulated for the case of nonstochastic
boundaries but one can easy to check that the result does hold for stochastic boundaries h(t) ∈ Lip(κ)
as well.
Further we assume: T = 1, ξt has the form (6) with a standard BM W
(r)
t ,
h(t) = g(t) −
Z t
0
ξsds,
g(t) is a twice continuously diﬀerentiable deterministic function, b is a constant.
Let ˆ g(t) be a piece-wise linear function such that ˆ g(tj) = g(tj) and Rt be a piece-wise linear process,
the stochastic boundary ˆ h is deﬁned in (9).
Theorem 1. Let tj =
j
m. Then
|P(b,ˆ h) − P(b,h)| = O(
log(m)
m3/2 ).
For the proof we will need the following
Proposition 3. Let xm = C
p
log(m) with large enough constant C. Then
P{max
t≤T
|
Z t
0
ξsds − Rt| >
xm
m3/2} = o(
1
m3/2).
Proof of Theorem 1. Since g(t) is a twice continuously diﬀerentiable function, for the linear piece-wise
function ˆ gt such that ˆ g(tj) = g(tj) we have 1
|ˆ g(t) − g(t)| ≤
C
m2.
By virtue of Proposition 3 proved below with xm = C
p
log(m) we have
P(b,ˆ h −
xm
m3/2 −
C
m2) − o(
1
m3/2) ≤ P(b,g) ≤ P(b,ˆ h +
xm
m3/2 +
C
m2) + o(
1
m3/2).
Now applying Proposition 2, for any κ > 0 we obtain
|P(b,g) − P(b,ˆ h)| ≤ (5κ/2 + 1)(
2xm
m3/2 +
C
m2) + P{max
s≤1
|ξs| > κ} + o(
1
m3/2). (11)
Furthermore, using properties of Gaussian distribution one can show that
P(max
s≤1
|ξs| > κ) ≤ Ce−λκ
2
/κ
with some λ > 0 . Choosing κ = κ(m) =
q
2log(m)
λ we obtain
P{max
s≤1
|ξs| > κ(m)} = O(
1
m3/2).
Combining this estimate with (11) we complete the proof of Theorem 1.
Proof of Proposition 3. Due to the deﬁnition of Rt, we have
max
t≤1
|
Z t
0
ξsds − Rtj| = max
j
Z tj
tj−1
|ξs − m
Z tj
tj−1
ξudu|ds ≤
2
m
max
j
( max
tj−1≤s≤tj
|ξs − ξtj−1|).
1C is a generic constantKordzakhia, Novikov
Applying Girsanov transform, for any x > 0 we have
P{max
j
( max
tj−1≤s≤tj
|ξs − ξtj−1|) > x} =
E[I{max
j
( max
tj−1≤s≤tj
|Ws − Wtj−1|) > x}e
−ar
R 1
0
WsdWs−
a2
r
2
R 1
0
W
2
s ds]
≤ ear/2P{max
j
( max
tj−1≤s≤tj
|Ws − Wtj−1|) > x}, (as
Z 1
0
WsdWs = (W2
1 − 1)/2).
The random variables maxtj−1≤s≤tj |Ws − Wtj−1|, j = 1,...,m, are iid and by scaling and homogeneity
properties we have
max
tj−1≤s≤tj
|Ws − Wtj−1|
d = max
0≤s≤1
|Ws|/
√
m,
Note that
max
s≤1
|Ws| ≤ max
s≤1
(Ws) + max
s≤1
(−Ws)
and
max
0≤s≤1
(−Ws)
d = max
0≤s≤1
(Ws)
d = |W1|.
It implies that for any x > 0
P{max
j
( max
tj−1≤s≤tj
|Ws − Wtj−1|) > x} ≤ 2P{max
j
γj > x/2}
where iid r.v.’s γj
d = |W1|. and, hence,
P{max
t≤1
|
Z t
0
ξsds − Rt| >
x
m3/2} ≤ 2ear/2P{max
j
(γj) > x/2} =
= 2ear/2(1 − (2Φ(x/2) − 1)m) ≤ 2ear/2(1 − (1 −
2e− x2
8
x
√
2π
)m),
here we use the well-known inequality 1 − Φ(x) ≤ e
− x2
2
x
√
2π for standard normal distribution Φ(x).
For x = xm → ∞
(1 −
4e−
x2
m
8
xm
√
2π
)m = e
− 4me
−
x2
m
8
xm
√
2π (1+o(1))
Choose now xm = C
p
log(m) with large enough C. Then 4me
−
x2
m
2
xm
√
2π = o( 1
m3/2) and so
1 − e
− 2e
−
x2
m
2
xm
√
2π m(1+o(1)) = o(
1
m3/2).
That completes the proof of Proposition 3.
5. Numerical examples. Here we consider model (5) with parameters
T = 1,ar = 1, ¯ r = 0.1,r0 = 0.15.
For the asset price St we consider the same set of parameters as in [12], namely,
S0 = 10, σ = 0.1, K = 11,G+(t) = 12.
In terms of our notation used in Section 2
rt = a(t) + σrξt,E(rt) = a(t) = 0.1 + 0.05e−t,
Z t
0
a(s)ds = 0.1t + 0.05(1 − e−t),
dξt = −ξtdt + σrdW
(r)
t ,ξ0 = 0,
where W
(r)
t is a standard BM.Pricing of Defaultables
5.1. Deterministic interest rate. In [12] the authors have set σr = 0 and then using some proba-
bilistic arguments obtained the following bounds for a fair price Cf of Up-and-In European barrier call
option
0.516758 ≤ Cf ≤ 0.517968. (12)
Denote by Cd
f(m) the fair prices of the discretely monitored options with m monitoring dates and let
ˆ Cd
f(m) be a Monte Carlo estimation for Cd
f(m).
With the number of paths N = 107 we obtained the following results
ˆ Cd
f(1000) = 0.5125,
ˆ Cd
f(100) = 0.5036,
ˆ Cd
f(50) = 0.4985. (13)
The signiﬁcant diﬀerence of 0.8%, even for m = 1000, compared to bounds of the continuously monitored
barrier price from (12) can be explained by a slow rate of convergence of discretely monitored prices to
continuous monitored ones. The magnitude of convergence rate is known to be of order O( 1 √
m), (see
e.g. Nagaev (1970), Borovkov (1982)).
Denote by ˆ Cf(m) a Monte Carlo estimation for the fair prices of the continuously monitored options
obtained through the numerical procedure described in Section 3; here m is a number of node points
for a piece-wise linear boundary. With the number of paths N = 107 we obtained the following results
ˆ Cf(1000) = 0.5167,
ˆ Cf(100) = 0.5167,
ˆ Cf(50) = 0.5168. (14)
The contrast diﬀerence with the results for discretely monitoried options is due to a better rate of
convergence which, accordingly to Theorem 1, is at least of order O(
log(m)
m3/2 ). As a matter of fact, for
smooth deterministic boundaries the rate of convergence is even faster, namely, it is of order O( 1
m2) as
shown in [11], [3].
In practice, the price Cd
f(m) as a function of m is also of interest. Based on the known rate of convergence
Cd
f(m) to the limit as m → ∞, we suggest to use for any m a simple approximation ˜ Cd
f(m) deﬁned by
the following formula:
˜ Cd
f(m) = ˆ Cf(50) +
A
√
m
, (15)
where A is a constant; further we deﬁne the constant A from the equation ˆ Cd
f(50) = ˆ Cf(50) + A √
50.
The following table contains the simulated prices ˆ Cd
f(m) and the relative error of approximation (15) in
the range m ∈ [100,250].
Table 1. Prices of discretely monitored option, σr = 0, ˆ Cf(50) = 0.51680
Price Relative
m ˆ Cd
f(m) Error(%)
100 0.5036 0.06
150 0.5058 0.09
200 0.5073 0.08
250 0.5083 0.07Kordzakhia, Novikov
5.2. Stochastic interest rate. For comparison, we included results of simulation of prices ˆ Cd
f(m) and
the relative error of approximation (15) for σr = 0.2. Note that the relative errors of approximating
formula (15) in the range [10,250] is consistently less than 0.1%.
Table 2. Prices of discretely monitored option, σr = 0.2, ˆ Cf(50) = 0.62524
Price Relative
m ˆ Cd
f(m) Error(%)
100 0.6148 0.05
150 0.6172 -0.04
200 0.6183 -0.03
250 0.6191 -0.03
Acknowledgement
The authors are very grateful to Dr. Thorsten Schmidt for his valuable comments, which improved
the exposition of the paper.
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Pricing of Defaultable Securities under Stochastic Interest

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