EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 15, No. 3, 2022, 948-970
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

British call option on stocks under stochastic interest
rate

Kreanne Falcasantos1,∗, Felipe R. Sumalpong Jr.2

1 Mathematics Department, College of Science and Information Technology,
Ateneo de Zamboanga University, 7000, Zamboanga City, Philippines
2 Department of Mathematics and Statistics, College of Science and Mathematics, Mindanao
State University-Iligan Institute of Technology, 9200, Iligan City, Philippines

Abstract. The closed form expression for the price of the British put and call options have long
been established where both interest rate and volatility are assumed to be constant. In reality,
these assumptions do not fully reflect the variable nature of the financial markets. In this paper, we
derived a closed form expression for the arbitrage-free price of the British call option by assuming
stochastic interest rate which follows the Cox-Ingersoll-Ross model and constant volatility as

V (t, rt, x) = p(t, rt;T ) +
∫ T

t
J(t, rt, x, v, bD(v, rv))dv,

where the first term is the arbitrage-free price of the European call option under stochastic interest
rate and the second term is the early-exercise premmium. We have also shown that the price
function of the British call option satisfies the partial differential equation given by

∂V

∂t
+

1

2
σ2
1X

2
t

∂2V

∂x2
+ ρσ1σ2

√
rt

∂2V

∂x∂r
+

1

2
σ2
2rt

∂2V

∂r2
+

∂V

∂x
rtXt

+ [aθ − (a+ λσ)r]
∂V

∂r
− rtV = 0.

Moreover, we have shown that the contract drift satisfies µc < rt+ρσ1σ2
√
rtλ(0, t+u) for u ∈ [0, τ ]

and t ∈ [0, T ].

2020 Mathematics Subject Classifications: 35A01, 35A02, 35C05, 35C15, 35R35, 60H15,
60H35, 60J65

Key Words and Phrases: British call option, American call option, European call option,
Arbitrage-free price, Cox-Ingersoll-Ross model, Rational exercise boundary, Geometric Brownian
motion or Wiener process, Optimal stopping time, Free boundary problem

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v15i3.4385

Email addresses: falcasantoskrel@adzu.edu.ph (K. Falcasantos),
felipejr.sumalpong@g.msuiit.edu.ph (R. Sumalpong)

https://www.ejpam.com 948 © 2022 EJPAM All rights reserved.



K. Falcasantos, F. Sumalpong / Eur. J. Pure Appl. Math, 15 (3) (2022), 948-970 949

1. Introduction

Over the years, derivatives have become increasingly important in the global financial
market, with great impact on national economics. They are embedded in capital invest-
ment opportunities, added to bond issues[5], used as price discovery and price stabilizer
[14] and so on. The most common forms of derivatives is an option. An option is defined
to be a contract between two parties granting one party the opportunity to buy or sell a
security from or to the other party at a specified price also known as strike price on or
before a specified maturity date [7]. The two parties involved are called the buyer and
seller of the option. The buyer has the right but not the obligation to exercise the option.
In order to acquire the option, the buyer should pay the option price, which is also known
as premium to the seller [7].

In 2013, G.Peskir and F. Samee introduced a new type of call option called British
call options where the holder enjoys the early exercise feature of the American option
whereupon his payoff is the best prediction of the European payoff under the hypothesis
that the true drift of the stock price equals the contract drift [11]. British option provides
its holder with a protection mechanism against unfavorable stock price movements as well
as securing higher returns when the movements are favourable. The motivation for the
British call option starts from the difference between the paid premium and the expected
payoff when the true drift deviates from the risk-free rate. An added feature is built
into this instrument which aim at both providing protection against unfavourable price
movements as well as securing higher returns when these movements are favourable [11].
Accordingly, the value function of the British call option is similar to American call option
but they differ on their respective boundary functions.

The closed form expression for the price of the British call option has been derived
by Peskir and Samee (2013) by assuming constant interest rate and constant volatility.
However, the assumptions of constant interest rate and constant volatility fail to reflect
the fact that these market rates are stochastic in the real world. This paper extends the
results in [11] to address the mentioned shortcomings by considering British call option
under stochastic interest rate and constant volatility. In particular, it will be assumed
that the short rate follows the Cox-Ingersoll-Ross (CIR) model. Furthermore, this paper
focuses on the theoretical framework of British option pricing. Actual implementation
through simulation and numerical approximation will not be included.

2. Setting of the Problem

Let us consider the financial market consisting of a risky stock with price process
X = (Xt : t ∈ [0, T ]) and a zero-coupon bond with price process P = (Pt : t ∈ [0, T ])
where the prices respectively evolve as

dXt = µXtdt+ σ1XtdWt, (1)

dPt = rtPtdt− λ(t, T )σ2PtdW̄t, (2)



K. Falcasantos, F. Sumalpong / Eur. J. Pure Appl. Math, 15 (3) (2022), 948-970 950

where µ ∈ R is the appreciation rate, σ1 and σ2 are volatility coefficients, W = (Wt)t≥0

and W̄ = (W̄t)t≥0 are standard Wiener processes defined on a probability space (Ω,F ,P)
with

cov(dWt · dW̄t) = ρdt , (|ρ| < 1), (see [1])

rt follows the Cox-Ingersoll-Ross (CIR) model

drt = a(θ − rt)dt+ σ2
√
rtdW̄t (3)

where a, θ, σ2 are positive constants and the market price of risk is given by λ(t, T ) = λ
√
rt.

In this model, the standard deviation of the stochastic term σ2
√
rtdW̄t is proportional to

the square root of the interest rate, that is, as the rate increases, the standard deviation
also increases and as the interest rate approaches zero, σ2

√
rtdW̄t also approaches zero.

Moreover this model is a mean reverting process, that is, when rt > θ the drift is negative
and when rt < θ, the drift is positive where a is the speed of the mean reversion and θ is
the equilibrium level. The deterministic part of the solution of (3) is given by

rt = θ + (r0 − θ)e−at. (4)

Note that the price Pt = P (t, r;T ) of a zero coupon bond satisfies the partial differential
equation (see [4])

∂P

∂t
+ [aθ − (a+ λσ)r]

∂P

∂r
+

1

2
σ2
2r

∂2P

∂r2
− rP = 0 (5)

such that P (T, r;T ) = 1 for all r ∈ R and t ∈ [0, T ]. The price of a zero-coupon bond at
time t with maturity time T using the risk neutral valuation framework is given by

P (t, rt, T ) = E[e−
∫ T
t rudu|Ft], (see [8]) (6)

where Ft denotes the natural filtration generated by the price process X. The natural
filtration Ft represents the information generated by the process X as time progresses.

Note that the interest rate ru is a Markovian process (see [8]). This implies that, ru is
dependent on rt for u > t, i.e., ru is a function of rt. We then have,

P (t, rt, T ) = E[e−
∫ T
t rudu|Ft].

= E[e−
∫ T
t rudu|rt]

where E is taken with respect to the probability measure P. The solution to the partial
differential equation (5) is given by

P (t, r, T ) = e−λ(t,T )r−A(t,T ) see [4] (7)

where

A(t, T ) =
2aθ

Γσ2

[
−Γ

(
T − t

h1

)
− h1 − h2

h1h2
ln

(
h1 − h2e

Γ(T−t)

h1 − h2

)]
(8)



K. Falcasantos, F. Sumalpong / Eur. J. Pure Appl. Math, 15 (3) (2022), 948-970 951

λ(t, T ) =
2

σ2

[
1− eΓ(T−t)

][
h1e−Γ(T−t) − h2

] (9)

Γ =
√
(a+ γ)2 + 2σ2 (10)

h1 =
−a+ γ

σ2
+

Γ

σ2
(11)

h2 =
−a+ γ

σ2
− Γ

σ2
(12)

Let us now consider the British call option on stock given the financial market described
above. Moreover, we assume that the stock does not pay dividends and there are no
transaction costs involved in its trade. In 2013, Peskir and Samee defined the british call
option with strike price K > 0 and maturity time T > 0 in years as follows:

Definition [11] The British call option is a financial contract between a seller/hedger
and a buyer/holder entitling the latter to exercise at any (stopping) time τ prior to T
whereupon his payoff (deliverable immediately) is the ‘best prediction’ of the European
payoff (XT − K)+ given all the information up to time τ under the hypothesis that the
true drift of the stock price equals the contract drift µc.

In [11], the price of the British call option is derived under the hypothesis that the
risk-free rate is constant, that is, rt = r for all t ∈ [0, T ]. Hence, this paper presents an
extension of the results in Peskir and Samee (2013) by considering stochastic interest rate.

Let (Ω,F , P ) denote the probability space for which all the succeeding processes are
defined. Let Xt denote the price of a risky stock at time t ∈ [0, T ] which satisfies dXt =
µXtdt + σ1XtdWt. We note that the coefficients depend on the maturity time T as well
as the current time t. Let (F)0≤t≤T denote the natural filtration generated by the process
Xt. All local martingales involved are with respect to the filtration Ft. Let µc > 0 be the
risk-free rate and µ be the expected rate of return (appreciation rate) such that µc ̸= µ.

Let

Zt := exp

[∫ t

0
β(u)dWu − 1

2

∫ t

0
β2(u)du

]
for 0 ≤ u ≤ t ≤ T where

β(t) :=
µc − µ

σ1

is independent of T . By Itö’s formula [2], we have

dZt

Zt
= β(t)dWt.

This shows that Zt is a local martingale with E [Zt] = 1 for 0 ≤ t ≤ T .



K. Falcasantos, F. Sumalpong / Eur. J. Pure Appl. Math, 15 (3) (2022), 948-970 952

Moreover, define the Radon-Nikodym derivative of Pµc with respect to P via the fol-
lowing:

dPµc

dP
= ZT . (13)

By the generalized Girsavanov’s theorem ( see Lemma 1 in [15]), we have

Wµc := Wt −
∫ t

0
β(u)du

is a Pµc-Brownian motion. Let Eµc be an expectation with respect to the measure Pµc .
Under the probability Pµc , the stock price process in equation (1) becomes

dXt = µcXtdt+ σ1XtdW
µc
t (14)

where 0 ≤ t ≤ T with X0 = x ∈ (0,∞). Thus,

Law(X(µ)|P) = Law(X(µc)|Pµc)

where Law means “distribution”. Furthermore, making use of the property of ZT , we have

Eµc(X) = E(ZTX) = E(ZT )E(X) = E(X)

for any random variable X.

Thus, the payoff of the British call option at a given stopping time t = τ can be written
as

Eµc [(XT −K)+|Fτ ] (15)

where the conditional expectation is taken with respect to a new (equivalent) probability
measure Pµc . Clearly, when we exercise the British call option, we just substitute the
contract drift µc to the true (unknown) drift µ of the stock price for the remaining term
of the contract.

3. Payoff, Premium and the Price Process of the British Call Option

Using the properties of the Wiener process Wµc on X (stationary and independent
increments of Wµc on X), implies that

Eµc [(XT −K)+|Ft] = Gµc(t,Xt) (16)

where Gµc is the payoff function given by

Gµc(t, x) = E(xZµc

T−t −K)+ (see [11]) (17)

for t ∈ [0, T ] and x > 0 where Gµc(t, x) represents the value of the investment and
E(xZµc

T−t −K)+ represents the expected value of its payoff, and Zµc

T−t is given by

Zµc

T−t = exp

[
(µc −

σ2
1

2
)(T − t) + σ1WT−t

]
(18)

for t ∈ [0, T ] and x ∈ (0,∞). It can be verified that (17) can be expressed as follows



K. Falcasantos, F. Sumalpong / Eur. J. Pure Appl. Math, 15 (3) (2022), 948-970 953

Gµc(t, x) = xeµc(T−t)Φ(d1)−KΦ(d2)

where

d1 =
ln x

K + (µc +
1
2σ

2
1)(T − t)

σ1
√
T − t

, d2 = d1 − σ1
√
T − t

for t ∈ [0, T ], x > 0 and Φ(·) is the standard normal cumulative distribution function.

By applying the standard hegding scheme based on self-financing portfolios (with con-
sumption), the arbitrage-free price of the British call option at deal time (time 0) is given
by

V = sup
0≤τ≤T

Ẽ
[
e−

∫ τ
0 ruduEµc

[
(XT −K)+|Fτ

]]
, (19)

where the supremum is taken over all stopping time τ ∈ [0, T ] of X and Ẽ is taken
with respect to the (unique) equivalent martingale measure P̃ under which the stock price
process evolve as

dXs = rsXsds+ σ1XsdW
P̃
s , (20)

s ≥ t, with Xt = x > 0 and W P̃
t is a Brownian motion under P̃. Since the price is given by

Xs = xe

(∫ s
t rudu− 1

2
σ2
1(s−t)+σ1W P̃

s−t

)
,

then the discounted price given by

e(
∫ s
t rudu)Xs = xe

(
− 1

2
σ2
1(s−t)+σ1W P̃

s−t

)

is a martingale. Note that in (19), E
[
e−

∫ τ
0 rudu

]
is the discounting factor which brings

the payoff [Eµc(XT −K)+|Fτ ] from exercise date τ to time 0.

Now, we fix t ∈ [0, T ]. We want to solve for a general expression for the price of the
British call option at any time t with stock price Xt = x and short rate rt = r. We denote
this by V (t, r, x). Extending the argument in (19), if the exercise date is at any time
t ∈ [0, T ], then using (16) and the optimal sampling theorem, we have

V (t, r,Xt) = sup
0≤τ≤T−t

Ẽt,x

[
e−

∫ t+τ
t ruduGµc(t+ τ,Xt+τ )|Fτ

]
, (21)

where the supremum is taken over all stopping times τ ∈ [0, T − t] of X and Ẽt,x is taken
with respect to the (unique) equivalent martingale measure P̃t,x under which Xt = x and
rt ∈ R. Note that V (0, r0, x) = V . Since the supremum in (21) is attained in the first entry
time of X to the closed set where V = Gµc and Law(X(µ)|P) is the same as Law(X(rt)|P̃),
then from the well-known structure of the geometric Brownian motion X,

V (t, r, x) = sup
0≤τ≤T−t

E
[
e−

∫ t+τ
t ruduGµc(t+ τ, xXτ )|Xt = x, rt = r

]
, (22)



K. Falcasantos, F. Sumalpong / Eur. J. Pure Appl. Math, 15 (3) (2022), 948-970 954

where the process X = (Xt(t, rt))t∈[0,T ] under P evolves as

dXt = rtXtdt+ σ1XtdWt (X0 = 1). (23)

Moreover, the British call option at maturity time T is given by

V (T, rT , XT ) = Gµc(T,XT ) = E(XT −K)+. (24)

This implies that the price of the British call option at maturity time T coincides
with the payoff of the European call option whose stock dynamics follows the stochastic
differential equation (20) above. Let Grt(t, x) = E(XT −K)+, where X follows (20) with
Xt = x. Thus

V (T, rT , XT ) = Grt(t, x).

Note that when the interest rate is constant (rt = r for all t ∈ [0, T ]), i.e., the coef-
ficients a and σ2 in equation (3) are all equal to zero, then the expression for Grt(t, x)
multiplied by e−r(T−t) coincides with the Black-Scholes formula for the arbitrage-free price
of the European call option at time t with maturity time T .

We can directly see from (17) that x 7→ Gµc(t, x) is convex. To clearly verify this, we
present the proposition below.

Proposition 1. For any t ∈ [0, T ] given and fixed, the mapping

x 7→ Gµc(t, x) (25)

is convex on (0,∞).

Proof. Let t ∈ [0, T ] be given and fixed. Moreover, let 0 ≤ λ ≤ 1 and x2 =
λx1 + (1− λ)x3 for some x1, x2, x3 ∈ (0,∞) with x1 < x3. Then, we have

Gµc(t, x2) = Gµc(t , λx1 + (1− λ)x3)

= Eµc

[(
λx1Z

µc

t,T + (1− λ)x3Z
µc

t,T −K
)+

| Xt = x2

]
= Eµc

[ (
λx1Z

µc

t,T + (1− λ)x3Z
µc

t,T − (λK + (1− λ)K)
)+

| Xt = x2

]
= Eµc

[ (
λx1Z

µc

t,T + (1− λ)x3Z
µc

t,T − λK − (1− λ)K
)+

| Xt = x2

]
= Eµc

[ (
λx1Z

µc

t,T − λK + (1− λ)x3Z
µc

t,T − (1− λ)K
)+

| Xt = x2

]
= Eµc

[(
(λx1Z

µc

t,T − λK) + ((1− λ)x3Z
µc

t,T − (1− λ)K)
)+

|Xt = x2

]
≤ λEµc

[(
x1Z

µc

t,TK
)+

|Xt = x1

]
+ (1− λ)Eµc

[(
x3Z

µc

t,T −K
)+

|Xt = x3

]
= λGµc(t, x1) + (1− λ)Gµc(t, x3).

Therefore, the mapping x 7→ Gµc(t, x) is convex on (0,∞).



K. Falcasantos, F. Sumalpong / Eur. J. Pure Appl. Math, 15 (3) (2022), 948-970 955

It can also be verified that the mapping in (25) is strictly increasing on (0,∞) with
Gµc(t, 0) = 0 and Gµc(t,∞) = ∞ for any t ∈ [0, T ] given and fixed. Define the set

D :=
{
(t, rt, Xt) ∈ [0, T ]× R× (0,∞)|V (t, rt, Xt) = Gµc(t,Xt)

}
. (26)

By (20), we say that {T} × {rT } × (0,∞) ⊂ D, which is consistent with the fact that the
supremum in (17) is taken over (F)t∈[0,T ] - stopping times τ ∈ [t, T ]. Furthermore, by a
Corollary in [12], the (F)t∈[0,T ] - stopping time defined by

τD(t, rt, x) := inf {s ∈ [0, T − t] : (t+ s, rt+s, Xt+s) ∈ D} , (27)

with Xt+s = x ∈ (0,∞) and rt+s ∈ R, is an optimal stopping time for the option price in
(17) since both x 7→ V (t, rt, x) and x 7→ Gµc(t, x) are continuous on (0,∞) and Gµc(t, x) ≤
K for all t ∈ [0, T ] and rt ∈ R. Consequently, we call the set D as stopping set. Thus, we
can define the continuation set C = Dc as

C = {(t, rt, x) ∈ [0, T ]× R× (0,∞)|V (t, rt, x) > Gµc(t, x)} . (28)

4. Stopping Set and Boundary Function

In order to deal with the existence of an optimal stopping time for (26) above, we
define

F (t, rt, x) = V (t, rt, x)−Gµc(t, x) ≥ 0,

which is nonnegative for t ∈ [0, T ], rt ∈ R and x ∈ (0,∞), so that we define the stopping
set

D = {(t, rt, x) ∈ [0, T ]× R× (0,∞)|F (t, rt, x) = 0}.

By Lemmas 2 and 3, we say that the set D is closed. Thus, the continuation set
defined by

C := Dc = {(t, rt, x) ∈ [0, T ]× R× (0,∞)|F (t, rt, x) > 0}

is open. Let (T, rt, x) ∈ {T} × {rT } × (0,∞) ⊂ D, which is consistent with the fact that
the supremum in (21) is taken over (F)t∈[0,T ] - stopping times τ ∈ [t, T ]. Furthermore, by
a Corollary in [12], the (F)t∈[0,T ] - stopping time defined by

τD(t, rt, x) := inf {s ∈ [0, T − t] : (t+ s, rt+s, Xt+s) ∈ D} (29)

with Xt+s = x ∈ (0,∞) and rt+s ∈ R, is an optimal stopping time for the option price in
(21) since both x 7→ V (t, rt, x) and x 7→ Gµc(t, x) are continuous on (0,∞) and Gµc(t, x) ≤
K for all t ∈ [0, T ] and rt ∈ R.

Lemma 1. For any (t, rt, x) ∈ D, we have

lim sup
ϵ↘ 0

F (t, rt, x+ ϵ)− F (t, rt, x)

ϵ
≤ 0.



K. Falcasantos, F. Sumalpong / Eur. J. Pure Appl. Math, 15 (3) (2022), 948-970 956

Proof. For all x ∈ (0,∞) and ϵ > 0, consider the (Fs)s∈[t,T ]-stopping time

τ+ϵ = τD(t, rt, x+ ϵ) ∈ [0, T − t] (30)

defined by τD(t, rt, x) := inf {s ∈ [0, T − t] : (t+ s, rt+s, Xt+s) ∈ D}. Note that
τD(t, rt, x) solves the optimal stopping problem given by

V (t, rt, x+ ϵ) = sup
0≤τ≤T−t

Ẽ

[
e−

∫ t+τ
t ruduGµc(t+ τ,Xt+τ )

]

= Ẽ

[
e−

∫ t+τ
t ruduGµc(t+ τ+ϵ , Xt+τ+ϵ

)

] (31)

Claim: τ+ϵ → 0 as ϵ → 0. From the definition τD(t, rt, x) of τD(t, rt, x+ ϵ),

τD(t, rt, x+ ϵ)

= inf {s ∈ [0, T − t] : (t+ s, rt+s, Xt+s) ∈ D}

= inf

{
s ∈ [0, T − t] : sup

0≤τ≤T−s
Ẽ

[
e−

∫ t+τ
t ruduEµc

[(
(x+ ϵ)xTZ

µc

T−(t+τ) −K

)+

|Ft+τ

]
|Fr

]
= Eµc

[(
(x+ ϵ)xTZ

µc

T−(t+τ) −K

)+

|Ft+τ

]}

≤ inf

{
s ∈ [0, T − t] : sup

0≤τ≤T−s
Ẽ

[
e−

∫ t+τ
t ruduEµc

[(
xxTZ

µc

T−(t+τ) −K

)+

|Ft+τ

]
|Fr

]
≥ Eµc

[(
(x+ ϵ)xTZ

µc

T−(t+τ) −K

)+

|Ft+τ

]}

≤ inf

{
s ∈ [0, T − t] : sup

0≤τ≤T−s
Ẽ

[
e−

∫ t+τ
t ruduEµc

[(
(x+ ϵ)xTZ

µc

T−(t+τ) −K

)+

|Ft+τ

]
|Fr

]
≥ Eµc

[
1

2

(
xZµc

T−(t+τ) + ϵZµc

T−(t+τ) −K

+ |xZµc

T−(t+τ) + ϵZµc

T−(t+τ) −K|

)+

|Ft+τ

]}
.

This implies that

lim
ϵ→0

τD(t, rt, x+ ϵ)

≤ lim
ϵ→0

inf

{
s ∈ [0, T − t] : sup

0≤τ≤T−s
Ẽ

[
e−

∫ t+τ
t ruduEµc

[(
(x+ ϵ)xTZ

µc

T−(t+τ) −K

)+

|Ft+τ

]
|Fr

]
≥ Eµc

[
1

2

(
xZµc

T−(t+τ) + ϵZµc

T−(t+τ) −K



K. Falcasantos, F. Sumalpong / Eur. J. Pure Appl. Math, 15 (3) (2022), 948-970 957

+ |xZµc

T−(t+τ) + ϵZµc

T−(t+τ) −K|

)+

|Ft+τ

]}

= inf

{
s ∈ [0, T − t] : sup

0≤τ≤T−s
Ẽ

[
e−

∫ t+τ
t ruduEµc

[(
xxTZ

µc

T−(t+τ) −K

)+

|Ft+τ

]
|Fr

]
≥ Eµc

[
1

2

(
xZµc

T−(t+τ) −K + |xZµc

T−(t+τ) −K|

)+

|Ft+τ

]}

= inf

{
s ∈ [0, T − t] : sup

0≤τ≤T−s
Ẽ

[
e−

∫ t+τ
t ruduEµc

[(
xxTZ

µc

T−(t+τ) −K

)+

|Ft+τ

]
|Fr

]
≥ Eµc

[
1

2

(
(xZµc

T−(t+τ) −K)+

]}

= inf

{
s ∈ [0, T − t] : sup

0≤τ≤T−s
Ẽ

[
e−

∫ t+τ
t ruduEµc

[(
xxTZ

µc

T−(t+τ) −K

)+

= inf

{
s ∈ [0, T − t] : sup

0≤τ≤T−s
Ẽ

[
e−

∫ t+τ
t ruduEµc

[(
xxTZ

µc

T−(t+τ) −K

)+

|Ft+τ

]
|Fr

]
= Eµc

[
1

2

(
(xZµc

T−(t+τ) −K)+

]}

= inf

{
s ∈ [0, T − t] : (t, rt, x) ∈ D

}
.

Now to show that lim sup
ϵ↘ 0

F (t, rt, x+ϵ)−F (t, rt, x)
ϵ ≤ 0, we use the optimal stopping problem.

Hence, we have

lim sup
ϵ↘ 0

V (t, rt, x+ ϵ)− V (t, rt, x)

ϵ
= lim sup

ϵ↘ 0

1

ϵ

{
Ẽ

[
e−

∫ t+τ
t ruduGµc(t+ τ+ϵ , x+ ϵ)|Ft

]

− sup
0≤τ≤T−t

Ẽ

[
e−

∫ t+τ
t ruduGµc(t+ τ, x)

]}

≤ lim sup
ϵ↘ 0

1

ϵ

{
Ẽ

[
e−

∫ t+τ
t ruduGµc(t+ τ+ϵ , x+ ϵ)|Ft

]

− Ẽ

[
e−

∫ t+τ
t ruduGµc(t+ τ, x)

]}

≤ lim sup
ϵ↘ 0

1

ϵ

{
Gµc(t+ τ+ϵ , x+ ϵ)−Gµc(t+ τ+ϵ , x)

}



K. Falcasantos, F. Sumalpong / Eur. J. Pure Appl. Math, 15 (3) (2022), 948-970 958

=
∂Gµc

∂x
(t, x).

Therefore, lim sup
ϵ↘ 0

F (t, rt, x+ϵ)−F (t, rt, x)
ϵ ≤ 0.

Now, we characterize the stopping set defined above in terms of the boundary function
bD(t, rt).

Proposition 2. For any (t, rt, x) ∈ [0, T ]× R× (0,∞) such that (t, rt, x) ∈ D we have

{t} × {rt} × (0,∞) ⊂ D (32)

and

D =

{
(t, rt, x) ∈ [0, T ]× R× (0,∞)|V (t, rt, x)−Gµc(t, x) ≥ 0

}
. (33)

Proof.
Let (t, rt, x) ∈ {t} × {rt} × (0, x). Since (t, rt, x) ∈ D and V (t, rt, x) ≥ Gµc(t, x) for all

x ∈ (0,∞), we have

V (t, rt, x)− V (t, rt, y)

x− y
=

Gµc(t, x)− V (t, rt, y)

x− y

≤ Gµc(t, x)−Gµc(t, y)

x− y
.

Taking the limit on both sides as x− y → 0 and by Lemma 1, we have (t, rt, y) ∈ D and
conclude that {t} × {rt} × (0,∞) ⊂ D. Furthermore, by the definition of the optimal
stopping boundary

bD(t, rt) = sup

{
x ∈ (0,∞) : (t, rt, x) ∈ D

}
,

we have the equivalence

(t, rt, x) ∈ D ⇐⇒ {t} × {rt} × (0,∞) ⊂ D ⇐⇒ x ≥ bD(t, rt).



K. Falcasantos, F. Sumalpong / Eur. J. Pure Appl. Math, 15 (3) (2022), 948-970 959

5. Continuity Lemmas

We next discuss the following continuity results on the payoff and option price functions
to show that the stopping set D given by D = {(t, rt, x) = V (t, rt, x)−Gµc(t, x) ≥ 0} is
closed.

Lemma 2. The mapping (t, x) 7→ Gµc(t, x) is jointly continuous on [0, T ]× (0,∞).

Proof. The continuity of the mapping x 7→ Gµc(t, x) follows from the fact that
Gµc(t, x) is convex with respect to x ∈ (0,∞) for any t ∈ [0, T ] given and fixed. It remains
to show the uniform continuity of the mapping t 7→ Gµc(t, x). Let x ∈ (0,∞) be given
and fixed and 0 ≤ t1 ≤ t2 ≤ T . Thus

0 ≤
∣∣∣∣Gµc(t2, x)−Gµc(t1, x)

∣∣∣∣ by definition of absolute value

=

∣∣∣∣Eµc

[
(xZµc

T−t2
−K)

+|Ft2

]
− Eµc

[
(xZµc

T−t1
−K)

+|Ft1

] ∣∣∣∣ by equation (17)

≤
∣∣∣∣Eµc

[
(xZµc

T−t2
−K)

+ − (xZµc

T−t1
−K)

+|Ft2

] ∣∣∣∣ by monotonicity

≤ Eµc

[∣∣∣(xZµc

T−t2
−K)

+ − (xZµc

T−t1
−K)

+
∣∣∣ |Ft2

]
since |E(X)| ≤ E(|X|)

= xEµc

[∣∣∣(Zµc

T−t1
− Zµc

T−t2
)
+
∣∣∣ |Ft2

]
since −K −−K = 0.

Therefore, as |t2 − t1| → 0, we have Gµc(t2, x) − Gµc(t, x) → 0, which completes our
proof.

Lemma 3. The mapping (t, rt, x) 7→ V (t, rt, x) is jointly continuous on [0, T ]×R×(0,∞).

Proof.
For t ∈ [0, T ] and rt ∈ R given and fixed, the continuity of the mapping x 7→ V (t, rt, x)

on (0,∞) follows from the fact that it is convex on (0,∞). Also, the mapping rt 7→
V (t, rt, x) is continuous on R since the zero-coupon bond price P (t, rt; t+ τ) is continuous
with respect to rt on R for t ∈ [0, T ] and x ∈ (0,∞) given and fixed. It remains to show
that t 7→ V (t, rt, x) is continuous. Let 0 ≤ t1 ≤ t2 ≤ T , τ1 = τD(t, rt, x) be the optimal
stopping time for equation (19) and τ2 = τ1 ∧ (T − t2). Then

0 ≤ |V (t1, rt1 , x)− V (t2, rt2 , x)|

= |Ẽt1,x

[
e−

∫ t1+τ1
t1

ruduGµc(t1 + τ1, Xt1+τ1)|Ft1

]
− Ẽt2,x

[
e−

∫ t2+τ2
t2

ruduGµc(t2 + τ2, Xt2+τ2)|Ft2

]
|

= |Ẽt1,x

[
e−

∫ t1+τ2
t1

ruduGµc(t1 + τ1, Xt1+τ1)|Ft1

]



K. Falcasantos, F. Sumalpong / Eur. J. Pure Appl. Math, 15 (3) (2022), 948-970 960

− Ẽt2,x

[
e−

∫ t2+τ2
t2

ruduGµc(t2 + τ2, Xt2+τ2)|Ft2

]
|

≤
∣∣∣∣Ẽt2,x

[
e−

∫ t2+τ2
t2

rudu {Gµc(t1 + τ1, Xt1+τ1)−Gµc(t2 + τ2, Xt2+τ2)} |Ft2

]∣∣∣∣
≤ Ẽt2,x

[
e−

∫ t2+τ2
t2

rudu |{Gµc(t1 + τ1, Xt1+τ1)−Gµc(t2 + τ2, Xt2+τ2)}| |Ft2

]
.

By the continuity of the mapping t 7→ Gµc(t, x) in Lemma 2 above, the mapping
t 7→ V (t, rt, x) is continuous on [0, T ], uniformly in x ∈ (0,∞).

Note that from the paper of Peskir and Samee (2013) both x 7→ Gµc(t, x) and x 7→
V (t, rt, x) are convex. Furthermore, recall that every convex function on the open interval
I is differentiable almost everywhere.

6. Boundary Value Problem

Applying the Itö ’s formula on the price function V = V (t, rt;x) of the British call
option, we have

dV =

(
∂V

∂t
+

1

2
σ2
1X

2
t

∂2V

∂x2
+ ρσ1σ2

√
rtXt

∂2V

∂x∂r
+

1

2
σ2
2r

∂2V

∂r2

)
dt+

∂V

∂x
dXt

+
∂V

∂r
drt.

(34)

We now derive the partial differential equation satisfied by V (t, rt, x) by creating a risk
neutral portfolio. Consider the time interval [t, t+∆t] and the following portfolio at time
t: long 1 unit of derivative + short ϵ1 units of stock + short ϵ2 units of zero-coupon bond

Let
∏

t be the value of the portfolio at time t. Then∏
t

= V (t, rt;x)− ϵ1Xt − ϵ2P. (35)

The change in the value of the portfolio from time t to t+∆t is then

∆
∏
t

= ∆V − ϵ1∆Xt − ϵ2∆P. (36)

Using the discrete approximation of (34) for ∆V , we have

∆
∏
t

=

(
∂V

∂t
+

1

2
σ2
1X

2
t

∂2V

∂x2
+ ρσ1σ2

√
rtXt

∂2V

∂x∂r
+

1

2
σ2rt

∂2V

∂r2

)
∆t+

∂V

∂x
∆Xt

+
∂V

∂r
∆rt − ϵ1∆Xt − ϵ2

(
∂P

∂t
∆t+

∂P

∂r
∆rt +

1

2
σ2
2rt

∂2P

∂r2
∆t

) (37)



K. Falcasantos, F. Sumalpong / Eur. J. Pure Appl. Math, 15 (3) (2022), 948-970 961

=

(
∂V

∂t
+

1

2
σ2
1X

2
t

∂2V

∂x2
+ ρσ1σ2

√
rtXt

∂2V

∂x∂r
+

1

2
σ2
2r

∂2V

∂r2

)
∆t

+

(
∂V

∂x
− ϵ1

)
∆Xt +

(
∂V

∂r
− ϵ2

∂P

∂r

)
∆rt − ϵ2

(
∂P

∂t
+

1

2
σ2
2r

∂2P

∂r2

)
∆t

To eliminate the risk, let ϵ1 =
∂V
∂x and ϵ2 =

∂V
∂r /

∂P
∂r . Then equation (37) becomes

∆
∏
t

=

(
∂V

∂t
+

1

2
σ2
1X

2
t

∂2V

∂x2
+ ρσ1σ2

√
rtXt

∂2V

∂x∂r
+

1

2
σ2rt

∂2V

∂r2

)
∆t

+

(
∂V

∂x
− ∂V

∂x

)
∆Xt +

(
∂V

∂r
− ∂V

∂r
/
∂P

∂r

∂P

∂r

)
∆rt

− ϵ2

(
∂P

∂t
+

1

2
σ2
2rt

∂2P

∂r2

)
∆t

=

(
∂V

∂t
+

1

2
σ2
1X

2
t

∂2V

∂x2
+ ρσ1σ2

√
rtXt

∂2V

∂x∂r
+

1

2
σ2
2r

∂2V

∂r2

)
∆t

− ϵ2

(
∂P

∂t
+

1

2
σ2
2r

∂2P

∂r2

)
∆t

(38)

By equation (5),

∆
∏
t

=

(
∂V

∂t
+

1

2
σ2
1X

2
t

∂2V

∂x2
+ ρσ1σ2

√
rtXt

∂2V

∂x∂r
+

1

2
σ2rt

∂2V

∂r2

)
∆t

− ∂V

∂r
/
∂P

∂r

(
[aθ − (a+ λσ)r]

∂P

∂r
− 1

2
σ2
2r

∂2P

∂r2
+ rtP +

1

2
σ2
2rt

∂2P

∂r2

)
∆t

=

(
∂V

∂t
+

1

2
σ2
1X

2
t

∂2V

∂x2
+ ρσ1σ2

√
rtXt

∂2V

∂x∂r
+

1

2
σ2rt

∂2V

∂r2

)
∆t

− ∂V

∂r
/
∂P

∂r

(
rtP − [aθ − (a+ λσ)r]

∂P

∂r

)
∆t

=

(
∂V

∂t
+

1

2
σ2
1X

2
t

∂2V

∂x2
+ ρσ1σ2

√
rtXt

∂2V

∂x∂r
+

1

2
σ2
2r

∂2V

∂r2
− ∂V

∂r
/
∂P

∂r
rP

+ [aθ − (a+ λσ)r]
∂V

∂r

)
∆t.

(39)

Note that the change in the value of the portfolio ∆
∏

t, is deterministic. That is, the
portfolio is riskless during the interval [t, t+∆t]. Hence, it must instantaneously earn the
same rate of return as other short-term risk-free securities. That is,∏

t+∆t

=
∏
t

ert∆t (40)



K. Falcasantos, F. Sumalpong / Eur. J. Pure Appl. Math, 15 (3) (2022), 948-970 962

Since ert∆t ≈ 1 + rt∆t, then approximately

∆
∏
t

≈ rt
∏
t

∆t (41)

Substituting equations (39) and (35) to (41), we have(
∂V

∂t
+

1

2
σ2
1X

2
t

∂2V

∂x2
+ ρσ1σ2

√
rtXt

∂2V

∂x∂r
+

1

2
σ2rt

∂2V

∂r2
− rtP

∂V

∂r
/
∂P

∂r

+ [aθ − (a+ λσ)r]
∂V

∂r

)
∆t = rt

(
V − ∂V

∂x
Xt −

∂V

∂r
/
∂P

∂r
P

)
∆t.

(42)

Simplifying this expression, we have

(
∂V

∂t
+

1

2
σ2
1X

2
t

∂2V

∂x2
+ ρσ1σ2

√
rtXt

∂2V

∂x∂r
+

1

2
σ2
2r

∂2V

∂r2
+ rtXt

∂V

∂x

+ [aθ − (a+ λσ)r]
∂V

∂r

)
− rtV = 0.

(43)

Now, we define the infinitesimal generator

Lf(s, y1, y2) =

 ∂

∂s
+
∑
i

∂

yi
ai +

1

2

∑
i

∑
j

∂2

∂yiyj
bibjρij

 f(s, y1, y2) (44)

for any differentiable function f(s, y1, y2) on [0, T ]× R× (0,∞), where the Itö’s
processes Y1 = (Y1,s)s≥0 and Y2 = (Y2,s)s≥0 defined on probability space (Ω,Fs,P)
satisfy the stochastic differential equations

dY1,s = a1Y1,sds+ b1Y1,sdW1,s (Y1,0 = y1)

dY2,s = a2Y2,sds+ b2Y2,sdW2,s (Y2,0 = y2),

respectively, and ρij = cov(dWi,s, dWj,s) (see [1]).

Lemma 4. We have

D =

{
(t, rt, x) ∈ [0, T ]× R× (0,∞)|LX(t, rt, x) > 0

}
⊂ C (45)

where C = DC is the continuation set.

Proof. Let (t, rt, x) ∈ [0, T ]× R× (0,∞) be such that LXV (t, rt, x) > 0. By
Lemma 1 in [15], Dynkin’s formula holds. Thus we have,

P (t, rt; t+ s)Gµc(t+ s,Xt+s) = Gµc(t, x)

+

∫ τ

0
LP (t, rt; t+ s)Gµc(t+ s,Xt+s)ds

+Ms

(46)



K. Falcasantos, F. Sumalpong / Eur. J. Pure Appl. Math, 15 (3) (2022), 948-970 963

where Ms =
∫ τ
0 σ1x

∂Gµc

∂x dWs +
∫ τ
0 σ2

√
rsG

µc ∂P
∂rs

dW̄s defines a continuous martingale for

s ∈ [0, T − t] with t ∈ [0, T ]. By Lemma 5.2 and the fact that Gµc ∈ C1,2 (see [11]),
the infinitesimal generator LP (t, ru; t + s)Gµc(t, x) is continuous with respect to (t, x) ∈
[0, T ]× (0,∞). Thus, there exists an open neighborhood U × V ⊂ [0, T ]× (0,∞) of (t, x)
such that LXV (t, rt, x) > 0 for all (s, rs, y) ∈ U × V × 0. Let

τU = inf

{
τ : (t+ τ, rt+τ , Xt+τ ) ∈ U × V × 0, Xt = x, rt = r

}
.

By Optimal Sampling Theorem, the relation equation (46) with s = τU shows that

E [P (t, rt; t+ s)Gµc(t+ s,Xt+s)] = Gµc(t, x)

+ E

[∫ T

0
LP (t, rt; t+ s)Gµc(t+ s,Xt+s)ds

]
.

(47)

Since LXV (t, rt, x) > 0, the right hand side of (47) is strictly greater than Gµc , while from
(19), we have

V (t, rt, x) ≥ E[P (t, rt; t+ s)Gµc(t+ s,Xt+s)|Ft]

showing that V (t, rt, x) > Gµc(t, x), which implies that (t, rt, x) ∈ C. This completes the
proof.

We now proceed to the solution of the free boundary problem in (21). Going back to the
option price in (17), note that it can be written as

V = sup
0≤τ≤T

Ẽ
[
P (0, rt; τ)E

µc(XT −K)+|Fτ

]
(48)

Similarly, (20) can be written as

V = sup
0≤τ≤T−t

E [P (0, rt; τ)G
µc(t+ τ, xXτ )|Xt = x, rt = r] (49)

Now, let
Γ(s, r, x) = P (0, r; s)Gµc(s, x). (50)

Then

∂Γ
∂s = P s

0
∂Gµc

∂s +Gµc ∂P
s
0

∂s R, ∂2Γ
∂x2 = P s

0
∂2Gµc

∂x2 , ∂Γ
∂r = Gµc ∂P

s
0

∂r ,

∂Γ
∂x = P s

0
∂Gµc

∂x , ∂2Γ
∂x∂r = ∂Gµc

∂x
∂P s

0
∂r and ∂2Γ

∂r2
= Gµc ∂

2P s
0

∂r2
.

By Itö’s formula,

dΓ =
∂Γ

∂s
ds+

∂Γ

∂r
dr +

∂Γ

∂x
dx+

1

2
σ2
1

∂2Γ

∂r2
+ ρσ1σ2

√
r
∂2Γ

∂x∂r
+

1

2
σ2r

∂2Γ

∂x2
ds. (51)



K. Falcasantos, F. Sumalpong / Eur. J. Pure Appl. Math, 15 (3) (2022), 948-970 964

Thus,

d [P (0, r; s)Gµc(s,Xs)] = P s
0

∂Gµc

∂s
+Gµc

∂P s
0

∂s
ds+ [aθ − (a+ λσ)r]Gµc

∂P s
0

∂r
ds

+ rXsP
s
0

∂Gµc

∂x
ds+

1

2
σ2
1X

2
sP

s
0

∂2Gµc

∂x2
ds

+ ρσ1σ2
√
rXs

∂Gµc

∂x

∂P s
0

∂r
ds+

1

2
σ2
2rG

µc
∂2P s

0

∂r2
ds

+ σ1Xs
∂Gµc

∂x
dWs + σ2

√
rGµc

∂P s
0

∂r
dWs.

(52)

Integrating both sides from 0 to τ :

P (t, rt; t+ s)Gµc(t+ s,Xt+s)

= Gµc(t,Xt) +

∫ τ

0

{
P (t, rt; s)

[∂Gµc

∂s
+ rsXs

∂Gµc

∂x
+

1

2
σ2
1X

2
s

∂2Gµc

∂x2

]
+

1

P (0, r0; t)

[
Gµc(s,Xs)

∂P

∂s
+ [aθ − (a+ λσ)rt+u]G

µc(s,Xs)
∂P

∂r

+ ρσ1σ2
√
rsXs

∂Gµc

∂x
(s,Xs)

∂P

∂r
+

1

2
σ2
2rsG

µc(s,Xs)
∂2P

∂r2

]}
d

+

∫ τ

0

1

P (0, r0; t)

[
σ1Xs

∂Gµc

∂x
dWsσ2

√
rsG

µc
∂P

∂r
dWs

]

+

∫ τ

0

1

P (0, r0; t)

[
σ1Xs

∂Gµc

∂x
dWs + σ2

√
rsG

µc
∂P

∂r
dWs

]

(53)

Since P (t, rt; t+ τ)Gµc(t+ τ,Xt+τ ) is a martingale, we have

P (t, rt; t+ u)

[
∂Gµc

∂s
+ rsXs

∂Gµc

∂x
+

1

2
σ2
1x

2∂
2Gµc

∂x2

]
+

Gµc

P (0, r0; t)

∂P

∂t

+ [aθ − (a+ λσ)r]
Gµc

P (0, r0; t)

∂P

∂r
+ ρσ1σ2

√
rsXs

x

P (0, r0; t)

∂Gµc

∂x

∂P

∂r

+
1

2
σ2
2rs

Gµc

P (0, r0; t)

∂2P

∂r2
= 0.

Since the supremum in (19) is taken overall stopping time τ ∈ [0, T − t], we have

LxV (t, rt, x) =

[
∂V

∂t
+

1

2
σ2
1X

2
t

∂2V

∂x2
+ ρσ1σ2

√
rtXt

∂2V

∂x∂r
+

1

2
σ2rt

∂2V

∂r2
+

∂V

∂x
rtXt

− [aθ − (a+ λσ)r]
∂V

∂r

]
V (t, rt, x) = 0.



K. Falcasantos, F. Sumalpong / Eur. J. Pure Appl. Math, 15 (3) (2022), 948-970 965

This shows that there is a continuous (smooth) function h : [0, T ]×R → R+ such that

LXP (t, rt; t+ τ)Gµc(t+ τ, h(t, rt)Xτ ) = 0 (54)

for all t ∈ [0, T ]. Linearity of LXP (t, rt; t+ τ)Gµc(t+ τ,Xt+τ ) in terms of Xt+τ > 0 shows
that LXP (t, rt; t + τ)Gµc(t + τ,Xt+τ ) > 0 when Xt+τ < h(t + τ, rt) and LXP (t, rt; t +
τ)Gµc(t+τ,Xt+τ ) < 0 when Xt+τ > h(t+τ, rt). This shows in particular that it is optimal
to exercise immediately when x > h(t, rt) and t < T is sufficiently close to maturity T for
the same reason mentioned in [10]. From here, we define the optimal stopping boundary
(the early-exercise premium representation) as follows:

bD(t, rt) = sup {x ∈ (0,∞) : (t, rt, x) ∈ D} .

The result below characterizes the stopping set in terms of the boundary function
bD(t, rt).

Proposition 3. The boundary function bD(t, rt) is continuous in t ∈ [0, T ] for all rt ∈ R.

Proof.
Let rt ∈ R be fixed. We first show the right continuity. Suppose that bD(t, rt) is not

continuous at t = t0. We consider two cases:
Case 1: bD(t0, rt0) < bD(t0+ , rt0+ )
Let (t′, rt.x

′) ∈ (t0, T ) × R × (bD(t0, rt0), bD(t0+ , rt0+ )) be a point in the stopping set D
with t′ close to t0 and t′ ↓ t0. By Newton-Leibniz formula and Lemma 3 we have

0 <

∫ x′

bD(t0,rt)
[Vx(t

′, rt, u)−Gµc
x (t′u)]du

= V (t′, rt, x
′)−Gµc(t′, x′)

as t′ → t0. This implies that V (t0, rt, x
′) − Gµc(t0, x

′) > 0, i.e., (t0, rt, x
′) ∈ D, which

contradicts the fact that x′ > bD(t0, rt), i.e., (t0, rt, x
′) ∈ C.

Case 2: bD(t0, rt0) > bD(t0+ , rt0+ )
Let (t∗, rt.x∗) ∈ (t0, T )×R× (bD(t0, rt0), bD(t0+ , rt0+ )) be a point in the continuation set
D with t∗ close to t0 and t∗ ↓ t0. By Newton-Leibniz formula and Lemma 3 we have

0 <

∫ bD(t0,rt)

x∗
[Vx(t∗, rt, u)−Gµc

x (t∗, u)]du

= Gµc(t∗, x∗)− V (t∗, rt, x∗)

as t∗ → t0. This implies that V (t0, rt, x∗) > Gµc(t0, x
′) > 0, i.e., (t0, rt, x

′) ∈ D, which
contradicts the fact that (t0, rt, x∗) ∈ D.



K. Falcasantos, F. Sumalpong / Eur. J. Pure Appl. Math, 15 (3) (2022), 948-970 966

7. The Arbitrage-free Price and the Rational Exercise Boundary

In this section, we present the main result which is a derivation of the closed form
expression for the arbitrage-free price V (t, rt, x) of the British call option in terms of the
early-exercise premium over the European option counterpart.

First, we introduce the following functions:

F (t, rt, x) := Gµc(t, x)− P (t, rt;T )G
rt(t, x) (55)

J(t, rr, x, v, z) := −
∫ ∞

z
L[P (t, rt; v)G

µc(v, y)]f(v − t, x, y)dy (56)

for t ∈ [0, T ], x > 0, v ∈ [t, T ] and z > 0, where y 7→ f(v − t, rt, x, y) is the probability
density function of xZrt

v−t (see [3], [13] ) given by

f(v − t, x, y) =
2a

σ2

(
1− e−a(v−t)

)(yea(v−t)

x

) aθ
σ2−

1
2
exp

{
2a(x+ yea(v−t))

σ2(1− ea(v−t))

}

× Iq

(−4a
√
xyea(v−t)

σ2(1− ea(v−t))

)
Iq(·) is the modified Bessel function of the first kind of order q = 2aθ

σ2 − 1. Note that
f(v − t, x, y) can be transformed as

f(v − t, x, y) = eHδ(x− y) (57)

where

H =
∂

∂r
a(θ − rt) +

∂2

∂r

σ2
2

2
rt

and δ(x− y) is the delta function (see [13]). Thus, J(t, rr, x, v, z) > 0 for all (t, rt, x) ∈ D.
The function L[P (t, rt; v)G

µc(v, y)] is given by

L[P (t, rt; v)G
µc(v, y)] = [2rt − µc + ρσ1σ2

√
rtλ(0, t+ u)]xP (t, rt, t+ u)

eµc(T−(t+u))Φ(d1)− rtP (t, rt, t+ u)KΦ(d2).

It can be verified that (see Appendix)

J(t, rt, x, T, z) = rte
−xH

z P (t, rt, T )KΦ(d2)

− [2rt − µc + ρσ1σ2
√
rtλ]xe

−xH+µc
z P (t, rt, T )Φ(d1)

(58)

where



K. Falcasantos, F. Sumalpong / Eur. J. Pure Appl. Math, 15 (3) (2022), 948-970 967

d1 =
ln x

K +A(t, T ) +
σ2
1
2 (T )

σ1
√
T

d2 =
ln x

K +A(t, T )− σ2
1
2 (T )

σ1
√
T

A(t, T ) =
r0 − θ

a

(
1− e−aT

)
+ θ(T )

Φ(x) =
1√
2π

∫ x

−∞
e

−y2

2 dy

ϕ(x) =
1√
2π

e−
x2

2

P (t, rt, T ) =

[[
xΦ(d1)−Ke−A(t,T )Φ(d2)

]
+ σC0

[
xΦ(d1)−Ke−A(t,T )

(
ϕ(d2)−

σ1
√
T − tΦ(d2)

)]
+ σC1

[
d2xϕ(d1)− d1Ke−A(t,T )

)
ϕ(d2)

]
+ 0(σ)

]
where 0(σ) is a zero-mean error (see [6])

Theorem 1. The arbitrage free-price of the British call option with stochastic interest
rate admits the following early-exercise premium representation

V (t, rt, x) = P (t, rt;T ) +

∫ T

t
J(t, rt, x, v, bD(v, rv))dv (59)

for all (t, rt, x) ∈ [0, T ] × R × (0,∞), where the first term is the arbitrage-free price of
the European call option under stochastic interest rate and the second term is the early-
exercise premium.
The rational exercise boundary of the British call option can be characterized as the unique
continuous solution bD : [0, T ]× R → R+ to the nonlinear integral equation

F (t, rt, bD(t, rt)) =

∫ T

t
J(t, rt, x, v, bD(v, rv))dv (60)

which satisfies bD(t, rt) ≥ h(t, rt) for all t ∈ [0, T ] where h is defined as a continuous
(smooth) function h : [0, T ] × R → R+ such that LP (t, rt; t + u)Gµc(t + u,Xt+u) = 0 for
all t ∈ [0, T ].

Proof :
For any (t, rt, x) ∈ C, we have



K. Falcasantos, F. Sumalpong / Eur. J. Pure Appl. Math, 15 (3) (2022), 948-970 968

V (t, rt, x) = E [P (t, rt; t+ τD)G
µc(t+ τD, Xt+τD)] (61)

with Xt = x ∈ (0,∞) and rt = r ∈ R where τD = τD(t, rt, x) is the optimal stopping
defined as τD(t, rt, x) := inf {s ∈ [0, T − t] : (t+ s, rt+s, Xt+s) ∈ D} . It can easily be

verified from (7) that P (t, rt, x) has continuous
∂2P
∂r2

(t, rt, x). This implies that V (t, rt, x)

has also continuous ∂2V
∂r2

(t, rt, x). Moreover, it is well-known from the theory of Markov
processes that V in (61) is C1,2; hence is C1,2,2, and it solves the Cauchy-Dirichlet free
boundary problem

LXV (t, rt, Xt) = 0 (t, rt, Xt) ∈ C (62)

V (t, rt, Xt) = Gµc(t,Xt)(t, rt, Xt) ∈ ∂C (63)

where ∂C ⊂ D denotes the open set C. By applying the change of variable formula with
local time on surfaces in [9] to

(s, rt, y) 7→ P (t, rt; t+ s)V (t+ s, rt+s, y)

with t ∈ [0, T ] and Xt = x ∈ (0,∞) given and fixed, we have

E[P (t, rt; t+ s)V (t+ s, rt+s, Xt+s)|rt] = V (t, rt, x)

+ E[

∫ s

0
LxP (t, rt; t+ v)V (t+ v, rt+v, xXv)I(Xt+v ̸= b(t+ v, rt+v))dv|rt]

+ E[M b
s |rt]

+
1

2
E[

∫ s

0
P (t, rt; t+ v)[Vx(t+ v, rt+v, Xt+v+)− Vx(t+ v, rt+v, Xt+v−)]

I(Xt+v = b(t+ v, rt+v))dℓ
b
v(X

x)|rt]

(64)

where M b
s =

∫ s
0

[
σ1Xt+vP

∂Gµc

∂x dWt+v + σ2G
µc ∂P

∂r dW̄t+v

]
defines a continuous local mar-

tingale for s ∈ [0, T − t] and ℓb = (ℓbv)0≤v≤s is the local time of Xx = (Xt+v)0≤v≤s on the
curve bD for s ∈ [0, T − t]. Since the coefficients of the respective Wiener processes of M b

s

are finite (and so are their respective squares) and that Gµc and P are Ft-adapted, hence
we have E

[
M b

s

]
= 0. By the smooth-fit property [11] or convexity of V , the last term in

(64) vanishes. Hence,

E [P (t, rt; t+ s)V (t+ s, rt+s, Xt+s)|rt] = V (t, rt, x)

+

∫ s

0
E [LxP (t, rt; t+ v)Gµc(t+ v,Xt+v)I(Xt+v > b(t+ v, rt+v))|rt] dv,

(65)

where we use (62) above and the fact that V = G in the stopping set D to obtain the
second term above. By replacing s by T − t, we have

E
[
P (t, rt;T )(XT −K)+

]
= V (t+ rt, x)

+

∫ T−t

0
E [LP (t, rt; t+ v)Gµc(t+ v,Xt+v)I(Xt+v > b(t+ v, rt+v))|rt] dv,

(66)



REFERENCES 969

where we used the fact that V (T, rT , x) = Gµc(T, x) = (x − K)+. Recognizing the left-
hand side of (66) above as the price p(t, rt;T ) of the European call option under stochastic
interest rate, we have

V (t, rt, x) = p(t, rt;T ) +
∫ T
t J(t, rt, x, v, bD(v, rv))dv.

Moreover, since V (t, rt, x) = Gµc(t, x) for all (t, rt, x) ∈ D, we have V (t, rt, bD(t, rt)) =
Gµc(t, bD(t, rt)). This implies that the boundary functiom bD solves equation (60). This
establishes the existence of the solution to (60).

The uniqueness of this solution can be shown parallel to the proof in [11].

Acknowledgements

This research is funded by the Commission on Higher Education (CHED) under the
CHED K-12 Transition Program and the Ateneo de Zamboanga University Faculty De-
velopment Program.

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