A Discrete-time Valuation of Callable Financial Securities with Regime

Switches

Kimitoshi Sato

and Katsushige Sawaki

Graduate School of Finance, Accounting and Law, Waseda University,

1-4-1, Nihombashi, Chuo-ku, Tokyo, 103-0027, Japan

Graduate School of Business Administration, Nanzan University, 18 Yamazato-cho, Showa-ku, Nagoya, 466-8673, Japan

Keywords:

Optimal Stopping, Game Option, Markov Chain, Regime Switching, Callable Securities, Stopping Bound-

aries.

Abstract:

In this paper, we consider a model of valuing callable ﬁnancial securities when the underlying asset price

dynamic is modeled by a regime switching process. The callable securities enable both an issuer and an

investor to exercise their rights to call. We show that such a model can be formulated as a coupled stochastic

game for the optimal stopping problem with two stopping boundaries. We provide analytical results of optimal

stopping rules of the issuer and the investor under general payoff functions deﬁned on the underlying asset

price, the state of the economy and the time. In particular, we derive speciﬁc stopping boundaries for the both

players by specifying for the callable securities to be the callable American put option.

1 INTRODUCTION

The purpose of this paper is to develop a dynamic

valuation framework for callable ﬁnancial securities

with general payoff function by explicitly incorporat-

ing the use of regime switches. Such examples of

the callable ﬁnancial security may include game op-

tions (Kifer, 2000), (Kyprianou, 2004), convertible

bond (Yagi and Sawaki, 2005), (Yagi and Sawaki,

2007), callable put and call options (Black and Sc-

holes, 1973), (Brennan and Schwartz, 1976), (Geske

and Johnson, 1984). Most studies on these securities

have focused on the pricing of the derivatives when

the underlying asset price processes follow a Brown-

ian motion deﬁned on a single probability space. In

other words the realizations of the price process come

from the same source of the uncertainty over the plan-

ning horizon.

The Markov regime switching model make it pos-

sible to capture the structural changes of the under-

lying asset prices based on the macro-economic en-

vironment, fundamentals of the real economy and ﬁ-

nancial policies including international monetary co-

operation. Such regime switching can be presented

by the transition of the states of the economy, which

follows a Markov chain. Recently, there is a grow-

ing interest in the regime switching model. (Naik,

1993), (Guo, 2001), (Elliott et al., 2005) address the

European call option price formula. (Guo and Zhang,

2004) presents a valuation model for perpetual Amer-

ican put options. (Le and Wang, 2010) study the op-

timal stopping time for the ﬁnite time horizon, and

derive the optimal stopping strategy and properties of

the solution. They also derive the technique for com-

puting the solution and show some numerical exam-

ples for the American put option.

In this paper we show that there exists a pair of op-

timal stopping rules for the issuer and of the investor

and derive the value of the coupled game. Should the

payoff functions be speciﬁed like options, some an-

alytical properties of the optimal stopping rules and

their values can be explored under the several assump-

tions. In particular, we are interested in the cases of

callable American put option in which we may derive

the optimal stopping boundaries of the both of the is-

suer and the investor, depending on the state of the

economy. Numerical examples are also presented to

illustrate these properties.

The organizationof our paper is as follows: In sec-

tion 2, we formulate a discrete time valuation model

for a callable contingent claim whose payoff func-

tions are in general form. And then we derive opti-

mal policies and investigate their analytical properties

by using contraction mappings. Section 3 discusses

a case of the payoff functions to derive the speciﬁc

stop and continue regions for callable put. In Sec-

Sato K. and Sawaki K..

A Discrete Time Valuation of Callable Financial Securities with Regime Switches.

DOI: 10.5220/0004201402250229

In Proceedings of the 2nd International Conference on Operations Research and Enterprise Systems (ICORES-2013), pages 225-229

ISBN: 978-989-8565-40-2

 2013 SCITEPRESS (Science and Technology Publications, Lda.)

tion 4 we present numerical results for the American

callable put option using binomial model. Finally, last

section concludes the paper with further comments. It

summarize results of this paper and raises further di-

rections for future research.

2 A GENETIC MODEL OF

CALLABLE-PUTABLE

FINANCIAL COMMODITIES

In this section we formulate the valuation of callable

securities as an optimal stopping problem in discrete

time. Let T be the time index set {0, 1,···}. We con-

sider a complete probability space (Ω,F ,P ), where

P is a real-world probability. We suppose that the

uncertainties of an asset price depend on its ﬂuctu-

ation and the economic states which are described

by the probability space (Ω,F ,P ). Let {1,2,··· ,N}

be the set of states of the economy and i or j de-

note one of these states. We denote Z := {Z

}

t∈T

be the ﬁnite Markov chain with transition probabil-

ity P

= Pr{Z

t+1

= j | Z

= i}. A transition from i to

j means a regime switch. Let r be the market interest

rate of the bank account. We suppose that the price

dynamics B := {B

}

t∈T

of the bank account is given

= B

t−1

, B

= 1.

Let S := {S

}

t∈T

be the asset price at time t. We sup-

pose that {X

} be a sequence of i.i.d. random variable

having mean µ

with the probability distribution F

(·)

and its parameters depend on the state of the economy

modeled by Z. Here, the sequence {X

} and {Z

} are

assumed to be independent. Then, the asset price is

deﬁned as

t+1

= S

. (1)

The Esscher transform is well-known tool to de-

termine an equivalent martingale measure for the val-

uation of options in an incomplete market ((Elliott

et al., 2005) and (Ching et al., 2007)). (Ching et al.,

2007) deﬁne the regime-swiching Esscher transform

in discrete time and apply it to determine an equiva-

lent martingale measure when the price dynamics is

modeled by high-order Markov chain.

We deﬁne Y

= logX

and Y := {Y

}

t∈T

. Let F

and F

denote the σ-algebras generated by the val-

ues of Z and Y, respectively. We set G = F

∨ F

for t ∈ T . We assume that θ

be a F

-measurable

random variable for each t = 1,2,···. It is interpreted

as the regime-switching Esscher parameter at time t

conditional on F

. Let M

(t,θ

) denote the moment

generating function of Y

given F

under P , that is,

(t,θ

) := E[e

| F

]. We deﬁne P

as a equiva-

lent martingale measure for P on G

associated with

(θ

,θ

,··· ,θ

The next proposition follows from (Ching et al.,

2007).

Proposition 2.1. The discounted price process

}

t∈T

is a (G , P

)-martingale if and only if θ

satisﬁes

(t + 1,θ

t+1

+ 1)

(t + 1,θ

t+1

)

= e

. (2)

If the dynamics Y is governed by the following

Markov-modulated binomial model:

P(Y

= y) =

(

p(Z

), if y = b(Z

1− p(Z

), if y = a(Z

(3)

then the following proposition provides the Esscher

transform of this process. For simplicity of notation,

we write p

, a

and b

instead of p(Z

), a(Z

) and

b(Z

), respectively.

Proposition 2.2. The Esscer transform of the Markov

modulated binomial model with parameter p

is again

a binomial model with the parameter

−e

A callable contingent claim is a contract between

an issuer I and an investor II addressing the asset with

a maturity T. The issuer can choose a stopping time

σ to call back the claim with the payoff function f

and the investor can also choose a stopping time τ to

exercise his/her right with the payoff function g

any time before the maturity. Should neither of them

stop before the maturity, the payoff is h

. The payoff

always goes from the issuer to the investor. Here, we

assume

0 ≤ g

≤ h

≤ f

, 0 ≤ t < T

and

= h

. (4)

The investor wishes to exercise the right to maximize

the expected payoff. On the other hand, the issuer

wants to call the contract to minimize the payment to

the investor. Then, for any pair of the stopping times

(σ,τ), deﬁne the payoff function by

R(σ,τ) = f

{σ<τ≤T}

+ g

{τ<σ≤T}

+ h

{σ∧τ=T}

(5)

When the initial asset price S

= s, our stopping prob-

lem becomes the valuation of

(s,i) = min

σ∈J

0,T

max

τ∈J

0,T

s,i

[β

σ∧τ

R(σ,τ)], (6)

where β ≡ e

−r

, 0 < β < 1 is the discount factor, J

is the ﬁnite set of stopping times taking values in

ICORES2013-InternationalConferenceonOperationsResearchandEnterpriseSystems

{0,1,··· ,T}, and E

[·] is an expectation under P

Since the asset price process follows a random walk,

the payoff processes of g

and f

are both Markov

types. We consider this optimal stopping problem as

a Markov decision process. Let v

(s,i) be the price

of the callable contingent claim when the asset price

is s and the state is i. Here, the trading period moves

backward in time indexed by n = 0, 1,2,· ·· ,T. It is

easy to see that v

(s,i) satisﬁes

n+1

(s,i) ≡ (U v

)(s,i)

≡ min{ f

n+1

(s,i),max(g

n+1

(s,i),A v

)}

(7)

with the boundary conditions are v

(s,i) = h

(s,i) for

any s, i and v

(s,0) ≡ 0 for any n and s. A is the

operator deﬁned by

(A v

)(s,i) ≡ β

∑

j=1

∞

(sx, j)dF

(x). (8)

Remark 2.1. The equation (7) can be reduced to the

non-switching model when we set P

= 1 for all i, or

(s,i) = f

(s), g

(s,i) = g

(s), h

(s,i) = h

(s) and

= µ for all i, n and s.

Let V be the set of all bounded measurable func-

tions with the norm kvk = sup

s∈(0,∞)

|v(s,i)| for any

i. For u,v ∈ V, we write u ≤ v if u(s,i) ≤ v(s,i) for

all s ∈ (0,∞). A mapping U is called a contraction

mapping if

kU u− U vk ≤ βku − vk

for some β < 1 and for all u,v ∈ V.

Lemma 2.1. The mapping U as deﬁned by equation

(7) is a contraction mapping.

Corollary 2.1. There exists a unique function v ∈ V

such that

(U v)(s,i) = v(s, i) for all s,i. (9)

Furthermore, for all u ∈ V,

u)(s,i) → v(s,i) as T → ∞,

where v(s,i) is equal to the ﬁxed point deﬁned by

equation (9), that is, v(s,i) is a unique solution to

v(s,i) = min{ f(s,i),max(g(s,i),A v)}.

Since U is a contraction mapping from Corol-

lary 2.1, the optimal value function v for the perpetual

contingent claim can be obtained as the limit by suc-

cessively applying an operator U to any initial value

function v for a ﬁnite lived contingent claim.

To establish an optimal policy, we make some as-

sumptions;

Assumption 2.1.

(i) F

(x) ≥ F

(x) ≥ ·· · ≥ F

(x) for all x.

(ii) f

(s,i) ≥ f

(s, j), g

(s,i) ≥ g

(s, j) and

(s,i) ≥ h

(s, j) for each n and s, and states i,

j, 1 ≤ j < i ≤ N.

(iii) f

(s,i), g

(s,i) and h

(s,i) are monotone in s for

each i and n, and are non-decreasing in n for

each s and i.

(iv) For each k ≤ N,

∑

j=k

is non-decreasing in i.

Assumption (i) means X

i+1

ﬁrst order stochasti-

cally dominates X

for any i and n. That is, as the

state i increases, the economy is going well. Thus,

the state N represents that the most ”Good” econ-

omy. Assumption (ii) implies that the payoff values

increase as the economy is getting better. In addition,

by Assumption (iii), the payoff values decreases as

the maturity approaches. Assumption (iv) asserts that

the probability of a transition into any block of states

{k, k + 1,·· ·} is an increasing function of the present

state.

Lemma 2.2. Suppose Assumption 2.1 holds.

(i) For each i, (U

v)(s,i) is monotone in s for v ∈ V.

(ii) v satisfying v = U v is monotone in s.

(iii) Suppose v

(s,i) is monotone non-decreasing in

s, then v

(s,i) is non-decreasing in i.

(iv) v

(s,i) is non-decreasing in n for each s and i.

(v) For each i, there exists a pair (s

∗

(i),s

∗∗

(i)),

∗∗

(i) < s

∗

(i), of the optimal boundaries such

that

(s,i) ≡ (U v

n−1

)(s)







(s,i), if s

∗

(i) ≤ s,

A v

n−1

, if s

∗∗

(i) < s < s

∗

(i),

(s,i), if s ≤ s

∗∗

(i),

with v

(s,i) = h

(s,i).

Corollary 2.2. The relationship between g

, f

and

(s,i) is given by

(s,i) ≤ v

(s,i) ≤ f

(s,i).

We deﬁne the stopping regions S

for the issuer

and S

for the investor as

(i) = {(s,n, i) | v

(s,i) ≥ f

(s,i)}, (10)

(i) = {(s,n, i) | v

(s,i) ≤ g

(s,i)}. (11)

Moreover, the optimal exercise boundaries for the is-

suer and the investor are deﬁned as

∗

(i) = inf{s ∈ S

(i)}, (12)

∗∗

(i) = inf{s ∈ S

(i)}. (13)

ADiscreteTimeValuationofCallableFinancialSecuritieswithRegimeSwitches

3 A SIMPLE CALLABLE

AMERICAN PUT OPTION

WITH REGIME SWITCHING

Interesting results can be obtained for the special

cases when the payoff functions are speciﬁed. In

this section we consider callable American put option

whose payoff functions are speciﬁed as a special case

of callable contingent claim. If the issuer call back

the claim in period n, the issuer must pay to the in-

vestor g

(s,i) + δ

. Note that δ

is the compensate

for the contract cancellation, and varies depending on

the state and the time period. If the investor exercises

his/her right at any time before the maturity, the in-

vestor receives the amount g

(s,i). We discuss the

optimal cancel and exercise policies both for the is-

suer and investor and show the analytical properties

under some conditions.

We consider the case of a callable put op-

tion where g

(s,i) = max{K

− s,0} and f

(s,i) =

(s,i) + δ

. The stopping regions for the issuer S

(i)

and the investor S

(i) with respect to the callable put

option are given by











(i) = {s | v

(s,i) ≥ (K

− s)

+ δ

}, for n ≥ 1,

(i) = φ, for n = 0,

(i) = {s | v

(s,i) ≤ (K

− s)

}, for n ≥ 0.

For each i and n, we deﬁne the optimal exercise

boundaries for the issuer ˜s

∗

(i) and the investor ˜s

∗∗

(i)

˜s

∗

(i) = inf{s | v

(s,i) = (K

− s)

+ δ

},(14)

˜s

∗∗

(i) = inf{s | v

(s,i) = (K

− s)

}. (15)

Assumption 3.1.

(i) βµ

≤ 1

(ii) 0 ≤ K

≤ K

≤ ··· ≤ K

(iii) 0 ≤ δ

≤ δ

≤ ··· ≤ δ

for each n.

(iv) δ

= 0 and δ

is non-decreasing and concave in

n > 0 for each i.

(v) β

∑

j=1

− K

is non-decreasing in i.

Assumption (i) means the expected rate of vari-

ability for the asset price is less than or equal to

= e

. Assumption (ii) and (iii) imply that the strike

price and the compensate increase as the economy is

getting better. These assumptions consistent with the

Assumption 2.1 (ii). Assumption (iv) shows that the

compensate becomes smaller and smaller as the matu-

rity approaches. Assumption (v) asserts that the dif-

ference between the discounted expected value of a

strike price when the state transits to any state and the

strike price of present state is an non-decrasing func-

tion of the present state.

Theorem 3.1. Suppose that Assumption 3.1 (i)-(v)

holds. The stopping regions for the issuer and in-

vestor can be obtained as follows;

(i) The optimal stopping region for the issuer:

(

(i) = {K

}, if n

∗

≤ n ≤ T,

(i) = φ, if 0 ≤ n < n

∗

(16)

where 0 ≤ K

≤ K

≤ ··· ≤ K

, and n

∗

≡ inf{n |

≤ v

,i)} which is non-decreasing in i.

Here, v

(s,i) = max{(K

− s)

,A v

n−1

(s,i)}.

(ii) The optimal stopping region for the investor:

(

(i) = [0, ˜s

∗∗

(i)], if n > 0,

(i) = {K

}, if n = 0,

(17)

where ˜s

∗∗

(i) is non-increasing in n and i. More-

over, ˜s

∗∗

(i) ≤ ˜s

∗

(i) for each i and n.

4 NUMERICAL EXAMPLES

In this section we provide a numerical example for a

callable American put option by using the binomial

tree model. We assume that the transition probability

matrix is given by

P =



1− p



. (18)

For a ﬁxed T, let us divide the interval [0,T] into M

subintervals such that T = hM. By Proposition 2.2,

the probability of upward in the state i is given by

− d

, i = 1,2, (19)

where u

= e

, d

= e

−b

. Let u

i, j

and d

i, j

be the up-

ward and downward rate when the state changes from

i to j, respectively. The probability distribution func-

tion of X

is described by

P(X

= x) =











, if x = u

i,i

(1− p

), if x = u

i, j

(1− q

, if x = d

i,i

(1− q

)(1− p

), if x = d

i, j

(20)

where i = 1, 2, i 6= j. It is easy to show that the process

is a martingale. The asset price after n periods on tree

can be obtained by

= S

(21)

where n

+ n

= n.

We set the parameters as T = 1, M = 300, r = 0.1,

= 0.03, b

= 0.01, p

= 0.7, p

= 0.8, K

= K

100, δ

= δ

= 5 and δ

= δ

= 6 for all n. These

parameters satisfy and Assumption 2.1 (i), (iv) and

Assumption 3.1. The optimal exercise regions for the

issuer and the investor is represented in Figure 1.

ICORES2013-InternationalConferenceonOperationsResearchandEnterpriseSystems

100

105

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

˜s

∗

(2)

˜s

∗

(1)

˜s

∗∗

(1)

˜s

∗∗

(2)

∗

Asset price

Time

Figure 1: Optimal exercise boundaries for the callable

American put.

5 CONCLUDING REMARKS

In this paper we consider the discrete time valuation

model for callable contingent claims in which the

asset price depends on a Markov environment pro-

cess. The model explicitly incorporates the use of

the regime switching. It is shown that such valuation

model with the Markov regime switches can be for-

mulated as a coupled optimal stopping problem of a

two person game between the issuer and the investor.

In particular, we show under some assumptions that

there exists a simple optimal call policy for the issuer

and optimal exercise policy for the investor which can

be described by the control limit values. If the distri-

butions of the state of the economy are stochastically

ordered, then we investigate analytical properties of

such optimal stopping rules for the issuer and the in-

vestor, respectively, possessing a monotone property.

ACKNOWLEDGEMENTS

This paper was supported in part by the Grant-in-Aid

for Scientiﬁc Research (No. 20241037) of the Japan

Society for the Promotion of Science in 2008-2012.

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ADiscreteTimeValuationofCallableFinancialSecuritieswithRegimeSwitches