ghosh - semi - markov

7/26/2019 Ghosh - Semi - Markov

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SIAM J. CONTROL OPTIM. c 2009 Society for Industrial and Applied MathematicsVol. 48, No. 3, pp. 15191541

RISK MINIMIZING OPTION PRICING IN A SEMI-MARKOV

MODULATED MARKET

MRINAL K. GHOSH AND ANINDYA GOSWAMI

Abstract. We address risk minimizing option pricing in a semi-Markov modulated marketwhere the floating interest rate depends on a finite state semi-Markov process. The growth rate andthe volatility of the stock also depend on the semi-Markov process. Using the FollmerSchweizerdecomposition we find the locally risk minimizing price for European options and the correspondinghedging strategy. We develop suitable numerical methods for computing option prices.

Key words. semi-Markov modulated market, minimal martingale measure, locally risk mini-mizing option price, BlackScholes equations

AMS subject classifications. 91B28, 91B70

DOI. 10.1137/080716839

1. Introduction. We study risk minimizing option pricing in a semi-Markovmodulated market. We assume that the state of the market is governed by a finitestate semi-Markov process{Xt}t0 taking values inX ={1, 2, . . . , k}. IfXt = i, thelocally risk-free floating interest rate is r(i). The stock price process{St}t0 followsa semi-Markov modulated geometric Brownian motion, i.e.,{St}satisfies

dSt = St((Xt)dt+ (Xt)dWt),

where {Wt} is a standard Brownian motion and(i), (i) are the drift and the volatil-ity ofStgivenXt = i. Since the stock price is governed by two sources of uncertaintiesarising due to the driving Brownian motion and also due to the regime switching de-termined by the semi-Markov process, the resulting market becomes incomplete. Thisleads to the nonuniqueness of an arbitrage-free option price on the stock St. At thesame time every contingent claim in such a market is associated with its intrinsic risk.

Therefore, the writer of the option cannot hedge himself perfectly.The option pricing in a regime switching framework has been studied by severalauthors using different approaches: Buffington and Elliott [4], DiMasi, Kabanov, andRunggaldier [7], Elliott, Chan, and Siu [8], Guo [13], Guo and Zhang [14], Mamonand Rodrigo [20], Jobert and Rogers [17], and Tsoi, Yang, and Yeung [31]. In allthese works it is assumed to be that the state of the market{Xt}is a continuous timeMarkov chain. For a continuous time Markov chain{Xt}, the sojourn time in eachregime i is necessarily exponentially distributed. If the distribution of so journ timein each regime is not exponential, then the corresponding process{Xt} is no moreMarkov. This is precisely what we have done in this paper. We assume that in eachstatei the conditional holding time distribution F( |i) is a general distribution. Inother words, the process{Xt}is semi-Markov.

In [11], [25], [26], [27], [28] Follmer, Sondermann, and Schweizer have addressed

option pricing in an incomplete market via risk minimizing hedging. By introducing aReceived by the editors February 27, 2008; accepted for publication (in revised form) January 7,

2009; published electronically May 1, 2009. This work was supported in part by DST projectSR/S4/MS: 379/06; also supported in part by a grant from UGC via DSA-SAP Phase IV and inpart by a CSIR Fellowship.

http://www.siam.org/journals/sicon/48-3/71683.htmlDepartment of Mathematics, Indian Institute of Science, Bangalore-12, India ([email protected].

ernet.in, [email protected]).

1519


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1520 MRINAL K. GHOSH AND ANINDYA GOSWAMI

quadratic risk function, they have obtained an abstract formula for the risk minimizingoption price via the minimal martingale measure. In [11], Follmer and Sondermannstudy the case when the discounted stock price process is a martingale under themarket measure. They show that if the discounted price ofSt is a martingale, then

a trading strategy is risk minimizing if and only if the corresponding discounted costprocess is a martingale orthogonal to the discounted stock price. Schweizer [26],[27] studies the case when the discounted stock price is a semimartingale. He showsthat a trading strategy is locally risk minimizing if and only if the cost process is amartingale under the minimal martingale measure. Using a certain decompositionwhich is referred to as FollmerSchweizer decomposition, it is shown that for anycontingent claimH(attainable or nonattainable), the locally risk minimizing price attimet with maturityTis given by

BtE[B1T H| Ft],

whereE is the expectation under minimal martingale measure P,Bt is the amountin the money market account due to one unit of initial investment, andFt is thefiltration generated by

{Xt

}and

{St

}. For attainable claim, the residual risk is zero

whereas for nonattainable claim, the risk is strictly positive. After the pioneering workof Follmer, Sondermann, and Schweizer, option pricing under minimal martingalemeasure has been studied by various authors for several models. In particular, optionpricing under a very general stochastic volatility model has been studied by Hofmann,Platen, and Schweizer in [15] using the minimal martingale measure. In [5], Colwelland Elliot have used the minimal martingale measure in option pricing where stockfollows a diffusion with jump. Under certain conditions they have shown that therisk minimizing option price satisfies a certain integro-differential equation. Optionpricing for a general marked point process has been studied by Prigent in [23]. Forvarious results and developments of option pricing using a quadratic hedging in anincomplete market, we refer to an excellent survey paper by Schweizer [28].

In this paper we compute the minimal martingale measure P for the semi-Markovmodulated market model. Using the results of Follmer and Schweizer [10], Schweizer

[26], [28], we show that the risk minimizing price of a European call option satisfiesa system of BlackScholes equations, which turns out to be a system of nonlocalequations given by

t(t,s,i,y) +

y(t,s,i,y) + r(i)s

s(t,s,i,y) +

1

22(i)s2

2

s2(t,s,i,y)

+ f(y|i)

1 F(y|i)j=i

pij[(t,s,j, 0) (t,s,i,y)] = r(i) (t,s,i,y),(1.1)

t[0, T), s R, i X, 0y < t, with the terminal condition(T,s,i ,y) = (s K)+, sR, 0yT , i= 1, 2, . . . , k ,(1.2)

where K is the strike price at maturity time T, (pij ) is the transition probability

matrix associated with{Xt}, andf( |i) is the conditional probability density func-tion of holding time. If{Xt} is a Markov chain, then the conditional holding timedistribution is exponential and thus memoryless. In this situation the risk minimizingoption price is independent ofy , and (1.1) reduces to

t(t,s,i) + r(i)s

s(t,s,i) +

1

22(i)s2

2

s2(t,s,i) +

j

ij (t,s,j)

=r(i)(t,s,i),(1.3)


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OPTION PRICING IN A SEMI-MARKOV MODULATED MARKET 1521

where = (ij ) is theQ-matrix of the Markov chain {Xt}. Note that (1.3) is a systemof PDE, whereas (1.1) is not a PDE due to the presence of the term (t,s,j, 0). Infact (1.1) can be shown to be an integro-partial differential equation (see (3.9)). Thus,for (1.3), standard theory of PDE can be used to establish the existence of a unique

solution satisfying boundary condition (1.2) in an appropriate class of functions [4],[6], [19]. The same cannot be carried out for the semi-Markov case.

In this paper we show the existence and uniqueness of a classical solution of(1.1) with boundary condition (1.2) using a probabilistic method. We use this toobtain the optimal hedging strategy. For both Markov and semi-Markov cases, thecorresponding equations (1.1) or (1.3) with boundary condition (1.2) do not admita closed form solution. Thus we have to resort to numerical methods for computingoption prices. Since the governing equation for the Markovian case is a system of PDE,a finite difference scheme along the line of CrankNicholson method can be used. Thishas been done in [3]. But the situation is quite different for the semi-Markov case.Though the CrankNicholson method for (1.1) can be developed, its implementationis extremely slow due to the extra dimensionality and nonrectangular nature of thedomain. To overcome this problem we have developed a new numerical method based

on discretization procedure of a certain integral equation. This procedure saves ahuge amount of computational time.

Our paper is structured as follows. The model description is presented in section2, wherein we also establish a representation of a class of semi-Markov processes asa stochastic integral with respect to a Poisson random measure. Following Schweizer[28] we present a brief description ofquadratic hedging or risk minimizing hedgingstrategy for a European call option in section 3. This section also contains the riskminimizing price for the European call option. In section 4, we obtain the locallyrisk minimizing price of European barrier options. In section 5 we present locallyrisk minimizing price of second order compound option. We generalize the resultsfor basket option in section 6. Section 7 deals with the computational aspects of thetheoretical results. We conclude our paper in section 8 with a few remarks.

2. Model description. Let (, F, P) be the underlying (complete) probabilityspace. Let the state of the market X ={Xt}t0 be a semi-Markov process on afinite state spaceX ={1, 2, . . . , k}with transition probabilities (pij) and conditionalholding time distributionsF(t|i). That is, if 0 =T0 < T1 < T2


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Let {St}t0be the stock price process which is governed by a semi-Markov modulatedgeometric Brownian motion. The evolution ofSt is given by

dSt = St ((Xt)dt+ (Xt)dWt), S0 > 0,(2.3)

where W ={Wt}t0 is a standard Wiener process independent of X ={Xt}t0, :X R is the drift coefficient, and :X (0, ) describes the volatility. LetFt = (Su, Xu, u t). Without loss of generality we assume that{Ft} is rightcontinuous. This would be the basic filtration for our model.

We first establish a representation of the semi-Markov process {Xt} as a stochasticintegral with respect to a Poisson random measure, which would play an importantrole later. We make the following assumptions which will be in effect throughout thepaper.

(A1) (i) The transition matrix (pij) is irreducible.(ii) Let G( |i) := 1 F( |i). For eachi, G(t|i)> t[0, ).(iii) For each i, F( |i) has continuously differentiable and bounded densityf(

|i).

EmbedX in Rk by identifying i withei Rk. Fory[0, ),i, j X let

ij (y) := pijf(y|i)

1 F(y|i) 0 for i=j,ii(y) :=

jX,j=i

ij (y) for i X.(2.4)

For a Polish spaceS, letB(S) denote its Borel -field andM(S) the set of all non-negative integer valued -finite measures onB(S). LetM(S) be the smallest -field onM(S), with respect to which the maps B :M(S) N

{} defined byB() := (B) are measurable for all B B(S);M(S) is assumed to be endowedwith the -fieldM(S).

Fori=j X, y R+, let ij (y) be consecutive (with respect to the lexicographicordering on

X X) left-closed right-open intervals of the real line, each having length

ij (y). Define the functions h :X R+R Rk and g :X R+R R+ by

h(i ,y,z) :=

j i ifzij (y),0 otherwise,

(2.5)

g(i ,y,z) :=

y ifzij (y) for somej=i,0 otherwise.

(2.6)

Consider the process {Xt, Yt}described by the following stochastic integral equations:

Xt= X0+

t0

R

h(Xu, Yu, z)(du,dz)

Yt = t t

0

R g(X

u, Yu, z)(du,dz)

(2.7)

where the integrations are over the interval (0, t] and(dt,dz) is an M(R+R)-valuedPoisson random measure with intensitydtdz, independent ofX0, aX-valued randomvariable.

Theorem 2.1. The process{Xt} defined in(2.7) is a semi-Markov process withtransition probability matrix(pij) and conditional holding time distributionsF(y|i).


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Proof. From (2.7) it is clearly seen that Xt is a right continuous (since theintegrations are over (0, t]) jump process taking values inX. Again from (2.5), (2.6),and (2.7), for a fixed ,{Xt()} has a jump at t0 to a state j if and only if{

t0

} X

t0()j (Yt0()) ()

= 0. Using this inductively for each jump, we see

that,Yt0() = 0 if and only if{Xt()} has a jump att0. LetTndenote the time ofnthjump ofXt, T0 := 0 and n := Tn Tn1. For a fixed t, letn(t) := max{n: Tnt}.Thus Tn(t) t < Tn(t)+1 and Yt = tTn(t). Hence, using the property of Poisonrandom measure,

P

no jump in (Tn, Tn+ y] |XTn =i

= exp

y

0

j=i

ij (s)ds

.

From (2.4), we have

d

dyF(y

|i) = (1

F(y

|i))

j=i

ij (y),

which gives

exp

y

0

j=i

ij (s)ds

= 1 F(y|i).

Thus

P(n+1y|XTn =i) = F(y|i).(2.8)

Again

P(XTn+1 =j, n+1y|XTn =i) = y0

exp

u

0

j=i

ij (s)ds

ij (u)du

=

y0

(1 F(u|i))pij f(u|i)1 F(u|i) du

=pij F(y|i).(2.9)

Equations (2.8) and (2.9) together show {Xt} is a semi-Markov processwith transition probability matrix (pij ) and conditional holding time distributionsF(y

|i).

In view of Theorem 2.1 we write Xt =Xt from now on. We make the followingassumption.

(A2) (dt, dz), X0, W, and S0, defined on (, F, P) are independent.Under (A2), (2.3) has a pathwise unique strong solution. From (2.3) and (2.7) it isclearly seen that the process (St, Xt, Yt), defined on (, F, P) is jointly Markov. Wenow compute the infinitesimal generator of the Markov process (St, Xt, Yt) from (2.3)and (2.7).


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Let : R X R+ R be a smooth function. Then by Itos formula [16],

d(St, Xt, Yt) =

R

St, Xt+ h(Xt, Yt, z), Yt g(Xt, Yt, z)

(St, Xt, Yt)(dt,dz) + s (St, Xt, Yt)dSt+

1

2

2

s2(St, Xt, Yt)dS, St+

y(St, Xt, Yt)dt

=(Xt)St

s(St, Xt, Yt)dt+

1

22(Xt)S

2t

2

s2(St, Xt, Yt)dt

+

y(St, Xt, Yt)dt

+

j=Xt

[(St, j, 0) (St, Xt, Yt)]Xt,j (Yt)dt+ d Mt,(2.10)

where Mt is a martingale given by

Mt = M0 +

t

0

Su(Xu) s

(Su, Xu, Yu)dWu+

t

0

R

Su, Xu

+ h(Xu, Yu, z), Yu g(Xu, Yu, z)

(Su, Xu, Yu)

(du,dz),(2.11)

where (dt,dz) := (dt,dz) dtdz is the compensated Poisson random measure. IfA denotes the infinitesimal generator of (St, Xt, Yt), then from (2.10) we obtain

A(s,i,y) = y

(s,i,y) + (i)s

s(s,i,y) +

1

22(i)s2

2

s2(s,i,y)

+ f(y|i)1 F(y|i)

j=i

pij[(s,j, 0) (s,i,y)],(2.12)

where :R

X R

R

is aC

2,1

function.We now discuss the option pricing problem. LetT >0 be the maturity time andHa European-type contingent claim maturing at time T. We wish to determine thearbitrage-free price of the claim at any time 0 t < T. To this end we first showthat there is an equivalent martingale measure for discounted stock price St, whereSt :=B

1t St; St andBt are given by (2.3), (2.1). Note that

St= S0exp

t0

(Xu)dWu +

t0

(Xu) 1

22(Xu)

du

is the solution of (2.3). Also St satisfies

dSt =(Xt)St dWt+ ((Xt) r(Xt))St dt.(2.13)

For i X

, let (i) := (i)r(i)

(i)

, i.e., (i) is the market price of risk in regime (or

mode) i. Let

t:= exp

t0

(Xu)dWu12

t0

(Xu)2du

, tT

dP :=T dP,

and Wt :=Wt+

t0

(Xu)du.

(2.14)


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Note thatP, as defined above, is a probability measure and is equivalent to P. Underthe new measure P,{Wt}0tT is anFt-Wiener process, and the stock process Stsatisfies

dSt = St (r(Xt)dt+ (Xt)dW

t

).

Also under P

St =S0exp

t0

(Xu)dW

u 1

2

t0

2(Xu)du

.(2.15)

Therefore,{St } is a martingale under P. Thus P is an equivalent martingalemeasure(EMM). This establishes that the market considered is arbitrage free. We noteat this point that in [8], a regime-switching random Esscher transform is employed toobtain the P for the Markovian regime switching model.

We next show that the P constructed in (2.14) has some special properties, viz.,it is the minimal martingale measure [10], [28]. To see this, note that{St } is asemimartingale under P, and

Mt := t0

(Xu)SudWu(2.16)

is the martingale part ofSt underP. Following [10] we define the minimal EMM ofSt using the P-martingaleM={Mt}0tT.

Definition 2.1. An EMMP P is said to be minimal ifP = P onF0 andif any square integrableP-martingale which is orthogonal toM under P remains amartingale underP.

The following result follows from Theorem 3.5 of [10].Lemma 2.2. The EMMP (as in 2.14) is the unique minimal martingale mea-

sure.If the contingent claimHis attainable, then using the EMMP, an arbitrage-free

price forHat time t is given by

BtE[B1T H| Ft] =E[eTt r(Xu)duH| Ft],(2.17)

whereE denotes the expectation under P. Since the semi-Markov modulated mar-ket described above is not complete, the claim Hmay not be attainable. Hence thewriter of an option cannot hedge himself perfectly. Therefore, to price an option thewriter must take care of the risk associated with a hedging strategy. Hence he wouldlike to seek a risk minimizing option price described in the next section.

3. Risk minimizing option price. Following [28] we first present a brief de-scription of quadratic hedging. Let theT-maturity contingent claim Hbe in L2(, FT,P). In order to replicate this claim at T in the above-described Market model, let ={t}0tT ={t, t}0tT be a strategy, where t and t denote the amountsinvested in St andBt, respectively, at timet. The process={t}0tTis assumedto be predictable, and it satisfies

E

T

0

2t 2(Xt)S

2t dt+

T0

|t| |(Xt)| dt2


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The value of the portfolio at time t under the strategy is given by

Vt() := tSt+ tBt.

Hence the discounted value of the portfolio becomesVt () =tS

t + t.

Let Gt () :=t0

udSu be the discounted gain process. A right continuous square

integrable process Ct(), defined by

Ct() :=V

t () t0

udSu, 0tT ,(3.2)

is called thediscounted cost process, which measures the accumulated additional cashflow up to time t. A strategy is self-financing ifCt() = constant. A strategy issaid to beadmissible ifVT() =HandP(Vt () m t) = 1 for some positive m.A contingent claim is called attainable if there is a self-financing admissible strategy.

Since the market under consideration is not complete, every contingent claim Hmaynot be attainable. Hence, instead of a self-financing admissible strategy, we look foran admissible strategy which minimizes, at each time t, the residual risk, given by

Rt() :=E[(CT() Ct())2 | Ft],(3.3)

over all admissible strategies. We say that an admissible strategy isrisk-minimizingif

0Rt()Rt()

for any other admissible strategy. Clearly,His attainable if and only if there existsan admissible strategy such thatC()= constant or equivalentlyRt() = 0t.Though the notion of risk minimizing hedging strategy is quite natural, it is technicallydifficult to work with it when the market measure Pis not itself a martingale measure(see [26, p. 347]). This leads us to define a weaker notion of risk minimizing strategy[26].

Definition 3.1. A strategy ={t, t}0tT is called a small perturbation ifit satisfies the following conditions:

(i) is bounded, (ii)T0 |u|d|A|u is bounded, where At := StS0Mt, (iii)

T =T = 0.Let be a strategy, a small perturbation, andPn = (ti)0in be a partition of

[0, T]. Then we define theR-quotient

QPn[, ] := tiP

Rti

+ (tI(ti,ti+1], tI[ti,ti+1))

Rti()

E[ti+1ti

2(Xt)S2t

dt| F

ti ]I(ti,ti+1](t).

An admissible strategy is calledlocally risk minimizing if

liminfn

QPn[, ]0 a.e.

for every small perturbation and every increasing sequencePn of partitions of[0, T],withPn 0.


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It is shown in [26] that an admissible strategy islocally risk minimizingif andonly if{Ct()}is a square integrable martingale orthogonal to{Mt}. This leads tothe following definition.

Definition 3.2. An admissible strategy is said to be optimal (i.e., locally

risk minimizing) if the corresponding discounted cost{Ct(

)} as in(3.2)is a squareintegrable martingale orthogonal to{Mt} (as in2.16).

In this situation the locally risk minimizing option price of the claim H is givenby the expression (2.17) [10], [28].

We now focus on a European call option on {St} with strike priceKand maturitytimeT. In this case the contingent claim is given by

H= (ST K)+.Let H = B1T H. It is shown in [10] that the existence of an optimal strategy isequivalent to the existence ofFollmerSchweizer decomposition ofH in the form

H = H0+

T0

H

u dSu + L

HT under P,(3.4)

where H0 L2(, F0, P), H ={Ht } satisfies (3.1), and LH ={LHt , 0tT}is a square integrable martingale orthogonal to the martingale{Mt, 0tT} (asin (2.16)). For the decomposition (3.4), the associated optimal strategy = (t, t)is given by

t = Ht , t = V

t tSt ,(3.5)

with

Vt =H0+

t0

H

u dSu + L

Ht , 0tT .(3.6)

Thus the discounted optimal cost Ct() is given by

Ct() =H0+ LHt .(3.7)

Since P is the minimal martingale,{LHt } is also a martingale under P. Thus Vt

as in (3.6) is a martingale under P, which is the risk minimized discounted price oftheH att. This immediately implies that the locally risk minimizingoption price ofthe claimHis given by (2.17). Thus the expression for locally risk minimizing optionprices for attainable and nonattainable claims remain the same. For an attainableclaim, the residual risk is zero, whereas for a nonattainable claim, the residual risk isstrictly positive.

To find an optimal hedging strategy for the semi-Markov market model we con-sider the system of (integro-partial) differential equations given by (1.1) defined on

D:={(t,s,i,y) (0, T) R X (0, T) |y(0, t)}(3.8)with the terminal condition (1.2). Equation (1.1) is the generalization of BlackScholes PDE for the semi-Markov modulated market. We observe that (1.1) is an

integro-partial differential equation given by

t(t,s,i,y) +

y(t,s,i,y) + r(i)s

s(t,s,i,y) +

1

22(i)s2

2

s2(t,s,i,y)

+

R

[(t,s,i + h(i ,y,z), y g(i ,y,z)) (t,s,i,y)]dz = r(i) (t,s,i,y),(3.9)

whereh and g are as in (2.5), (2.6).


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Theorem 3.1. Under assumption(A1),(1.1)with terminal condition(1.2)has aunique solution in the class of functions belonging to C(D)C1,2,1(D), with at mostpolynomial growth whereD is as in(3.8).

Proof. Let (, F, P) be a probability space which holds a standard Brownian mo-tion Wand a semi-Markov process X independent ofW; the transition probabilitiesand holding time distribution ofXare as same as X. Let St be given by

dSt = St(r(Xt)dt+ (Xt)dWt).

Let Yt represents the amount of time the process Xt is at the current state after thelast jump. Then (St, Xt, Yt) is jointly Markov with generator A given by

A(s,i,y) = y

(s,i,y) + r(i)s

s(s,i,y) +

1

22(i)s2

2

s2(s,i,y)

+ f(y|i)1 F(y|i)

j=i

pij[(s,j, 0) (s,i,y)].(3.10)

Let

(t, St, Xt, Yt) := E[eT

t r(Xu)du(ST K)+ | St, Yt, Xt].

Then(t,s,i,y) is a mild solution of (1.1) with terminal condition (1.2) [22]. Let Tndenote the time instant ofnth jump. Hence Tn(t) = t Yt. Now

(t, St, Xt, Yt)

= E[eT

t r(Xu)du(ST K)+ | St, Xt, Yt]

= E[E[eT

t r(Xu)du(ST K)+ | St, Xt,Yt, Tn(t)+1]| St, Xt, Yt]

=P(

Tn(t)+1 > T)E[e

Tt r(Xu)du(

ST K)

+

| St,

Xt,

Yt,

Tn(t)+1 > T]

+

Tt0

E[eT

t r(Xu)du(ST K)+ | St, Xt, Yt, Tn(t)+1 Tn(t)= Yt+ v]

f(Yt+ v| Xt)1 F(Yt| Xt)

dv

=1 F(T t +Yt| Xt)

1 F(Yt| Xt)Xt(t,

St) +

Tt0

er(Xt)v


j

pXtj

0

E[eT

t+vr(Xu)du(ST K)+ | St+v = x, St, Yt+v = 0,

Xt+v =j, Tn(t)+1 = t + v]e12 ((ln(

x

St

)(r(Xt)2(Xt)2 )v)

1(Xt)

v)2

2(Xt)vxdxdv

=1 F(T t +Yt| Xt)

1 F(Yt| Xt)Xt(t,

St) +

Tt0

er(Xt)v

(Xt)

v


j

pXtj

0

(t+ v,x,j, 0)e12 ((ln(

x

St

)(r(Xt)2(Xt)2 )v)

1(Xt)

v)2

2x

dxdv.


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Thus

(t,s,i,y) = 1 F(T t + y|i)

1 F(y|i) i(t, s) + Tt0

er(i)v

(i)

v

f(y+ v|i)1 F(y|i)

j

pij

0

(t+ v,x,j, 0)e

12 ((ln(

x

s )(r(i)2(i)

2 )v) 1(i)v )

2

2x

dxdv,

(3.11)

where i(t, s) is the standard BlackScholes price of call option with fixed interestrate and volatility r(i) and (i), respectively. Hence the first term on the right-hand side (R.H.S.) is in C1,2,1(D). Under the assumption (A1), the second termis in C1,2,1(D). Hence (t,s,i,y) is in C1,2,1(D). The uniqueness follows from thestochastic representation of(t,s,i,y).

Using Itos formula and the results in [28] summarized in (3.4)(3.7), the followingTheorem can be proved; we omit the details.

Theorem 3.2. Let(t,s,i,y) denote the unique solution of the Cauchy problem(1.1), (1.2). Then

(i) (t, St, Xt, Yt) is the locally risk minimizing option price at timet;(ii) an optimal strategy ={t , t } is given by

t =

s(t, St, Xt, Yt),(3.12)

t =V

t t St ,(3.13)where

Vt =(0, S0, X0, Y0) +

t0

s(u, Su, Xu, Yu)dS

u

+

t

0 Re

u

0 r(Xv)dv

u, Su, Xu+ h(Xu, Yu, z),

Yu g(Xu, Yu, z) (u, Su, Xu, Yu)(du,dz);

(iii) the corresponding discounted cost process is given by

Ct() = (0, S0, X0, Y0) +

t0

R

eu

0 r(Xv)dv

u, Su, Xu

+ h(Xu, Yu, z), Yu g(Xu, Yu, z)

(u, Su, Xu, Yu)

(du,dz).

Remark 3.1. (i) It can be shown that under P, (St, Xt, Yt) is jointly Markov,and its generator A is given by

A(s,i,y) =

y

(s,i,y) + r(i)s

s

(s,i,y) +1

2

2(i)s22

s2

(s,i,y)

+ f(y|i)1 F(y|i)

j=i

pij[(s,j, 0) (s,i,y)].(3.14)

Hence (1.1) can be rewritten as

t(t,s,i,y) + A()(t,s,i,y) = r(i) (t,s,i,y).


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Therefore, by FeynmanKac formula, the solution of (1.1), (1.2) is given by

(t, St, Xt, Yt) = E[e

T

t r(Xu)du(ST K)+ | Ft].(3.15)

(ii) Using Itos isometry, the residual risk at time t is given by

Rt() = E

Tt

R

eu

0 r(Xv)dv{(u, Su, Xu+ h(Xu, Yu, z),

Yu g(Xu, Yu, z)) (u, Su, Xu, Yu)}(du,dz)2Ft

=E

Tt

R

e2u

0 r(Xv)dv{(u, Su, Xu+ h(Xu, Yu, z),

Yu g(Xu, Yu, z)) (u, Su, Xu, Yu)}2 (du,dz)Ft

=E

T

t

e2u

0 r(Xv)dv f(Yu|Xu)

1 F(Yu|Xu)

j=Xu

pXuj((u, Su, j, 0)

(u, Su, Xu, Yu))2du| Ft

.(3.16)

(iii) In particular, for exponential (memoryless) holding time distribution,{Xt}becomes a Markov chain. That is, iff(y|i) = ieiy for somei> 0, i = 1, 2, . . . , k,then

f(y|i)1 F(y|i) =i.

Let

ij :=

ipij ifi=j,i otherwise.(3.17)

The matrix := (ij )kk is the rate matrix of the Markov chain{Xt}. In otherwords, is the generator of{Xt}. Thus for this particular case, (3.14) becomes

A(s, i) = r(i)ss

(s, i) +1

22(i)s2

2

s2(s, i) +

j

ij (s, j).

Therefore (1.1) reduces to (1.3).This case has been studied in [4], [7], [8], [13], [14], [20], [17], [31]. In particular,

for a two-state Markov chain, analytical solutions are obtained for European call in

[13] and perpetual American put in [14].In the next section we obtain the locally risk minimizing price of European barrier

options in semi-Markov modulated market.

4. European barrier options. In this section we study European barrier op-tion on {St} of different types. The basic forms of barrier options are down-and-out,up-and-out, down-and-in, and up-and-in. We first recall the various barrier op-tions. Let the barrier be given by b >0.


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down-and-out: The option becomes worthless if the barrierS= b is reached fromabove before expiry.

up-and-out: The option becomes worthless if the barrierS= b is reached frombelow before expiry.

down-and-in: The option becomes worthless unless the barrier S= b is reachedfrom above before expiry.

up-and-in: The option becomes worthless unless the barrierS= bis reached frombelow before expiry.

These four different barrier conditions can be implemented to any kind of options,either European call option or European put option. Letdoc ,

uoc ,

dic , and

uic denote

the locally risk minimizing prices of down-and-out, up-and-out, down-and-in, and up-and-in call prices, respectively.

It is important to note that the sum of two options, one of down-and-out typeand another of down-and-in type with identical barrier, strike price and maturity isequivalent to one option with same parameters with no barrier. In particular,

doc + dic =c,

wherec is the price of European call option. The same holds for up- barrier options.That is,

uoc + uic =c.

In view of the above relations we describe the evaluation ofuoc . The rest can be doneanalogously. In this case the contingent claim is given by

H= (ST K)+I

max[0,T]

St< b

,

whereKis the strike price, b is the barrier, and maturity time is T. Let

:= min{

t: St= b}

.

Note thatis anFt-stopping time, finite almost surely. IfS0 < b,Hcan be rewrittenas

H= (ST K)+I{ >T}.Consider (1.1) defined on

D:={(t,s,i,y) (0, T) (0, b) X (0, T) |y(0, t)},(4.1)with the terminal condition

(T,s,i ,y) = (s K)+, s < b, i= 1, 2, . . . , k(4.2)

and the boundary condition

(t,b,i,y) = 0 t(0, T], i= 1, 2, . . . , k .(4.3)Theorem 4.1. The problem(1.1), (4.2), (4.3)has a unique solution in the class

of functions belonging toC(D)

C1,2,1(D) (whereD is as in(4.1)), given by

uoc (t,s,i,y) = E[e

T

0 r(Xu)du(ST K)+I{ >T}|St = s, Xt = i, Yt= y].


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Proof. We can mimic the proof of Theorem 3.1 to show that the problem (1.1),(4.2), (4.3) has a mild solution uoc (t,s,i,y) which satisfies the following integralequation:

uo

c

(t,s,i,y)

=1 F(T t + y|i)

1 F(y|i) uoi (t, s) +

Tt0

er(i)v

(i)

v

f(y+ v|i)1 F(y|i)

ln( b

s) (r(i) 2(i)2 )v

(i)

v

e

2r(i)

2(i)1ln( b

s)

ln( bs

) (r(i) 2(i)2 )v(i)

v

j

pij

b0

uoc (t+ v,x,j, 0)e12 ((ln(

x

s)(r(i)

2(i)2 )v)

1(i)

v)2

2x

dxdv,

where uo

i (t, s) is the standard BlackScholes price of up-and-out barrier call optionwith fixed interest rate and volatility r(i) and (i), respectively, and () is thecumulative distribution function of a standard normal random variable. Hence thefirst term of the right side is in C1,2,1(D). The second term is also in C1,2,1(D) bythe assumption (A1). Hence uoc (t,s,i,y) is in C

1,2,1(D). The uniqueness followsfrom the stochastic representation ofuoc (t,s,i,y).

Theorem 4.2. Letuoc (t,s,i,y)denote the unique solution of(1.1), (4.2), (4.3).Then

(i) uoc (t, St, Xt, Yt)I{ >t} is the locally risk minimizing option price at timet forup-and-out European call option with strike priceK, barrierb, and maturitytimeT;

(ii) an optimal strategy ={t , t } is given by

t =

s uoc (t, St, Xt, Yt)I{ >t}

t =V

t t St

,(4.4)

where

Vt =uoc (0, S0, X0, Y0) +

t0

suoc (u, Su, Xu, Yu)I{ >u}dS

u

+

t0

R

eu

0 r(Xv)dv

uoc

u, Su, Xu+ h(Xu, Yu, z), Yu

g(Xu, Yu, z)

uoc (u, Su, Xu, Yu)

I{ >u}(du,dz);

(iii) the residual risk at timet is given by

Rt() = E

T

t

e2u

0 r(Xv)dv

f(Yu|Xu)1 F(Yu|Xu)

j=Xu

pXuj (uoc (u, Su, j, 0)

uoc (u, Su, Xu, Yu))2I{ >u}du| Ft

.(4.5)


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Proof. Let 0tT,Nt := e

t

0 r(Xu)duuoc (t, St, Xt, Yt)I{ >t}

= e t0

r(Xu)duuoc (t

, St, Xt, Yt),

since uoc (, S, X, Y) = 0. Sinceuoc is a C(D)

C1,2,1(D) function satisfying

(1.1), (4.2), (4.3), using ItoDynkin formula and (2.10), (2.12), (2.13), and (1.1), weobtain (under P)

Nt = uoc (0, S0, X0, Y0) +

t0

suoc (u, Su, Xu, Yu)I{ >u}dS

u

+

t0

R

eu

0 r(Xv)dv

uoc


Yu g(Xu, Yu, z)

uoc (u, Su, Xu, Yu)

(du,dz).(4.6)

By optional sampling theorem, the last term on the R.H.S. of (4.6) is an{Ft}-

martingale under Pwhich is orthogonal to{Mt} (by the independence of{Wt} and(, )). Hence by letting t T, (4.6) gives the FollmerSchweizer decomposition ofNT(the discounted contingent claim). Hence the results (i), (ii), and (iii) of Theorem4.2 follow immediately.

5. Compound options. In this section we discuss second order options, i.e.,an option on an option. In practice the price of the second order option depends onthe market price of the underlying option. But for the modeling purpose, we assumethat the theoretical price and market price of an underlying option are the same. Thesecond order options can be of four types: (1) call-on-a-call, (2) call-on-a-put, (3)put-on-a-call, and (4) put-on-a-put.

We consider only the call-on-a-call option; the other three types can be pricedsimilarly. LetT1, K1 be the expiry time and the strike price of the underlying calloption, respectively. Let T2(< T1), K2 be the expiry time and the strike price of the

compound option, viz., a call on a call. Therefore, the contingent claim atT2 is givenby

H= (c(T2, ST2 , XT2, YT2) K2)+,where c(t, St, Xt, Yt) is the locally risk minimizing price of underlying call optionwith maturity T1 and strike price K1. In other words, c(t,s,i,y) is the uniquesolution of the Cauchy problem (1.1) defined on

D1 :={(t,s,i,y) (0, T1) R X (0, T1)|y(0, t)},(5.1)with the terminal condition

c(T1,s ,i ,y) = (s K1)+ i,y.(5.2)To find the optimal hedging strategy for a European call-on-a-call option, we considerthe Cauchy problem (1.1) defined on

D2 :={(t,s,i,y) (0, T2) R X (0, T2)|y(0, t)},(5.3)with the terminal condition

(T2, s , i , y) = (c(T2, s , i , y) K2)+ i,y.(5.4)


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One can prove that the above Cauchy problem has a unique solution in the classofC(D2)

C1,2,1(D2) functions having at most polynomial growth using analogous

argument as in the proof of Theorem 3.1. Thus as before, we obtain the followingresult.

Theorem 5.1. Letcc(t,s,i,y) denote the unique solution of(1.1), (5.3), (5.4).Then

(i) cc(t, St, Xt, Yt) is the locally risk minimizing option price at time t for theEuropean call-on-a-call option with strike priceK2 and maturity timeT2;

(ii) an optimal strategy ={t , t }tT2 is given by

t =

scc(t, St, Xt, Yt)

t =V

t t St

,(5.5)

where

Vt =cc(0, S0, X0, Y0) +

t0

scc(u, Su, Xu, Yu)dS

u

+ t0

R

eu0 r(Xv)dv

cc


Yu g(Xu, Yu, z)

cc(u, Su, Xu, Yu)

(du,dz);

(iii) the residual risk at timet is given by

Rt() = E

T2

t

e2u

0 r(Xv)dv

f(Yu|Xu)1 F(Yu|Xu)

j=Xu

pXuj

cc(u, Su, j, 0)

cc(u, Su, Xu, Yu)2

du| Ft .

Remark5.1. The underlying asset for a call-on-a-call option is a call option. Butfor a pricing or hedging call-on-a-call option, we do not use the primary underlying ascall, instead we use the primary underlying as St. This is clearly reflected in (5.5).

6. Basket options. We have studied the risk minimizing option price in theframework of Follmer and Schweizer [10]. Assuming the existence of only one riskyasset in the market, we have obtained the locally risk minimizing option price as asolution of BlackScholes equations (1.1), (1.2). We have also obtained the corre-sponding hedging strategy. Using a slightly generalized notion of pseudooptimalitywhich is defined by Definition 3.2 (see [28]), our method can be generalized to themultidimensional case where there are n stocks which are correlated. As before weassume that the states of economy is given by a semi-Markov process {Xt} takingvalues inX ={1, 2, . . . , k}. The instantaneous interest rate is r : [0, T] X [0, ),which is assumed to be continuous. Thus the amount

{Bt

} in the money market

satisfies

dBt= r(t, Xt)Btdt.

The price of the ith stock{Sit} is given by

dSit =Sit

i(t, Xt)dt + n

j=1

ij (t, Xt)dWj

t

,


17/23



where W :={W1t, . . . , W nt} is a standard n-dimensional Brownian motion indepen-dent of{Xt}and i : [0, T] X R, ij : [0, T] X R+. We assume thati andij are continuous. Let a(t, x) :=(t, x)

(t, x). We assume that a(t, x) is uniformly

positive definite, i.e., there exists a >0 such that

a(t, x)I x = 1, 2, . . . , k ,whereI is then n identity matrix. Set

T:= exp

ni=1

T0

fi(t, Xt)dWi

t1

2

ni=1

T0

f2i(t, Xt)dt

,

where

fi(t, Xt) :=n

j=1

(1(t, Xt))ij (j (t, Xt) r(t, Xt)).

Let dP

dP

=T.Then as before, we can show thatP is the minimal martingale measureto this model.

Consider a European call option on a combination of stocksn

j=1 j Sjt with strike

priceKand terminal timeT. Then we can show that the pseudooptimal option priceis given by

E

e Tt r(u,Xu)du

n

j=1

j SjT K

+

| Ft

,

whereFt = (Xs, S1s , . . . , S ns , st).Let (t, s1, . . . , sn, x , y) denote the pseudooptimal option price of the basket

n

j=1 jSjt at time t when Xt = x, Yt = y, S

it = s

i, i = 1, 2, . . . , n . Then again we

can show that(t, s1, . . . , sn, x , y) is the unique solution (in appropriate class of func-

tions on appropriate domain) of the system of partial integro-differential equationsgiven by

t(t,s,x,y) +

y(t,s,x,y) +

ni=1

r(t, x)si

si(t,s,x,y)

+1

2

ni,l=1

ail(t, x)sisl

2

sisl(t,s,x,y) +

f(y|x)1 F(y|x)

x=x

pxx [(t,s,x, 0)

(t,s,x,y)] = r(t, x)(t,s,x,y) x X(6.1)

(T,s,x,y) =

n

j=1j s

j K

+

,(6.2)

wheres = (s1, . . . , sn).The existence of a solution of (6.1), (6.2) under (A1) can be established as before.

7. Numerical method. The standard method for computing option prices in-volves solving the relevant PDEs numerically by a CrankNicholson discretizationprocedure [29]. For the semi-Markov modulated market, we showed that the cor-responding equations are systems of (nonlocal) integro-partial differential equations


18/23



on some nonrectangular domains. For such systems, one can develop a stable finitedifference implicit scheme which extends CrankNicholson discretization procedure[12]. But the scheme is computationally expensive due to the extra dimensionalityand nonrectangular nature of the domain. In order to overcome this difficulty we

develop an alternative numerical scheme based on the discretization of integral equa-tion (3.11). We have already shown in the proof of Theorem 3.1 that the Cauchyproblem (1.1), (1.2) has a unique solution, and the solution satisfies (3.11). We findan approximate solution of (3.11) using a step-by-step quadrature method. We referto [1], [2], and [24] for quadrature method for linear integral equations.

Puttingy = 0 in (3.11) we obtain

(t,s,i, 0) = (1 F(T t|i))i(t, s) + Tt0

er(i)vf(v|i)

j

pij

0

(t+ v,x,j, 0)e12

ln( x

s)

r(i)

2(i)2

v

1(i)

v

2

2(i)x

vdxdv.(7.1)

Note that the integrand in (7.1) vanishes at v = 0. Therefore, the dependence ofthe function (t, , , 0) on (t, , , 0) for t (t, T] is explicit. Thus even an im-plicit quadrature method to discretize (7.1) actually results in an explicit quadraturemethod. Hence we are able to solve (7.1) in step-by-step manner by taking the ter-minal condition

(T,s,i, 0) = (s K)+, sR, i= 1, 2, . . . , k .(7.2)

Let t and s be the time step and stock step sizes, respectively. For m, m, lpositive integers and i X, set

G(m, m, l , i) :=

e12 ((ln(

m

m)(r(i)

2(i)2 )lt)

1(i)

lt

)2

2(i)mslt,

nm(i) (T nt, ms,i, 0).

Now we use the following quadrature rule over successive intervals [0 , nt]: For afunction on this interval, we use

nt0

(v)dvtn

l=0

n(l)(l),

wheren(l) are weights to be chosen appropriately. Applying the above discretizationprocedure in (7.1) we obtain the following set of equations:

n

m(i) = (1 F(nt|i))i(T nt, ms) + t

nl=1

n(l)e

r(i)lt

f(lt|i)j

pij sm

nlm (j)G(m, m, l , i),(7.3)

with

0m(i) = (K ms)+.


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We choose a repeated trapezium rule, that is, the weights n are given by

n(l) = 1 for l = 1, 2, . . . , n 1, n(0) =n(n) = 12

.

We now show that for sufficiently small t, the above scheme (7.3) is strictly stablewith respect to an isolated perturbation. We also show that the scheme displaysbounded error propagation (i.e., the accumulated effect of isolated perturbations ,added at each step, in nm(i), is uniformly bounded by a constant multiple of). Werefer to [2] for definitions and other details.

Theorem 7.1. Under(A1) leta:= maxX[0,T] er(i)vf(v|i). For

teaT

a ,(7.4)

the scheme(7.3) is strictly stable with respect to an isolated perturbation. Moreover,the scheme displays uniformly bounded error propagation.

Proof. We first note that (i)G(m, m, l , i) corresponds to a lognormal density,and (ii) under (A1), the holding time densities f(

| ) are bounded. Let n be an

additive error innm(i) mandi. Now it is easy to show that the effect of the isolatedperturbation n in Nm(i) (N := [

Tt ]) is

n= at(1 + at)Nnn.

If t is sufficiently small satisfying (7.4), we get n < n, i.e., the scheme is strictlystable with respect to an isolated perturbation. Letnbe bounded by a fixed constant. Now the total effects of the perturbations in the valueNm(i) is given by

:=N1n=0

n < (eaT 1).

Hence the result follows.

Having established the stability of the above scheme,nm(i) (Tnt, ms,i, 0)is computed for n = 0, 1, 2, . . . , [ Tt ]; m = 1, 2, . . . ; i = 1, 2, . . . , k, using the step-by-step quadrature method (7.3). Next it is straightforward to compute (Tnt, ms,i,y) for a giveny using

(T nt, ms,i,y) = (1 F(nt+ y|i))(1 F(y|i)) i(T nt, ms)

+ tn

l=1

n(l)er(i)lt f(lt + y|i)

(1 F(y|i))j

pijsm

nlm (j)G(m, m, l , i).(7.5)

The system of equations (7.5) gives an approximation for European call price.To illustrate the results we next consider an example of a semi-Markov modulated

market with three regimes. The state spaceX ={1, 2, 3}. The drift coefficient,volatility, and instantaneous interest rate at each regime are chosen as follows:

(i), (i), r(i)

:=

(0.2, 0.2, 0.2) if i= 1,(0.6, 0.4, 0.5) if i= 2,(0.8, 0.3, 0.7) if i= 3.


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0

0.5

1

1.5

2

2.5

0 0.5 1 1.5 2 2.5 3

Stock

Option

R eg im e1 R eg im e2 R eg im e3 M at ur it y B -S ( Re gi me 2 )

Fig. 7.1. Price of European call option.

The transition probability matrix is assumed to be given by

(pij ) =

0 2/3 1/31/2 0 1/2

1/3 2/3 0

.

In this example the holding time in each regime is assumed to be (2, 1). That is,

f(y|i) = yey, y0, and i = 1, 2, 3.For this semi-Markov modulated market, we compute the price functions of Europeancall and barrier options numerically. Figure 7.1 describes the risk minimizing priceof European call option for three different initial regimes. The result is obtained bysolving the integral equation (3.11) numerically. We take strike price K = 1 and

maturity T = 1. The time step length t is taken as 0.02, the space (stock) meshlength s is taken as 0.04. We take s along horizontal axis and plot the (0, s , i)along vertical axis for i = 1, 2, 3. In the plot (Figure 7.1) the thick line shows theEuropean call option prices at the maturity T, whereas the other three curves showthe European call option prices at time zero for three different initial regimes. Thethin line shows the BlackSholes price of call option with fixed interest rate r(2) andvolatility(2).

Corresponding to the optimal hedging strategy , the quadratic residual risk attime zero is given by

R0() = E[(CT() C0 ())2 |S0, X0, Y0].

Let

R0()(s, i) :=E[(CT() C0 ())2 |S0 = s, X0 = i, Y0 = 0].

We simulateR0()(s, i) using the expression in (3.16) for 0.3s1.3 andi = 1, 2, 3.The expression (3.16) involves a conditional expectation. We have taken 101 equi-spaced grid points on the interval [0.3, 1.3], which contains the strike price K = 1.For each grid point s and each i = 1, 2, 3, we compute the conditional expectationby simulating the process (St, Xt, Yt) 105 times. In Figure 7.2 for each i, we plot


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0

0.04

0.08

0.12

0.16

0.2

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3

Stock

Risk

Regime1 Regime2 Regime3

Fig. 7.2. Quadratic risks of hedging European call at different initial conditions.

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 0.5 1 1.5 2 2.5 3 3.5 4Stock

Option

Regime 1 Regime 2 Regime 3 B-S

Fig. 7.3. Price of up-out barrier option.

the function R0()(s, i) along the vertical axis against S0 = s along the horizontalaxis. The plots ofR0()(s, 1), R0()(s, 2), and R0()(s, 3) are put together in oneframe for clear comparison. One obvious observation is that, due to incompletenessof the market, the quadratic residual risk at t = 0 is nonzero. Beside this, theplot leads to another important observation regarding relative behavior ofR0() atdifferent regimes. One numerical example of barrier option price in the semi-Markovmodulated market is also presented here. As before the option prices at time zero foreach regime are plotted. Figure 7.3 shows up-out barrier option prices at differentregimes. Barrier is taken as b = 4. Therefore, up-out option price vanishes at s = 4.That is why the interval taken along the x-axis is [0, 4]. Strike price is 1. Therefore,the price increases almost linearly at s = 1. For s >1/2 b, the chance of knock outincreases significantly, thus the up-out barrier option price decreases. This featureoccurs in standard BlackScholes (B-S) model as well. Since it is better to comparethe regime switching case with the standard B-S model, one graph corresponding to


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up-out option price for standard B-S model (for the fixed regime i = 2) is inserted inthe same frame. This is calculated using a formula given in [32, p. 408].

8. Conclusions. In this paper we have studied risk minimizing option pricing

in a semi-Markov modulated market in the framework of Follmer and Schweizer [10],[26], [28]. We have developed suitable numerical methods for computing various optionprices. In [30] Svishchuk has studied option pricing with semi-Markov volatility. In[30] the option price and the hedging strategies have no dependence on the variable y.This has led to an incorrect form of the BlackScholes equation leading to incorrectresults. We have assumed that the state of the market modeled by the semi-Markovprocess{Xt} is completely observable. In a more realistic situation, the process{Xt} would be unobservable. Then{Xt} needs to be estimated from the availableinformation. If we assume that the mapi(i) is injective, then Xt can be obtainedusing the estimation on the volatility(Xt). For general cases, we have to use filteringtechniques to estimate the semi-Markov process Xt. For the Markov case, certainschemes to estimate the parameters r(i),(i),(i), etc., are developed in [9]. Similarwork needs to be done for the semi-Markov case.

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