EULER-MARUYAMA APPROXIMATIONS IN MEAN-REVERTING STOCHASTIC VOLATILITY MODEL UNDER REGIME-SWITCHING

(1)

EULER-MARUYAMA APPROXIMATIONS IN MEAN-REVERTING STOCHASTIC VOLATILITY MODEL UNDER REGIME-SWITCHING

XUERONG MAO, AUBREY TRUMAN, AND CHENGGUI YUAN

Received 28 December 2005; Revised 9 February 2006; Accepted 9 February 2006

Stochastic differential equations (SDEs) under regime-switching have recently been developed to model various financial quantities. In general, SDEs under regime-switching have no explicit solutions, so numerical methods for approximations have become one of the powerful techniques in the valuation of financial quantities. In this paper, we will concentrate on the Euler-Maruyama (EM) scheme for the typical hybrid mean-reverting θ-process. To overcome the mathematical difficulties arising from the regime-switching as well as the non-Lipschitz coefficients, several new techniques have been developed in this paper which should prove to be very useful in the numerical analysis of stochastic systems.

Copyright © 2006 Xuerong Mao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

In the well-known Black-Scholes model, the asset price is described by a geometric Brow- nian motion

dX(t)=μX(t)dt+νX(t)dw1(t), (1.1) wherew1(t) is a scalar Brownian motion,μis the rate of return of the underlying asset, andνis the volatility. In this classical model, Black and Scholes [2] assumed that the rate of return and the volatility are constants. However, it has been proved by many authors (see, e.g., [5,14,16,20]) that the volatility is itself an It ˆo process in many situations.

For instance, Hull and White [16] assume that the instantaneous varianceV=ν²obeys another geometric Brownian motion

dV(t)=αV(t)dt+βV(t)dw2(t), (1.2) whereα,βare constants whilew2(t) is another Brownian motion andw1(t) andw2(t)

Hindawi Publishing Corporation

Journal of Applied Mathematics and Stochastic Analysis Volume 2006, Article ID 80967, Pages1–20

DOI10.1155/JAMSA/2006/80967

(2)

have correlationρ. Heston [14] assumes that the varianceV obeys the mean-reverting square root process

dV(t)=αλ−V(t)dt+βV(t)dw2(t) (1.3) while the mean-reverting process

dV(t)=αλ−V(t)dt+βV(t)dw2(t) (1.4) is also proposed as the volatility process by others. In particular, Lewis [18] proposes the mean-revertingθ-process

dV(t)=αλ−V(t)dt+βV^θ(t)dw2(t), (1.5) whereθ≥1/2. This process unifies processes (1.3) and (1.4).

On the other hand, the rate of returnμis not a constant either and there is a strong evidence to indicate that it is a Markov jump process (see, e.g., [4,6,7,10,17,22,23,25]).

Of course, when the rate jumps, the volatility will jump accordingly. For example, the hybrid geometric Brownian motion

dX(t)=μr(t)X(t)dt+νr(t)X(t)dw1(t) (1.6) has been proposed by several authors (see [27,28] among others). Here,r(t) is a Markov chain with a finite state spaceᏹ= {1, 2,...,N}andμ,νare mappings fromᏹto [0,∞).

Equation (1.6) is also known as the geometric Brownian motion under regime-switching.

We observe that in this model, the volatility is also assumed to obey a Markov jump process. Recalling the stochastic volatility models mentioned above, we may more reasonably assume that the volatility process obeys a stochastic diﬀerential equation (SDE) under regime-switching, for example, the hybrid mean-revertingθ-process

dV(t)=αr(t)λr(t)−V(t)dt+βr(t)V^θ(t)dw2(t). (1.7) Such stochastic models under regime-switching have recently been developed to model various financial quantities, for example, option pricing [4,10–13,17], stock returns [6, 7,23], and portfolio optimization [22,25]. In particular, the mean-reverting square root process under regime-switching or, more generally, (1.7) has found its considerable use as a model for volatility and interest rate. In general, SDEs under regime-switching have no explicit solutions so the Monte Carlo simulations have become one of the powerful techniques in valuation of financial quantities, for example, option price (see [9,15,24]).

However, there is currently a lack of theory that guarantees the convergence of the Monte Carlo simulations for SDEs under regime-switching in finance. This is due to the fact that most of SDEs under regime-switching in finance are nonlinear and non-Lipschitzian so we cannot appeal to the standard convergence theory for numerical simulations, as typ- ified by [26], to deduce that the numerically computed paths are accurate for small step sizes.

In this paper, we will concentrate on the Euler-Maruyama (EM) scheme for the typical hybrid mean-revertingθ-process (1.7) but the theory established here can certainly be developed to cope with other SDEs under regime-switching in finance. InSection 2, we will

(3)

introduce necessary notations and investigate the global positive or nonnegative solutions to the mean-revertingθ-process under regime-switching. The EM numerical scheme will be defined inSection 3, where we will explain how to simulate discrete Markov chains, and hence the EM approximate solutions. InSection 4, we will show that the EM solutions converge to the exact solution. The path-dependent option with the volatility described by the hybrid mean-revertingθ-process will be discussed inSection 5, while Section 6contains applications to other financial quantities.

2. Nonnegative solutions

Throughout this paper, we let (Ω,Ᏺ,{Ᏺt}_t≥0,P) be a complete probability space with a filtration{Ᏺt}t≥0 satisfying the usual conditions (i.e., it is increasing and right continuous whileᏲ0contains allP-null sets). Letw(t) be a scalar Brownian motion defined on the probability space. Let| · |denote the Euclidean norm. Letr(t),t≥0, be a right- continuous Markov chain on the probability space taking values in a finite state space ᏹ= {1, 2,...,N}with the generatorΓ=(γi j)_N_×_Ngiven by

P

r(t+δ)=j|r(t)=i=

⎧⎪

⎨

⎪⎩

γ_{i j}δ+o(δ) ifi=j,

1 +γ_{i j}δ+o(δ) ifi=j, (2.1) whereδ >0. Hereγ_{i j}is the transition rate fromitojandγ_{i j}>0 ifi=jwhile

γii= −

j=i

γi j. (2.2)

We assume that the Markov chainr(·) is independent of the Brownian motionw(·). It is well known that almost every sample path ofr(·) is a right-continuous step function with a finite number of sample jumps in any finite subinterval ofR+:=[0,∞).

Let 1/2≤θ≤1. Consider the mean-revertingθ-process under regime-switching of the form

dS(t)=λr(t)μr(t)−S(t)dt+σr(t)S^θ(t)dw(t), t≥0, (2.3) with initial dataS(0)=S0>0 andr(0)=i0∈ᏹ. Hereλ(i),μ(i),σ(i),i∈ᏹ, are positive constants. The initial dataS0andi0 could be random, but the Markov property ensures that it is suﬃcient to consider only the case when bothS0andi0are constants. We note that the case whenθ=1/2 and the state space of the Markov chainᏹ= {1}corresponds to the classical mean-reverting square root process (1.3) (without regime-switching).

Since (2.3) is mainly used to model stochastic volatility or interest rate or an asset price, it is critical that the solutionS(t) will never become negative. The following lemma reveals this nonnegative property.

(4)

Lemma 2.1. For given any initial dataS(0)=S0>0 andr(0)=i0∈ᏹ, the solutionS(t) of (2.3) will never become negative with probability 1.

Proof. Clearly, the statement of the lemma is equivalent to that the solution of equation

dS(t)=λr(t)μr(t)−S(t)dt+σr(t)S(t)^θdw(t), t≥0, (2.4) will never become negative with probability 1 for any initial dataS(0)=S0>0 andr(0)= i0∈ᏹ. To show this, leta0=1, and for each integerk=1, 2,...,

ak=

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

e⁻^k⁽^k⁺¹⁾ ifθ=1 2, (2θ−1)k(k+ 1)

2

1/(1−2θ)

if1

2 < θ≤1,

(2.5)

so that

_a_k₋₁

ak

du

u²^θ ⁼k. (2.6)

For each k=1, 2,..., there clearly exists a continuous function ψk(u) with support in (a_k,a_k−1) such that

0≤ψk(u)≤ 2

ku²^θ forak< u < ak−1 (2.7) and_a^a_k^k⁻¹ψ_k(u)du=1. Defineϕ_k(x)=0 forx≥0 and

ϕ_k(x)= ₋_x

0 dy _y

0 ψ_k(u)du forx <0. (2.8) Thenϕ_k∈C²(R,R) and has the following properties:

(i)−1≤ϕ_k(x)≤0 forak< x < ak−1, or otherwiseϕ_k(x)=0;

(ii)|ϕ_k(x)| ≤2/k|x|²^θforak< x < ak−1, or otherwiseϕ_k(x)=0;

(iii)|x| −a_k−1≤ϕ_k(x)≤ |x|for allx∈R.

Let ¯λ=max_i_∈_ᏹλ(i), let ¯μ=max_i_∈_ᏹμ(i), and let ¯σ=max_i_∈_ᏹσ(i). Now for anyt≥0, by the well-known It ˆo formula (see [19,21]), we can derive that

Eϕ_kS(t)=ϕ_kS0

+E

_t

0

λr(u)μr(u)−S(u)ϕ_kS(u)+σ²r(u)

2 S(u)²^θϕ_kSr(u)du

≤σ¯²t k .

(2.9) Hence

−a_k−1≤ES⁻(t)−a_k−1≤σ¯²t

k , (2.10)

(5)

whereS⁻(t)= −S(t) ifS(t)<0, or otherwiseS⁻(t)=0. Lettingk→ ∞, we get thatES⁻(t)

=0 for allt≥0. This implies thatS(t)≥0 for allt≥0 with probability 1 as required.

Furthermore, the following lemma reveals the (strictly) positive property of the solution.

Lemma 2.2. For given any initial dataS(0)=S0>0 andr(0)=i0∈ᏹ, the solutionS(t) of (2.3) will remain positive with probability 1, namelyS(t)>0 for allt≥0 almost surely, if one of the following two conditions holds:

(i) 1/2< θ≤1;

(ii)θ=1/2 andσ²(i)≤2λ(i)μ(i) for alli∈ᏹ.

To show this lemma, let us first invoke the standard results, for example, results of G¯ıhman and Skorohod [8], to establish the following result.

Lemma 2.3. Consider the mean-revertingθ-process

dX(t)=λμ−X(t)dt+σX^θ(t)dw(t) (2.11) ont≥0 with initial valueX(0)=x0>0, where 1/2≤θ≤1 andλ,μ,σ are all positive constants. Then

(i) with probability 1, the solutionX(t) takes an infinite time to reach origin 0 if either 1/2< θ≤1 orθ=1/2 withσ²≤2λμ;

(ii) with positive probability, the solutionX(t) reaches the origin in finite time ifθ=1/2 andσ²>2λμ.

Proof. The coeﬃcients of (2.11)

a(z)=λ(μ−z), b(z)=σz^θ (2.12)

obey the linear growth condition onz∈R+, so the solution will never explode to infinity in any finite time with probability 1. We therefore need only to consider if it reaches the origin in finite time or not.

Consider

L1= 1

0exp

− _x

1

2a(z) b²(z)dz

dx. (2.13)

When 1/2< θ≤1, this gives L1=C1

1 0exp

− 2λμx¹⁻^2θ

σ²(1−2θ)+ 2λx²⁻²^θ σ²(2−2θ)

dx, (2.14)

whereC1is a positive constant. A simple inspection shows thatL1diverges. Hence, the required result when 1/2< θ≤1 follows from G¯ıhman and Skorohod [8, Chapter 21].

Similarly, whenθ=1/2, L1=C2

₁

0exp

−2λμ σ² log(x)

dx=C2

₁

0x⁻²^λμ/σ²dx, (2.15)

(6)

whereC2is another positive constant. It is then easy to see thatL1= ∞if 2λμ≥σ²while L1<∞if 2λμ < σ². The required results corresponding to 2λμ≥σ² or 2λμ < σ² when θ=1/2 follow from G¯ıhman and Skorohod [8, Chapter 21] again.

UsingLemma 2.3, we can now proveLemma 2.2.

Proof ofLemma 2.2. It is well known (see, e.g., [1]) that there is a sequence of stopping times 0=τ0< τ1<···< τk→ ∞such that the Markov chainr(t) has the representation

r(t)= ∞ k=0

rτk

I[τk,τk+1)(t), t≥0, (2.16)

whereI_Ais the indicator function of setA. Hence, fort∈[0,τ1], (2.3) becomes dS(t)=λr0

μr0

−S(t)dt+σr0

S^θ(t)dw(t) (2.17)

withS(0)>0. This is a mean-revertingθ-process of type (2.11). ApplyingLemma 2.3, we observe that S(t)>0 for all t∈[0,τ1] with probability 1 under the conditions of Lemma 2.2. Now, fort∈[τ1,τ2], (2.3) becomes

dS(t)=λrτ1

μrτ1

−S(t)dt+σrτ1

S^θ(t)dw(t) (2.18)

with initial valueS(τ1)>0 a.s. Again, this is a mean-revertingθ-process of type (2.11).

ByLemma 2.3, we see thatS(t)>0 for allt∈[τ1,τ2] with probability 1. Repeating this procedure, we see thatS(t)>0 for allt≥0 with probability 1 as required.

It is still remaining open whether the solutionS(t) will reach the origin in finite time with positive probability in the case whenθ=1/2 whileσ²(i)≤2λ(i)μ(i) does not hold for alli∈ᏹ.

3. The Euler-Maruyama method

To define the Euler-Maruyama approximate solution, we will need the following lemma (see [1]).

Lemma 3.1. GivenΔ >0, letr_k^Δ=r(kΔ) fork=0, 1, 2,....Then{r_k^Δ, k=0, 1, 2,...}is a discrete-time Markov chain with the one-step transition probability matrix

P(Δ)=

Pi j(Δ)_N_×_N=e^ΔΓ. (3.1) Given a step size Δ >0, the discrete-time Markov chain {r_k^Δ, k=0, 1, 2,...} can be simulated as follows: compute the one-step transition-probability matrix

P(Δ)=

P_{i j}(Δ)_N_×_N=e^ΔΓ. (3.2)

(7)

Letr0^Δ=i0 and generate a random numberξ1 which is uniformly distributed in [0, 1].

Define

r1^Δ=

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎩

i1 ifi1∈ᏹ− {N}is such that

i1−1 j=1

P_i0,j(Δ)≤ξ1<

i1

j=1

P_i0,j(Δ), N if

N−1 j=1

P_i0,j(Δ)≤ξ1,

(3.3)

where we set⁰_i₌1P_i0,j(Δ)=0 as usual. Generate independently a new random number ξ2which is again uniformly distributed in [0, 1] and then define

r2^Δ=

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎩

i2 ifi2∈ᏹ− {N}is such that

i2−1 j=1

P_r₁^Δ,j(Δ)≤ξ2<

i2

j=1

P_r₁^Δ,j(Δ), N if

N−1 j=1

P_r₁^Δ,j(Δ)≤ξ2.

(3.4)

Repeating this procedure, a trajectory of{r_k^Δ,k=0, 1, 2,...}can be generated. This procedure can be carried out independently to obtain more trajectories.

After explaining how to simulate the discrete-time Markov chain{r_k^Δ,k=0, 1,...}, we can now define the EM approximate solution for (2.3). Given a step sizeΔ >0, lett_k=kΔ fork=0, 1, 2,.... Compute the discrete approximationss_k≈S(t_k) by settings0=S0,r0^Δ= i0and forming

sk+1=sk+λr_k^Δμr_k^Δ−sk

Δ+σr_k^Δsk^θΔwk, k=0, 1, 2,..., (3.5) whereΔwk=w(t_k+1)−w(t_k). Let

s(t)¯ =sk, r(t)¯ =r^Δ_k fort∈ tk,tk+1

,k=0, 1, 2,..., (3.6)

and define the continuous EM approximate solution by s(t)=s0+

_t

0λr(u)¯ μr(u)¯ −¯s(u)du+ _t

0σr(u)¯ s(u)¯ ^θdw(u). (3.7) Note thats(t_k)=s(t¯ _k)=s_k, that is,s(t) and ¯s(t) coincide with the discrete approximate solution at the grid points.

4. Convergence of the EM approximate solution

Since the coeﬃcients of (2.3) satisfy the linear growth condition, by [26], we have the following lemma.

Lemma 4.1. LetS(t) be the solution of (2.3). Then for anyp≥1, there is a constantK, which is dependent on onlyp,T,S0but independent ofΔ, such that the exact solution and the EM

(8)

approximate solution to (2.3) have the property that

E sup

0≤t≤T

S(t)^p

∨E sup

0≤t≤T

s(t)^p

≤K. (4.1)

From this follows easily the following useful result.

Lemma 4.2. There is a constantC, which is independent ofΔ, such that

Es(t)−s(t)¯ ²^θ≤CΔ^θ, ∀t∈[0,T]. (4.2) Proof. From now on,Cused in the proofs below will be a generic positive number independent ofΔbut may have diﬀerent values where it appears.

For anyt∈[0,T], letk_t=[t/Δ], the integer part oft/Δ. ByLemma 4.1, we then derive that

Es(t)¯ −s(t)²≤4(¯λ∨λ¯μ¯∨σ)¯ E

1 +s_kt²

Δ²+w(t)−wk_tΔ²≤CΔ. (4.3) So, since 1/2≤θ≤1, by the Lyapunov inequality, we get

Es(t)¯ −s(t)^2θ≤

Es(t)¯ −s(t)²^θ≤CΔ^θ (4.4)

as required.

We can now state one of our main results.

Theorem 4.3. For each integerk=1, 2,...,

sup

0_≤t≤TES(t)−s(t)≤e^λT^¯

e⁻^k⁽^k⁻¹⁾^/²+4¯σ²T k +

1 ka²_k^θ+ 1

CΔ^θ+o(Δ)

, (4.5)

whereCis a constant which is independent of the step sizeΔand ¯λ, ¯σhave been defined in the proof ofLemma 2.1.

Proof. Note that

S(t)−s(t)= _t

0

λr(u)μr(u)−λr(u)¯ μr(u)¯ −λr(u)S(u) +λr(u)¯ s(u)¯ du

+ _t

0

σr(u)S(u)^θ−σr(u)¯ ¯s(u)^θdw(u).

(4.6)

(9)

Letϕkbe the same as defined in the proof ofLemma 2.1. Applying the It ˆo formula gives Eϕ_kS(t)−s(t)

=E _t

0ϕ_kS(u)−s(u)λr(u)μr(u)−λr(u)¯ μr(u)¯

−λr(u)S(u) +λr(u)¯ s(u)¯ du +1

2^E _t

0ϕ_kS(u)−s(u)σr(u)S(u)^θ−σr(u)¯ s(u)¯ ^θ²du=:I(t) +1 2J(t).

(4.7) By property (i) ofϕk,

I(t)≤E _t

0

ϕ_kS(u)−s(u)λr(u)μr(u)−λr(u)¯ μr(u)¯

−λr(u)S(u) +λr(u)¯ s(u)¯ du

≤E _t

0

λr(u)μr(u)−λr(u)¯ μr(u)¯ du

+E _t

0

λr(u)S(u)−λr(u)¯ s(u)¯ du.

(4.8)

Letn=[T/Δ], the integer part ofT/Δ. Then E

_T

0

=ⁿ

k=0

E _t_k+1

tk

(4.9)

withtn+1being now set to beT. LetIGbe the indicator function of setGand compute E

_t_k+1

tk

λr(u)μr(u)−λr(u)¯ μr(u)¯ du≤2¯λμE¯ _t_k+1

tk

I{r(u)=r(tk)}du

≤2¯λμ¯ _t_k+1

tk

P

r(u)=rtk

du=2¯λμ¯ _t_k+1

tk

i∈ᏹ

P rtk

=iP

r(u)=i|rtk

=idu

=2¯λμ¯ _t_k+1

tk

i∈ᏹ

P

rt_k=i

j=i

γ_{i j}u−t_k+ou−T_k

≤2¯λμ¯

1max_≤i≤N

−γii

Δ+o(Δ)

Δ.

(4.10)

(10)

Therefore, E

_T

0

λr(u)μr(u)−λr(u)¯ μr(u)¯ du≤2¯λμ¯

1max_≤i≤N

−γ_iiΔ+o(Δ). (4.11)

On the other hand, E

_t

0

λr(u)S(u)−λr(u)¯ ¯s(u)du

≤E _t

0

λr(u)−λr(u)¯ s(u)¯ du+E _t

0λr(u)S(u)−s(u)¯ du.

(4.12)

But E

_t

0

λr(u)−λr(u)¯ s(u)¯ du

=ⁿ

k=0

_t_k+1

tk

E

Eλr(u)−λrtksk|I{r(u)=r(tk)} du

= n k=0

_t_k+1

tk

E

Eλr(u)−λrtk|I{r(u)₌r(tk)_}

Esk|I{r(u)₌r(tk)_}

,

(4.13)

where in the last step we use the fact thats_kandI{r(u)=r(tk)}are conditionally independent with respect to theσ-algebra generated byr(tk). In the same way as in (4.10), we have

E _t

0

λr(u)−λr(u)¯ ¯s(u)du

≤2¯λμ¯

1max_≤i≤N

−γii

Δ+o(Δ) _T

0 Es(u)¯ du.

(4.14)

So, byLemma 4.1, E

_t

0

λr(u)−λr(u)¯ s(u)¯ du≤2(1 +K)¯λμ¯

1max_≤i≤N

−γ_iiΔ+o(Δ)

. (4.15) Substituting (4.15) into (4.12) and usingLemma 4.2, we obtain

E _t

0

λr(u)S(u)−λr(u)¯ ¯s(u)du

≤CΔ+o(Δ) + ¯λE _t

0

S(u)−s(u)¯ du

≤CΔ+o(Δ) + ¯λE _t

0

S(u)−s(u)du,

(4.16)

(11)

whereCis a positive constant independent of Δand it may change line by line. This, together with (4.11), yields

I(t)≤CΔ+o(Δ) + ¯λE _t

0

S(u)−s(u)du. (4.17)

In the following, we will estimateJ(t):

J(t)≤2¯σ²E _t

0

ϕ_kS(u)−s(u)S(u)^θ−¯s(u)^θ²du

+ 2E _t

0

ϕ_kS(u)−s(u)σr(u)−σr(u)¯ ²S(u)^2θdu.

(4.18)

Using property (ii) ofϕkandLemma 4.2, we have E

_t

0

ϕ_kS(u)−s(u)S(u)^θ−s(u)¯ ^θ²du

≤E _t

0

ϕ_kS(u)−s(u)S(u)−¯s(u)²^θdu

≤2²^θ⁻¹E _t

0

ϕ_kS(u)−s(u)S(u)−s(u)²^θdu + 2²^θ⁻¹E

_t

0

ϕ_kS(u)−s(u)s(u)−s(u)¯ ²^θdu

≤2E _t

0

2

kI{ak<|S(u)₋s(u)_|<ak−1}du+ 2 _t

0

2

ka^2θ_k ^Es(u)−¯s(u)²^θdu≤4T k +CΔ^θ

ka^2θ_k . (4.19) In the same way as (4.15) was proved, we can show that

E _t

0

ϕ_kS(u)−s(u)σr(u)−σr(u)¯ ²S(u)²^θdu

≤E _t

0

2 ka²_k^θ

σr(u)−σr(u)¯ ²S(u)²^θdu≤CΔ+o(Δ) ka²_k^θ .

(4.20)

Substituting (4.20) and (4.19) into (4.18), we have J(t)≤8¯σ²T

k +CΔ^θ+o(Δ)

ka²_k^θ . (4.21)

Therefore, Eϕk

S(t)−s(t)≤4¯σ²T

k +CΔ^θ+o(Δ)

ka²_k^θ +CΔ+o(Δ) + 2¯λE _t

0

S(u)−s(u)du. (4.22)

(12)

Noting that

Eϕk

S(t)−s(t)≥ES(t)−s(t)−ak−1 (4.23)

gives

ES(t)−s(t)≤ak−1+4¯σ²T k +

1

ka²_k^θ + 1CΔ^θ+o(Δ)+ ¯λ _t

0ES(u)−s(u)du.

(4.24) The required assertion follows finally from the Gronwall inequality.

Next, we derive a bound for a stronger form of the error. This version uses anL²- distance and places the supremum over time inside the expectation operation. The result below involves theL¹-error which is explicitly bounded inTheorem 4.3, and hence is also computable.

Theorem 4.4. One has E

sup

0≤t≤T

S(t)−s(t)²

≤e^(8¯^σ²^+2¯^λ²⁾^T²

CΔ+o(Δ) + 8¯σ²T sup

0≤u≤TES(u)−s(u)

. (4.25) Proof. For any 0≤t≤T, using the Cauchy-Schwarz inequality, we have

S(t)−s(t)²

≤T _t

0

λr(u)μr(u)−λr(u)¯ μr(u)¯ −λr(u)S(u) +λr(u)¯ s(u)¯ ²du

+ _t

0

σr(u)S(u)^θ−σr(u)¯ s(u)¯ ^θdw(u) 2

.

(4.26) In the same way as (4.11) and (4.16) were proved, we derive

E _t

0

λr(u)μr(u)−λr(u)¯ μr(u)¯ −λr(u)S(u) +λr(u)¯ ¯s(u)²du

≤2E _t

0

λr(u)μr(u)−λr(u)¯ μr(u)¯ ²du

+E _t

0

λr(u)S(u)−λr(u)¯ ¯s(u)²du

≤CΔ+o(Δ) + 2¯λ²E _t

0

S(u)−s(u)²du.

(4.27)

(13)

Using the Doob martingale inequality (see [19]), we find that for anyt1∈[0,T], E

sup

0≤t≤t¹

_t

0

σr(u)S(u)^θ−σr(u)¯ s(u)¯ ^θdw(u) 2

≤4E _t₁

0

σr(u)S(u)^θ−σr(u)¯ s(u)¯ ^θ²du

≤CΔ+o(Δ) + 8¯σ²E _t₁

0

S(u)−s(u)²^θdu

≤CΔ+o(Δ) + 8¯σ²E _t₁

0

S(u)−s(u)du+ 8¯σ²E _t₁

0

S(u)−s(u)²du.

(4.28)

Therefore, E

sup

0≤t≤t1

S(t)−s(t)²

≤CΔ+o(Δ) +8¯σ²+ 2¯λ²E _t₁

0

S(u)−s(u)²du+ 8¯σ²E _t₁

0

S(u)−s(u)du

≤CΔ+o(Δ) +8¯σ²+ 2¯λ²^t

1

0 E

sup

0_≤u≤v

S(u)−s(u)²

dv + 8¯σ²T sup

0_≤u≤TES(u)−s(u).

(4.29) An application of the Gronwall inequality completes the proof.

5. Options under stochastic volatility and regime-switching

In this section, we study the Heston stochastic volatility model under regime-switching, namely

dX(t)=λ1

r(t)μ1

r(t)−X(t)dt+σ1

r(t)X(t)V(t)dw1(t), (5.1) dV(t)=λ2

r(t)μ2

r(t)−V(t)dt+σ2

r(t)V^θ(t)dw2(t), 0≤t≤T. (5.2) HereV(t) is the volatility that feeds into the asset priceX(t). The Brownian motions w1(t) andw2(t) may be correlated. Naturally, we assume that the initial valuesX(0) and V(0) are both positive constants. Moreover,λ1,σ1, and so forth are all mappings fromᏹ toR+.

We begin with a lemma showing that the positivity in the initial data leads to the positive solutionX(t).

Lemma 5.1. IfV(t),t∈[0,T], is given by (5.2), then P

X(t)>0∀0≤t≤T=1. (5.3)