option pricing and hedging under a stochastic volatility l

Option pricing and hedging under a stochastic volatility

Levy process model

February 4, 2011

Young Shin Kim

Department of Statistics, Econometrics and Mathematical Finance,

School of Economics and Business Engineering, Karlsruhe Institute of Technology, Germany

E-mail: [email protected]

Frank J. Fabozzi

Professor in the Practice of Finance, Yale School of Management, USA


Zuodong Lin

HECTOR School of Engineering & Management, International Department,

Karlsruhe Institute of Technology

E-mail: zuodong [email protected]

Svetlozar T. Rachev

Department of Applied Mathematics & Statistics, Stony Brook University, USA

School of Economics and Business Engineering, Karlsruhe Institute of Technology, Germany

and FinAnalytica


1

Abstract

In this paper, we construct a new stochastic volatility model with a Levy driv-ing process and then apply the model to option pricing and hedging. The stochasticvolatility in our model is defined by the continuous Markov chain. We use the Es-scher transform to find the equivalent martingale measure. The option price usingthis model is obtained by the Fourier transform method. We obtain the closed-formsolution for the hedge ratio by applying the local risk minimizing hedging.

Keywords: Option pricing; Hedging; Stochastic volatility; Continuous Markov chain;Regime-switching; Levy process, Esscher transform

JEL Classifications: C6, G11, G12, G13

1 Introduction

The skewness and heavy-tail property observed for return distributions and the time-

varying volatility of the return process are two major issues for modeling underlying return

processes and option pricing. The skewness and heavy-tail property can be described by

the non-Gaussian infinitely divisible distributions (see Rachev et al., 2011). The time-

varying volatility models for option pricing has been studied for the following three major

classes of models: (1) continuous time and continuous market volatility, (2) discrete time

and continuous market volatility, and (3) continuous time and finite market volatility.

The stochastic volatility model by Heston (1993) belongs to the first class since the

model is a continuous-time model and the volatility in this model is defined on all positive

real numbers. The stochastic volatility Levy process model by Carr et al. (2003) is also

included in the first class. Option pricing with the GARCH model as proposed by Duan

(1995) is a discrete-time model, but volatility is defined on the all positive real numbers.

Hence, the model is included in the second class. The model was subsequently enhanced

by several researchers. For example, Menn and Rachev (2009), Kim et al. (2009), and

Kim et al. (2010) replace the Gaussian innovation in Duan’s model with the α-stable and

2

the tempered stable innovation process. If market volatility is modeled by the finite states

continuous Markov chain, then this model is referred to as a regime-switching model (see

Buffington and Elliott (2002) and Elliott and Kopp (2010)) and belongs to the third class.

The main issue associated with option pricing is finding a risk-neutral measure; that

is, finding an equivalent martingale measure (EMM) corresponding to the market proba-

bility measure1 of the underlying stock return process. The Girsanov theorem has been

successfully applied to find the EMM in time-varying volatility models having Brownian

motion as a driving process such as the stochastic volatility model by Heston (1993) and

the GARCH option pricing model by Duan (1995). However, the Girsanov theorem can-

not be applied to general time-varying volatility models with Levy driving processes. One

classical method to find an EMM for non-Gaussian Levy process models is the Esscher

transform presented by Gerber and Shiu (1994); another reasonable method is finding the

“minimal entropy martingale measure” presented by Fujiwara and Miyahara (2003). Kim

and Lee (2007) discuss the EMM on a tempered stable process model with the generalized

Girsanov theorem by Sato (1999). Finding an EMM with the Esscher transform for a

regime-switching model with Brownian motion is addressed in Elliott et al. (2005).

In the Black-Scholes model (Black and Scholes, 1973), the market is assumed to be

complete so that a perfect hedge can be attained by delta hedging. However, generally

in an incomplete market, a perfect hedge is not attainable. Follmer and Schweizer (1990)

propose what they refer to as a local risk-minimizing method for constructing a portfolio

containing options and underlying stocks that minimizes the variance based on the physical

market measure. Boyarchenko and Levendorskii (2002) discuss the local risk-minimizing

hedging under the exponential Levy stock price model and find an explicit form for the

hedge ratio.

1The market probability measure is the probability measure for the physical price process of the un-derlying stock. Another name for the market probability measure is the physical measure. Typically themeasure is estimated using the underlying stock’s historical returns.

3

The purpose of this paper is threefold. First, we construct a stochastic volatility model

with a Levy driving process and then discuss the option pricing and hedging problems using

this model. The stochastic volatility in our model is defined by the continuous Markov

chain, the same approach as the regime-switching model. Although Jackson et al. (2007)

also discuss regime-switching with Levy driving process, their approach does not focus

on stochastic volatility and therefore differs from our model. Second, we explain how to

find the equivalent martingale measure (EMM) using the Esscher transform under the

new stochastic volatility model and provide the European option pricing formula under

this measure. Finally, we apply local risk-minimizing hedging for the stochastic volatility

model with a Levy driving process and present an explicit form for the hedge ratio.

The remainder of this paper is organized as follows. In Section 2, The Markov chain

stochastic volatility Levy process model is constructed and then Esscher transform are

discussed. we provide the option pricing formula for the model in Section 3. Local risk

minimizing hedging under the model is presented in Section 4. In Section 5, we summarize

our principal findings.

2 Markov chain stochastic volatility Levy process model

and Esscher transform

Suppose (L(t))t≥0 is a Levy process on a complete probability space (Ω,P,F). We assume

that the states of the economy are modeled by a continuous-time hidden Markov chain

process (X(t))t≥0 generated by a generater matrix A on (Ω,P,F) with a finite state space

which is a finite set of unit vectors e1, e2, · · · , eN, where en = (0, · · · , 1, · · · 0)′ ∈ RN .

Then, we have the following semi-martingale representation theorem for X(t)t≥0:

X(t) = X(0) +

∫ t

0AX(s)ds+M(t), (1)

4

where (M(t))t≥0 is an RN -valued martingale increment process with respect to the fil-

tration generated by X(t)t≥0. We assume that X(t)t≥0 and L(t)t≥0 are indepen-

dent and EP[exp(uL(t))] < ∞ if u ∈ I for some real interval I. We denote Φ(z) =

logE[exp(zL(1))]. The domain of Φ(z) is extended to a complex subset z ∈ C : <(z) = I

by the analytic continuation in complex analysis (see Chruchill and Brown, 1990). Let

(FXt )t≥0 and (FLt )t≥0 be the natural filtrations generated by (X(t))t≥0 and (L(t))t≥0, re-

spectively. We define a filtration (Gt)t≥0 such that Gt is the σ-algebra generated by FXt

and FLt for all t ≥ 0.

The stock price process (S(t))t≥0 is referred to as the Markov chain stochastic volatility

Levy process model if S(t) is given by

S(t) = S(0) exp

(∫ t

0〈µ,X(s)〉ds−

∫ t

0Φ(〈σ,X(s)〉)ds+

∫ t

0〈σ,X(s)〉dL(s)

), (2)

where µ = (µ1, µ2, · · · , µN )′ ∈ RN and σ = (σ1, σ2, · · · , σN )′ ∈ RN with σn > 0 and σn ∈ I

for each n = 1, 2, · · · , N . The value µ(t) = 〈µ,X(t)〉 and σ(t) = 〈σ,X(t)〉 are referred

to as the expected return and the market volatility at time t, respectively. The process

(S(t))t≥0 is (Gt)t≥0-adapted.

Lemma 1. Let 0 ≤ s ≤ t and y = (y1, y2, · · · , yN )′ ∈ RN . The conditional characteristic

function of∫ ts 〈y,X(u)〉du under the σ-field FXs is equal to

E

[exp

(i

∫ t

s〈y,X(u)〉du

) ∣∣∣FXs ] = 〈exp[(t− s)(A+ diag(iy))]X(s),1〉 (3)

where (FXt )t≥0 is the natural filtrations generated by (X(t))t≥0 and 1 = (1, 1, · · · , 1)′ ∈ RN .

Proof. Consider a process

Z(t) = exp

(i

∫ t

s〈y,X(u)〉du

)X(t).

5

Then

dZ(t) = exp

(i

∫ t

s〈y,X(u)〉du

)dX(t) + i〈y,X(t)〉 exp

(i

∫ t

s〈y,X(u)〉du

)X(t)dt

By (1), we obtain

Z(t)− Z(s) =

∫ t

s(A+ idiag(y))Z(u)du+

∫ t

sexp

(i

∫ u

s〈y,X(v)〉dv

)dMu.

Since (M(t))t≥0 is martingale, idiag(y) = diag(iy), and Z(s) = X(s), we obtain the

following equation by taking the conditional expectation

E[Z(t)|FXs ] = X(s) +

∫ t

s(A+ diag(iy))E[Z(u)|FXs ]du,

and hence

E[Z(t)|FXs ] = exp ((t− s)(A+ diag(iy)))X(s).

Therefore we obtain (3) as follows

E

[exp

(i

∫ t

s〈y,X(u)〉du

) ∣∣∣FXs ] = E

[⟨exp

(i

∫ t

s〈y,X(u)〉du

)X(t),1

⟩ ∣∣∣FXs ]= 〈exp[(t− s)(A+ diag(iy))]X(s),1〉.

Because the Markov process has a countable state space, the amount of time that the

volatility state stays on each state en for n = 1, 2, · · · , N from 0 to time t is given as:

τ tn =

∫ t

0〈en, Xu〉du, (4)

where∑N

n=1 τtn = t. By substituting s = 0 in equation (3) of Lemma 1, the characteristic

6

function of the joint distribution of the random vector (τ t1, τt2, · · · , τ tN ) is obtained by

E

[exp

(iN∑k=1

ykτtn

)]= 〈exp[t(A+ diag(iy))]X(0),1〉, (5)

where y = (y1, y2, · · · , yN )′, 1 = (1, 1, · · · , 1)′ ∈ RN , and X(0) is the initial state. For

0 ≤ s ≤ t, we have

E

[exp

(i

N∑k=1

yk(τtn − τ sn)

)∣∣∣FXs]

= 〈exp[(t− s)(A+ diag(iy))]X(s),1〉.

Lemma 2. Let 0 ≤ s ≤ t, v = (v1, v2, · · · , vN )′ ∈ RN , and w = (w1, w2, · · · , wN )′ ∈ RN

such that Φ(wn) <∞ for all n = 1, 2, · · · , N . Then

∫ t

s〈v,X(u)〉du =

N∑n=1

vn(τ tn − τ sn), 0 ≤ s ≤ t (6)

∫ t

s〈w,X(u)〉dL(u) =

N∑n=1

wnL(τ tn − τ sn), 0 ≤ s ≤ t. (7)

and

EP

[exp

(N∑n=1

vn(τ tn − τ sn) +N∑n=1

wnL(τ tn − τ sn)

)∣∣∣Gs]

= 〈exp[(t− s)(A+ diagγ(x))]X(s),1〉, (8)

where γ(x) = (v1 + Φ(w1), v2 + Φ(w2), · · · , vN + Φ(wN ))′.

Proof. Equations (6) and (7) are easy to prove. Equation (8) is proved using properties

7

of the conditional expectation and independence between L(t) and τ tn − τ sn as follows:

EP

[exp

(N∑n=1

vn(τ tn − τ sn) +

N∑n=1


)∣∣∣Gs]

= EP

[EP

[exp

(N∑n=1

vn(τ tn − τ sn) +

N∑n=1


)∣∣∣τ tn − τ sn] ∣∣∣Gs]

= EP

[exp

(N∑n=1

(vn + Φ(wn))(τ tn − τ sn)

)∣∣∣Gs]

= 〈exp[(t− s)(A+ diagγ(x))]X(s),1〉,

where γ(x) = (v1 + Φ(w1), v2 + Φ(w2), · · · , vN + Φ(wN ))′.

By (6) and (7), we have

S(t) = S(0) exp

(N∑n=1

µnτtn −

N∑n=1

Φ(σn)τ tn +N∑n=1

σnL(τ tn)

). (9)

We consider a financial model consisting of two risky underlying assets, namely a bank

account and a stock, which are tradable continuously. The risk-free rate of return, denoted

by r(t), at time t ≥ 0 is given by r(t) = 〈r,X(t)〉, where r = (r1, r2, · · · , rN )′ ∈ RN with

rn > 0 for each n = 1, 2, · · · , N and 〈·, ·〉 denotes the inner product in RN . The discounted

stock price process (S(t))t≥0 is defined by

S(t) = exp

(−∫ t

0〈r,X(s)〉ds

)S(t).

By (6) and (7), we have

S(t) = S(0) exp

(N∑n=1

(µn − rn)τ tn −N∑n=1

Φ(σn)τ tn +

N∑n=1

σnL(τ tn)

)(10)

Proposition 1. Let Z(t) =∫ t0 〈σ,Xs〉dLs, t ≥ 0. If there exist θ = (θ1, θ2, · · · , θN ) ∈ RN

8

which satisfies the condition

Φ(θnσn) <∞

Φ((1 + θn)σn) <∞

µn − rn = Φ(σn) + Φ(θnσn)− Φ((1 + θn)σn)

for all n = 1, 2, · · · , N, (11)

then the measure Qθ equivalent to the measure P with a Radon-Nikodym derivative

dQθ

dP∣∣Gt =

exp(∫ t

0 〈θ,X(s)〉dZ(s))

EP

[exp

(∫ t0 〈θ,X(s)〉dZ(s)

)|FXt

] , for t ≥ 0,

is an equivalent martingale measure (EMM).

Proof. Let ξt = dQθdP∣∣Gt . Then we have

ξTξt

=exp

(∫ Tt 〈θ,X(s)〉dZ(s)

)exp

(∑Nn=1 Φ(θnσn)(τTn − τ tn)

) =exp

(∑Nn=1 θnσnL(τTn − τ tn)

)exp

(∑Nn=1 Φ(θnσn)(τTn − τ tn)

)and ξT

ξtis independent to Gt. By (10), we have

S(T )

S(t)= exp

(N∑n=1

(µn − rn − Φ(σn))(τTn − τ tn) +N∑n=1

σnL(τTn − τ tn)

)

and S(T )

S(t)is independent to Gt. Hence we have

EP

[S(T )

ξTξt|Gt]

= S(t)EP

[S(T )

S(t)· ξTξt|Gt

]

= S(t)EP

[exp

(N∑n=1

(µn − rn − Φ(σn)− Φ(θnσn))(τTn − τ tn) +

N∑n=1

(1 + θn)σnL(τTn − τ tn)

)|Gt

]

= S(t)EP

[exp

(N∑n=1

(µn − rn − Φ(σn)− Φ(θnσn) + Φ((1 + θn)σn))(τTn − τ tn)

)|Gt

], by (8).

9

Therefore, if

µn − rn = Φ(σn) + Φ(θnσn)− Φ((1 + θn)σn), for n = 1, 2, · · · , N,

then

S(t) = EP

[S(T )

ξTξt|Gt], 0 ≤ t ≤ T, (12)

and hence Qθ is an EMM corresponding to P.

The method to find an EMM using Proposition 1 is referred to as the Esscher transform

under the Markov chain stochastic volatility Levy process model. In the remainder of this

paper, we denote ξt = dQθdP∣∣Gt . By the definition of dQθ

dP∣∣Gt in Proposition 1, we obtain that

ξt = exp

(N∑n=1

(θnσnL(τ tn)− Φ(θnσn)τ tn)

). (13)

3 Option pricing under the Markov chain stochastic volatil-

ity Levy process model

In this section, we assume that (S(t))t≤0 is given by the Markov chain stochastic volatility

Levy process model and Qθ is the EMM obtained by the Esscher transform. The expected

return and and the market volatility for (S(t))t≤0 is supposed to be µ = (µ1, µ2, · · · , µN )′

and σ = (σ1, σ2, · · · , σN )′, respectively, the risk-free rate of return is supposed to be

r = (r1, r2, · · · , rN )′, and we denote Y (t) = log(S(t)/S(0)) in this section.

The conditional characteristic function of Y (T )− Y (t) under measure P and the con-

dition Gt is given by

φPY (T )−Y (t)(u|Gt) = EP[exp(iu(Y (T )− Y (t)))|Gt]

= 〈exp[(T − t)(A+ diag(yP(u)))]X(t),1〉

10

where

yP(z) =

iu(µ1 − Φ(σ1)) + Φ(iuσ1)

iu(µ2 − Φ(σ2)) + Φ(iuσ2)

...

iu(µN − Φ(σN )) + Φ(iuσN )

. (14)

Lemma 3. Let 0 ≤ t ≤ T . The conditional characteristic function of Y (T )− Y (t) under

measure Qθ and the condition Gt is given by

φQθY (T )−Y (t)(u|Gt) = EQθ [exp(iu(Y (T )− Y (t)))|Gt]

= ξt〈exp[(T − t)(A+ diag(yQθ(u)))]X(t),1〉

where ξt is given by (13) and

yQθ(u) =

iu(r1 − Φ((1 + θ1)σ1)) + Φ((iu+ θ1)σ1)

iu(r2 − Φ((1 + θ2)σ2)) + Φ((iu+ θ2)σ2)

...

iu(rN − Φ((1 + θN )σN )) + Φ((iu+ θN )σN )

. (15)

Proof. We have

EQθ [exp(iu(Y (T )− Y (t)))|Gt] = ξtEP[exp(iu(Y (T )− Y (t)))ξT /ξt|Gt]

= ξtEP

[exp

(iu

N∑n=1

((µn − Φ(σn))(τTn − τ tn) + σnL(τTn − τ tn)

))

× exp

(N∑n=1

(θnσnL(τTn − τ tn)− Φ(θnσn)(τTn − τ tn)

)) ∣∣∣Gt].

11

By the condition (11), we obtain

EQθ [exp(iu(Y (T )− Y (t)))|Gt]

= ξtEP

[exp

(N∑n=1

iu(rn − Φ((1 + θn)σn))(τTn − τ tn) +N∑n=1

(iu+ θn)σnL(τTn − τ tn)

)∣∣∣Gt].By (8), we complete the proof.

For convenience, we define a function on C× [0,∞) where C is the complex field that

Mr,σ,θ(z, t, T ) = 〈exp[(T − t)(A+ diag(yQθ(z)))]X(t),1〉, z ∈ C, 0 ≤ t ≤ T,

where yQθ(z) is given by (15).

We obtain the option pricing formula under the Markov chain stochastic volatility

Levy process model by the Fourier transform method discussed in Carr and Madan (1999)

and Lewis (2001) as follows.2 Let Π(ST ) be a payoff function of one European option

with time to maturity T , h(x) = Π(ex) with x = logS(T ), and h(η)) =∫∞−∞ e

−iηxh(x))dx.

Suppose r = r1 = · · · = rN and h(η) is defined for all η ∈ Rh where Rh = z ∈ C :

Im(z) ∈ Ih for some real interval Ih. The characteristic function φQθY (T )−Y (t)(η) is defined

for all η ∈ Rφ where Rφ = z ∈ C : Im(z) ∈ Iφ for some real interval Iφ. Based on the

no-arbitrage pricing framework, the European option price V with time to maturity T is

given by

V (t) = EP

[exp(−r(T − t))Π(S(T ))

ξTξt|Gt]

=1

ξtEQθ

[e−r(T−t)Π(elogS(t)+Y (T )−Y (t))|Gt

]=e−r(T−t)

ξtEQθ [h(logS(t) + Y (T )− Y (t))|Gt] .

2This method is discussed in Liu et al. (2006) under the regime-switching model with Brownian motion.

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Since the probability density function fQθY (T )−Y (t)(x|Gt) of Y (T )− Y (t) under measure Qθ

and the condition Gt can be obtained by

fQθY (T )−Y (t)(x|Gt) =1

2π

∫ ∞−∞

exp(−i(u+ iρ)x)φQθY (T )−Y (t)(u+ iρ|Gt)du

using the complex inversion formula, we compute

EQθ [h(Y (T )− Y (t))|Gt]

=

∫ ∞−∞

h(logS(t) + x)1

2π

∫ ∞−∞

e−i(u+iρ)xφQθY (T )−Y (t)(u+ iρ|Gt)du dx

=1

2π

∫ ∞−∞

∫ ∞−∞

e−i(u+iρ)(x−logS(t))h(x)dxφQθY (T )−Y (t)(u+ iρ|Gt)du,

and hence

V (t) =e−r(T−t)

2πξt

∫ ∞−∞

ei(u+iρ) logS(t)h(u+ iρ)φQθY (T )−Y (t)(u+ iρ|Gt)du, (16)

where ρ ∈ Ih∩Iφ. By the Lemma 3, we obtain the European option pricing formula under

the Markov chain stochastic volatility Levy process model:

V (t) =e−r(T−t)

2π

∫ ∞−∞

ei(u+iρ) logS(t)h(u+ iρ)Mr,σ,θ(u+ iρ, t, T )du. (17)

In particular, the payoff function of a European call option with time to maturity T

and strike price K is given by Π(S(T )) = (S(T ) − K)+ and hence h(x) = (ex − K)+,

where (·)+ = max(·, 0). Therefore, we have

h(η) = −Ke−iη logK

η(i+ η), (18)

where η ∈ u+ iρ|ρ < −1, u ∈ R. By substituting (18) into (17), a European call option

13

pricing formula at time t is equal to

V (t) =e−r(T−t)K1+ρ

2πS(t)ρ

∫ ∞−∞

eiu log(S(t)/K)

(ρ− iu)(1 + ρ− iu)Mr,σ,θ(u+ iρ, t, T )du, (19)

where ρ is real number such that ρ < −1 and yn(u+ iρ) <∞ for all n ∈ 1, 2, · · · , N and

u ∈ R. Moreover, by the property that h(−η;S(t)) = h(η;S(t)) for the payoff function of

a European call option, (19) becomes

V (t) =e−r(T−t)K1+ρ

πS(t)ρRe

∫ ∞0

eiu log(S(t)/K)

(ρ− iu)(1 + ρ− iu)Mr,σ,θ(u+ iρ, t, T )du. (20)

We can compute a European put option’s price in the same way as for a European call

option. A European put option’s price can be obtained by the same formula as (20),

but the condition of ρ is different; that is, ρ is a strictly positive real number such that

yn(u+ iρ) <∞ for all n ∈ 1, 2, · · · , N and u ∈ R.

4 Local risk-minimizing hedge ratio under the Markov chain

stochastic volatility Levy process model

The Markov chain stochastic volatility Levy process model implies incomplete market.

In an incomplete market, the perfect hedge for a general claim is no longer possible.

In this section, we present the locally risk-minimizing hedge ratio which is discussed in

Boyarchenko and Levendorskii (2002) and deduce the explicit form of the hedge ratio in

the Markov chain stochastic volatility Levy process model.

Consider a European option with maturity T > 0 and let V (t) be the European option’s

price at t. Let Wt+∆t(ϕ) = V (t+∆t)− ϕSt+∆t at time t < T . Let ϕ∆t(t) be the share of

14

the underlying stock which minimizes variance of Wt+∆t(ϕ). That is

ϕ∆t(t) = arg minϕEP[(Wt+∆t(ϕ)− EP[Wt+∆t(ϕ)|Gt])2 |Gt]. (21)

The locally risk-minimizing hedge ratio ϕt at time t is defined as

ϕ(t) = lim∆t↓0

ϕ∆t(t).

In this section, we assume that X(t+∆t) = X(t) for small ∆t.

Proposition 2. Let θ = (θ1, θ2, · · · , θN )′ ∈ RN which satisfies the condition (11) and

Φ(2σn) for all n ∈ 1, 2, · · · , N and let ρ ∈ R satisfy Φ((1−ρ)σn) <∞ and Φ(−ρσn) <∞

for all n ∈ 1, 2, · · · , N. The locally risk-minimizing hedging ratio is equal to

ϕ(t) =e−r(T−t)

2πS(t)〈(diag(d(σ))−A)X(t),1〉

×∫ ∞−∞

ei(u+iρ)Y (t)h(u+ iρ)Mr,σ,θ(u+ iρ, t, T )〈(diag(yH(u+ iρ))−A)X(t),1〉du,

(22)

where

d(σ) =

Φ(2σ1)− 2Φ(σ1)

Φ(2σ2)− 2Φ(σ2)

...

Φ(2σN )− 2Φ(σN )

and yH(z) =

Φ((iz + 1)σ1)− Φ(izσ1)

Φ((iz + 1)σ2)− Φ(izσ2)

...

Φ((iz + 1)σN )− Φ(izσN )

for z ∈ C.

15

and

(EP[S(t+∆t)|Gt])2 = S(t)2(EP[eY (t+∆t)−Y (t)|Gt]

)2= S(t)2 (〈exp[∆t(diag(µ) +A)]X(t),1〉)2

= S(t)2(〈X(t),1〉+ 2∆t〈(diag(µ) +A)X(t),1〉) +O(∆t2).

Hence we have

Gt(∆t) = S(t)2∆t〈(diag(d(σ))−A)X(t),1〉+O(∆t2), (24)

where

d(σ) =

Φ(2σ1)− 2Φ(σ1)

Φ(2σ2)− 2Φ(σ2)

...

Φ(2σN )− 2Φ(σN )

.

By equation (16), we have

EP[S(t+∆t)V (t+∆t) |Gt]

=S(t)e−r(T−t)

2π

∫ ∞−∞

ei(u+iρ)Y (t)h(u+ iρ)

EP

[ei(u+i(ρ−1))∆Y (t)+r∆tMr,σ,θ(u+ iρ, t+∆t, T ) |Gt

]du,

where ∆Y (t) = Y (t+∆t)− Y (t). Since we can prove that

Mr,σ,θ(u+ iρ, t+∆t, T )

= Mr,σ,θ(u+ iρ, t, T )(1−∆t〈(diag(yQθ(u+ iρ)) +A)X(t),1〉) +O(∆t2).

under the assumption X(t+∆t) = X(t) for small ∆t where yQθ is given by the equation

17

(15), we obtain that

EP[S(t+∆t)V (t+∆t) |Gt]

=S(t)e−r(T−t)

2π

∫ ∞−∞

ei(u+iρ)Y (t)h(u+ iρ)Mr,σ,θ(u+ iρ, t, T )

×(1 + r∆t−∆t〈(diag(yQθ(u+ iρ)) +A)X(t),1〉

+∆t〈(diag(yP(u+ i(ρ− 1)) +A)X(t),1〉)du

+O(∆t2),

where yP is given in the equation (14). By the same argument, we obtain the second part

that

EP[S(t+∆t) |Gt]EP[V (t+∆t) |Gt]

=S(t)e−r(T−t)

2π

∫ ∞−∞


× (1 + r∆t+∆t〈(diag(µ) +A)X(t),1〉 −∆t〈(diag(yQθ(u+ iρ)) +A)X(t),1〉

+∆t〈(diag(yP(u+ iρ)) +A)X(t),1〉)du+O(∆t2)

Therefore we obtain

Ft(∆t) =S(t)e−r(T−t)

2π∆t

∫ ∞−∞


× 〈(diag(yP(u+ iρ− i)− yP(u+ iρ)− µ)−A)X(t),1〉du

+O(∆t2). (25)

18

By the definition of yP, we have

yP(z − i)− yP(z)− µ =

Φ((iz + 1)σ1)− Φ(izσ1)

Φ((iz + 1)σ2)− Φ(izσ2)

...

Φ((iz + 1)σN )− Φ(izσN )

for z ∈ C.

Substituting (24) and (25) into (23), we obtain

ϕ∆t(t) =e−r(T−t)

∫∞−∞ e

i(u+iρ)Y (t)h(u+ iρ)Mr,σ,θ(u+ iρ, t, T )Ψ(u+ iρ,X(t))du+O(∆t)

2πS(t)〈(diag(d(σ))−A)X(t),1〉+O(∆t),

where Ψ(z, x) = 〈(diag(yH(z))−A)x,1〉. Taking ∆t→ 0, we obtain (22).

5 Conclusion

In this paper we construct the Markov chain stochastic volatility Levy process model. The

model is stochastic volatility model with a Levy driving process and a continuous Markov

chain volatility process. The EMM of the model is defined by the Esscher transform

and option price formula is obtained by the Fourier transform method. The locally risk-

minimizing hedging ratio is also discussed and finally the closed form solution of the hedge

ratio is presented.

References

Black, F. and Scholes, M. (1973). The pricing of options and corporate liabilities. The

Journal of Political Economy , 81 (3), 637–654.

19

Boyarchenko, S. I. and Levendorskii, S. Z. (2002). Non-Gaussian Merton-Black-Scholes

Theory . World Scientific.

Buffington, J. and Elliott, R. J. (2002). American options with regime switching. Inter-

national Journal of Theoretical and Applied Finance, 5 (5), 497–514.

Carr, P., Geman, H., Madan, D., and Yor, M. (2003). Stochastic volatility for Levy

processes. Mathematical Finance, 13 , 345–382.

Carr, P. and Madan, D. (1999). Option valuation using the fast fourier transform. Journal

of Computational Finance, 2 (4), 61–73.

Chruchill, R. V. and Brown, J. W. (1990). Complex Variables and Applications. McGraw-

Hill.

Duan, J.-C. (1995). The GARCH option pricing model. Mathematical Finance, 5 (1),

13–32.

Elliott, R. J., Chan, L., and Siu, T. K. (2005). Option pricing and Esscher transform

under regime switching. Annals of Finance, 1 (4), 423–432.

Elliott, R. J. and Kopp, P. E. (2010). Mathematics of Financial Markets. Springer, 2nd

ed.

Follmer, H. and Schweizer, M. (1990). Hedging of contingent claims under incomplete

information. In M. H. A. Davis and R. J. Elliott (Eds.), Applied Stochastic Analysis,

New York: Gordon and Breach, vol. 5. 389–414.

Fujiwara, T. and Miyahara, Y. (2003). The minimal entropy martingale measures for

geometric Levy processes. Finance & Stochastics, 7 , 509–531.

Gerber, H. and Shiu, E. (1994). Option pricing by Esscher transforms. Transactions of

the Society of Actuaries, XLVI , 99–140.

20

Heston, S. L. (1993). A closed form solution for options with stochastic volatility with

applications to bond and currency options. Review of Financial Studies, 6 , 327–343.

Jackson, K. R., Jaimungal, S., and Surkov, V. (2007). Option pricing with regime switching

Levy processes using fourier space time stepping. Journal of Computational Finance,

12 (2), 1–29.

Kim, Y. and Lee, J. H. (2007). The relative entropy in CGMY processes and its applica-

tions to finance. Mathematical Methods of Operations Research, 66 (2), 327–338.

Kim, Y. S., Rachev, S. T., Bianchi, M. L., and Fabozzi, F. J. (2010). Tempered stable

and tempered infinitely divisible GARCH models. Journal of Banking & Finance, 34 ,

2096–2109.

Kim, Y. S., Rachev, S. T., Chung, D. M., and Bianchi, M. L. (2009). The modified tem-

pered stable distribution, GARCH-models and option pricing. Probability and Mathe-

matical Statistics, 29 (1), 91–117.

Lewis, A. L. (2001). A simple option formula for general jump-diffusion and other expo-

nential Levy processes. avaible from http://www.optioncity.net .

Liu, R. H., Zhang, Q., and Yin, G. (2006). Option pricing in a regime switching model us-

ing the fast fourier transform. Journal of Applied Mathematics and Stochastic Analysis,

1–22. Article ID: 18109, DOI: 10.1155/JAMSA/2006/18109.

Menn, C. and Rachev, S. T. (2009). Smoothly truncated stable distributions, GARCH-

models, and option pricing. Mathematical Methods in Operation Research, 69 (3).

Rachev, S. T., Kim, Y. S., Bianchi, M. L., and Fabozzi, F. J. (2011). Financial Models

with Levy Processes and Volatility Clustering . John Wiley & Sons.

21

Sato, K. (1999). Levy Processes and Infinitely Divisible Distributions. Cambridge Univer-

sity Press.

22

option pricing and hedging under a stochastic volatility l

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