a black-scholes model with markov switching
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Option Pricing in a Black-Scholes Model with Markov Switching
Fuh, Cheng-Der1 and Wang, Ren-Her
Institute of Statistical Science, Academia Sinica, Taipei, Taiwan, R.O.C.
Cheng, Jui-Chi
Graduate Institute of Finance, National Taiwan University, Taipei, Taiwan, R.O.C.
ABSTRACT
The theory of option pricing in Markov volatility models has been developed in recent
years. However, an efficient method to compute option price in this setting remains
lacking. In this article, we present a way of modeling time-varying volatility; to generalize
the classical Black-Scholes model to encompass regime-switching properties. Specifically,
the unobserved state variables for stock fluctuations are governed by a first-order Markov
process, and the drift and volatility parameters take different values depending on the
state of this hidden Markov model. Standard option pricing procedure under this model
becomes problematic. We first provide a closed-form formula for the arbitrage-free price
of the European call option. In the case of the underlying Markov process has two
states, we have an explicit analytic formula for option price. When the number of states
for the underlying Markov process is bigger than two, we propose an approximation
formula for option price. The idea of the approximation formula is that we replace
the distribution of occupation time for a given state by its corresponding stationary
distribution multiplies the whole time period. Numerical methods, such as Monte-Carlo
simulation and Markovian tree, to compute European call option price are presented
for comparison. Our numerical results show that the approximation formula provide
an efficient and reliable implementation tool for option pricing. Implied volatility and
implied volatility surface are also given.
Keywords: Arbitrage, Black-Scholes model, European call option, hidden Markov model,
implied volatility, Laplace transform, Monte Carlo, tree.
1. Research partially supported by NSC 91-2118-M-001-016.
1
1. Introduction
It is well-known that the volatility of financial data series tends to change over time,
and changes in return volatility of stock returns tend to be persistent. The statistical
properties of return volatility have been deeply studied and uncovered in the financial
economic literatures, for instance, the extensive work on modeling stock fluctuations
with stochastic volatility (cf. Anderson, 1996; Hull and White, 1987; Stein, 1991; Wig-
gins, 1987; Heston, 1993), and uncertain volatility (Avellaneda, Levy, and Paras, 1995).
Many efforts have been made to study financial markets with different information levels
among investors (cf. Duffie and Huang, 1986; Ross, 1989; Anderson, 1996; Karatzas and
Pikovsky, 1996; Guilaume et. al., 1997; Grorud and Pontier, 1998; Imkeller and Weisz,
1999). Like the Markov switching model of Hamilton (1988, 1989) may provide more
appropriate modeling of volatility. In this regard, Hamilton and Susmel (1994) proposed
the Markov switching ARCH model. In a partial equilibrium model, Turner, Startz and
Nelson (1989) formulate a switching model of excess returns in which returns switch ex-
ogenously between a Gaussian low variance regime and a Gaussian high variance regime.
So, Lam and Li (1998) generalizes the stochastic volatility model to incorporate Markov
regime switching properties. Diebold and Inoue (2001) considers the properties of long
memory and regime switching.
In deed, recent research (cf. Bittlingmayer, 1998) has shown that investors’ un-
certainty over some important factors affecting the economy may greatly impact the
volatility of stock returns. More generally, there is evidence that investors tend to be
more uncertain about the future growth rate of the economy during recessions, thereby
partly justifying a higher volatility of stock returns. In this paper, to encompass the
empirical phenomena of the stock fluctuations relate to business cycle, we introduce a
model of an incomplete market by adjoining the Black-Scholes exponential Brownian
motion model for stock fluctuations with a hidden Markov process. Specifically, we
assume that stock of asset pricing are generated by realization of a Gaussian diffusion
process, and the drift and volatility parameters take different values depending on the
state of a hidden Markov process. That is, we assume that identical investors cannot
observe the drift rate, as well as the volatility of course, of the dividend process, but
they have to infer it from the observation of past dividends. We call this model a Black-
Scholes model with Markov switching, or a hidden Markov model in brief. Related works
in this type have been studied and documented in the literatures, for instance, Detemple
(1991) and David (1997) encompass the regime switching properties of the drift to the
classical Cox-Ingersoll-Ross model, while Veronesi (1999) studies the Gaussian diffusion
model. These papers focus on the issue of modeling stock returns and their empirical
phenomena. They also study investors’ optimal portfolio allocation and show that equi-
librium marketwise excess returns display changing volatility, negative skewness, and
negative correlation with future volatility.
2
For the concern of option pricing in Markov volatility models. Di Masi, Kabanov
and Runggaldier (1994) considers the problem of hedging an European call option for
a diffusion model where drift and volatility are function of a two state Markov jump
process. Guo (2001) also considers the same model and gives a closed-form formula for
the European call option. However, her formula is not appropriate stated and has a
mistake. The contribution of this paper is to give a closed-form formula in the case of
a two state hidden Markov model, and to propose an approximation formula for the
arbitrage-free price of the European call option in general case. The idea of the approxi-
mation formula is that we replace the distribution of occupation time for a given state by
its corresponding stationary distribution multiplies the whole time period. Numerical
methods, such as Monte-Carlo simulation and Markovian tree, to compute European
call option price are presented for comparison. Our numerical results show that the
approximation formula provide an efficient and reliable implementation tool for option
pricing.
This article is organized as follows. In Section 2, we first define the Black-Scholes
model with Markov switching that capture the phenomenon of business cycle in stock
fluctuations. Then, we provide a closed-form formula for the price of the European call
option, and in particular, we give an explicit analytic formula for option price when the
underlying Markov process has two states. In Section 3, we propose an approximation
formula for option price in a finite state hidden Markov model. To illustrate the perfor-
mance of the approximation formula, in Section 4, we presents numerical and simulation
results for comparison. These outcomes are reported in terms of Monte-Carlo simulation
and Markovian tree, respectively. Implied volatility and implied volatility surface are
also given to support our results. Section 5 is conclusions. The derivation of the option
price formula is in the Appendix.
2. The hidden Markov model and option price
We incorporate the existence of business cycle by modeling the fluctuations of a single
stock price Xt, using an equation of the form
dXt = Xtµε(t)dt+Xtσε(t)dWt, (1)
where ε(t) is a stochastic process representing the state of business cycle, Wt is the
standard Wiener process which is independent of ε(t). For each state of ε(t), the drift
parameter µε(t) and volatility parameter σε(t) are known, and take different values when
ε(t) is in different state.
We also assume that the supply of the risky asset is fixed and normalized to 1. The
risky free asset is supplied and has an instantaneous rate of return equal to r. Note that
3
this assumption not only simplify the analysis, in particular for option pricing, but also
matches the empirical finding that the volatility of the risk-free rate is much lower than
the volatility of market returns.
Assume that ε(t) is a Markov process with a few states (maybe two or three states).
In financial economic, we know that business cycle can divide into two different states
called expansion and contraction. A growing economy is described as being in expansion.
In this state, let ε(t) = 0, µε(t) = µ0 and σε(t) = σ0. On the other hand, we can take
the value ε(t) = 1, µε(t) = µ1 and σε(t) = σ1 to represent the state in contraction. More
generally, one can use the state space Ω = 0, 1, · · · , N for ε(t) to model more complex
business cycle structures. In this section, for simplicity, we consider a two state hidden
Markov model for a single stock price Xt by using an equation of (1), where ε(t) is a
Markov process representing the state. Let
ε(t) =
0, when the business cycle in expansion,1, when the business cycle in contraction.
Assume σ0 6= σ1.
For the concern of the transition rate, let λi be the rate of leaving state i, and τi the
time of leaving state i. We assume that
P (τi > t) = e−λit, i = 0, 1.
Then the memoryless property of this process is plausible in that, from a practical
standpoint, the information flow be identified more easily otherwise.
It is conceivable that sometimes investors will try to manipulate their buying and
selling in such a way that the existence of such information is not detectable from the
change of volatilities, namely σ are identical, see Detemple (1991), David (1997), and
Veronesi (1999). The problem of detecting the state change of ε(t) when σ remain
unchanged appears to be hard to solve mathematically. It is plausible that change in
business cycle distribution, hence predictability, manifests itself in the diffusion coeffi-
cient in the form of both stochastic volatility and drift. If we assume that the σ are
distinct then it is no loss of generality to assume that ε(t) is actually observable, since
the local quadratic variation of Xt in any small interval to the left of t will yield σε(t)
exactly. (For details, see McKean, 1969.) Hence, even if Xt is not Markovian, the joint
distribution (Xt, ε(t)) is so.
Despite the success of the classical Black-Scholes model, some empirical phenom-
ena have received much attention recently. An important assumption in the Black-
Scholes model is that the underlying asset distribution is assumed to be Normal, and
the volatility is a fixed constant. However, empirical evidences suggest that it has clus-
ter phenomenon, leptokurtic and unsymmetrical feature. Due to the adjoin of the stock
4
fluctuations with a hidden Markov process (the drift and volatility parameters take dif-
ferent values depending on the state of this hidden Markov process) in the Black-Scholes
exponential Brownian motion model, David (1997) shows that model (1) displays the
asymmetric leptokurtic features, negative skewness, and negative correlation with future
volatility. Veronesi (1999) shows that model (1) is better than the Black-Scholes model
to explain features of stock returns, including volatility clustering, leverage effects, ex-
cess volatility and time-varying expected returns. They both show that model (1) can
be embedded into a rational expectation equilibrium framework. For the concern of the
empirical phenomenon on volatility smile, we will present implied volatility and implied
volatility surface in the end of Section 4.
To get the option price formula, we know model (1) is arbitrage-free but incomplete
(cf. Harrison and Pliska (1981), Harrison and Kreps (1979)). One way to treat this
situation can be found in Follmer and Sondermann (1986), and Schweizer (1991), based
on the idea of hedging under a mean-variance criterion. Here, we follow the method by
D. Duffie to complete the market: at each time t, there is a market for a security that
pays one unit of account (say, a dollar) at the next time τ(t) = infu > t|ε(u) 6= ε(t)that the Markov chain ε(t) changes state. One can think of this as an insurance contract
that compensates its holder for any losses that occur when the next state change occurs.
Of course, if one wants to hedge a given deterministic loss C at the next state change,
one holds C of the current change-of-state (COS) contracts. For the detail, see Guo
(2001). It can be shown that the absence of arbitrage is effectively the same as the
existence of a probability Q, equivalent to P , under which the price of any derivative is
the expected discounted value of its future cash flow.
The same exercise applied to the underlying risky-asset implies that its price process
X must have the formdXt
Xt= (r − dε(t))dt+ σε(t)dW
Qt , (2)
where WQt is the standard Brownian motion under the risk-neutral probability Q, and
dε(t) = r − µε(t). Note that dε(t) is the cost of change of state at the time t. When the
state change at time t, the drift is different from the riskless interest rate r by d0 − d1,
and the arbitrage opportunity emerges at this moment.
Denote Ti as the total time between 0 and T during which ε(t) = 0, starting from
state i for i = 0, 1. Let fi(t, T ) be the probability distribution function of Ti.
Theorem 1 Under hidden Markov model (1). Given Equation (2), COS, and riskless
interest rate r, the arbitrage free price of a European call option with expiration date T
and strike price K is
Vi(T,K, r) = E[e−rT (XT −K)+|ε(0) = i]
5
= e−rT∫ ∞
0
∫ T
0
y
y +Kρ(ln(y +K), m(t), v(t))fi(t, T )dtdy, (3)
where ρ(x,m(t), v(t)) is the normal density function with expectation m(t) and variance
v(t),
m(t) = ln(X0) + (d1 − d0 −1
2(σ2
0 − σ21))t+ (r − d1 −
1
2σ2
1)T,
v(t) = (σ20 − σ2
1)t+ σ21T,
and
ρ(x,m(t), v(t)) =1
√
2πv(t)exp
− (x−m(t))2
2v(t)
,
f0(t, T ) = e−λ0T δ0(T − t) + e−λ1(T−t)−λ0t[λ0I0(2(λ0λ1t(T − t))1/2)
+(λ0λ1t
T − t)1/2I1(2(λ0λ1t(T − t))1/2)], (4)
f1(t, T ) = e−λ1T δ0(T − t) + e−λ1(T−t)−λ0t[λ1I0(2(λ0λ1t(T − t))1/2)
+(λ0λ1(T − t)
t)1/2I1(2(λ0λ1t(T − t))1/2)], (5)
where I0 and I1 are modified Bessel functions such that
Ia(z) = (z
2)a
∞∑
k=0
(z/2)2k
k!Γ(k + a+ 1)!.
Remarks: 1. When µ0 = µ1, σ0 = σ1, m(t) and v(t) are independent of t and
Equation (3) reduces to the classical Black-Scholes formula for European call options.
2. By using the properties of order statistics, Di Masi, Kabanov and Runggaldier
(1994) obtains the distribution of the Kac process, which is a key step for option pricing
in a two state hidden Markov model. In this paper, we apply the method of Laplace
transform to get the distribution. The proof of Theorem 1 will be given in the Appendix.
3. Option pricing under a finite state hidden Markov
model
In this section, we extend our results to a finite state hidden Markov model. When
the number of states is two, Equation (3) gives a closed-form formula of the European
call option price, and Equations (4) and (5) provide the explicit form to compute the
distributions of the occupation times. Note that the idea of duality to compute fi(t, T )
for i = 0, 1 in a two state hidden Markov process can not be applied to general states,
and the computation of the probability distribution function fi(t, T ) with three or more
6
states becomes complicated and infeasible. However, detailed observation of Equation
(3) reveals that there are two important features in the option price formula. First, it
depends only the occupation time Ti, but not depend on the sample path of visiting each
particular state. Secondly, Equation (3) is valid in no respect to the number of states.
Therefore, instead of direct computation of fi(t, T ) as those in Equations (4) and (5),
we can approximate Ti as follows: Note that Ti/T → πi in probability as T → ∞ for
any finite state ergodic Markov process, hence we can give an approximation formula by
simply substituting Ti in Equation (3) by πiT . Since the expiration time T for a market
option is half to one year, the approximation should be accurate enough in such time
period.
In the case of a two state hidden Markov model, we also have the following empirical
evidence. Note that in Equations (3) to (5), there are two different states, 0 and 1, and
the option prices depend on the initial state. To the extremely case, we may consider the
situation to which the state is the same in whole time period. That is, the price given by
the classical Black-Scholes models at state 0 (or at state 1), respectively. For example, in
Figure 1, the dash lines represent the option prices given by the classical Black-Scholes
model for each given state, and the real lines represent the prices by Equation (3). Note
that the option prices by Equation (3) are within the two classical Black-Scholes models
and converge to a fixed option price when T is large enough. This empirical phenomenon
also provide the rational of the approximation formula for large enough time period.
Figure 1: Option pricing with Black-Scholes model and two state hidden Markov model.The parameter values: X0 = 100, K = 110, λ0 = λ1 = 10, d0 = d1 = 0, r = 0.1, σ0 =0.2, σ1 = 0.3.
Next, we will explain how to compute the stationary distribution π as follows: Let
ε(t), t ≥ 0 be the underlying continuous time Markov process in (1). Denote the
7
transition probability from state i to state j in the time period t as
pt(i, j) = P (ε(t) = j|ε(0) = i).
Let
q(i, j) = limh→0
ph(i, j)
h
be the jump rates for i 6= j. We have the stationary distribution π by solving the balance
equation πQ = 0, where Q is the matrix of transition rates
Q(i, j) =
q(i, j) j 6= i,−λi j = i,
where λi =∑
j 6=i q(i, j) is the total rate of transitions out of state i.
As the expiration date T tends to infinity, Equations (4) and (5) become
f0(t, T ) = f1(t, T ) =
0, when t = T,1, when t 6= T.
(6)
Therefore the approximation formula of Equation (3) in Theorem 1 can be written as
Vi (7)
= e−rT∫ ∞
0
y
y +K(ρ(ln(y +K), m(π0T ), v(π0T ))(1− e−λ0T δ0(i− 0)− e−λ1T δ0(1− i))
+ρ(ln(y +K), m(T ), v(T ))e−λ0T δ0(i− 0)) + ρ(ln(y +K), m(0), v(0))e−λ1T δ0(1− i))dy.
Figure 2 displays option prices under the same parameters. The dash lines represent
the prices given by the approximation formula (7), and the real lines represent the prices
given by Equation (3) under different initial state. We note that two dash lines are very
close to the two real lines, and they are within the two real lines. Hence they will
converge to the fixed option price faster than the real lines. The reason to describle this
phenomenon is simply that we use the stationary distribution π in (7).
By using the same idea as that in (7), we have the approximation formula for Euro-
pean call option in hidden Markov model ε(t) with finite state space Ω = 0, 1, · · · , Nas
Vi = e−rT∫ ∞
0
y
y +K(ρ(ln(y +K), m(t0, t1, · · · , tN), v(t0, t1, · · · , tN))
(1− e−λ0T δ0(0− i)− · · · − e−λN T δ0(N − i))
+ρ(ln(y +K), m(T, 0, · · · , 0), v(T, 0, · · · , 0))e−λ0T δ0(0− i) + · · ·+ρ(ln(y +K), m(0, 0, · · · , T ), v(0, 0, · · · , T ))e−λN T δ0(N − i))dy, (8)
8
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Figure 2: Option pricing for a two states hidden Markov model. The parameter values:X0 = 100, K = 110, λ0 = λ1 = 1, d0 = d1 = 0, r = 0.1, σ0 = 0.2, σ1 = 0.3.
where ti = πiT , i = 0, · · · , N and
m(t0, t1, · · · , tN) = ln(X0) +N
∑
i=0
(r − di −1
2σ2
i )ti,
v(t0, t1, · · · , tN) =N
∑
i=0
σ2i ti.
In practice, a three state hidden Markov chain is good enough to capture the empir-
ical phenomena of most financial data series, and the computational method is the same
for each finite state Markov process. Hence, we only consider the case of three state
hidden Markov model in our simulation study. Figure 3 displays option prices under the
same parameters. Here, the dash lines represent the prices given by the approximation
formula, and the real lines represent the prices given by Black-Scholes formula under dif-
ferent initial state. In principle, the number of states exceed three can be implemented
in the same matter, but to discriminate and find out the matrix of transition rates Q in
some time interval might be computational demanding.
9
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3
4
5
6
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Figure 3: Option pricing for a three states hidden Markov model. The parameter values:X0 = 100, K = 110, λ0 = λ1 = λ2 = 10, d0 = d1 = d2 = 0, r = 0.1, σ0 = 0.1, σ1 =0.2, σ2 = 0.3.
4. Numerical performance
4.1 Design of the simulation
Numerical performance of the approximation formula (8) for option pricing is presented
in this section. We compare European call option prices obtained by Monte-Carlo sim-
ulation, Markovian tree method and the approximation formula. Two state and three
state hidden Markov models are considered in the simulation study, respectively. The
performance of this comparison is based on the “difference” of the option prices. The
definition of difference will be given later. We use the exact analytic formula as the
benchmark in the case of a two state hidden Markov model, and the Monte-Carlo sim-
ulation result as the benchmark in the case of a three state hidden Markov model. The
results are shown in Tables 1 to 10, in which four factors are used:
(1) Low and high volatilities.
(2) Different transition rates in the case of a three state hidden Markov model.
(3) Different strike prices. Specifically, we consider the cases of in the money, at the
money and out of the money.
(4) Different expiration date; T = 0.1, 0.2, 0.5, 1, 2, 3, with unit of year.
Since model (1) is a continuous time model, we discretize this continuous time model
in the vein of Cox, Ross, and Rubinstein (1979). A corresponding two state hidden
Markov model based on Cox, Ross and Rubinstein (1979) can be found in David (1997).
It is worth pointing out that this methodology applies also to the general case where
the hidden Markov process ε(t) takes more than three states, i.e., the state space can
10
be Ω = 0, 1, · · · , N, when, for example, more complex information patterns can be
imposed.
We can apply Equation (2) for Monte-Carlo simulation of the option price. First,
rewrite Equation (2) in the discrete time form
Xn+h = Xne(r−
1
2σ2
εn− dεn
)h+ σεn
√hη, (9)
where η is simulated from the standard normal random variable N(0, 1). Second, con-
sider the random variable εn with transition probability of changing state by (δi,j +
(−1)δi,je−q(i,i)h)(|q(i, j)/λi|). Repeat the steps for M times, where M is large enough to
guarantee the convergence of the simulation.
To illustrate the idea of discretization in Markovian tree method, we first construct
the structure of a two state hidden Markov model as follows: Suppose the time interval
[0, t] is divided into n sub-intervals such that t = nh. Let X = (Xk, k ≥ 0) and Xk is a
price at time kh. Define
Xεk
k = (Xk, εk) = (X(kh), ε(kh)).
Let ηi,jn be independent and identically distributed (i.i.d.) random variables, taking
values uj with probability pj(δi,1−j + (−1)δi,1−je−λih), and 1/uj with probability (1 −pj)(δi,1−j + (−1)δi,1−je−λih), i, j = 0, 1, respectively, where
ui = eσi
√h, pi =
µih + σi
√h− 0.5σ2
i h
2σi
√h
.
Then, we have the following recurrence relation,
(Xn, εn) = ηε(n),ε(n−1)n (Xn−1, εn−1). (10)
By the memoryless property of τi, (Xεnn , n ≥ 0) is a Markov chain. The Markov chainXεn
n
with initial state X0 = x is a random walk on the set Ex = xur|r = σ0n0+σ1n1, n0, n1 ∈Z, u = e
√h.
In general, we can apply the same idea to the hidden Markov model with state
space Ω = 0, 1, · · · , N. Denote Q as the matrix of transition rates. Note that the
ηi,jn in Equation (9) are independent and identically distribute random variables, taking
values uj with probability pj(δi,j+(−1)δi,je−q(i,i)h)(|q(i, j)/λi|), and 1/uj with probability
(1 − pj)(δi,j + (−1)δi,je−q(i,i)h)(|q(i, j)/λi|), where i represents the current state and j
represents the oncoming state in one sub-interval. Similarly, the Markov chain X εnn with
initial state X0 = x is a random walk on the set Ex = xur|r = σ0n0 + σ1n1 + · · · +σNnN , n0, n1, · · · , nN ∈ Z, u = e
√h.
11
Here we consider n, the number of sub-interval, to be 30, since our simulation
shows that this number is large enough to provide accurate results. The Monte-Carlo
replication size for both Monte-Carlo simulation and Markovian tree method is B =
50, 000, 000. Computations were performed using Visual Basic programs on the personal
computer system with a Pentium 4 CPU, 1.6G and 256MB of RAM, of the Institute
of Statistical Science, Academia Sinica, Taipei, Taiwan, R.O.C.. The pseudo-random
numbers were generated by using IMSL routines. All tests were compared on the basis of
the same random numbers, samples of different size were nested. The reported running
time is given the CPU time in seconds.
4.2 Simulation results
We will use the following abbreviations in Tables 1 to 10. B-S i refers to the clas-
sical Black-Scholes formula with state i = 0, 1; Vi refers to the exact price of a two
state hidden Markov model; Vi refers to the approximation price of a two states hidden
Markov model; sim i and tree i refer to Monte-Carlo simulation for Equation (2) and
Markovian tree method, respectively.
We first consider the case of a two state hidden Markov model, with the same
transition rates. Tables 1 and 2 report the numerical and simulation results according
to low and high volatilities. The other parameters are the same. Note from Tables 1 and
2 that the numerical values given by the Monte-Carlo simulation and the Markov tree
method are very close to the same values for various time period. In the case of short
time period, the performance of the approximation formula is better than Monte-Carlo
simulation and Markovian tree method, in the sense to compare the results obtained
by the exact formula. And the computational time is reduced significantly. Therefore,
the approximation formula (7) gives a rather promising result in the case of a two state
hidden Markov model.
Table 1: The parameters are: X0 = 100, K = 90, λ0 = λ1 = 1, d0 = d1 = 0, r =0.1, σ0 = 0.2, σ1 = 0.3, n = 30.
T (year) B-S 0 B-S 1 V0 V1 sim 0 sim 1 tree 0 tree 1 V0 V1
0.1 10.975 11.381 10.993 11.360 10.986 11.346 10.910 11.314 10.993 11.3600.2 12.088 12.968 12.164 12.889 12.151 12.883 12.257 12.698 12.166 12.8860.5 15.288 17.034 15.614 16.718 15.592 16.719 15.497 16.960 15.639 16.6981.0 19.988 22.510 20.721 21.812 20.681 21.796 20.740 21.747 20.811 21.7392.0 28.037 31.281 29.287 30.085 29.256 30.051 29.166 30.060 29.478 29.9173.0 34.996 38.504 36.476 37.061 36.426 37.057 36.219 36.877 36.689 36.863
runningtime(sec) < 10−5 < 10−5 1.51 1.45 240.55 251.32 140.24 144.23 0.061 0.063
12
Table 2: The parameters are: X0 = 100, K = 90, λ0 = λ1 = 1, d0 = d1 = 0, r =0.1, σ0 = 0.1, σ1 = 1.0, n = 30.
T (year) B-S 0 B-S 1 V0 V1 sim 0 sim 1 tree 0 tree 1 V0 V1
0.1 17.007 18.100 17.059 18.050 17.048 18.045 16.842 18.050 17.060 18.0500.2 21.612 23.180 21.752 23.045 21.731 23.050 21.860 23.135 21.757 23.0420.5 31.027 33.429 31.478 32.996 31.468 32.994 31.765 32.995 31.513 32.9691.0 41.547 44.688 42.460 43.818 42.455 43.760 42.662 44.106 42.570 43.7262.0 55.470 59.236 56.934 57.858 56.945 57.871 56.817 57.616 57.158 57.6683.0 64.952 68.853 66.638 67.286 66.671 67.415 66.080 66.550 66.884 67.078
runningtime(sec) < 10−5 < 10−5 1.44 1.52 249.46 252.55 100.23 101.17 0.060 0.063
Let Q1 be the matrix of transition rate
Q1 =
−1 1 01/2 −1 1/20 1 −1
,
and let Q2 be the matrix of transition rate
Q2 =
−1 1 05/2 −5 5/20 10 −10
.
The results of a three state hidden Markov model are recorded in Tables 3 to 6. Note
that the matrix of transition rates Q is the same for a two state hidden Markov model;
while the matrix of transition rates Q may be different in a three or more state hidden
Markov models. Since the difference of the matrix of transition rates in a three or more
states hidden Markov model may also play an important role for option pricing, we will
present our simulation results accordingly. Tables 3 and 4 displays the option prices for
a three states hidden Markov model, with matrix of transition rate Q1 (with the same
transition rate), the same parameters, but different volatilities. Tables 5 and 6 displays
the option prices for a three states hidden Markov model, with matrix of transition rate
Q2 (with different transition rate), the same parameters, but different volatilities.
Note from Tables 3 to 6 that we have three numerical consequences. First, the
option prices obtained by the approximation formula (8) are less sensitive to the initial
state than those obtained by Monte-Carlo simulation and Markovian tree method. The
rational behind this phenomenon is that we use the stationary distribution π, instead of
exactly probability distribution, in the approximation formula (8). Secondly, the option
prices in Tables 5 and 6 with bigger λi values, i.e. λ1 and λ2, are closer to the values given
by the Monte-Carlo simulations than the price with λ0. This result is reasonable, since
13
Table 3: The parameters are: X0 = 100, K = 90, λ0 = λ1 = λ2 = 1, d0 = d1 = d2 =0, r = 0.1, σ0 = 0.1, σ1 = 0.2, σ2 = 0.3, n = 30.
T (year) sim 0 sim 1 sim 2 tree 0 tree 1 tree 2 V0 V1 V2
0.1 10.898 10.973 11.346 10.904 10.988 11.309 10.906 10.978 11.3450.2 11.806 12.078 12.886 11.840 12.199 12.694 11.854 12.103 12.8230.5 14.572 15.282 16.721 14.639 15.084 17.050 14.833 15.358 16.4171.0 19.061 19.978 21.761 18.987 20.167 21.744 19.656 20.155 21.0832.0 27.201 28.028 29.736 27.067 27.409 29.776 28.103 28.327 28.7663.0 34.333 34.953 36.426 34.057 34.667 36.386 35.250 35.331 35.506
runningtime(sec) 269.25 272.84 276.63 171.25 175.22 178.32 0.061 0.062 0.061
Table 4: The parameters are: X0 = 100, K = 90, λ0 = λ1 = λ2 = 1, d0 = d1 = d2 =0, r = 0.1, σ0 = 0.8, σ1 = 0.9, σ2 = 1.0, n = 30.
T (year) sim 0 sim 1 sim 2 tree 0 tree 1 tree 2 V0 V1 V2
0.1 15.984 16.993 18.021 15.683 16.797 18.025 16.036 17.010 17.9990.2 20.170 21.584 23.062 20.105 21.725 23.145 20.344 21.620 22.9040.5 29.118 31.025 32.906 29.027 31.345 32.939 29.590 31.053 32.5101.0 39.464 41.555 43.674 39.726 42.048 43.887 40.427 41.602 42.7582.0 53.760 55.456 57.202 53.621 56.074 57.049 55.033 55.563 56.0723.0 63.782 64.822 66.494 63.304 64.395 65.646 64.851 65.058 65.252
runningtime(sec) 267.11 282.29 288.13 164.54 170.91 178.30 0.062 0.063 0.062
the one with bigger λi means that the Markov process will leave state i in a short period
of time, and hence the effect of such initial state is smaller. Thirdly, the option prices
in a longer time period, for instance a 3-year contract in Tables 3 to 6, have different
outcome compared to the results in a smaller time period. Note that the option price
converges to the true value by the λi’s order, and the prices with two larger λ values will
emerge faster than the third one. Therefore, the simulation region for option prices can
not cover the whole approximation prices. In other words, the option prices with many
initial states will merge to a fewer one in a reasonable time period. Hence, one does
not need too many states for modeling stock returns from the perspective of computing
option prices, and it is reasonable to replace a finite state hidden Markov model by a
two or three state hidden Markov model.
For the concern of the effect of the time to maturity T , in Tables 7 to 10, we provide
14
Table 5: The parameters are: X0 = 100, K = 90, λ0 = 1, λ1 = 5, λ2 = 10, d0 = d1 =d2 = 0, r = 0.1, σ0 = 0.1, σ1 = 0.2, σ2 = 0.3, n = 30.
T (year) sim 0 sim 1 sim 2 tree 0 tree 1 tree 2 V0 V1 V2
0.1 10.901 10.937 11.175 10.906 10.925 11.191 10.896 10.949 11.0820.2 10.768 12.083 12.669 11.845 12.178 12.473 11.799 11.947 12.0150.5 14.740 14.798 15.261 14.636 15.187 15.632 14.541 14.771 14.7411.0 18.862 19.171 19.389 18.904 19.508 19.873 18.949 19.139 19.1342.0 26.840 27.131 27.281 26.777 27.260 27.542 26.891 26.971 26.9713.0 33.745 34.059 34.168 33.692 34.047 34.245 33.884 33.910 33.910
runningtime(sec) 263.01 264.90 263.45 172.78 171.44 174.20 0.059 0.062 0.062
Table 6: The parameters are: X0 = 100, K = 90, λ0 = 1, λ1 = 5, λ2 = 10, d0 = d1 =d2 = 0, r = 0.1, σ0 = 0.8, σ1 = 0.9, σ2 = 1.0, n = 30.
T (year) sim 0 sim 1 sim 2 tree 0 tree 1 tree 2 V0 V1 V2
0.1 15.979 16.993 17.671 15.689 16.790 17.653 15.974 16.761 17.0140.2 20.186 21.564 22.231 20.111 21.599 22.366 20.172 21.042 21.0440.5 29.109 31.008 31.278 28.953 30.714 31.353 29.016 29.748 29.6591.0 39.518 41.553 41.434 39.541 40.607 41.218 39.209 39.719 39.7072.0 53.729 55.569 55.113 52.986 53.893 54.170 53.000 53.226 53.2263.0 63.525 64.918 64.695 62.244 62.905 63.142 62.498 62.586 62.586
runningtime(sec) 261.34 265.69 267.45 171.34 182.22 183.20 0.061 0.061 0.062
approximate European call option prices for various T in a hidden Markov model with
three states. Tables 7 and 9 reports the case of low volatilities, and Tables 8 and 10
reports the case of high volatilities. The matrix of transition rates in Tables 7 and 8 is
Q2, and the matrix of transition rate in Tables 9 and 10 is Q3, which is defined as
Q3 =
−1 1 00 −5 510 0 −10
.
For each strike price K, we consider three different strike-to-stock price ratios K/X0.
They are 1.1, 1.0 and 0.9. Note that in Tables 7 to 10, the first row in each panel is
the approximate analytical prices with different initial state, whereas the second and
forth rows report Monte-Carlo tree prices and their standard deviation, respectively.
The number in the third row is error estimate between the approximation value and
15
Monte-Carlo simulation. Specifically, in Tables 7 to 10, Vi refers the approximate price
for a three state hidden Markov model with state i = 0, 1, 2.; tree i refers to Monte-Carlo
method for tree with Q2; difference is the ratio between approximate price and tree price,
i.e. difference:= (Vi − tree i)/tree i; and Std is the standard deviation of Monte-Carlo
tree prices.
Note that the option prices Vi for i = 0, 1, 2 in Table 7 is higher than the corre-
sponding option prices in Table 9. The reason can be described as follows: Although
the parameters in both tables are the same, the matrix of transition rates are different.
We use Q2 in Table 7; while Q3 in Table 9. By the definition of Q2 and Q3, it is easy to
see that the occupation time of state 0 in Q3 is longer than that of Q2, and state 0 is
the worst state. By using the same argument, we have the option prices Vi for i = 0, 1, 2
in Table 8 is higher than the corresponding option prices in Table 10. Note from Tables
1 to 10, the convergence rate of option price in approximation formula depends on the
value of λ and difference of prices from initial state.
Furthermore, we observe that the option price obtained by the approximation for-
mula (8) depends on the strike price K. It is more accurate in the cases of in the money
and at the money; while it is less accurate in the case of out of the money for each time
period. A heuristic argument based on Equation (3) can be described as follows:
We consider the case of a two state hidden Markov model for simplicity. Let K(1)
and K(2) be two given strike prices with K(1) < K(2). Without loss of generality, we
may assume σ1 > σ0; that is the volatility at state 1 is bigger than the volatility at state
0. Then from Equation (3), we have V1(T,K, r) > V0(T,K, r). It is also known from
Equation (3) again that Vi(T,K(1), r) > Vi(T,K(2), r), for i = 0, 1. Under the condition
V1(T,K(1), r)− V0(T,K(1), r)
V1(T,K(1), r)<V1(T,K(2), r)− V0(T,K(2), r)
V1(T,K(2), r), (11)
which is correct for T is not too large. Note also that |Vi(T,K(1), r)− Vi(T,K(1), r)| ≈V1(T,K(1), r) − V0(T,K(1), r), and |Vi(T,K(2), r) − Vi(T,K(2), r)| ≈ V1(T,K(2), r)−V0(T,K(2), r) for i = 0, 1. Therefore,
|V1(T,K(1), r)− V1(T,K(1), r)|V1(T,K(1), r)
<|V1(T,K(2), r)− V1(T,K(2), r)|
V1(T,K(2), r). (12)
Note that (12) is the definition of “difference” in Tables 7 to 10, and it is an increasing
function of K.
It is easy to see that under the condition (11), we also have
|V0(T,K(1), r)− V0(T,K(1), r)|V0(T,K(1), r)
<|V0(T,K(2), r)− V0(T,K(2), r)|
V0(T,K(2), r). (13)
16
Tab
le7:
The
par
amet
ers
are:
X0
=10
0,λ
0=
1,λ
1=
5,λ
2=
10,d
0=d
1=d
2=
0,r
=0.
1,σ
0=
0.1,
σ1
=0.
2,σ
2=
0.3,n
=30
. T=
0.1
T=
0.2
T=
0.5
T=
1.0
K/X
01.
11.
00.
91.
11.
00.
91.
11.
00.
91.
11.
00.
9
V0
0.01
061.
8759
10.8
960.
1600
3.08
8211
.799
1.66
406.
3350
14.5
415.
5403
11.2
2118
.949
tree
00.
0128
1.88
2910
.906
0.16
413.
0783
11.8
451.
6462
6.31
8614
.636
5.44
2111
.159
18.9
04diff
eren
ce-0
.172
-0.0
04-0
.001
-0.0
250.
0032
-0.0
040.
0108
0.00
26-0
.006
0.01
800.
0056
0.00
24Std
0.00
010.
0010
0.00
160.
0004
0.00
160.
0022
0.00
180.
0031
0.00
370.
0040
0.00
510.
0058
V1
0.19
482.
8094
10.9
490.
6870
4.09
9511
.947
2.57
537.
1804
14.7
716.
3232
11.7
6319
.139
tree
10.
2884
3.01
4010
.925
1.01
734.
5460
12.1
783.
3714
7.95
8215
.187
7.16
5612
.484
19.5
08diff
eren
ce-0
.325
-0.0
680.
0022
-0.3
25-0
.098
-0.0
19-0
.236
-0.0
98-0
.027
-0.1
18-0
.058
-0.0
19Std
0.00
060.
0018
0.00
280.
0013
0.00
270.
0036
0.00
310.
0044
0.00
540.
0053
0.00
650.
0074
V2
0.41
553.
1278
11.0
820.
7567
4.14
6512
.015
2.49
817.
1080
14.7
416.
3108
11.7
5319
.134
tree
20.
7362
3.84
1111
.191
1.70
755.
4005
12.4
734.
2281
8.76
9315
.632
8.01
1113
.153
19.8
73diff
eren
ce-0
.436
-0.1
86-0
.010
-0.5
57-0
.232
-0.0
37-0
.409
-0.1
89-0
.057
-0.2
12-0
.106
-0.0
37Std
0.00
110.
0025
0.00
350.
0020
0.00
340.
0045
0.00
380.
0051
0.00
610.
0059
0.00
710.
0081
17
Tab
le8:
The
par
amet
ers
are:
X0
=10
0,λ
0=
1,λ
1=
5,λ
2=
10,d
0=d
1=d
2=
0,r
=0.
1,σ
0=
0.8,
σ1
=0.
9,σ
2=
1.0,n
=30
. T=
0.1
T=
0.2
T=
0.5
T=
1.0
K/X
01.
11.
00.
91.
11.
00.
91.
11.
00.
91.
11.
00.
9
V0
6.69
9410
.570
15.9
7411
.300
15.2
0120
.172
20.9
9824
.684
29.0
1632
.222
35.5
0639
.209
tree
06.
6165
10.4
8615
.689
11.4
0815
.064
20.1
1121
.188
24.4
0328
.953
32.4
6735
.419
39.5
41diff
eren
ce0.
0125
0.00
800.
0182
-0.0
090.
0091
0.00
30-0
.009
0.01
150.
0022
-0.0
070.
0025
-0.0
08Std
0.00
640.
0079
0.00
940.
0106
0.01
210.
0134
0.02
090.
0222
0.02
350.
0366
0.03
770.
0388
V1
7.60
7511
.481
16.7
6112
.295
16.1
6821
.042
21.8
3425
.481
29.7
4832
.804
36.0
5739
.719
tree
17.
7310
11.6
7816
.790
12.8
9816
.617
21.5
9922
.944
26.2
7930
.714
33.8
5636
.866
40.6
07diff
eren
ce-0
.016
-0.0
16-0
.001
-0.0
46-0
.027
-0.0
25-0
.048
-0.0
30-0
.031
-0.0
31-0
.021
-0.0
21Std
0.00
750.
0089
0.01
030.
0122
0.01
360.
0148
0.02
330.
0246
0.02
580.
0397
0.04
080.
0418
V2
7.89
6011
.766
17.0
1412
.297
16.1
6921
.044
21.7
3325
.385
29.6
5932
.790
36.0
4439
.707
tree
28.
6757
12.4
9417
.653
13.8
2417
.506
22.3
6623
.687
27.0
6931
.353
34.6
3237
.508
41.2
18diff
eren
ce-0
.089
-0.0
58-0
.036
-0.1
10-0
.076
-0.0
59-0
.082
-0.0
62-0
.054
-0.0
53-0
.039
-0.0
36Std
0.00
830.
0097
0.01
100.
0131
0.01
450.
0157
0.02
440.
0257
0.02
690.
0411
0.04
210.
0432
18
Tab
le9:
The
par
amet
ers
are:
X0
=10
0,λ
0=
1,λ
1=
5,λ
2=
10,d
0=d
1=d
2=
0,r
=0.
1,σ
0=
0.1,
σ1
=0.
2,σ
2=
0.3,n
=30
. T=
0.1
T=
0.2
T=
0.5
T=
1.0
K/X
01.
11.
00.
91.
11.
00.
91.
11.
00.
91.
11.
00.
9
V0
0.00
851.
8658
10.8
960.
1448
3.06
1811
.795
1.57
486.
2516
14.5
115.
3237
11.0
5618
.873
tree
00.
0148
1.88
9410
.904
0.18
053.
1023
11.8
491.
7180
6.38
0314
.663
5.50
4511
.196
18.9
09diff
eren
ce-0
.426
-0.0
12-0
.001
-0.1
98-0
.013
-0.0
05-0
.083
-0.0
20-0
.010
-0.0
33-0
.013
-0.0
02Std
0.00
010.
0010
0.00
160.
0004
0.00
160.
0022
0.00
180.
0031
0.00
380.
0040
0.00
510.
0058
V1
0.18
632.
7678
10.9
470.
6339
4.00
7411
.932
2.36
716.
9859
14.7
025.
9828
11.5
0419
.021
tree
10.
3497
3.18
1710
.959
1.19
854.
7978
12.2
493.
5432
8.11
0215
.230
7.10
3812
.423
19.4
18diff
eren
ce-0
.467
-0.1
30-0
.001
-0.4
71-0
.165
-0.0
26-0
.332
-0.1
39-0
.035
-0.1
58-0
.074
-0.0
20Std
0.00
060.
0020
0.00
290.
0015
0.00
290.
0039
0.00
320.
0045
0.00
550.
0051
0.00
630.
0073
V2
0.40
173.
0609
11.0
790.
6841
4.02
0511
.994
2.27
296.
8976
14.6
665.
9682
11.4
9219
.015
tree
20.
5988
3.53
1411
.127
1.21
904.
7581
12.2
233.
0319
7.60
7215
.118
6.61
7711
.989
19.2
48diff
eren
ce-0
.329
-0.1
33-0
.004
-0.4
39-0
.155
-0.0
19-0
.250
-0.0
93-0
.030
-0.0
98-0
.041
-0.0
12Std
0.00
100.
0023
0.00
330.
0016
0.00
300.
0039
0.00
290.
0042
0.00
510.
0048
0.00
600.
0068
19
Tab
le10
:T
he
par
amet
ers
are:X
0=
100,
λ0
=1,
λ1
=5,
λ2
=10,d
0=d
1=d
2=
0,r
=0.
1,σ
0=
0.8,
σ1
=0.
9,σ
2=
1.0,n
=30
. T=
0.1
T=
0.2
T=
0.5
T=
1.0
K/X
01.
11.
00.
91.
11.
00.
91.
11.
00.
91.
11.
00.
9
V0
6.68
8510
.559
15.9
6411
.270
15.1
7220
.146
20.8
9724
.589
28.9
2832
.009
35.3
0439
.022
tree
06.
6217
10.4
8715
.683
11.4
2815
.091
20.1
2721
.189
24.4
9329
.009
32.4
6635
.392
39.4
59diff
eren
ce0.
0101
0.00
690.
0179
-0.0
130.
0054
0.00
09-0
.013
0.00
39-0
.002
-0.0
14-0
.002
-0.0
11Std
0.00
640.
0079
0.00
940.
0106
0.01
210.
0134
0.02
100.
0223
0.02
360.
0365
0.03
770.
0387
V1
7.56
2411
.436
16.7
2212
.191
16.0
6720
.951
21.6
0025
.259
29.5
4332
.469
35.7
4039
.425
tree
17.
8958
11.8
5316
.950
13.1
5716
.858
21.7
8422
.882
26.4
0530
.697
33.6
5036
.707
40.3
99diff
eren
ce-0
.042
-0.0
35-0
.013
-0.0
73-0
.046
-0.0
38-0
.056
-0.0
43-0
.037
-0.0
35-0
.026
-0.0
24Std
0.00
770.
0091
0.01
040.
0124
0.01
380.
0150
0.02
330.
0247
0.02
580.
0392
0.04
020.
0412
V2
7.82
3611
.693
16.9
5212
.154
16.0
3020
.920
21.4
8025
.144
29.4
3832
.453
35.7
2539
.411
tree
28.
2894
12.1
4117
.328
12.9
1816
.743
21.5
9522
.257
25.8
1930
.159
33.1
0536
.277
40.0
32diff
eren
ce-0
.056
-0.0
36-0
.021
-0.0
59-0
.042
-0.0
31-0
.034
-0.0
26-0
.023
-0.0
19-0
.015
-0.0
15Std
0.00
790.
0093
0.01
070.
0123
0.01
370.
0149
0.02
250.
0237
0.02
490.
0380
0.03
920.
0402
20
¿From the results appeared in Tables 3 to 10, we note that the performance of ana-
lytic approximation is very promising, compare to the examined numerically outcomes
obtained by Monte-Carlo simulation and Markovian tree method. The advantage of the
analytical approximation is that it accelerates the computation time of the option prices
in hidden Markov models.
4.3 Volatility smiles and surfaces
If the Black-Scholes model is correct, then the implied volatility should be constant.
But it is widely recognized that the volatility has a “smile” feature. For instance, Figure
4 illustrates our notion of “volatility smile” that is presented in the Microsoft call option
with the Black-Scholes model and the hidden Markov model. Model (1) can reveal the
phenomena of volatility smile as shown in Figure 4.
We can show implied volatility against both maturity and strike in a three-dimensional
plot. That is we consider σ(X, t) as a function of X and t. One is shown in Figure 5 that
IBM call option with Black-Scholes model and hidden Markov model from 2002/3/15
with five expiry of nine months. There are call option traded with an expiry of five
months and strikes of 100, 105, 110, 115, 120, 125, 130, 135 and 140. This implied
surface represents the constant value of volatility that gives each traded option a theo-
retical value equal to the market value. We can see how the time dependence in implied
volatility could be turned into a volatility of the underlying that was time dependent.
In the case of σ(X, t) can be deduced from volatility surface at a specific time t∗, we
might call it the local volatility surface. This local volatility surface can be thought of
as the market’s view of the future value of volatility when the asset price is X at time t.
We should emphasize that the examples pressed in Figures 4 and 5 are not an
empirical test of the model (1), it is only an illustration to show that the model can
produce a close fit to the empirical phenomenon.
D=E FGFHI@JLKNM OGPGQ@RSPFGHTE RVUT E HFVQPGT KNW E T E W X
YZ [YZ [G\YZ ]YZ ]\YZ \
]\ \G\ ^G\ _\ `\a
b cdef gh ije klfef l m
no pSqSrsGtuv qVwu v tsVxrGu yNz v u v z
| ~| ~| ~| ~G| ~| | | G|
G
Figure 4: Implied volatility of Black-Scholes model and two-state hidden Markov model
21
-C+ " =
=- == " ==
Figure 5: Implied volatilities against expiry and strike price. The parameter values:X0 = 107, λ0 = λ1 = 1, d0 = d1 = 0, r = 0.0261.
5. Conclusions
A closed form formula for the Black-Scholes with Markov switching option pricing model
has been developed in this paper. In the case of a two state hidden Markov model, we
have an exact analytic formula; while in the case of a three or more state hidden Markov
model, we have an approximation formula. Numerical evaluation of the formula is
studied for both two state and three state cases. The performance of the approximation
was examined numerically using option prices obtained from Monte-Carlo simulation
and Markovian tree method. The pricing error, in terms of ratio, by the analytical
approximation is shown to be small except in extreme cases (out of the money).
An exact closed formula provides useful insight for the European option pricing in
the Black-Scholes model with Markov switching. It not only explains the effect of regime
switching for option pricing, but also gives the idea of an analytical approximation, in
which it accelerates the computation of the European option pricing in the Black-Scholes
model with Markov switching. It has a number of further applications, for instance, it
can be used to compute hedge ratios and implied hidden Markov models parameters,
i.e. calibrate the parameters by using implied volatility surface. The approximation can
facilitate empirical studies of the index options that are, in many cases, of European
style. This feature is worthy of further exploration and may have many implications in
other applications.
22
References
Anderson, T. G. (1996). Return volatility and trading volume: an information
flow interpretation of stochastic volatility. Journal of Finance, 51, 169-204.
Avellaneda, M., Levy, P., and Paras, A. (1995). Pricing and hedging derivative
securities in markets with uncertain volatilities. Applied Mathematical Finance, 2,
73-88.
Bittlingmayer, G. (1998). Output, stock volatility and political uncertainty in a
natural experiment: Germany, 1880-1940. Journal of Finance, 53, 2243-2257.
Black, F. and Scholes, M. (1973). The pricing of options and corporate liabilities.
Journal of Political Economy, 81, 637-654.
Cox, J., Ross, S., and Rubinstein, M. (1979). Option pricing, a simplified approach.
Journal of Financial Economics, 7, 229-263.
David, A. (1997). Fluctuating confidence in stock markets: Implications for re-
turns and volatility. Journal of Financial and Quantitative Analysis, 32, 427-462.
Detemple, J. B. (1991). Further results on asset pricing with incomplete informa-
tion. Journal of Economic Dynamics and Control, 15, 425-453.
Di Masi, G. B., Kabanov, Yu, M., and Runggaldier, W. J. (1994). Mean-variance
hedging of options on stocks with Markov volatility. Theory of Probability and Its
Applications, 39, 173-181.
Diebold, F. X., and Inoue, A. (2001). Long memory and regime switching. Journal
of Econometrics, 105, 131-159.
Duffie, D., and Huang, C. F. (1986). Multiperiod security markets with differential
information. Journal of Mathematical Economics, 15, 283-303.
Follmer, H. and Sondermann, D. (1986). Hedging of nonredundant contingent
claims. In: Hildenbrand and Mas-Colell, eds., Contributions to Mathematical
Economics, 205-223.
Grorud, A., and Pontier, M., (1998). Insider trading in a continuous time market
model. International Journal of Theoretical and Applied Finance, 1, 331-347.
Guilaume, D. M., Dacorogna, M., Dave, R., Muller, U., Olsen, R., and Pictet, P.
(1997). ¿From the bird’s eye to the microscope, a survey of stylized facts of the
intra-daily foreign exchange market. Finance and Stochastics, 1, 95-129.
Guo, X. (2001). Information and option pricing. Journal of Quantitative Finance,
1, 38-44.
23
Hamilton, J. D. (1988). Rational-Expectations econometric analysis of changes in
regime: an investigation of term structure of interest rates. Journal of Economic
Dynamics and Control, 12, 385-423.
Hamilton, J. D. (1989). A new approach to the economic analysis of nonstationary
time series and the business cycle. Econometrica, 57, 357-384.
Hamilton, J. D., and Susmel, R. (1994). Autoregressive conditional heteroscedas-
ticity and changes in regime. Journal of Econometrics, 64, 307-333.
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.
Hull, J. and White, A. (1987). The pricing of options on assets with stochastic
volatility. Journal of Finance, 2, 281-300.
Karatzas, I., and Pikovsky, I. (1996). Anticipative portfolio optimization. Ad-
vances in Applied Probability, 28, 1095-1122.
McKean, H. P. (1969). Stochastic Integrals. Academic Press, New York.
Ross, S. A. (1989). Information and volatility, the no-arbitrage martingale ap-
proach to timing and resolution irrelevancy. Journal of Finance, 44, 1-8.
Schweizer, M. (1991). Option hedging for semimartingales. Stochastic Processes
and their Applications, 37, 339-363.
So, M. K. P., Lam, K., and Li, W. K. (1998). A stochastic volatility model with
Markov switching. Journal of Business & Economic Statistics, 16, 244-253.
Stein, E. M., and Stein, C. J. (1991). Stock prices distribution with stochastic
volatility, an analytic approach. Review of Financial Studies, 4, 727-752.
Turner, C., Startz, R., and Nelson, C. (1989). A Markov model of heteroscendas-
ticity, risk, and learning in the stock market. Journal of Financial Economics, 25,
3-22.
Veronesi, P. (1999). Stock market overreaction to bad news in good times: a
rational expectations equilibrium model. Review of Financial Studies, 12, 5, 975-
1007.
Wiggins, J. B. (1987). Option values under stochastic volatility. Theory and
empirical evidence. Journal of Financial Economics, 19, 351-372.
Appendix
24
Proof of Theorem 1.
Since the arbitrage price of the European option is the discounted expected value of
Xt under the equivalent martingale measure Q, we have
Vi(T,K, r) = EQ[e−rT (XT −K)+|ε(0) = i].
Recalling that
Xt = X0 exp(∫ t
0(r − dε(s) −
1
2σ2
ε(s))ds+∫ t
0σε(s)dW
Qs ),
the key point is to calculate the instantaneous distribution of Xt. Let Yt = lnXt, then
Yt = Y0 +∫ t
0(r − dε(s) −
1
2σ2
ε(s))ds+∫ t
0σε(s)dW
Qs .
Denote Ti as the total time between 0 and T during which ε(t) = 0, starting from
state i, and we consider the probability distribution function fi(t, T ), which is defined
as the probability distribution function of Ti.
Vi(T,K, r) = E[e−rT (XT −K)+|ε(0) = i]= e−rTE[E[(XT −K)+|Ti]|ε(0) = i]= e−rTEi[E[(XT −K)+|Ti]|Fε(0) = i]
= e−rT∫ ∞
0
∫ T
0
y
y +Kρ(ln(y +K), m(t), v(t))fi(t, T )dtdy,
where ρ(x,m(t), v(t)) is the normal density function with expectation m(t) and variance
v(t),
m(t) = ln(X(0)) + (d1 − d0 −1
2(σ2
0 − σ21))t + (r − d1 −
1
2σ2
1)T,
v(t) = (σ20 − σ2
1)t+ σ21T,
and
ρ(x,m(t), v(t)) =1
√
2πv(t)exp(−(x−m(t))2
2v(t)).
Then
fi(t, T )dt = P (∫ T
0χ0(εs)ds ∈ dt),
where χ0 is the indicate function at state 0.
Let
ψi(r, T ) = E[e−r∫ T
0χ0(εs)ds|ε(0) = i]
=∫ ∞
0e−rtfi(t, T )dt
:= Lr(fi(·, T )),
then we haveψi(r, T ) = e−rT e−λiT δ0(i− 0) + e−λiT δ0(1− i)
+∫ T
0e−λiuλiψ1−i(T − u)e−ruδ0(i−0)du.
25
That is,
ψ0(r, T ) = e−rTe−λ0T +∫ T
0e−λ0uλ0ψ1(T − u)e−rudu,
ψ1(r, T ) = e−λ1T +∫ T
0e−λ1uλ1ψ0(T − u)du.
Taking Laplace transforms on both sides, and writing
Ls(ψi(r, ·)) = Ls[Lr(fi(·, T ))(r, ·)]=
∫ ∞
0e−sTψi(r, T )dT
:= ψi(r, s),
then
ψ0(r, s) =1
r + s+ λ0+
λ0
r + s+ λ0ψ1(r, s),
ψ1(r, s) =1
s+ λ1+
λ1
s + λ1ψ0(r, s).
Solving these equations, we get
ψ0(r, s) =s+ λ0 + λ1
s2 + sλ1 + sλ0 + rs+ rλ1,
ψ1(r, s) =r + s+ λ0 + λ1
s2 + sλ1 + sλ0 + rs+ rλ1
.
Taking r the inverse Laplace transform with represent w
L−1r (ψ0(r, s))(w, ·) = e
−sws+ λ0 + λ1
s+ λ1s+ λ0 + λ1
s+ λ1
,
L−1r (ψ1(r, s))(w, ·) = e
−sws+ λ0 + λ1
s+ λ1λ1(s+ λ0 + λ1)
(s+ λ1)2+δ0(w)
s+ λ1.
Again, tacking s the inverse Laplace transform with represent v
L−1s [L−1
r (ψ0(r, s))(w, ·)](·, v) = e−λ0vδ0(v − w) + e−λ1(v−w)−λ0w[λ0I0(2(λ0λ1w(v − w))1/2)
+ (λ0λ1w
v − w)1/2I1(2(λ0λ1w(v − w))1/2)],
L−1s [L−1
r (ψ1(r, s))(w, ·)](·, v) = e−λ1vδ0(v − w) + e−λ1(v−w)−λ0w[λ1I0(2(λ0λ1w(v − w))1/2)
+ (λ1λ0(v − w)
w)1/2I1(2(λ0λ1w(v − w))1/2)].
Thus, we get f0(w, v) and f1(w, v), the distribution function of T0 and T1, such that
f0(w, v) = e−λ0vδ0(v − w) + e−λ1(v−w)−λ0w[λ0I0(2(λ0λ1w(v − w))1/2)
+ (λ0λ1w
v − w)1/2I1(2(λ0λ1w(v − w))1/2)],
f1(w, v) = e−λ1vδ0(v − w) + e−λ1(v−w)−λ0w[λ1I0(2(λ0λ1w(v − w))1/2)
+ (λ1λ0(v − w)
w)1/2I1(2(λ0λ1w(v − w))1/2)].
26
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