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Are Regime Shift Sources of Risk priced in theMarket?
Kyriakos Chourdakis, Yiannis Dendramis and Elias Tzavalis
October, 2012
Abstract
In this paper we suggest a discrete-time option pricing model of Euro-pean calls when the log-return of the underlying asset (stock) is subjectto discontinuous regime shifts in its mean or volatility. The risk of theseshifts is allowed to be priced in the options market. The paper shows howto estimate the model and, then, it employes it to empirically examine ifregime-shift risks are priced in the options market as separate sources ofrisk. The results of the paper clearly indicate that shifts from the bull tobear and from bear to crash regimes carry substantial prices of risk. Ignor-ing this risk will lead to substantial option pricing errors, across di¤erentmoneyness levels.
Keywords: European call prices, stock market regime shifts, Markovregime switching model, risk neutral transition probabilities, occupationtime of a regime.
JEL Classi�cation: G10, G12, G13
� � � � � � � �The authors are grateful to Karim Abadir, Ian Cooper, Chenghu Ma,
Nikolaos Panigirtzoglou, Erric Renault, Michael Rockinger and the seminarparticipants at the Bank of England and the European Central Bank foruseful comments on an earlier version of the paper.
Chourdakis K., Department of Economics, Queen Mary University ofLondon, London E1 4NS, UK. [email protected] Y., Department of Economics, Athens University of Eco-
nomics & Business, Greece; [email protected] E. Department of Economics, Athens University of Economics
& Business, Greece; [email protected]
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1 Introduction
Many recent studies have documented that discrete-time, Markov chain basedof regime-switching (MRS) processes (see, e.g., Hamilton (1989) can capture thedynamics of asset prices and explain main features of their log-return distribu-tions, e.g. negative skewness and excess kurtosis (see, e.g., Ceccheti et al (1990),Whitelaw (2000), Ang and Bekaert (2002), Dai et al (2007), Chabi-Yo et al (2008),and Ghosh and Constantinides (2010), inter alia). The intuition behind regime-switching asset pricing models is clear. A simple two regime MRS asset pricemodel can re�ect the bull and bear states of asset markets. The bull regime isoften identi�ed by stock market data as one which has the highest mean and lowestvolatility, while the bear as that with the lowest mean and highest volatility. MRSasset pricing models allowing for a higher than two market regimes have been alsoconsidered in the literature. Some of these models allow for a crash regime relatedto collapsing asset price bubbles (see, e.g., Dri¢ ll and Sola (1998)), Brooks andKatsaris (2005)).Modelling asset price dynamics based on MRS processes brings us to ask an
important question for option pricing. Are regime shifts in asset (or stock) marketspriced as independent sources of risk? Answering this question has importantimplications for option pricing and portfolio management, as regime shifts in assetmarkets can not be completely hedged out and, thus, should be priced in thesemarkets. The price of a regime-shift source of risk must compensate risk aversemarket participants towards possible adverse regime shifts, i.e. from the bull tothe bear, or the crash, regime. Assuming that this price of risk is zero (see, e.g.,Bollen et al (2000), Chourdakis and Tzavalis (2000), Chourdakis (2004), Edwards(2005), Aingworth et al (2006)) may lead to serious option mispricing. To answerthe above questions, in this paper we suggest a discrete-time MRS option pricingmodel for European calls. This model considers regime shifts in the mean orvolatility of the log-return (or log-price) of the underlying asset.1 The discrete-time nature of the model facilitates its estimation and implementation in optionpricing, by employing Hamilton�s (1989) EM method. To estimate the marketprice of a regime-shift risk, the paper puts forward a joint estimation procedureof the MRS process and option pricing model suggested, utilising both option andstock data.
1Note that allowing for a regime shift in the conditional mean of stock return distribution cancapture mean reversion in stock prices (see, e.g. Ceccheti et al (1990)). As aptly noted by Loand Wang (1995), ignoring this mean reversion will lead to substantial biases in pricing options.
2
The option pricing model that the paper derives calculates a European call op-tion price, analytically, as a weighted average of Black-Scholes (BS) European callprices written on stocks. This formula uses as weights the risk neutral probabili-ties of the number of periods that stock market will spent in each of its di¤erentregimes over the entire life of the option, referred to as occupation time. Thisweighting means that, if, for instance, stock market is currently in the bear regimeand is expected to stay in this regime for some time, then our option pricingformula will put more weight on the BS option price corresponding to the bearregime, compared to that corresponding to the bull regime. This will lead to ahigher option price of a European call written on a stock than that predicted bythe standard BS formula, which does not allow for regime shifts. Intuitively, thiscan be attributed to the higher volatility or lower mean of the bear regime thanthose implied by the bull regime.Based on option and aggregate stock market price data for US from year 1996
to 2010, the paper provides a number of interesting results. First, it shows thatthe stock market can be characterized by three di¤erent regimes: the bull, bearand crash. The joint estimation employed to obtain values of the parameters ofthe MRS option pricing model and underlying log-price MRS process helps tobetter identify the di¤erent regimes of the stock market over time and to trace outthe regime-switching dates. For the crash regime, these dates are associated withthose of the Asian and Russian �nancial crises, the terrorist attack of September16, 2001, the collapses of Enron, WorldCom and Lehman brothers. Second, thepaper clearly shows that regime shifts are priced in the options market, and thusshould not be treated as risk neutral in the literature. These can magnify thesteepness of the smirk and the curvature of the implied volatility smile, explainedby MRS option pricing models.In particular, the paper shows that the prices of regime shifts from bull to bear
and from bear to crash markets, which mainly concern investors in the optionsmarket, are substantial. The paper also documents important regime-shift sourcesof risks from crash to bear and from bear to bull markets, despite the fact thattransition probabilities of these regime switches are very small. Investors in theoptions markets price these regime-shifts by assigning smaller risk neutral tran-sition probabilities to them than their physical counterparts. This can be alsoattributed to their risk aversion behaviour, which decreases the physical transitionprobability of a shift to a better market regime. Finally, both the physical and riskneutral transition probabilities indicate that the most likely to happen sequenceof regime shifts is that from bull to bear and from that to crash, directly.The paper is organized as follows. In Section 2, we present a simple version
of the MRS option pricing model assuming that stock market consists of tworegimes: the bull and bear, as often assumed in practice. This version of the
3
model will help us to better understand the option pricing formula implied by theMRS process of the underlying stock log-price process. In Section 3, this modelis extended to the case of N distinct stock market regimes. In Section 4, weimplement the MRS option pricing model suggested by the paper to actual datawith the main aim of estimating the prices of regime-shift sources of risks impliedby options market participants. In this section, we also examine if the MRS modelcan explain patterns of the BS implied volatility observed in practice and we assessthe magnitude of the option pricing errors if regime-shift risks are assumed that arenot priced in option markets. Finally, we also compare the pricing performance ofthe MRS model with the discrete-time option pricing model of Nandi and Heston(2000), assuming a GARCH volatility function of the underlying log-price process.This model can be considered as competent to the MRS model, given evidence thatGARCH e¤ects can be generated by regime shifts in volatility (see, e.g., Moranaand Beltratti (2004), Kramer and Tameze (2007), and Andreou and Ghysels (2009),for a survey), or they can approximate these shifts (see Duan et al (2002)). Section5 concludes the paper. All proofs of the paper are given in its Appendix.
2 The MRS Option Pricing Model
2.1 The MRS data generating process
Let the logarithm of the stock price, de�ned as yt = log Yt, obey the followingMarkov regime-switching (MRS) process under the physical probability measure,allowing for two regimes of stock market:
yt+1 = yt + � (St+1)�1
2�2 (St+1) + � (St+1) �t+1, (1)
where
� (St) = �0St, � = (�1; �2)0, St = (e1; e2)0;
� (St) = �0St, � = (�1; �2)0 and
�t+1 � NIID (0; 1) :
In the above speci�cation of the MRS process (1), St is a binary vector processwhich follows a discrete-time state space Markov chain. That is, St 2 fe1; e2g, wherefei ;i 2 (1; 2)g are unit vectors that span the two dimension space R2. The i-th el-ement of ei is 1 for the i-th regime (state) of stock market, and zero otherwise. Thetransition probability matrix between the two regimes of stock market considered
4
by (1), denoted as �1�and �2�, from time t to t+ 1 is de�ned as follows:
P � [pij] =�1� p12 p21p12 1� p21
�, (2)
where pij declares the transition probability of moving from regime �i� to �j�.Regime �1�is often identi�ed by stock market data as the bull regime. This is char-acterized as one having the highest mean and lowest volatility of stocks returnsdistributions between the two regimes. On the other hand, regime �2�, referred toas bear, is often identi�ed as one having the highest volatility and lowest mean.2
2.2 Option pricing
To derive a European call option pricing formula for the MRS process (1), we will�rst de�ne the risk neutralized equivalent process of (1) of the � -period ahead log-price yt+� (or its implied log-return, de�ned as ��yt+� � yt+� � yt). Then, we willobtain the risk neutral density of this process conditional on the current, t-timemarket information set, denoted as It. To this end, we make the following twoassumptions: First, we will consider that information set It includes, in additionto all historical stock prices, the present and past values of the state vector processSt, i.e. It = fyt;St; yt�1;St�1; : : :g. This is a standard assumption made in theliterature (see, e.g., Garcia et al (2003) and Dai et al (2007)), which means thatoption market investors know the current regime of the stock market. Second, wewill assume that vector process St is independent of the innovation term �t of MRSprocess (1), for all t. This is also a standard assumption made in the literaturewhich means that stock market investors are surprised by future regime changes(see, e.g., Turner et al (1989), Cecceti et al (1990), and Chabi-Yo et al (2008),more recently).To derive the risk-neutral process of the � -period log-return ��yt+� , write its
physical counterpart, by solving forward MRS process (1), as follows:
��yt+� (3)
=
��1 �
1
2�21
�� +
�(�2 � �1)�
1
2
��22 � �21
��O2� +
�Xi=1
� (St+i) �t+i,
2Recent studies (see, e.g., Barro and Ursua (2009)) associate the bull and bear stock marketregimes to expansionary and recessionary conditions of the economy, respectively. Deep reces-sionary conditions of the economy are often associated with stock market crashes, which can beconsidered as a separate regime.
5
or, by substituting volatility function of � (St+i) into (3), as
��yt+� (4)
=
��1 �
1
2�21
�� +
�(�2 � �1)�
1
2
��22 � �21
��O2� +
q�21� + (�
22 � �21)O2�!t;� ,
where O2� is a random variable, de�ned as
O2� =�Xk=1
e02St+k, (5)
which represents the number of times that the stock market will stay in regime�2�over maturity horizon � , and !t;� is a normally distributed variable capturingthe innovation terms "t+i of process ��yt+� . Given that St+� is a binary vectorprocess taking values e1 = (1; 0)0 and e2 = (0; 1)0, random variable O2� takesvalues �2 2 f0; :::; �g. These will re�ect the occupation time of regime �2�, overinterval � . Thus, O2� will be henceforth referred to as occupation time variable.In the appendix, we present an algorithm calculating the probabilities of the setof values of O2� , �2. These probabilities are needed to calculate the expectation oflog-return process ��yt+� conditional on current information set It and, then, toobtain European call option prices with maturity intervals bigger than one period.The risk-neutral counterpart of (3) can be derived under the equivalent risk
neutral measure, denoted as Q , implying that the discounted stock price Yt+� willform a martingale process, i.e.3
EQt (Yt+� ) = Yt expn�rft
o; (6)
where rft is the � -period risk-free interest rate at time t and EQt (�) is the conditional
expectation Et(�) under risk neutral measure Q . Under this measure, the followingrisk neutralized measure of �tyt+� can be de�ned
�tyt+�
= rft �1
�lnEQt
�exp
�(�2 � �1)O2;Q�
��� 12�21 +
�(�2 � �1)�
1
2
��22 � �21
��O2;Q�
+
q�21� + (�
22 � �21)O
2;Q� !Qt;� , (7)
where !Qt;�and O2;Q� constitute the risk neutral counterparts of random variables
!t;� .and O2� , respectively, under measure Q (see Appendix). Under this measure,
3See, e.g. Pliska (1999).
6
!Qt;� will be assumed that is a NIID(0; 1) random variable, while the transitionprobability matrix of the binary state vector St, P, determining the probabilitiesof occupation time variable O2;Q� , will be de�ned as follows:
PQ � [pQij] =�1� pQ12 pQ21pQ12 1� pQ21
�;
where
pQij = �ijpij.
These relationships between risk neutral transition probabilities pQij and their phys-ical counterparts, pij, re�ect the risk averse behaviour of investors, and thus suggestthat the risk of a regime-shift is priced in options market. Such an assumptionare often made in the literature. For instance, Jarrow et al (1997) and Whitelaw(2000) assume that regime shifts from the bull to the bear regime must be pricedin the stock and options markets, since they have adverse e¤ects on the value andthe income of investments.Based on an equilibrium pricing approach, Dai et al (2007) show that even
regime shifts from the bear to the bull regime should be priced by the markets,due to the risk aversion behavior of investors. This behavior implies that a regimeshift from the bull to the bear regime will imply a value of the price of regime-shiftrisk coe¢ cient �ij bigger than unity, implying that p
Qij > pij, for i = 1 < j = 2.
On the other hand, a regime shift from the bear to the bull regime will imply avalue of �ij < 1, implying that p
Qij < pij, for i = 2 < j = 1. A value of �ij bigger
than unity, for fi; jg = f1; 2g, will increase the risk neutral probability that thestock market will stay in the bear regime more than one times over the maturityinterval � , i.e. �2 > 0. This will be done at the expense of the probability that themarket will stay in the bull regime over � , i.e. �2 = 0. The opposite will happenfor a value of �ij less than unity, for fi; jg = f2; 1g. Finally, note that a value of�ij equal to unity, implying p
Qij = pij, means that regime shifts are considered as
risk risk neutral for investors in the options markets, and thus they are not pricedin the options markets. That is, they have zero price.The risk-neutral process of log-return �tyt+� , given by equation (7), implies
that, in addition to the risk-free interest rate rft , its conditional on informationset It mean also depends on the di¤erence of the means across the two regimes�2 � �1, as well as the conditional mean of the occupation time random variableO2;Q� . These terms a¤ect the conditional mean of�tyt+� and, hence, its risk neutraldistribution, since regime shifts are not traded in the stock market. Thus, theire¤ects can not be hedged out under the risk neutral measure Q. The dependenceof the risk neutral distribution of �tyt+� on the di¤erence of means �2 � �1 can
7
explain the skewness of this distribution and the smirk (skew) of the Black-Scholesimplied volatility smile, observed in practice (see, e.g., Bakshi et al (1997)). Thecurvature of this volatility smile can be explained by regime shifts in the volatilityfunction of MRS process (1), implying �22 6= �21. These shifts can explain evidenceof excess kurtosis of the risk neutral distribution of �tyt+� . Based on the aboverisk-neutral speci�cation of the MRS process of log-return �tyt+� (or log-priceyt+� ), in the next proposition we derive a closed form solution of a European calloption price based on Cox�s et al (1976) risk neutral pricing framework.
Proposition 1 Let the log-price of the underlying stock, yt, follow MRS process(1). Then, the price of a European call option with strike price K and maturityinterval � , denoted as Ct(�), can be analytically derived by the following formula:
Ct(�) =�X
�2=0
CBSt��2 (�2) ,�(�2); �
�� (�2) Pr
�O2;Q� = �2jIt
�, (8)
where
CBSt��2 (�2) ; � (�2) ; �
�= Yt� (d1(�2))�
K
� (�2)e��r
ft � (d2(�2)) , (9)
d1 (�2) =� ln K
Yt �(�2)+ �rft +
�2(�2)2
� (�2),
d2 (�2) =� ln K
Yt �(�2)+ �rft �
�2(�2)2
� (�2),
� (�2) =e(�2��1)�2
EQt [e(�2��1)�2 ]
,
� (�2) =q�21� + (�
22 � �21) �2,
and �(�) denotes the cumulative normal distribution. See Appendix, for the proofof the proposition.
Proposition 1 demonstrates that the European call option price implied by theMRS process (1) can be written as a weighted average of the Black-Scholes (BS)European call option prices CBSt (�2 (�2) ; � (�2) ; �) with strike prices
K�(�2)
, de�nedby formula (9). These option prices are conditional on the values of the risk neutralequivalent measure of the occupation time variable O2;Q� , given by set �2, re�ecting
8
the number of periods that stock market will stay in the bear regime, "2", overits maturity interval � . Each of these conditional option prices is weighted by itscorresponding risk neutral probability of the values of set �2 to occur, given asPr�O2;Q� = �2jIt
�. Calculating these probabilities is necessary to price European
call options with a maturity horizon bigger than one period, � > 1.The risk neutral probabilities Pr
�O2;Q� = �2jIt
�, by which the BS conditional
option prices CBSt (�2 (�2) ; � (�2) ; �) are weighted, capture the e¤ects of a possibleadverse regime shift of the stock market from the bull to the bear regime onEuropean call option prices, over maturity horizon � . For instance, if the stockmarket lies in the bear regime at current time t , they will tend to weight morethe BS prices CBSt (�2 (�2) ; � (�2) ; �) considering that the stock market will stayin this regime for more than one times until the expiration date of the option,compared to those assuming that the stock market will stay more times in thebull regime, over � . Obviously, the values of probabilities Pr
�O2;Q� = �2jIt
�will
depend on the persistency of each regime.There are two interesting special cases where the MRS option pricing formula
(8) can be reduced. The �rst is when there is no regime change, for all t. Then, �2 =0 and, thus, equation (8) reduces to the standard BS option pricing formula, whichassumes no regime-switching. The second case is when there is no regime shift inthe mean of log-return yt (i.e., �1 = �2 = �), but only in its volatility function(i.e., �1 6= �2). In this case, (8) reduces to a formula which is a weighted averageof the standard BS European call option prices conditional on the occupation timevalues �2, given as
CBSt��2 (�2) ; � (�2) ; �
�= Yt� [d
01 (�2)]� e��r
ftK� [d02 (�2)] , (10)
where d01 (�2) =log(Yt=K)+�r
ft
�(�2)+ 1
2� (�2) and d
02 (�2) = d
01 (�2)� � (�2). This formula
corresponds to that suggested by Hull andWhite�s (1987) assuming that log-returnyt follows a stochastic volatility model. This model can only explain BS impliedvolatility smiles which are symmetric, as it does not consider stochastic changesin the conditional mean of yt+1.
2.3 Generalization to N di¤erent market regimes
In this section, we generalize the MRS option pricing model provided in the pre-vious section to the case where the number of stock market regimes is bigger thantwo, given by a �nite number N . In particular, we assume that the stock log-priceprocess yt, given by MRS process (1), allows for N di¤erent regimes of the mar-ket. The mean and volatility functions of this process will be de�ned as the innerproduct of the (N � 1)-dimension vectors of the N di¤erent mean and volatilityparameters �i and �i, respectively, with state vector St, which now is de�ned as
9
St = (e1; e2; ::; eN)0, where fei ;i 2 (1; 2; ::; N)g are the unit vectors spanning the
N -dimension space RN . That is, we will have � (St) = �0St and � (St) = �0St,where � = (�1; �2; ::; �N)
0 and � = (�1; �2; ::; �N)0. This multi-regime speci�cationof MRS process (1) implies that the transition matrix of the physical probabilitiesP among all the di¤erent stock market regimes considered has dimension (N�N).It is de�ned as follows:
P =
2664p11 p21 : pN1p12 p22 : pN2: : : :p1N p2N : pNN
3775 .The above generalization of the MRS process (1) implies that the � -period
ahead, future log-return process �tyt+� and its risk neutral counterpart can berespectively written as follows:
�tyt+� =
��1 �
1
2�21
�� +
NXi=2
�(�i � �1)�
1
2
��2i � �21
��Oi�
+
vuut�21� + NXi=2
(�2i � �21)Oi� !t;� (11)
and
�tyt+� =
rft �
1
�lnEQt
"exp
(NXi=2
(�i � �1)Oi;Q�
)#� 12�21
!�
+
NXi=2
�(�i � �1)�
1
2
��2i � �21
��Oi;Q�
+
vuut�21� + NXi=2
(�2i � �21)Oi;Q� !Qt;� , (12)
where !Qt;� and Oi;Q� constitute the risk neutral counterparts of random variables
!t;� and Oi� , respectively, under measure Q (see Appendix A3). Note that, underthe above generalization of the MRS process (1), the occupation time variable ofa representative stock market regime i now is de�ned as Oi� =
P�k=1 e
0iSt+k. This
variable will represent the number of periods (times) that the stock market willstay in regime i, for i = 2; :::; N stock market regimes, over the maturity horizon
10
� . SincePN
i=1Oi� = � , the occupation time variable for regime �1�will be de�ned
as O1� = � �PN
i=2Oi� . The risk-neutral counterpart of occupation time variable O
i�
will be denoted as Oi;Q� . Both variables Oi� and Oi;Q� take values � i which satisfy
the following conditionPN
i=2 � i � � . As in the previous subsection, to calculatethe risk neutral probabilities of the values of occupation time variable Oi;Q� , � i, wewill assume that the elements of the risk neutral transition matrix PQ � [pQij] arerelated to those of its physical counterpart P � [pij] as follows:
pQij = �ijpij;
where �ij denotes the price of risk of a shift from regime i to j. For analyticconvenience, we will assume that, in terms of investors�preferences, stock marketregimes are ordered from regime "1" to "N", where "1" represents the best stockmarket regime (i.e., a bull regime which is characterized by the highest mean andlowest volatility of the log-return yt, across all di¤erent stock market regimes),while regime "N" denotes the worst market regime. For instance, "N" can betaken to represent the crash regime of the market. In this regime, yt will have thelowest mean and largest volatility.4
Based on the risk-neutral relationship (12) for log-return�tyt+� and its impliedunderlying risk neutral density, in the next proposition we present an analyticoption pricing formula of a European call in the case of N > 2 di¤erent stockmarket regimes.
Proposition 2 Let the log-price of the underlying stock yt follow a generalizationof MRS process (1) allowing for N > 2 di¤erent stock market regimes. Then,under risk neutral probability measure Q, the price of a European call option priceCt(�) with strike price K and maturity interval � can be analytically derived bythe following formula:
Ct(�) (13)
=X�N
::X�2
CBSt��2(�); � (�) ; �
�� (�) Pr
�O2;Q� = �2; ::; O
N;Q� = �N jIt
�,
where
CBSt��2(�); �
��N�1
�; ��= YtN (d1)�
K
� (�)e��r
ftN (d2) , (14)
� = (�2; ::; �N) is a (N � 1)-dimension vector collecting the values of occupationtime variables Oi;Q� corresponding to regimes i = 2; 3; :::; N , � (�) is a function of
4Note that the ranking of the alternative market regimes can be thought of as an empiricalissue. This will not change the results of our analysis.
11
the elements of vector � de�ned as
� (�) =exp
nPNi=2 (�i � �1) � i
oEQt
hexp
�PNi=2 (�i � �1) � i
�i ;and
d1 =� ln K
Yt �(�)+ �rft +
�2(�)2
�(�)
d2 =� ln K
Yt �(�)+ �rft � �2(�)
2
�(�)
�2(�) = �21� +
NXi=2
��2i � �21
�� i
The proof of proposition is given in the appendix.
The option pricing formula given by equation (13) has analogous interpretationto that of the case of two market regimes (N = 2), given by (8). It calculatesEuropean call price Ct(�) based on BS call option prices CBSt (�2(�); � (�) ; �),which are conditional on the values of risk-neutral occupation time variables ofall possible stock market regimes i, Oi;Q� , for i = 2; 3; :::; N . These values now arecollected in vector � = (�2; ::; �N). One di¤erence of formula (13) from (8) is that,in calculating option price Ct(�), it allows for the possibility that the stock marketwill stay in more than one regimes until the expiration date of the option.
3 Estimation of the MRS option pricing model
In this Section, we show how to estimate and implement the MRS option pricingmodel derived in the previous section. Our analysis is focused on examining ifregime shifts constitute signi�cant sources of risks which are priced in the optionsmarket. We will also evaluate the magnitude of the pricing errors encountered,if we assume that regime shifts are risk neutral for investors in the markets, andwe will compare the model to the discrete-time stochastic volatility option pricingmodel of Nandi and Heston (2000), henceforth HN. The latter is frequently used,in practice, for option pricing (see, e.g., Christofersen and Jacobs (2004), andMoyaert and Petitjean (2011)). This model assumes that the volatility functionof stock return distributions follows a GARCH(1,1) process. As noted in theintroduction, GARCH e¤ects in the volatility function of stock returns can be
12
generated by MRS type of shifts in this function. To price call option prices underMarkov regime-switching, Duan et al (2002) used GARCH volatility functions toapproximate regime type of shifts of this volatility function.In answering the above all questions, the paper suggests a joint estimation of
the option pricing model suggested by the paper and MRS process (1), based onoptions and stock markets data. This estimation enables us to obtain estimatesof the regime-shift price of risk coe¢ cients �ij and, hence, to calculate the riskneutral occupation time probabilities Pr
�Oi;Q� = � ijIt
�, for i = 2; ::; N di¤erent
regimes. The latter are necessary for the implementation of formula (13), or (8).The data sets used in our analysis consists of time series observations on theS&P500 stock market index and European call option prices written on this index.These series cover the period from 01/1996 to 10/2010. During this period, the USstock market experienced four �nancial crises. The �rst was related to Asian andRussian crises of years 1997 and 1998, respectively. The second and third followedthe terrorist attack on September 11, 2001, and the collapses of the Enron andWorldCom companies in year 2002, respectively. The fourth followed the collapseof Lehman Brothers on September 16, 2008. Our option price data set employedin the estimation consists of nonoverlapping at-the-money (ATM), in-the-money(ITM) and out-of-the-money (OTM) European call option prices, with maturityintervals of 5,10,15 and 20 trading days. In total, this set consists of 710 units ofoption prices and stock returns.Our empirical analysis proceeds as follows. First, we present estimates of uni-
variate process (1) with the aim of investigating the number of di¤erent regimesof the stock market identi�ed by our data. Given this, next we carry out the jointestimation of the MRS process and option pricing formula (13), using stock andoptions price data. Finally, we assess the consequences of assuming risk-neutralregime shifts on option pricing and we compare the pricing performance of themodel to that HN�s model.
3.1 Estimates of the univariate MRS process (1)
Table 1 presents estimates of MRS process (1) based on the Maximum Likeli-hood (ML) method, suggested by Hamilton (1989). The table presents two setsof results. The �rst assumes that the number of the stock market regimes istwo (N = 2), while the second that it is three (N = 3).5 To examine which ofthese two numbers of regime speci�cations constitutes a better description of thedata, the table presents the Akaike Information Criterion (AIC)6 and Ljung-Box
5A speci�cation of the MRS model with four regimes (N = 4) is also estimated. But, this isnot found to describe our data satisfactorily.
6This is considered as an appropriate criterion in choosing the number of regimes assumed bythe MRS process (1). Likelihood ratio (LR) based tests can not be used for this purpose. These
13
test statistics for serial correlation and conditional heteroscedasticity of the dis-turbance term of the MRS process �t. These statistics are denoted as LB(�) andLB2(�), respectively, where the number of lags are reported in parentheses. Theyare based on the normally distributed Rosenblatt transformation of the log-return�yt and they are distributed as chisquared. As shown by Smith (2006), the nor-mal transformation of �yt improves considerably the �nite sample properties ofstatistics LB(�) and LB2(�), compared to those based on the standardized gener-alized disturbance term of the MRS process, de�ned as E (�tjIt�1).7 The latter arenonnormally distributed.
su¤er from singularity problems of the information matrix under the null hypothesis that thetrue number of regimes is N (see, e.g., Psaradakis and Spangolo (2003)).
7The Rosenblatt transformation of a random variable implies that, if �yt has a distributionfunction F , then Zt = ��1 (F (�yt)) will be a standard normal random variable, where ��1 (:)denotes the inverse of the standard normal cumulative distribution function. For MRS process(1) allowing for N di¤erent market regimes, the normally distributed random variable Zt isde�ned as
Zt = ��1
NXi=1
Pr (St = ijIt�1)Z �yt
1f (vjSt = i; It�1) dv
!The Ljung-Box statistics LB1(�) and LB2(�) can rely on sample estimates of Zt and Z2t to testfor serial correlation and conditional heteroscedasticity of order-h in the disturbance term �t,respectively. For instance, the Ljung-Box statistic LB1(h) is calculated based on the followingformula:
LB(h) = T (T + 2)hXj=1
�2jn� j ,
where �j is the sample autocorrelation of random variable Zt:
14
Table 1: Estimates of MRS process (1)no. reg. N = 3 N = 2 N = 3 N = 2�1 0.0033 0.0026 p23 0.029
(1e-06) (1e-10) (6e-5)�2 0.0012 -0.0044 p31 4e-9
(0.002) (2e-9) (1e-9)�3 -0.010 p32 0.21
(3e-5) (0.001)�1 0.015 0.020
(1e-6) (1e-10) logL 1.63e+3 1.61e+3�2 0.027 0.027 AIC -3248.35 -3224.56
(2e-6) (7e-6) LB(1) 3.25 3.49�3 0.062 LB(5) 4.28 4.82
(3e-5) LB(10) 9.10 10.99p12 0.009 0.0034 LB2(1) 0.96 1.93
(7e-5) (1e-5) LB2(5) 3.00 12.48p13 5e-17 LB2(10) 7.58 22.89
(5e-17)p21 0.006 0.10
(2e-5) (4e-4)
Notes: The table presents estimates of (1) for N = 2 and N = 3 number of regimes(no. reg.). Quasi ML (maximum likelihood) estimates of the standard errors of theseestimates are in parentheses. AIC denotes the Akaike Information Criterion, and LB(h)and LB2(h) respectively denote the Ljung-Box test statistics for serial correlation andconditional heteroscedasticity of the disturbance term of MRS process (1) of order h.logL is the maximum log-likelihood value of (1):
The results of the table clearly indicate that the speci�cation of MRS (1)allowing for N = 3 regimes of the stock market describes better the data. Thiscan be justi�ed both by the values of LB(�) and LB2(�) statistics and the AICcriterion, reported in the table. The three regimes identi�ed by our data throughthis speci�cation of the MRS process capture bull, bear and crash conditions of thestock market. The �rst (bull) regime has the highest mean �1 and lowest volatility�1 across the three regimes. The second (bear) regime has a mean �2 which is notdi¤erent than zero. Its volatility �2 is almost twice that of the bull regime. Finally,the third (crash) regime has a negative mean �3 and the highest volatility �3, acrossall three regimes. Estimates of the probabilities that the US stock market lies inthese regime at each point of time t are graphically presented in Figures 1, overour sample. These are denoted as Pr [St = eijyt], for i = 1; 2 and 3. They are
15
obtained based on Hamilton�s (1989) �lter. The results of this �gure clearly showthat the crash regime is associated with the �nancial crises of the US stock market,mentioned before. The bull regime of the market seems to characterize the periodfrom year 2003 to 2007. The bear regime seems to characterize the period fromthe middle of nineties to year 2003, with the exception of the short periods of thecrashes mentioned above.
1996 1998 2001 2003 2006 2009
0.20.40.60.8
Pr[S
t=1|y
t]
1996 1998 2001 2003 2006 2009
0.20.40.60.8
Pr[S
t=2|y
t]
1996 1998 2001 2003 2006 2009
0.20.40.60.8
Pr[S
t=3|y
t]
Filtered probabilties Pr [St = eijyt] vs the S&P500 index
Another interesting conclusion that can be drawn from the results of the tableis that the most likely to happen sequence of regime-switching in the stock marketis that from bull to bear and from that to crash regime, and the inverse. Thiscan be supported by the values of the transition probabilities pij, reported in thetable. The transition probability from the bull to the crash regime is found to bealmost zero, which means that it is less likely to be a regime shift from the bull
16
regime to the crash. Also note that the value of the transition probability fromthe crash to the bear regime is quite high. This is associated with the fact thatthe crash regime tend to last for the shortest time-intervals during our sample.
3.2 Joint estimation of the MRS process and option pric-ing model (13)
The joint estimation of the MRS option pricing model given by formula (13)and MRS process (1) will provide estimates of the risk neutral parameters of thisprocess and regime shift price of risk coe¢ cients �ij�s, which are needed for theimplementation of the model.8 Since the stock market regime St, at time t, is notknown by the econometrician, in the estimation procedure we will infer this fromthe data based on Hamilton�s (1989, 1993) �lter. This is a standard procedureoften followed in the literature to estimate MRS models (see, e.g., Melino andYang (2003), and Chabi-Yo (2008). The information set It�1 assumed by thisestimation procedure consists of past values of yt and Ct(�)(obs), or their di¤erences,i.e. It�1 = f yt�1; Ct�1,�yt�2; :::g. As can be con�rmed by our empirical resultslatter on, exploiting joint information from both the stock and options marketswill better help in identifying the current stock market regime St by the data,given that options are priced based on market information about this regime.If we assume that the observed European call price Ct(�)obs, at time t, is mea-
sured with an NIID error term vt, i.e. Ct(�)obs = Ct(�)+ vt, where vt � N (0; �2v)(see, e.g., ), then the vector of parameters of the MRS process (1) augmented withthe risk-neutral counterparts of pij, collected in vector � = (�;�;p12; :::; pN�1N ,pQ12; :::; p
QN�1N), can be estimated by maximizing the following loglikelihood func-
tion:
L =TXt=1
log f��yt; Ct(�)
obs�� It�1;��
=
TXt=1
log
"NXi=1
f��yt; Ct(�)
obsjSt;��Pr [St = eijIt�1]
#,
over a sample of t = 1; 2; :::; T observations , where f��yt; Ct(�)
(obs)jSt;��is the
joint density function of log-return �yt and option price Ct(�)obs, at time t. Thisis the product of two univariate probability density functions, i.e.
8Joint estimation of option pricing models and the underlying asset price stochastic process isa standard procedure for retrieving risk neutral probabilities from the data (see, e.g., Jackwerth(2000)). It has been also suggested by Chernov and Ghysels (2000) to estimate the price of riskcoe¢ cients of the stochastic volatility option pricing model of Heston (1993).
17
f��yt; Ct(�)
obsjSt;��=
�2�p�2 (St)�2v
��1(15)
exp
(��yt �
�� (St)�1
2�2 (St)
��22�2 (St)
)
exp
(�Ct(�)
obs � Ct(�)�2
2�2v
);
since error terms �t and vt are assumed to be independent. Estimates of theregime-shift price of risk coe¢ cients �ij can be obtained from the estimates of thephysical and risk-neutral transition probabilities pij and p
Qij, respectively, based on
the following relationship pQij = �ijpij (see Sections 2 and 3).Table 2 presents estimates of vector � based on the above joint ML estimation
procedure. This is done for two sets of cross-section data. The �rst uses ATMEuropean call options, which is the most liquid category of options. The second setincludes also ITM and OTM European call options. Both sets of estimates assumethat the number of stock market regimes is N = 3 and consider maturity intervalsof � = f5; 10; 15; 20g trading days. The results of Table 2 are consistent withthose of Table 1. They show that the stock and options markets are characterizedby three distinct regimes: the bull, bear and crash. The values of the meanand volatility parameters of the MRS process estimated, as well as the transitionprobabilities across the three regimes are very close to those reported in Table 1.The transition probability of a shift from bull to crash regime p13 and that of itsinverse, p31, are are very close to zero. This means that the most likely sequenceof regime shifts is that from bull to bear and from that to crash, and its inverse.This is also implied by the risk neutral transition probabilities pQij, reported in thetable. The above results are robust across the two di¤erent sets of moneyness levelsconsidered in our analysis. This result shows that using ATM options, which isthe most liquid category of options prices, can precisely estimate the risk neutralparameters of the MRS option pricing model.
18
Table 2: Joint estimates of the MRS process and option pricingmodel for N = 3
Moneyness ATM ATM, ITM, ATM ATM, ITM,OTM OTM
�1 0.0042 0.0044 pQ12 0.007 0.072(1e-06) (2e-7) (2e-4) (1e-7)
�2 0.0020 0.0029 pQ13 9e-9 8e-9(0.002) (7e-7) (4e-11) (6e-11)
�3 -0.029 -0.028 pQ21 5e-14 5e-14(1e-4) (1e-6) (5e-16) (7e-17)
�1 0.015 0.015 pQ23 0.05 0.08(3e-6) (2e-8) (1e-5) (1e-6)
�2 0.027 0.026 pQ31 1e-11 1e-11(7e-6) (8e-8) (1e-13) (8e-15)
�3 0.057 0.056 pQ32 2e-10 2e-10(2e-4) (9e-8) (2e-10) (7e-14)
p12 0.0034 0.0034 �v;ATM 4.16 3.86(1e-5) (2e-8) (1e-3) (3e-6)
p13 1e-11 9e-12 �v;ITM 3.87(2e-13) (1e-16) (1e-6)
p21 0.003 0.006 �v;OTM 3.79(1e-5) (3e-8) (4e-6)
p23 0.068 0.0076(2e-6) (1e-7) logL 453.44 4407.87
p31 2-10 1.5e-11 AIC 944.89 8857.69(5e12) (1e-16)
p32 0.07 0.062(2e-6) (2e-6)
Notes: The table presents joint estimates of the parameters of MRS process (1),for N = 3 regimes, and those implied by the MRS option pricing model (13). QuasiML (maximum likelihood) estimates of the standard errors of these estimates are inparentheses. AIC denotes the Akaike Information Criterion, and logL is the maximumlog-likelihood value.
Given that the estimates of transition probabilities p13 and p31 are very closeto zero, in Table 3 we present joint estimates of the MRS option pricing model(13) and MRS process (1) under the following restrictions: p13 = 0 and p31 = 0.This is done for the two di¤erent sets of moneyness levels, considered in Table 2.The likelihood ratio (LR) test statistic reported in the table can not reject these
19
restrictions, implying that this speci�cation of the MRS model can satisfactorilydescribe the stock and options market. As was expected, the estimates of the MRSoption pricing model and its underlying process reported in the table are consistentwith those of its more general speci�cation reported in Table 2. In particular, thevalues of the mean and volatility parameters of the MRS process estimated, aswell the physical and risk neutral transition probabilities reported in Table 3 arevery close to those of Table 2.Figure 2 graphically presents estimates of the �ltered probabilities of regimes
Pr [St = eijyt], for i = 1; 2 and 3, obtained based on the parameter estimates ofTable 3 for the data set which includes ATM, ITM and OTM options.9 Inspectionof the graphs of the �gure indicates that these estimates of Pr [St = eijyt] are moresmoothed than those obtained based on the estimates of univariate MRS process,presented by Figure 1. They can distinguish more clearly the current regime ofthe stock market St, perceived by option market investors. This obviously can beattributed to the extra information exploited by the joint estimation of the MRSprocess and option pricing model, based on options and stock market data.
9Note that analogous graphs of Pr [St = eijyt] are obtained for the data set which includesonly ATM option prices.
20
Table 3: Joint estimates of the MRS process and option pricingmodel for N = 3 under restrictions p13 = p31 = 0
Moneyness ATM ATM, ITM ATM ATM, ITMOTM OTM
�1 0.0041 0.0040 pQ12 0.070 0.072(7e-7) (1e-6) (2e-6) (1e-7)
�2 0.0025 0.0028 pQ21 4e-14 4.6e-14(6e-7) (2e-6) (1e-13) 2e-19
�3 -0.029 -0.030 pQ23 0.050 5e-14(6e-7) (4e-6) (1e-6) (2e-19)
�1 0.014 0.014 �v;ATM 4.16 0.08(2e-7) (9e-9) (1e-4) (1e-6)
�2 0.027 0.026 �v;ITM 1e-11(3e-8) (4e-7) (8e-15)
�3 0.057 0.057 �v;OTM 2e-10(1e-6) (6e-7) (7e-14)
p12 0.0034 0.0036(3e-7) (3e-7) logL -453.44 -4407.84
p21 0.003 0.0083 AIC 936.897 8849.69(2e-7) (7e-8)
p23 0.006 0.0078(1e-6) (4e-7)
Notes: The table presents joint estimates of the parameters of MRS process (1), forN = 3 regimes, and those implied by the MRS option pricing model (13). The MRSmodel is estimated under the following restrictions: p13 = p31 = 0. Quasi ML (maxi-mum likelihood) estimates of the standard errors of these estimates are in parentheses.AIC denotes the Akaike Information Criterion, and logL is the maximum log-likelihoodvalue.
21
1996 1998 2001 2003 2006 2009
0.20.40.60.8
Pr[S
t=1|y
t]
1996 1998 2001 2003 2006 2009
0.20.40.60.8
Pr[S
t=2|y
t]
1996 1998 2001 2003 2006 2009
0.20.40.60.8
Pr[S
t=3|y
t]
Joint estimates of the �ltered probabilties Pr [St = eijIt]
The estimates of the physical and risk neutral transition probabilities of Table3 in the case that the data set includes ATM, ITM and OTM option prices, implythe following price of risk coe¢ cients �ij:10
�12 �21 �23 �3220.41 1e-11 8.50 4e-9
These values of �ij are consistent with the risk aversion attitude of investors inthe stock and options markets. That is, �ij > 1, if i < j, and �ij < 1, if i > j, asis predicted by the theory (see Section 2). They show that the regime shift fromthe bull to the bear regime constitutes a very important source of risk. The shiftfrom the bear to crash regime constitutes also an important source of risk, which
10Analogous estimates are obtained based on the data set considers only ATM options.
22
is priced in the options market. But, its price �23 is less than that of the shift fromthe bear to the bull regime, i.e. �12. The smaller value of �23 than that of �12means that the options�market investors consider as more critical a regime shift ofthe market from the bull to bear regime. As noted before, this can be attributed tothe transient nature of this regime to the crash regime. The less than unity valuesof risk-price coe¢ cients of �21 than �32, capturing the reverse of the above regimeshifts (i.e. from the bear regime to the bull and from the crash regime to the bear,respectively) indicate that investors in the options market assign signi�cant riskpremia also to these regime shifts, despite the fact that the transition probabilitiesof these regime shifts are very small. The risk neutral transition probabilities ofthem are much smaller than those of their physical counterparts, due their riskaversion behaviour.The reported price of risk coe¢ cients �ij imply that there is a substantial
di¤erence between the physical and risk-neutral probabilities of the occupationtime variable Pr[Oi� = � ijIt] and Pr[Oi;Q� = � ijIt], for all regimes i = 1; 2 and 3.To see how di¤erent in magnitude are the values of Pr[Oi;Q� = � ijIt] from those ofPr[Oi� = � ijIt], in Figure 3 we graphically present estimates of both of them for thenumber of periods that the market will stay in the crash regime, i.e. for set �3 =f0; 1; 2; :::; 6g.11 Inspection of this �gure reveals the values of Pr
�O3;Q� = �3jIt
�are
much higher than those of Pr [O3� = �3jIt] when �3 > 0, as is expected by a riskaversion behavior of investors. This is done at the expense of the probability thatthe market will stay zero times in the crash regime, i.e. �3 = f0g. These resultsmean that, employing physical values of the occupation time probabilities in optionpricing formula (13), or (8), instead of risk neutral, i.e. assuming that regime-siftrisks are not price, will lead to serious option mispricing. It will underweight thevalues conditional on bull or bear BS prices CBSt
��2(�); �
��N�1
�; ��, employed
in formula (13).
11Analogous graphs can be obtained for occupation time probabilities of the bear regime.
23
0 1 2 3 40
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1RNPhys
Figure 3. Pr�O3;Q� = �3jIt
�vs Pr [O3� = �3jIt]
l
3.3 Assessing the e¤ects of assuming zero-price regime shiftsources of risks on option pricing
To further investigate the consequences of assumption that regime-shift risks arenot priced in the market on option pricing, implying �ij = 1, in this section wepresent estimates of the implied volatility smile of European call options, acrossdi¤erent moneyness levels K
Yt, for di¤rent values of �ij. See Figure 4. These are
generated based on the estimates of the MRS option pricing model parameters,reported in Table 3 in the case that our data set include ATM, ITM and OTMoptions.The upper set of graphs of the �gure presents values of the implied volatility
smile function in the case that regime shifts a¤ect both the mean and volatility of
24
MRS process (1), while the lower presents values of it in the case that regime shiftsa¤ect only the volatility function of the MRS process. The comparison of thesetwo di¤erent sets of graphs can show which features of the MRS process of returnyt, namely shifts in its mean or volatility function, can explain the pattern of theimplied volatility smile documented in practice. For the goals of our analysis, the�gure presents values of the implied volatility smile function for prices of regime-shift risk coe¢ cients �ij which are 2.5 times their estimates, and �ij = 1, whichmeans that the market regime-shift of risks are not priced in the market.Inspection of the graphs of Figure 4 (see upper set of graphs) indicates that the
pattern of the implied volatility smile generated by the MRS option pricing modelis consistent with that observed in practice (see, e.g., Bakshi et al (1997)). Theshifts in the mean a¤ect the skew (smirk) of the smile, while those in the volatilitya¤ect the curvature of the smile.12 The values of price of risk coe¢ cients �ij arefound to a¤ect positively both the skew and curvature of the smile. In particular,an increase in the value of �12 increases the skew of the smile, while an increasein the value of �23 moves upward the whole volatility smile function. The lastresult can be attributed to the very high value of volatility �3, under the crashregime. This mainly a¤ects the curvature of the volatility smile function. Theseresults imply assuming zero price of regime-shift risks (i.e., setting �12 = �23 = 1and �32 = �21 = 1) will lead to serious mispricing of European calls across alldi¤erent moneyness levels. Finally, a comparison between the upper and lowersets of graphs of Figure 4 clearly indicates that responsible for the smirk of theimplied volatility function is a regime shift in the mean of the MRS process of
12These results adds to those supporting the view that the MRS option pricing model canconsistently explain the shape of the BS implied volatility smile (see, Chourdakis and Tzavalis(2000) and Aingworth et al (2006)).
25
return yt, while for the volatility is a shift in its volatility.
0.92 0.94 0.96 0.98 1 1.02 1.04 1.06 1.080.17
0.18
0.19
0.2
0.21
0.22λ12=estimated λ12=1 λ12=2.5 estimated
0.92 0.94 0.96 0.98 1 1.02 1.04 1.06 1.08
0.18
0.2
0.22
0.24λ23=estimated λ23=1 λ23=2.5 estimated
Figure : BS implied volatilities generated by the MRS option pricing model
3.4 Option pricing performance
To quantitatively assess the pricing performance of the MRS option pricing model,in Table 4 we present values of the root mean square error (RMSE) and meanabsolute error (MAE) of its pricing errors, across di¤erent moneyness categoriesof our European call option price data set. These are obtained based on theparameter estimates of the model reported in Table 3, for the case of the ATMset of options price data which is the most liquid category of options. Thus, theresults of the table for the categories of ITM and OTM options can be thought ofas those of an out-of-sample evaluation exercise of the model, using cross-sectionsets of data.
26
To examine the performance of the model under di¤erent assumptions of inter-est, the table reports values of the above metrics in the case that the price of riskcoe¢ cients �ij are set equal to unity, i.e. �12 = �23 = �21 = �32 = 1, which meansthat regime shifts are considered as risk neutral by investors in the options mar-ket. Finally, to evaluate the relative pricing performance of the model to anotherdiscrete-time option pricing model, the table reports values of the above metricsfor the Heston and Nandi (2000) option pricing model, denoted HN . This modelassumes that the underlying stock follows the risk neutral stochastic process:
�yt+1 = rt �1
2�t+1 +
q�2t+1z
Qt+1;
where zQt � NIIID(0; 1) is the risk-neutral process of the log-return innovationzt+1, given as z
Qt+1 = zt+1 +
��+ 1
2
�p�2t+1, and �
2t+1 is the volatility function,
given as �2t+1 = $ + a�zQt �
p�2t
�2+ ��2t +
q�tz
Qt . The parameter estimates
used to calculate the pricing errors of this option price model are based on twodi¤erent estimation procedures. The �rst, often followed in practice, estimates theabove univariate process for �yt+1 with its volatility function, based only on stockmarket data. The second relies on a joint estimation of this and HN�s optionpricing formula, based on stock and options price data (e.g. ATM options). Theparameter estimates of these two procedures are given as follows:
$ = 8:3e�7; � = 4:5e�005; � = 0:79; = 57:12 and � = 0:39 and
$ = 2:7e�21; � = 5:3e�5; � = 0:80; = 48:71; � = 0:37 and ��;ATM = 4:34,
respectively. These are close to each other,especially those of �, and �. They arealso very close to those reported by Heston and Nandi (ibid).
Table 4: Option pricing performanceMRS (risk averse regime shifts) MRS (risk neutral regime shifts)OTM ATM ITM OTM ATM ITM
RMSE 4.2362 4.3358 4.4014 4.2684 4.4345 4.5826MAE 3.0270 3.0084 3.0506 2.9254 3.0388 3.0771
HN (joint estimates) HN (univariate estimates)RMSE 4.2519 4.3490 4.3935 4.2800 4.4033 4.5036MAE 3.1164 3.1901 3.1487 3.0852 3.1989 3.2047
27
A number of interesting conclusions can be drawn from the results of Table 4.First, they clearly indicate that ignoring the price of regime shift of sources of riskleads to higher pricing errors. These are higher for the ATM and ITM Europeancall options. Second, the pricing performance of the MRS model is better than thatof the HN model, which can be thought of as its competent given that GARCHe¤ects can mimic (or approximate) MRS type of shifts in volatility as mentionedbefore This result holds across all di¤erent moneyness levels examined. The aboveresults can be supported by the values of both the RMSE and MAE reported inthe table.
4 Conclusions
This paper has introduced a discrete time option pricing model for European callswhich assumes that the underlying asset (stock) follows a stochastic process whichallows for discontinuous regime shifts in the mean or volatility of stock marketreturns. These shifts re�ect switches among the alternative regimes of the stockmarket such as between the bull and bear regimes, or between the bear and crashregimes. We employ the model to assess if the risk of regime shifts is priced inthe options market, as many studies in the literature make the assumption thatinvestors are risk-neutral with regard to regime shifts and do price them explicitly.We apply the derived option pricing formula to a set of European call option
price data, and we show that regime shifts indeed constitute separate sources ofrisks which are priced in the market. We �nd that shifts from the bull to bearand from bear to crash regimes carry substantial prices of risk, a �nding whichis consistent with investors exhibiting signi�cant risk aversion to adverse regimechanges.
A Appendix
In this Appendix, we derive the main theoretical results of the paper. We alsopresent an algorithm calculating the occupation time probabilities.
A.1 Derivation of the mean of the risk neutral MRS process(7)
First, assume that the random variables of (4) are de�ned under measure Q. Then,add and subtract in this process interest rate rft and the yet undetermined com-pensating quantity A. This will yield
28
�tyt+� = �rft +
�(�2 � �1)�
1
2
��22 � �21
��O2;Q�
�A+q�21� + (�
22 � �21)O
2;Q� !Qt;� , (16)
where
!Qt;� = !t;� +
��1 � 1
2�21�� + 1
2��21 � �r
ftq
�21� + (�22 � �21)O
2;Q�
and
and O2;Q� denotes the occupation time variable O2� under risk neutral measuremeasure Q. To derive a closed-form solution for A, solve forwards equation (16)for the price Yt+� , i.e.
Yt+� = Yt exp
��rft +
�(�2 � �1)�
1
2
��22 � �21
��O2;Q� � A
+
q�21� + (�
22 � �21)O
2;Q� !Qt;�
�.
Taking the conditional on It expectation of the last relationship of Yt+� undermeasure Q yields
EQt (Yt+� ) = Yte�rft
�X�2=0
exp
��(�2 � �1)�
1
2
��22 � �21
���2 � A
+1
2�21� +
1
2
��22 � �21
��2
�P�O2;Q� = �2jIt
�. (17)
Substituting (17) into (6) gives the following relationship for A:
A = lnEQt�exp
�(�2 � �1)O2;Q�
�+1
2�21� : (18)
Substituting the last relationship into (16) yields the risk neutral representationof �tyt+� given by (7). The conditional on It mean of this process is given as
EQt (�tyt+� ) =
�rft �
1
�lnEQt
�exp
�(�2 � �1)O2;Q�
�� 12�21
�+
�(�2 � �1)�
1
2
��22 � �21
��O2;Q� .
29
Following analogous steps to the above, we can derive the risk neutral MRS rep-resentation of �tyt+� given by (12) and its mean, for the multi-regime case.
A.2 Proof of Proposition 1
To prove this proposition, �rst we will derive an analytic formula of the condi-tional on It probability density of log-price yt+� (or its implied log-return �tyt+� )under physical and risk neutral measures, denoted as fy (yt+� jIt) and fQy (yt+� jIt),respectively. The physical density fy (yt+� jIt) can be easily derived from process(3) by applying Bayes�rule. This will give
fy (yt+� jIt) =�X
�2=0
fy (yt+� jHt;� (�2; It)) Pr�O2� = �2jIt
�, (19)
where the information set Ht;� (�2; It), de�ned as
Ht;� (�2; It) = It [�O2� = �2
, (20)
constitutes an extension of It with the occupation time values �2, while the con-ditional on Ht;� (�2; It) density function fy (yt+� jHt;� (�2; It)) is normal given as
fy (yt+� jHt;� (�2; It)) =�2��2 (�2)
�� 12 exp
(� [yt+� � yt � � (�2)]
2
2�2 (�2)
). (21)
The mean and variance of this distribution are given as
� (�2) =
��1 �
1
2�21
�� +
�(�2 � �1)�
1
2
��22 � �21
���2
and �2 (�2) = �21� +��22 � �21
��2,
respectively.13
The risk neutral density fQy (yt+� jIt) has an analogous formula to its physical13These moments can be derived immediately by noticing that the conditional on the infor-
mation set Ht;� (�; It) distribution of the sum of random variablesP�
i=1 �(St+i)�t+i is givenas
�Xi=1
�(St+i)�t+ijHt;� (�; It) � N�0; ��21 + (�
22 � �21)�
�;
by �t � NIID (0; 1) and the assumption that �t and the vector os state variables St are inde-pendent, for all t.
30
counterpart, given by (19). It is de�ned as
fQy (yt+� jIt) =�X
�2=0
fQy (yt+� jHt;� (�2; It)) Pr[O2;Q� = �2jIt]; (22)
where
fQy (yt+� jHt;� (�2; It)) =�2��2 (�2)
�� 12 exp
(��yt+� � yt � �Q (�2)
�22�2 (�2)
);
�Q(�2) =
�rft �
1
�lnEQt [exp f(�2 � �1) �2g]�
1
2�21
��+
�(�2 � �1)�
1
2
��22 � �21
���2
and
�2(�2) = �21� +
��2i � �21
��2.
Having derived the analytic formula of the risk neutral density fQy (yt+� jIt),the option pricing formula (8), given by Proposition 1, can be derived using Cox,Ross and Rubinstein�s (1979) formula. According to this, a European call optionprice with � -periods to maturity at time t and strike price K satis�es the followingrelationship under measure Q:
Ct (�) = e��rft EQt [Yt+� �K]
+
= e��rft +ytEQt
�exp f�tyt+�g �
K
Yt
�+; (23)
where �tyt+� = yt+� � yt and [Yt+� �K]+ constitutes the payo¤ function of theoption. Using the analytic formula of the risk neutral density function fQy (yt+� jIt),given by (22), the last relationship can be written as follows:
Ct(�) = e��rft +yt
Z +1
�1
�exp f�tyt+�g �
K
Yt
�+fQ (yt+� jIt) dyt+�
=�X
�2=0
Yte��rft EQt [exp f�tyt+�g � ktjHt;� (�2; It)]
+ Pr�O2;Q� = �2jIt
�,
where kt = K=Yt and EQt [exp f�tyt+�g � ktjHt;� (�2; It)] is the expectation of con-
31
ditional density (21), i.e. �tyt+� jHt;� (�2; It) � N��Q(�2); �(�2)
2�.
Next, de�ne the random variable wt (�) as follows: �tyt+� = wt (�) + �Q(�2)�
�rft +�2(�2)2; where wt (�) jHt;� (�2; It) � N
��rft �
�2(�2)2; �2(�2)
�. Given this de�-
nition, the last relationship of the call option price Ct(�) can be rewritten as
Ct (�)
=
�X�2=0
� (�2)Yte��rft EQt
�exp fwt (�)g �
kt� (�2)
����Ht;� (�2; It)�+Pr�O2;Q� = �2jIt
�,
where
� (�2) = exp
��Q(�2)� �r
ft +
�2(�2)
2
�=
exp f(�2 � �1) �2gEQt [exp ((�2 � �1) �2)]
.
The last relationship proves Proposition 1. Under the assumptions of MRS process
(1) made in Section 2, Yte��rft EQt
hexp fwt (�)g � kt
�(�2)
���Ht;� (�2; It)i+constitutes
the conditional on �2 Black-Scholes option pricing formula which is denoted in theproposition as as CBSt (�2 (�2) ,�(�2); �) :
A.3 Proof of Proposition 2
To prove this proposition we will follow analogous steps to those of the proof ofProposition 1. In so doing, �rst note that the innovation error terms process !Qt;�and !t;� are linked through the following relationship:
!Qt;� = !t;� +
��1 � 1
2�21�� + lnEQ
hexp
nPNi=2(�i � �1)Oi;Q�
oi+ 1
2��21 � �r
ftq
�21� +PN
i=2 (�2i � �21)O
i;Q�
,
while the risk neutral density fQy (yt+� jIt) is given as
fQy (yt+� jIt) (24)
=X�N
::X�2
fQy�yt+� jHt;� (�
N�1; It)�Pr[O2;Q� = �2; ::; O
N;Q� = �N jIt],
32
where � = (�2; �3; ::; �N), Ht;� (�; It) =�O2� = �2; ::; O
N� = �N
[ It constitutes an
extension of the information set It with the vector of occupation time values �and
fQy (yt+� jHt;� (�; It)) =�2��2(�)
�� 12 exp
(���tyt+� � �Q(�)
�22 �2(�)
);
with
�Q(�) =
rft �
1
�lnEQt
"exp
(NXi=2
(�i � �1)� i
)#� 12�21
!�
+
NXi=2
�(�i � �1)�
1
2
��2i � �21
��� i
and
�2(�) = �21� +NXi=2
��2i � �21
�� i:
Using the analytic formula of the fQy (yt+� jIt) given by (24) and the fundamentaloption formula (23), we can derive the option pricing formula given by Proposition2.
A.4 An algorithm calculating the occupation time proba-bilities
We will describe this algorithm for the calculation of the physical values of theoccupation time probabilities Pr(Oi� ). An analogous procedure can be followed tocalculate the risk neutral values of them, denoted as Pr(Oi;Q� ).First, note that the joint probability of the occupation time values conditioned
on a value of the initial state S0 = ej can be written as
Pr�O1� = �1; O
2� = �2; ::; O
N�1� = �N�1jS0 = ej
�=
Pr�O1� = �1jS0 = ej
�� Pr
�O2� = �2jO1� = �1; S0 = ej
�:::
:::Pr�ON�1� = �N�1jO1� = �1; O2� = �2; ::; ON�2� = �N�2; S0 = ej
�,
where
Pr�O2� = �2jO1� = �1; S0 = ej
�=
(Pr�O2���1 = �2jS0 = ej
�if 0 � �2 � � � �1
0 otherwise
33
...
Pr�ON�1� = �N�1jO1� = �1; O2� = �2; ::; ON�2� = �N�2; S0 = ej
�=
(Pr�ON�1��PN�2i=1 �i
= �N�1jS0 = ej�
if 0 � �N�1 � � �PN�2
i=1 � i
0 otherwise.
Given the above relationships, the occupation time probabilities Pr(Oi� ) can becomputed recursively as follows.Consider the transition probability of moving from state i to state j in the
next period pij = Pr (St+1 = jjSt = i) and the transition probability matrix P �[p�j]1�i;j�N ; where the (j; i) element of matrix P equals the pij transition proba-bility. Using the Chapman-Kolmogorov identity, we can compute the u-th powerof the transition probability matrix P (u) � Pu, (u = 1; ::; �) ; where the (j; i)element of P (u) is de�ned as pij (u) = Pr (St+u = ejjSt = ei).Next, de�ne the �rst passage probability as
fij (t) = Pr (Yt = ej; Yt�1 6= ej; ::; Y1 6= ejjY0 = ei) ; i; j = 1; 2; ::; N ; t = 1; ::; � ;
where the �rst passage probability can be computed recursively using the followingequations:
fij = pij (t)�t�1Xk=1
pjj (t� k) fij (k) ; (t = 2; ::; �)
fij (1) = Pr (S1 = ejjS0 = ei) = pij:
Based on the above the de�nition of the �rst passage probability, we can obtainrecursive equations calculating the occupation time probability of i (i = f1; ::; Ng)given an initial state S0 = ej as follows:
Pr�Oi1 = 1jS0 = ej
�= pji
Pr�Oit = 0jS0 = ej
�= Pr (St 6= ei; ::; S1 6= eijS0 = ej) = 1�
tXu=1
fji (u) (t = 1; ::; �)
34
Pr�Oit = 1jS0 = ej
�=
t�1Xu=1
Pr (Su = ei; Su�1 6= ei; ::; S1 6= eijS0 = ej)
�Pr
tXl=u+1
e0iSl = 0
�����S0 = ej!
+Pr (St = ei; St�1 6= ei; ::; S1 6= eijS0 = ej)
=t�1Xu=1
fji (u) Pr�Oit�ujS0 = ej
�+ fji (t)
Pr�Oit = � ijS0 = ej
�=
t��i+1Xu=1
Pr (Su = ei; Su�1 6= ei; ::; S1 6= eijS0 = ej)
�Pr
tXl=u+1
e0iSl = � i � 1�����S0 = ej
!
=
t��i+1Xu=1
fji (u) Pr�Oit�u = � i � 1jS0
�; (t = 2; ::; � ; � i = 2; ::; t)
35
.
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