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A Regime-Switching Model for European Option PricingDavid D. Yao�, Qing Zhangy and Xun Yu ZhouzMarch 2003AbstractWe study the pricing of European-style options, with the rate of return and the volatilityof the underlying asset depending on the market mode or regime that switches among a�nite number of states. This regime-switching model is formulated as a geometric Brownianmotion modulated by a �nite-state Markov chain. With a Girsanov-like change of measure,we derive the option price using risk-neutral valuation, along with a system of partial dif-ferential equations that govern the option price, with smoothed boundary conditions. Wealso develop a numerical approach to compute the pricing formula, using a successive ap-proximation scheme with a geometric rate of convergence. Numerical examples demonstratethe presence of the volatility smile and volatility term structure with a simple, two-stateregime-switching model.Key words: Regime switching, option pricing, equivalent martingale measures, smoothedPDE, successive approximation, volatility smiles and term structure.
�Department of Industrial Engineering and Operations Research, Columbia University, New York, NY 10027;<[email protected]>, Fax: 212-854-8103. Research undertaken while on leave at the Department of SystemsEngineering and Engineering Management, The Chinese University of Hong Kong. Supported in part by NSFunder Grant DMI-00-85124, and by RGC Earmarked Grant CUHK4175/00E.yDepartment of Mathematics, University of Georgia, Athens, GA 30602; <[email protected]>, Fax: 706-542-2573. Supported in part by USAF Grant F30602-99-2-0548.zDepartment of Systems Engineering and Engineering Management, The Chinese University of Hong Kong,Shatin, Hong Kong; <[email protected]>, Fax: 852-2603-5505. Supported in part by RGC EarmarkedGrants CUHK4175/00E and CUHK4234/01E. 1
1 IntroductionThe classical Black-Scholes formula for option pricing is based on a geometric Brownian motionmodel to capture the price dynamics of the underlying security. The model involves two pa-rameters, the expected rate of return and the volatility, both are assumed to be deterministicconstants. It is well known, however, that the Black-Scholes model fails to re ect the stochasticvariability in the market parameters. Another widely acknowledged shortfall of this model isits failure to capture what is known as \volatility smile." That is, the implied volatility of theunderlying security (implied by the market price of the option on the underlying via the Black-Scholes formula), rather than being a constant, should change with respect to the maturity andthe exercise price of the option.Emerging interests in this area have focused on the so-called regime-switching model, whichstems from the need of more realistic models that better re ect random market environment.Since a major factor that governs the movement of an individual stock is the trend of thegeneral market, it is necessary to allow the key parameters of the stock to respond to the generalmarket movements. The regime-switching model is one of such formulations, where the stockparameters depend on the market mode (or, \regime") that switches among a �nite number ofstates. The market regime could re ect the state of the underlying economy, the general moodof investors in the market, and other economic factors. The regime-switching model was �rstintroduced by Hamilton [14] in 1989 to describe a regime-switching time series. Di Masi et al. [5]discuss mean-variance hedging for regime-switching European option pricing. To price regime-switching American and European options, Bollen [2] employs lattice method and simulation,whereas Bu�ngton and Elliott [3] use partial di�erential equations. Guo [13] and Shepp [24]use regime-switching to model option pricing with inside information. Duan et al. [6] establisha class of GARCH option models under regime switching. For the important issue of �ttingthe regime-switching model parameters, Hardy [15] develops maximum likelihood estimationusing real data from the S&P 500 and TSE 300 indices. In addition to option pricing, regime-switching models have also been formulated and investigated for other problems; see Zhang[30] for the development of an optimal stock selling rule, Zhang and Yin [31] for applicationsin portfolio management, and Yin and Zhou [28] for a dynamic Markowitz problem.In the regime-switching model, one typically \modulates" the rate of return and the volatil-ity by a �nite-state Markov chain �(�) = f�(t) : t � 0g, which represents the market regime.For example, �(t) 2 f�1; 1g with 1 representing the bullish (up-trend) market and -1 the bear-ish (down-trend) one. In general, we can take M = f1; 2; : : : ;mg. More speci�cally, let X(t),2
the price of a stock at time t, be governed by the following equation:dX(t) = X(t)[�(�(t))dt + �(�(t))dw(t)]; 0 � t � T ; X(0) = X0; (1)where X0 is the stock price at t = 0; �(i) and �(i), for each i 2M, represent the expected rateof return and the volatility of the stock price at regime i; and w(�) denotes the standard (one-dimensional) Brownian motion. Equation (1) is also called a hybrid model, where randomnessis characterized by the pair (�(t); w(t)), with w(�) corresponding to the usual noise involvedin the classical geometric Brownian motion model while �(t) capturing the higher-level noiseassociated with infrequent yet extremal events. For example, it is known that the up-trendvolatility of a stock tends to be smaller than its down-trend volatility. When the market trendsup, investors are often cautious and move slowly, which leads to smaller volatility. On the otherhand, during a sharp market downturn when investors get panic, the volatility tends to be muchhigher. (This observation is supported by an initial numerical study reported in Zhang [30] inwhich the average historical volatility of the NASDAQ Composite is substantially greater whenits price trends down than when it moves up.) Furthermore, when a market moves sidewaysthe corresponding volatility appears to be even smaller. Therefore, we can take the value of�(t) to bef�2 = severe downtrend (`crash');�1 = downtrend; 0 = sideways; 1 = rally; 2 = strong rallyg;then the sample paths of �(�) and X(�) is given in Figure 1. The `crash' state was reachedbetween sessions 70-80, which simulates steep downward movements in price (namely the dailyhigh is lower than last session's low).One objective of this paper is to price regime-switching European option in explicit forms.We propose two di�erent approaches to pricing the options. One is to derive a system of partialdi�erential equations (PDE's) that govern the option price. Note that the PDE's are derivedin Bu�ngton and Elliott [3] with a non-smooth boundary condition (the payo� function of theEuropean option). This non-smoothness leads to two major di�culties: First, the solution tothe system of PDE's may not be unique. This uniqueness is important, since without it thereis no guarantee that a solution returned by say, a numerical algorithm applied to the PDE's,is indeed the desired option price. Second, the solution to the system of PDE's may not bedi�erentiable, whereas the evaluation of option Greek letters (Delta, Gamma, Theta, Vega,etc.), which play an important role in sensitivity analysis and hedging strategies, requires thedi�erentiability (smoothness) of the pricing function. To address these issues, we introduce a3
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Figure 1: Sample Paths of �(�) and X(�).system of PDE's with a smoothed boundary conditions, and this system is proved to have aunique classical (smooth) solution. It is then shown that this smooth solution converges to theoriginal pricing function. As a result, one can use the approximate solution as the price of theoption for all practical purposes.While the above approach is perhaps mainly of theoretical interest, we also develop a secondapproach, which is easy to implement without having to solve di�erential equations. It is asuccessive approximation based on the �xed-point of a certain integral operator with a Gaussiankernel. Moreover, its convergence is shown to be geometric. Using this numerical approach,we demonstrate that our regime-switching model does generate the desired volatility smile andterm structure.We now brie y review other related literature. For derivative pricing in general, we referthe reader to the books by Du�e [7], Elliott and Kopp [9], Karatzas and Shreve [20], Musielaand Rutkowski [22], and Tompkins [25], among others. In recent years there has been exten-sive research e�ort in enhancing the classical geometric Brownian motion model. Merton [21]introduced additive Poisson jumps into the geometric Brownian motion, aiming at capturingdiscontinuities in the price trajectory. The drawback, however, is the di�culty in handlingthe associated dynamic programming equations, which become quasi-variational inequalities.Hull and White [18] studied stochastic volatility models, and showed that when the volatilityis uncorrelated with the asset price, the European option can be priced as the expected valueof the Black-Scholes price with respect to the distribution of the stochastic volatility. Thesemodels are revisited by Fouque et al. [11] using a singular perturbation approach, and by Heath4
and Platen [16] assuming a square root structure in volatility. Albanese et al. [1] studied amodel that is a composition of a Brownian motion and a gamma process, with the latter usedto rescale the time. In addition, the parameters of the gamma process are allowed to evolveaccording to a two-state Markov chain. While the model does capture the volatility smile, theresulting pricing formula appears to be quite involved and di�cult to implement.The rest of the paper is organized as follows. Our starting point is the hybrid model in(1) for the price dynamics of the underlying security, detailed in x2. We establish a Girsanov-like theorem (Lemma 1), which leads to an equivalent martingale measure, and therefore arisk-neutral pricing scheme. We then derive in x3 a system of partial di�erential equations(PDE's) that govern the option price, with smoothed boundary conditions. More precisely,this is a smoothed, �-approximation of the boundary conditions: as � ! 0, the approximationapproaches the exact solution { in the sense of the viscosity solution { to the PDE's. Inx4, we develop a numerical approach to compute the option price, a successive approximationtechnique with a geometric rate of convergence, based on the risk-neutral pricing scheme. In x5,we demonstrate via numerical examples that with a simple, two-state Markov chain modulatingthe volatility we can produce the anticipated volatility smile and volatility term structure.2 Risk-Neutral PricingA standard approach in derivative pricing is risk-neutral valuation. The idea is to derive asuitable probability space upon which the expected rate of return of all securities is equal tothe risk-free interest rate. Mathematically, this requires that the discounted asset price be amartingale; and the associated probability space is referred to as the risk-neutral world. Theprice of the option on the asset is then the expected value, with respect to this martingalemeasure, of the discounted option payo�. In a nutshell, this is also the route we are takinghere, with the martingale measure identi�ed in Lemma 1 below, and related computationalissues deferred to x4.Let (;F ;P) denote the probability space, upon which all the processes below are de�ned.Let f�(t)g denote a continuous-time Markov chain with state space M = f1; 2; : : : ;mg. Notethat for simplicity, we use each element inM as an index, which can be associated with a moreelaborate state description, for instance, a vector. Let Q = (qij)m�m be the generator of � withqij � 0 for i 6= j and Pmj=1 qij = 0 for each i 2 M. Moreover, for any function f on M wedenote Qf(�)(i) :=Pmj=1 qijf(j).Let X(t) denote the price of a stock at time t which satis�es (1). We assume that X0, �(�),and w(�) are mutually independent; and �2(i) > 0, for all i 2M.5
Let Ft denote the sigma �eld generated by f(�(u); w(u)) : 0 � u � tg. Throughout, all(local) martingales concerned are with respect to the �ltration Ft. Therefore, in the sequel weshall omit reference to the �ltration when a (local) martingale is mentioned. Clearly, fw(t)gand fw2(t)� tg are both martingales (since �(�) and w(�) are independent).Let r > 0 denote the risk-free rate. For 0 � t � T , letZt := exp�Z t0 �(u)dw(u) � 12 Z t0 �2(u)du� ;where �(u) := r � �(�(u))�(�(u)) : (2)Then, applying Ito's rule, we have dZtZt = �(t)dw(t);and Zt is a local martingale, with EZt = 1; 0 � t � T:De�ne an equivalent measure eP via the following:dePdP = ZT : (3)The lemma below is essentially a generalized Girsanov's theorem for Markov-modulatedprocesses. While results of this kind are considered well known, the authors did not �nd aconvenient reference which applied to the Markov switching mechanism used in this paper.Lemma 1 (1) Let ew(t) := w(t) � R t0 �(u)du: Then, ew(�) is a eP-Brownian motion.(2) X(0), �(�) and ew(�) are mutually independent under eP;(3) Dynkin's formula holds: for any smooth function F (t; x; i), we haveF (t;X(t); �(t)) = F (s;X(s); �(s)) + Z ts AF (u;X(u); �(u))du +M(t)�M(s);where M(�) is a eP-martingale and A is an generator given byAF = @@tF (t; x; i) + 12x2�2(i) @2@x2F (t; x; i) + rx @@xF (t; x; i) +QF (t; x; �)(i):This implies that (X(t); �(t)) is a Markov process with generator A.6
Proof. De�ne a row vector (t) = �If�(t)=1g; : : : ; If�(t)=mg� ;where IA is the indicator function of a set A. Letz(t) = (t)�(0)� Z t0 (u)Qdu:Note that both f(z(u); w(u)) : u � tg and f(�(u); w(u)) : u � tg generate the same sigma �eldFt. Thus, (z(t); w(t)) is a P-martingale. Let � denote a column vector and � a scaler. De�neV (t) = z(t)� + w(t)�:Then, V (t) is a P-martingale. Let�(t) = Z t0 �(u)dw(u) � 12 Z t0 �(u)2du:Then, for each � and �, eV (t) = V (t)� hV; �itis a eP-martingale, where hV; �it = � Z t0 �(u)dw(u):Thus, eV (t) = z(t)� + ew(t)�:Hence, in view of Elliott [8, Thm. 13.19] (z(t); ew(t)) is a eP-martingale. Moreover, since ew(�)and ( ew2(t)� t) are both eP-martingales, ew(�) is a eP-Brownian motion (see [8, Cor 13.25]).We next show the mutual independence of X0, �(�) and ew(�) under eP. Note that ZT is FTmeasurable, and EZT = 1. Let �1 denote a random variable, measurable with respect to X0.Then, making use of (3), we haveeE�1 = E(ZT �1) = (EZT )(E�1) = E�1:Furthermore, for any FT measurable random variable �2, we haveeE(�1�2) = E(ZT �1�2) = (E�1)E(ZT �2) = (eE�1)(eE�2):This implies the independence between X0 and (�(�); ew(�)) up to time T . To show the inde-pendence between �(�) and ew(�), for a given f(x; i), letA0f(x; i) = 12 @2@x2 f(x; i) +Qf(x; �)(i):7
Then, the associated martingale problem has a unique solution (see, e.g., Yin and Zhang [26,p. 199]). Using Ito's rule, we can show that (�(�); ew(�)) is a solution to the martingale problemunder eP. Since (�(�); w(�)) is also a solution to the same martingale problem under P, it mustbe equal in distribution to (�(�); ew(�)). The independence between �(�) and ew(�) then followsfrom the independence between �(�) and w(�).Under eP, (1) becomesdX(t) = X(t)[rdt+ �(�(t))d ew(t)]; 0 � t � T ; X(0) = X0:We now prove the Dynkin's formula. First, writeF (X(t); �(t)) = (t) �F (X(t));where �F (X(t)) = (F (X(t); 1); : : : ; F (X(t);m))0 . Applying [8, Cor. 12.22], we havedF (X(t); �(t)) = (t)d( �F (X(t)) + (d(t)) �F (X(t)) + d[; F ]t:Since is a pure jump process, we have [; F ]t = 0. In addition, we have(t)Q(F (X(t); 1); : : : ; F (X(t);m))0 = QF (X(t); �)(�(t)):Hence, Dynkin's formula follows.Finally, the Markov property of (X(t); �(t)) under eP can be established following the sameargument as in Ghosh et al. [12]. 2Therefore, in view of Hull [17] and Fouque et al. [11], (;F ; fFtg; eP) de�nes a risk-neutralworld. Moreover, e�rtX(t) is a eP-martingale.Consider a European-style call option with strike price K and expiration date T . Leth(x) = (x�K)+ := maxfx�K; 0g:The call option premium at time s, given the stock price X(s) = x and the state of the Markovchain �(s) = i, can be expressed as follows:c(s; x; i) = eE[e�r(T�s)h(X(T ))jX(s) = x; �(s) = i]: (4)Throughout, we shall focus on call options only. For European put options, the analysis issimilar, with the h function changed to h(x) = (K � x)+.8
3 Smoothed PDE'sThrough a Feynman-Kac type formula, we can verify that c(s; x; i) satis�es, formally, the fol-lowing system of PDE's:@@sc(s; x; i) + 12x2�2(i) @2@x2 c(s; x; i) + r[x @@xc(s; x; i) � c(s; x; i)]+Qc(s; x; �)(i) = 0; i = 1; 2; � � � ;m; (5)with the boundary condition:c(T; x; i) = h(x); i = 1; 2; � � � ;m: (6)We say \formally" as the function c is not di�erentiable in (s; x) due to the non-di�erentiabilityof the function h. However, it can be shown that c is a viscosity solution to (5), i.e., a weakversion of the solution that does not require di�erentiability; refer to Zhang and Yin [31] (alsosee Fleming and Soner [10] and Yong and Zhou [29]). On the other hand, the non-smoothboundary condition implies that the solution to the system of PDE's may not be unique. Asdiscussed in the introduction, the non-uniqueness and non-di�erentiability of solutions to thePDE's lead to di�culties in obtaining and analyzing the pricing function.To get around, we study the PDE's in (5) with a smoothed boundary condition; speci�cally,we replace the function h by h�, for a given � > 0, de�ned via a convolution:h�(x) = Z 1�1 h(y)k�(x� y)dy;with the kernel k�(x) = 8>>><>>>: a� exp �2x2 � �2! if jxj < �;0 if jxj � �;where a is a constant such that R1�1 k�(x)dx = 1. Then, it is easy to show that, for any � > 0,(i) h�(x) 2 C1;(ii) jh�(x)� h(x)j � � for all x;(iii) for n = 1; 2; 3; 4, the derivatives dnh�=dxn are bounded on IR.Let c�(s; x; i) = eE[e�r(T�s)h�(XT )jX(s) = x; �(s) = i]:9
We use c� to approximate c. Note that, from (ii) above, we havejc�(s; x; i) � c(s; x; i)j � e�r(T�s)eEjh�(XT )� h(XT )j � �;for all (s; x; i). This implies, in particular, thatlim�!0 c�(s; x; i) = c(s; x; i):In view of this, to evaluate c, we only need to �nd c� for � su�cient small. For example, if wetake � < 0:01, then jc�(s; x; i) � c(s; x; i)j < 0:01;i.e., c�(s; x; i) approximates c(s; x; i) up to one penny, which is su�cient for all practical pur-poses.We need two lemmas. The �rst one presents a technical result that is needed in the secondlemma.Denote the following sigma �elds:D1 = �f�(u) : s � u � Tg;D2 = �f ew(u)� ew(s) : s � u � Tg;and D = �f�(u); ew(u)� ew(s) : s � u � Tg:Then, D1 is independent of D2 under eP and D = �fD1;D2g.Lemma 2. Let g(y; i) = eE[F (y; �)j�(s) = i] for a given Borel function F and a D measurablerandom variable �. Then,g(Y (s); �(s)) = eE[F (Y (s); �)jY (s); �(s)]; eP a.s. (7)As a result, under the induced probability bP(�) = eP((Y (s); �(s)) 2 �),g(y; i) = eE[F (Y (s); �)jY (s) = y; �(s) = i]; bP a.s.Proof. It su�ces to show (7) for nonnegative F because one may always write F as thedi�erence of two nonnegative functions, e.g., F = F+ � F�. Let H denote a class of randomvariables as follows:H = nf(Y (s); �) � 0 : for Borel f such that (7) holds with F replaced by fo:10
Then we can show that H is a �-system [4]. Moreover, for random variables �1; �2; �3, we haveeE[�1�2j�3] = (eE�1)eE[�2j�3];provided that �1 is independent of (�2; �3). Using this fact together with the Markov propertyof (Y (�); �(�)) and the mutual independence of Y0, �(�) and ew(�), we can show that H containsall indicator functions of the form IAIB\C for A 2 �fY (s)g, B 2 D1 and C 2 D2. Therefore,H must contain all �fY (s);Dg-measurable random variables. This implies (7). 2To present the second lemma, �x � > 0, let y = log x, and denote�(s; y; i) := c�(s; x; i) = eE[e�r(T�s)h�(eY (T ))jY (s) = y; �(s) = i]: (8)Then, the di�erentiability of c� with respect to x is equivalent to that of � with respect to y.Lemma 3. For each i 2M, � 2 C1;2([s; T ]; IR). Moreover,�(s; y; i) + ���� @@y�(s; y; i)���� + ����� @2@y2�(s; y; i)����� � Ce2y;for some constant C.Proof. Note that, in view of Lemma 2, we have (s; y; i) = eEs;i[e�r(T�s)h�(ey+H(s;T ))]; (9)where eEs;i[�] := eE[�j�(s) = i].Recall that Y (t) = y +H(s; t). Given �y, we have1�y [h�(ey+�yeH(s;t))� h�(eyeH(s;t))]= eyeH(s;t)h0�(eY (t)) + �y2 [e�eH(s;t)h0�(e�+H(s;t)) + e2�e2H(s;t)h00� (e�+H(s;t))];for some � 2 (y; y +�y). It is readily veri�ed thateE exp � Z Ts �(�(u))d ew(u)! <1; for any �nite � > 0:From the boundedness of h0� and h00� , it follows thateEs;i ��y2 [e�eH(s;t)h0�(e�+H(s;t)) + e2�e2H(s;t)h00� (e�+H(s;t))]�! 0:Therefore, @�@y = lim�y!0 1�y [�(s; y +�y; i)� �(s; y; i)]= eEs;i[e�r(T�s)ey+H(s;T )h0�(ey+H(s;T ))]: (10)11
Similarly, we can show that @2�=@y2 exists and@2�@y2 = eEs;ife�r(T�s)[ey+H(s;T )h0�(ey+H(s;T )) + e2y+2H(s;T )h00� (ey+H(s;T ))]g:Next we prove the existence of @�=@s. First, for s � t � T , by conditioning on (Y (t); �(t)),we have �(s; y; i) = eEs;y;i[e�r(t�s)�(t; Y (t); �(t))]: (11)Similar to (9), we have (s; y; i) = eEs;i[e�r(t�s)�(t; y +H(s; t); �(t))]:It follows that @@y�(s; y; i) = eEs;i �e�r(t�s) @@y�(t; y +H(s; t); �(t))� ; (12)and @2@y2�(s; y; i) = eEs;i "e�r(t�s) @2@y2�(t; y +H(s; t); �(t))# : (13)Note that, for � small enough, jh�(x)j � jxj + 1. Thus, �(s; y; i) � Cey. Similarly, using(10) and the boundedness of h0�, we have@@y�(s; y; i) � Cey; @2@y2�(s; y; i) � Ce2y:We next show that �(s+�s; y; i)! �(s; y; i) as �s! 0:First, using (11), we have�(s; y; i) = e�r�seEs;y;i[�(s+�s; Y (s+�s); �(s+�s))]: (14)Moreover, applying the mean-value theorem with respect to y, we obtain�(s+�s; Y (s+�s); �(s+�s))= �(s+�s; y; �(s+�s))+(Y (s+�s)� y) @@y�(s+�s; �; �(s+�s));12
for some � between y and Y (s+�s). Taking conditional expectation on both sides, we haveeEs;y;i[�(s+�s; Y (s+�s); �(s+�s))]= eEs;y;i[�(s+�s; y; �(s+�s))] +O(p�s)= mXj=1�(s+�s; y; j)P(�(s +�s) = jj�(s) = i) +O(p�s)= mXj=1�(s+�s; y; j)(�ij +O(�s)) +O(p�s)= �(s+�s; y; i) +O(p�s);where �ij = 1 if i = j and �ij = 0 otherwise. Thus,�(s+�s; y; i)� �(s; y; i) = (1� e�r�s)�(s+�s; y; i) +O(p�s)! 0:Similarly, using (12) and (13), we can show, as �s! 0,@@y�(s+�s; y; i)! @@y�(s; y; i);and @2@y2�(s+�s; y; i)! @2@y2�(s; y; i):Finally, we show the di�erentiability of � with respect to s. Applying Dynkin's formula to�(t0; Y (t); �(t)) with a �xed t0, we obtaineEs;y;i[�(t0; Y (t); �(t))] = �(t0; y; i) +R(s; t); (15)where R(s; t) = eEs;y;i"Z ts [r � 12�2(�(u))] @@y�(t0; Y (u); �(u))+12�2(�(u)) @2@y2�(t0; Y (u); �(u)) +Q�(t0; Y (u); �)(�(u))!#du:Setting t = t0 = s+�s and using (14) and (15), we have1�s [�(s+�s; y; i)� �(s; y; i)] = er�s � 1�s !�(s; y; i)� R(s; s+�s)�s :It is easy to see that R(s; s+�s)�s13
has a limit. Therefore, @@s�(s; y; i) = lim�s!0 1�s [�(s+�s; y; i)� �(s; y; i)]exists. This completes the proof. 2Theorem 1. For a �xed small � > 0, c�(s; x; i) 2 C1;2([s; T ]; IR) and is the unique solution to(5) with boundary condition c�(T; x; i) = h�(x).Proof. Recall, from (8), that c�(s; x; i) = �(s; y; i)with y = log x. It su�ces to show that � is the unique solution to the following system ofequations @@s�(s; y; i) + 12�2(i) @2@y2�(s; y; i) + [r � 12�2(i)] @@y�(s; y; i)�r�(s; y; i) +Q�(s; y; �)(i) = 0; i = 1; 2; � � � ;m; (16)with boundary �(T; y; i) = h�(ey); i = 1; 2; � � � ;m:For any s � t1 < t2 � T , we have, in view of (11),eEt1;y;i[e�rt2�(t2; Y (t2); �(t2))] = e�rt1�(t1; y; i):Apply Dynkin's formula to e�rt�(t; Y (t); �(t)) to obtain0 = eEt1;y;i"Z t2t1 e�ruf�r�(u; Y (u); �(u)) + @@s�(u; Y (u); �(u))+[r � 12�2(�(u))] @@y�(u; Y (u); �(u)) + 12�2(�(u)) @2@y2�(u; Y (u); �(u))+Q�(u; Y (u); �)(�(u))gdu#:Since t1 and t2 are arbitrary, (16) follows.On the other hand, suppose (16) has another solution �1. Then, consideringd[e�rt�1(t; Y (t); �(t))];and integrating over [s; T ], we have�1(s; y; i) = eEs;y;i[e�r(T�s)h�(eY (T ))] = �(s; y; i):Thus the solution must be unique. 2. 14
4 A Numerical Approach via Successive ApproximationsThe smoothed PDE's developed in the last section are amenable to numerical techniques thatsolve this type of parabolic PDE's with smooth solutions, including the �nite di�erence approach(see e.g., John [19]), the domain decomposition technique (see e.g., Quarteroni and Valli [23]),among others.Here, we develop another numerical approach that is directly based on the risk-neutralvaluation in x2, without the need of solving any PDE's. LetH(s; t) := Z ts [r � 12�2(�(u))]du + Z ts �(�(u))d ew(u)and Y (t) := y +H(s; t); with y = log x:Ito's rule implies X(t) = exp[Y (t)]:Let (s; y; i) = e�yc(s; ey; i):Then, combining the above with (4) and taking into account h(x) � jxj, we have (s; y; i) � e�y�r(T�s)eE[ey+H(s;T )jY (s) = y; �(s) = i] � C;for some constant C. Let 0(s; y; i) = e�yeE[e�r(T�s)h(ey+H(s;T ))jY (s) = y; �(u) = i; s � u � T ];which corresponds to the case when �(�) has no jump in [s; T ]. Then, 0(s; y; i) = e�y�r(T�s) Z 1�1 h(ey+u)N(u;m(T � s; i);�2(T � s; i))du; (17)where N is the Gaussian density function with meanm(t; i) = [r � 12�2(i)]t;and variance �2(t; i) = �2(i)t:Note that, for each i, ey 0(s; y; i) gives the standard Black-Scholes price, as expected.If qii = 0, then (s; y; i) = 0(s; y; i). For qii 6= 0, let� = infft � s : �(t) 6= �(s)g;15
i.e., � is the �rst jump epoch of �(�). Then,P(� > uj�(s) = i) = eqii(u�s):For any bounded measurable function f on [0; T ]� IR�M, de�ne its norm as follow:jjf jj = sups;y;i jf(s; y; i)j:This induces a Banach space S of all the bounded measurable functions on [0; T ] � IR �M.Also, de�ne a mapping on S:(T f)(s; y; i)= Z Ts Xj 6=i e�r(t�s)�Z 1�1euf(t; y + u; j)N(u;m(t � s; i);�2(t� s; i))du�qijeqii(t�s)dt: (18)Theorem 2. (1) is a unique solution to the equation (s; y; i) = T (s; y; i) + eqii(T�s) 0(s; y; i): (19)(2) Let 0 = 0; and de�ne f ng, for n = 1; 2; : : :, recursively as follows: n+1(s; y; i) = T n(s; y; i) + eqii(T�s) 0(s; y; i): (20)Then, the sequence f ng converges to the solution .Proof. First, we note thate�yeEs;y;i[e�r(T�s)h(eY (T ))If�>Tg] = eqii(T�s) 0(s; y; i);where eEs;y;i[�] := eE[�jY (s) = y; �(s) = i]. Therefore, (s; y; i) = e�yeEs;y;i[e�r(T�s)h(eY (T ))If��Tg] + eqii(T�s) 0(s; y; i): (21)By conditioning on � = t, we write its �rst term as follows:Z Ts e�y�r(T�s)eEs;y;i[h(eY (T ))j� = t](�qiieqii(t�s))dt:Recall that � is the �rst jump time of �(�). Therefore, given f� = tg, the post-jump distributionof �(t) is equal to qij=jqiij, j 2M. Moreover,Y (t) = y + [r � 12�2(i)](t � s) + �(i)[ ew(t)� ew(s)]; (22)16
which has a Gaussian distribution and is independent of �(�). In view of these, it follows that,for s � t � T , eEs;y;i[h(eY (T ))j� = t]= eEs;y;i[eEs;y;i[h(eY (T ))jY (t); �(t)]j� = t]= eEs;y;i[eY (t)er(T�t) (t; Y (t); �(t))j� = t]= Xj 6=i qij�qii Z 1�1 ey+uer(T�t) (t; y + u; j)N(u;m(t � s; i);�2(t� s; i))du:Thus, the �rst term in (21) is equal toZ Ts 0@Xj 6=i e�r(t�s) Z 1�1 eu (t; y + u; j)N(u;m(t � s; i);�2(t� s; i))du1A qijeqii(t�s)dt:Let �(i) = Z Ts Xj 6=i e�r(t�s) �Z 1�1 euN(u;m(t� s; i);�2(t� s; i))du� qijeqii(t�s)dt:We want to show that 0 � �(i) < 1, for i = 1; 2; : : : ;m. In fact, letA(u; i) = Z 1�1 euN(u;m(t� s; i);�2(t� s; i))du:Then it is readily veri�ed thatA(u; i) = exp[m(u; i) + �2(u; i)] = exp (ru) :Thus, �(i) = Z Ts Xj 6=i e�r(t�s)A(t� s; i)qijeqii(t�s)dt = 1� eqii(T�s) < 1; (23)when qii 6= 0. Let � = maxf�(i) : i 2Mg. Then, 0 � � < 1 andjjT f jj � �jjf jj;i.e., T is a contraction mapping on S. Therefore, in view of the contraction mapping �xedpoint theorem, we know equation (19) has a unique solution. This implies also the convergenceof the sequence f ng to . 2Note that the convergence of n to is geometric. In fact, it is easy to see that n+1 � = T ( n � );17
which implies jj n+1 � jj � �jj n � jj:Therefore, jj n � jj � C�n;for some constant C. In addition, the convergence rate � depends on the jump rates of �(�).By and large, the less frequent the jumps, the faster is the convergence. This can be seen from(23).Therefore, to evaluate , we solve equation (19) via the successive approximations in (20).Finally, the call option price is c(s; x; i) = x (s; log x; i): (24)(Recall y = log x.)5 Volatility Smile and Term StructureWe now illustrate the volatility smile and volatility term structure implied in our model. Weshall focus on a special case, in which the volatility (as well as the return rate) is modulatedby a two-state Markov chain, i.e., M = f1; 2g andQ = 0BBB@ �� �0 0 1CCCA :That is, state 2 is an absorbing state. As it will be demonstrated the volatility smile presentsin even such a simple case. If we take �(s) = 1, then there exists a stopping time � such that(� � s) is exponentially distributed with parameter � and�(t) = 8>>><>>>: 1 if t < �2 if t � �:Therefore, the volatility process �(�(t)) jumps at most once at time t = � . Its jump size isgiven by �(2) � �(1), and the average sojourn time in state 1 (before jumping to state 2) is1=�. For instance, if the time unit is one year and � = 6, then it means that the expectedtime for the volatility to jump from �(1) to �(2) is two months. In this case, the volatility ischaracterized by a vector (�(1); �(2); �). 18
Applying the successive approximation in x4, we have (s; y; 1)= Z Ts e�r(t�s) �Z 1�1 eu (t; y + u; 2)N(u;m(t � s; 1);�2(t� s; 1))du� �e��(t�s)dt+e��(T�s) 0(s; y; 1); (25)and (s; y; 2) = 0(s; y; 2); (26)where 0(s; y; i); i = 1; 2, are de�ned in (17). Alternatively, following the PDE method in x3,from (5), we have@@sc(s; x; 1) + 12x2�2(1) @2@x2 c(s; x; 1) + r[x @@xc(s; x; 1) � c(s; x; 1)]+�(c(s; x; 2) � c(s; x; 1)) = 0;@@sc(s; x; 2) + 12x2�2(2) @2@x2 c(s; x; 2) + r[x @@xc(s; x; 2) � c(s; x; 2)] = 0;with the boundary condition c(T; x; 1) = c(T; x; 2) = h�(x) for some � small enough.Next, we present several numerical examples using the successive approximation approach.Let s = 0. Given the risk-free rate r, the current stock price x, the expiration date T , the strikeprice K, and the volatility vector (�(1); �(2); �), in view of (24), (25) and (26), the call optioncan be priced as follows:c(0; x; 1)= x Z T0 e�rt �Z 1�1 eu 0(t; u+ log x; 2)N(u;m(t; 1);�2(t; 1))du� �e��tdt+xe��T 0(0; log x; 1): (27)This pricing formula consists of two parts: the classical Black-Scholes part (with no jump)and a correction part. In addition, it is a natural extension to the classical Black-Scholes formulaby incorporating a possible volatility jump, which is usually adequate for near term options.Moreover, both of these two parts are given in analytic form which is extremely valuable forfurther analysis including evaluation of option Greeks, and various numerical comparisons.Applying the standard Black-Scholes formula to the above price, we can derive the so-calledimplied volatility using the standard Black-Scholes formula.19
The following numerical cases illustrate the volatility smile and volatility term structureimplied in our model. In all cases, we �x r = 0:04 and x = 50, while varying the otherparameters. First we consider the cases with �(1) � �(2). Let�� = f1; 2; : : : ; 20g;�K = f30; 35; : : : ; 70g;�T = f20=252; 40=252; : : : ; 240=252g;�� = f0; 0:1; : : : ; 2g:Case (1a): Here, we �x T = 60=252 (three months till expiration), �(1) = 0:3, �(2) = 0:8,� 2 ��, and K 2 �K . We plot the implied volatility against the strike price (K) and the jumprate (�) in Fig. 2 (a).As can be observed from Fig. 2 (a), for each �xed � 2 ��, the implied volatility reaches itsminimum at K = 50 (at money) and increases as K moves away from K = 50. This is thewell-known volatility smile phenomenon in stock options ([17]). In addition, for �xed K 2 �K ,implied volatility is increasing in �, corresponding to a sooner jump from �(1) to �(2).Case (1b): In this case, we take �(1) = 0:3 and �x � = 1, and replace the �-axis in Fig. 2(a) bythe volatility jump size �(2)��(1) 2 ��. As can be observed from Fig. 2 (b), the smile increasesin the jump size. In addition, the implied volatility is an increasing function of �(2)� �(1), foreach �xed K 2 �K .Case (1c): In this case, we take �(1) = 0:3, �(2) = 0:8 and K = 50, and replace the strike pricein Fig. 2(a) by the expiration date (T ). Then, Fig. 2 (c) shows that for �xed �, the impliedvolatility increases in T . Similarly, for �xed T , the implied volatility also increases in �.Case (1d): Here, we �x �(1) = 0:3, � = 1, K = 50, and continue with Case (1c), but replace �by the volatility increase. Then, implied volatility is also increasing in �(2)� �(1) 2 ��.Whereas in the above cases we have �(1) < �(2), in the next set of cases we consider�(1) � �(2). In these cases, the market anticipates a decline in volatility.Case (2a): We take �(1) = 0:8 and �(2) = 0:3, with T = 60=252, � 2 ��, and K 2 �K . Theimplied volatility against the strike price (K) and the jump rate (�) is plotted in Fig. 3 (a).This case is similar to Case (a). 20
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Figure 2: Volatility Smile and Term Structure (�(1) � �(2))21
Case (2b): Here, we let �(1) = 2:3, �x � = 1, and replace the �-axis in Fig. 3(a) by �(2)��(1) 2�1�, where �1� = f0;�0:1; : : : ;�2g. As can be seen from Fig. 3 (b), the smile increases in thejump size j�(2) � �(1)j.Case (2c): In this case, we take �(1) = 0:8, �(2) = 0:3 and K = 50. In contrast to the earlierCase (c), Fig. 3 (c) shows that for �xed �, the implied volatility decreases in (T ) (and also in(�) with �xed T ).Case (2d): In this last case, we �x �(1) = 2:3, � = 1, and K = 50. The implied volatilitydecreases in (T ) and also in j�(2) � �(1)j.The above examples illustrate the clear advantage of the Markov modulated volatility model| its striking simplicity. In particular, it requires fewer parameters than most stochasticvolatility models.To apply the model in practice, one needs to estimate the parameters �(i), �(i), and Q. Fora two-state Markov chain, a simple procedure to estimate these parameters is given in [30]. Formore general Markov chains, refer to the approach in [27] based on stochastic approximation.
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Figure 3: Volatility Smile and Term Structure (�(1) � �(2))23
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