Pricing exotic options under regime switching: A Fourier transform method

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1 Pricing exotic options under regime switching: A Fourier transform method Peter Hieber Lehrstuhl für Finanzmathematik M13, Technische Universität München, Parkring 11, Garching-Hochbrück, Germany, hieber@tum.de. Abstract This paper considers the valuation of exotic options i.e. digital, barrier, and lookback options in a Markovian, regime-switching, Black-Scholes model. In Fourier space, analytical expressions for the option prices are derived via the theory on matrix Wiener-Hopf factorizations. A comparison to numerical alternatives, i.e. the Brownian bridge algorithm or a finite element scheme, demonstrates that the given formulas are easy to implement and lead to accurate and unbiased price estimates. Keywords: regime switching, Markov switching, Wiener Hopf factorization, option pricing. Classification: 6G4, 6J, 6J7, 6J8. The natural and intuitive idea of regimes in financial time series has become a widely accepted feature. Following Hamilton [1989] s seminal work, regime switching model constitute a very simple extension of the Black-Scholes model to stochastic volatility. A continuous time, finite-state, Markov chain generates the switches between the model parameters and thus allows to incorporate the impact of structural changes in the economic conditions on the price dynamics. Typically, two or three regimes provide a good fit to monthly stock market returns. One reason for their popularity is the fact that regime switching models are conceptionally very simple conditional on the regimes, the innovations are normally distributed and thus analytically tractable. Nevertheless, they can generate many nonlinear effects like heavy tails or volatility clusters. Furthermore, they allow us to depart from the unsatisfactory assumption of stationary increments in Lévy models. The aim of this paper is to price exotic options in a regime switching model. The contributions are twofold: 1 We show how several exotic options i.e. digital, barrier, and lookback options can be priced using the theory on matrix Wiener Hopf factorizations. We compare the results to numerical alternatives, i.e. the Brownian bridge algorithm, an analytic approximation by Lo et al. [3], Elliott et al. [14], and a backward finite element scheme. Among others, Jiang and Pistorius [8] showed that the Fourier transform of the first-passage time in a regime switching model can be represented as a function of the matrix Wiener Hopf factorization see also London et al. [198], Kennedy and Williams [199], Barlow et al. [199], Rogers [1994], Asmussen [1995]. In a regime switching model, the matrix Wiener Hopf factorization is determined by a quadratic matrix equation. Generally, the factorization has to be computed numerically see, for 1

2 1 Model description example, Rogers [1994] 1. However, in the case of or 3 regimes or in the case of a zero interest rate, the matrix Wiener Hopf factorization can be derived analytically see, for example, Hieber [1], Escobar et al. [13]. If the Fourier transform of the first-passage time is known, this allows us to price many exotic derivatives i.e. digital, barrier, and lookback options, see also Jiang and Pistorius [8]. Alternatively, many numerical techniques have been developed to price exotic options in a regime switching environment. Lo et al. [3] and Elliott et al. [14] present semi-analytical approximations of barrier option prices in a regime switching model. Naik [1993], Boyle and Draviam [7], Kim et al. [8], Eloe et al. [9], Elliott et al. [14] and many others derive the coupled Black- Scholes type partial differential equations for several exotic options and solve them numerically. Metwally and Atiya [] and Hieber and Scherer [1] use a conditional Monte-Carlo technique called Brownian bridge algorithm to price digital and barrier options under regime switching. This paper is organized as follows: Section 1 introduces regime switching models. The main theoretical results on the pricing of exotic options under regime switching are given in Section. Section 3 compares the results to alternative numerical schemes. Finally, Section 4 concludes and discusses possible extensions to, for example, two-sided barriers or exponential jump-diffusions. 1 Model description On the filtered probability space Ω, F, F, Q, we consider the process S = {S t } t described by ds t S t = rdt + σ Zt dw t, S =: expx >, 1 where r is the risk-less interest rate, σ Zt the regime-dependent volatility, Z = {Z t } t {1,,...,M} a time-homogeneous Markov chain with intensity matrix Q, and W = {W t } t an independent Brownian motion. The initial value is S = e x R +. The filtration F = {F t } t is generated by the pair W, Z, i.e. F t = σ{w s, Z s : s t}. The model is fully determined if an initial state or, more generally, an initial distribution := QZ = 1, QZ =,...,QZ = M on the states is defined. The meaning of the entries q ij := Q i, j of the intensity matrix Q can easily be explained: The time spent in state i is exponentially distributed with intensity λ = q ii >. If a state change from the current state i occurs, the probability of moving to state j i is q ij /q ii note that, since Q is an intensity matrix, i j q ij = q ii. For a matrix Q C M M, we define the matrix exponential expq via the power series expq := 1 Numerical solutions to the matrix Wiener Hopf factorization are possible in more general model settings see, among others, Boyarchenko and Levendorskiĭ [8], Jiang and Pistorius [8], Kudryavtsev and Levendorskiĭ [1], Mijatović and Pistorius [13]. An intensity matrix has negative diagonal and non-negative off-diagonal entries. Each row sums up to zero. n= Q n n!.

3 Option pricing Then, the characteristic function ϕ t u, := E [ exp iulns t x ] of log-returns in a regime switching model is given by see, e.g., Buffington and Elliott []; Elliott et al. [5] ϕ t u, = exp Q t + iur Σ u /t 1, where Σ := diagσ 1, σ,...,σ M, exp denotes the matrix exponential function, transpose, and 1 a column vector of ones of appropriate size. The first-passage times on two constant barriers e b < S = e x < e a are defined as { inf { t : S t / e b, e a}, if such a t exists, T ab :=, if S t never hits the barriers. 3 Here, T ab is the first time the process S t hits one of the two barriers e a or e b. The corresponding first-passage time densities are for t, ] denoted f ab t, := Q T ab dt. 4 Option pricing In the following, we price exotic options in a regime switching model. Section.1 deals with the general case of M regimes: After pricing call options, we recall the Fourier transform of the first-passage time in regime switching models and introduce the matrix Wiener Hopf factorization. Digital, barrier, and lookback options can then be represented as Fourier integrals see also Jiang and Pistorius [8]. In Section. we exploit that the matrix Wiener Hopf factorization can be solved analytically in the case of regimes see, for example, Hieber [1]..1 M states As the characteristic function of log-returns in a regime switching model is known see Equation, it is possible to price vanilla call options via Fourier pricing. If there are multiple regimes, this is more convenient than the M-fold integrals presented in, for example, Elliott et al. [5]. Theorem 1 Vanilla call options Consider the regime switching model as defined in Equation 1. Under Q, the price of a call option with strike K and time to maturity T is recovered as CS, T, = e rt S e α iθ lnk/s ϕ T θ 1 + αi, α + α θ dθ, 5 + iα + 1θ where α [1, ] is an arbitrary constant and ϕ t u, the characteristic function of log-returns in a regime switching model given in. Proof See, for example, Carr and Madan [1999], Raible []. 3

4 .1 M states The first-passage time densities f ab t, can be expressed in terms of the matrix Wiener Hopf factorization see, for example, Jiang and Pistorius [8] and the references therein. The one-sided densities f a, t, and f, b t, are recalled in Theorem. Note that the class of irreducible M M generator matrices non-negative off-diagonal entries and non-positive row sums is denoted by Q M. Theorem First-passage time densities Consider the regime switching model as defined in Equation 1 with initial distribution on the states R 1 M. For two constant barriers e b < S = e x < e a, we get for t, 3 f a, t, = 1 f, b t, = 1 e iut exp Q + a x 1 du, 6 e iut exp Q x b 1 du, 7 where exp denotes the matrix exponential and 1 a column vector of ones of appropriate size. For u >, the matrices Q +, Q Q M are the matrix Wiener Hopf factorization of S, Z defined via Ξ Q + = ΞQ =, ΞQ := 1 Σ Q + V Q + Q iu I M, 8 Σ := diagσ 1, σ,...,σ M, V := diagr /, r σ /,...,r σ M /, and I M an M M identity matrix. Proof See, for example, Rogers [1994] and Jiang and Pistorius [8]. Similar results have been derived for two-sided first-passage times and in the more general case of regime switching exponential jump-diffusion models see, for example, Jiang and Pistorius [8]. If we are able to compute the matrix Wiener-Hopf factorization, Theorem demonstrates how to compute the first-passage time densities, a result that then allows us to price many exotic options in a regime switching model. For and 3 regimes and in the case where r =, closed-form solutions for the matrix Wiener Hopf factors Q +, Q are available see, for example, Section. and Hieber [1]. In the general case of M states, using diagonalization techniques, Rogers and Shi [1994] present an efficient algorithm to numerically compute the matrix Wiener Hopf factorization Q +, Q. As a first application of Theorem to option pricing, Theorem 3 presents prices for digital options, i.e. options that pay 1 if the barrier e b < S is hit during the lifetime of the option and otherwise. Theorem 3 Digital options Consider the regime switching model as defined in Equation 1 and a barrier e b < S. Under Q, the price of a digital option with payoff 1 {T, b T } at maturity T is given by DS, T, = e rt 1 e iut iu exp Q x b 1 du, 9 3 Note that there might be a positive probability that the barrier is never hit, i.e. f a, t, dt 1. 4

5 .1 M states where exp denotes the matrix exponential, 1 a column vector of ones of appropriate size, and Q the matrix Wiener Hopf factor as defined in Theorem. Proof Using Theorem and applying Fubini s theorem, we find that DS, T, :=e rt Q T, b T T = e rt f, b t, dt = e rt T = e rt 1 1 e iut iu e iut exp Q x b 1 du dt exp Q x b 1 du. Combining the results of digital options in the latter theorem and on call options in Theorem 1, it is possible to price barrier options in a regime switching model, see Theorem 4. Down-and-out barrier options pay maxs T K, at maturity T if the barrier e b K is not hit during the lifetime of the option and otherwise. Theorem 4 Barrier options Consider the regime switching model as defined in Equation 1. Under Q, the price of a barrier option with barrier e b < S and strike K := e k >, i.e. with payoff 1 {T, b T } maxs T K, at maturity T, is given by M T BS, T, = CS, T, e rt e iut exp Q x b e j du Ce b, T t, e j dt, j=1 where CS, T, is the price of a vanilla call option, exp denotes the matrix exponential, e j the j-th unit vector, and Q the matrix Wiener Hopf factor as defined in Theorem. Proof We denote for t, the first-passage time density conditional on Z T, b = j by f, b t,, e j, where e j is the j-th unit vector. Then, the price of the barrier option can be represented as BS, T, = CS, T, M T j=1 e rt f, b t,, e j Ce b, T t, e j dt, where the series sums over the possible states at the time of the first barrier crossing. Plugging in Equation 7, one obtains the stated results. 5

6 .1 M states Remark 5 Implementation of Theorem 4 To evaluate the barrier option price in Theorem 4 numerically, it might sense to further simplify the integrals. Plugging in Equations, 5, we obtain from Theorem 4 T e rt f, b t,, e j Ce b, T t, e j dt = e rt T = e rt S = e rt S S 1 T e iut exp Q x b e j du e α iθk b ϕ T t θ 1 + αi, e j α + α θ + iα + 1θ dθ dt [ e α iθk b α + α θ + iα + 1θ e iut ϕ T t θ 1 + αi, ej dt exp Q x b ] e j du dθ [ e α iθk b α + α θ + iα + 1θ exp Q x b ] e j e iut e j Φ T θ 1 + αi 1 du dθ, where Φ t v := Q + iu + ivr Σ v / 1 exp Q t + iu + ivr Σ v /t I M, α [1, ], and I M denotes the M M identity matrix. This then results in the numerically more convenient expression [ BS, T, = CS, T, e rt S e α iθk b α + α θ + iα + 1θ e iut exp Q x b ] Φ T θ 1 + αi 1 du dθ. Finally, we turn our attention to lookback options. Therefore, define the maximum of S = {S t } t on the time interval [, T] by MT := max t [,T] S t. A lookback strike put option then pays maxmt ST, at maturity T. Its price in a regime switching model is derived in Theorem 6. 6

7 .1 M states Theorem 6 Lookback strike put options Consider the regime switching model as defined in Equation 1. At maturity T, a lookback put option pays maxmt ST,. Under Q, its price is given by LS, T, = e rt x 1 e iut iu Q + exp Q + a x 1 du da S. where exp denotes the matrix exponential, 1 a column vector of ones of appropriate size, and Q + is the matrix Wiener Hopf factor as defined in Theorem. Proof Since MT ST, we can price lookback strike put options as LS, T, =e rt E Q [ MT ST ] = e rt E Q [ MT ] S. Using Theorem, the expected maximum can then be derived as [ ] T E Q MT = e a = = 1 x x e a T x a f a, t, dt da 1 1 e iut iu e iut Q + exp Q + a x 1 du Q + exp Q + a x 1 du da. dt da Remark 7 Implementation of Theorem 6 If one wants to numerically evaluate the integrals in Theorem 6, one can speed up the implementation by restricting the maximum MT to a bounded interval, i.e. choose a suitable a min, a max such that e x < e a min MT e amax <. Then LS, T, = 1 1 = 1 x 1 e iut 1 e iut iu iu e iut 1 iu Q + exp Q + a x 1 du da S [ Q+ + I M 1 Q+ exp Q + a x + I M a ] a max a min 1 du S Q+ + I M 1 Q+ exp Q + a max x + I M a max exp Q + a min x + I M a min 1 du S. We represented digital, barrier, and lookback options as functionals of the matrix Wiener Hopf factorization. Generally, the latter has to be solved numerically. However, in the case of M = regimes, analytical solutions of the matrix Wiener Hopf factors are available. This then gives us closed-form expressions for many exotic option prices. 7

8 . states. states First-passage times and option prices in the -state model depend on the roots of the so-called Cramér Lundberg equation given by 1 1 σ 1β + µ 1 β + q 11 u σ β + µ β + q u q 11 q =. 1 This quartic equation has 4 unique real roots < β 1,u < β,u < < β 3,u < β 4,u < see Guo [1b] for a proof. Those roots are available in closed-form, see, for example, Abramowitz and Stegun [1965], p. 17f. Depending on those roots, Theorem 8 presents the matrix Wiener Hopf factorization Q +, Q in the case of M = regimes. Theorem 8 -state model: Matrix Wiener Hopf factorization Consider the regime switching model as defined in Equation 1 with M = states and q 11 q. a The matrix Wiener Hopf factorization Q +, Q is given by Q + = β 3,u β 4,u + q 11 u β 3,u +β 4,u + µ 1 q σ β 3,u +β 4,u + µ σ q 11 β 3,u +β 4,u + µ 1 β 3,u β 4,u + q u σ β 3,u +β 4,u + µ σ, Q = β 1,u β,u q 11 u σ 1 β 1,u +β,u + µ 1 σ 1 q σ β 1,u +β,u + µ σ where < β 1,u < β,u < < β 3,u < β 4,u < are the roots of Equation 1. q 11 σ 1 β 1,u +β,u + µ 1 σ 1 β 1,u β,u q u σ β 1,u +β,u + µ σ b The matrix exponentials of Q +, Q are, for m R +, given by exp Q + m = β 3,ue β4,um β 4,u e β 3,um 1 + e β3,um e β 4,um Q β 3,u β 4,u 1 +, β 3,u β 4,u 11 exp Q m = β 1,ue β,um β,u e β 1,um 1 + eβ1,um e β,um Q β 1,u β,u 1. β 1,u β,u 1 Proof See, for example, Hieber [1]. If we apply Theorem 1 for m = x b, we obtain a simplified expression for exp Q x b in Theorems 3 and 4. The first-passage time densities can then also be expressed in terms of the roots of Equation 1. For = 1, we obtain 1 f a, t, 1 = 1 f, b t, 1 = e iut β 3,u β 4,u e iut β 1,u β,u β3,u β 3,u β 4,u u β 3,u + β 4,u + µ 1 β1,u β 1,u β,u u β 1,u + β,u + µ 1 e β 4,ua x e β,ux b β 4,u β 3,u β 4,u u β 3,u + β 4,u + µ 1 β,u β 1,u β,u u β 1,u + β,u + µ 1, e β 3,ua x e β 1,ux b see also Guo [1a], Hieber [13]. This result then yields analytical expressions for the prices of digital, barrier, and lookback options in Fourier space see Theorems 3 6. du, du, 8

9 3 Numerical comparison and applications Regime switching model Black Scholes Boyle, Draviam [7] Monte-Carlo Fourier analytic Fourier CS, T, CS, T, CS, T, CS, T, CS, T, S = ± S = ± S = ± S = ± S = ± S = ± S = ± Table 1 Vanilla call option prices CS, T, in a regime switching model comparing the backward finite elements scheme by Boyle and Draviam [7] left, t =.1, S min =, S max =, N = 1 1, an unbiased Monte-Carlo simulation 1 8 simulation runs, and the matrix Wiener Hopf factorization middle. The parameter sets are taken from Boyle and Draviam [7]: S = K = 1, Z = 1, Q 1, = Q, 1 =.5, r =.5, σ 1 =.15, σ =.5, T = 1. For the Monte-Carlo simulation 95% confidence intervals are given. The right column gives the corresponding prices CS, T, in a Black-Scholes model σ = σ 1. 3 Numerical comparison and applications We now compare our analytical solutions to a backward finite element scheme see, for example, Boyle and Draviam [7], the Brownian bridge algorithm see, for example, Hieber and Scherer [1], and an analytical approximation of barrier option prices by Lo et al. [3], Elliott et al. [14]. Additionally, we compare the results to Black-Scholes option prices 4 σ = σ 1, analytical Black-Scholes prices are marked by a D throughout the paper. In this special case σ = σ Zt for all t analytical expressions for the option prices allow us to assess the accuracy of our numerical techniques. In a first step, the Fourier integral in Theorem 1 is implemented to price vanilla call options. 4 In the Black-Scholes model, option prices are given by see, for example, Goldman et al. [1979], Reiner and Rubinstein [1991] where B := expb. lns/k + r + σ1/ T lns/k CS, T, = S Φ Ke rt + r σ1/ T Φ, σ 1 T σ 1 T BS, T, = CS, T, e r 1 lns /K CD /S, T,, lnb/s r σ1/ T DS, T, = Φ e σ 1 T LS, T, = S e rt 1 σ 1 r + σ 1 / T Φ r σ 1 T r 1 lns /K lnb/s + r σ1/ T Φ, σ 1 T + S 1 + σ 1 r σ 1 / T Φ r σ 1 T, 9

10 3 Numerical comparison and applications Regime switching model Black Scholes Brownian bridge matrix W. H. analytic matrix W. H. DS, T, DS, T, DS, T, DS, T, S ± S ± S ± S ± S ± S ± Table Digital option prices DS, T, in a regime switching model comparing the Brownian bridge algorithm left, 1 7 simulation runs and the matrix Wiener Hopf factorization middle. The parameter sets are taken from Hieber and Scherer [1]: S = Z = 1, r =.3, K = e b, and S 1 S 3 : σ 1 =.15, σ =.5, Q 1, =.8, Q, 1 =.6; K =.6 S 1, K =.8 S, K =.9 S 3. S 4 S 6 : σ 1 =.1, σ =.5, K =.8; Q 1, =., Q, 1 =.1 S 4, Q 1, = 1., Q, 1 =.6 S 5, Q 1, = 3., Q, 1 =. S 6. For the Brownian bridge algorithm 95% confidence intervals are given. The right column gives the corresponding prices DS, T, in a Black-Scholes model σ = σ 1. Table 1 uses the parameter set in Boyle and Draviam [7] and compares the Fourier technique to a finite element scheme see Table 1 in Boyle and Draviam [7] and to an unbiased Monte-Carlo simulation. We find that Fourier prices are within the 95% confidence intervals of the Monte-Carlo simulation. The results by Boyle and Draviam [7] seem to very slightly underestimate prices an indicator for a discretization bias. Then, we turn to digital options. This now allows to apply the matrix Wiener Hopf technique in Theorem 3. Using the dataset of Hieber and Scherer [1], we compare our results to the Brownian bridge algorithm see Table. Again the prices are within the 95% confidence intervals of the Brownian bridge algorithm. We realize that for those kind of options, regime switching seems to have a significant impact on option prices. Black Scholes prices σ = σ 1 are especially for the parameter sets S 1, S 4 S 6 significantly lower than the prices in a regime switching environment. Next, we use the same parameter sets to price barrier options. Prices computed via the matrix Wiener Hopf technique are given in Theorem 4. Table 3 compares those prices to the Brownian bridge algorithm see the results in Table 1 in Hieber and Scherer [1] and to an analytical approximation by Lo et al. [3], Elliott et al. [14]. Again we are well within the Monte-Carlo confidence intervals. The analytical approximation by Lo et al. [3], Elliott et al. [14] very slightly underestimates the option prices. For barrier options, the regime switching component does not seem to significantly influence option prices. Last, we price lookback options. Table 4 compares the prices from Theorem 6 to a backward fi- 1

11 3 Numerical comparison and applications Regime switching model Black Scholes Hieber, Scherer [1] Elliott et al. [14] matrix W. H. analytic matrix W. H. BS, T, BS, T, BS, T, BS, T, BS, T, S ± S.17 ± S ± S 4.3 ± S 5.33 ± S 6.5 ± Table 3 Barrier option prices BS, T, in a regime switching model comparing the Brownian bridge algorithm left, 1 7 simulation runs and the matrix Wiener Hopf factorization middle. The parameter sets are taken from Hieber and Scherer [1]: S = Z = 1, r =.3, K = e b, and S 1 S 3 : σ 1 =.15, σ =.5, Q 1, =.8, Q, 1 =.6; K =.6 S 1, K =.8 S, K =.9 S 3. S 4 S 6 : σ 1 =.1, σ =.5, K =.8; Q 1, =., Q, 1 =.1 S 4, Q 1, = 1., Q, 1 =.6 S 5, Q 1, = 3., Q, 1 =. S 6. For the Brownian bridge algorithm 95% confidence intervals are given. The right column gives the corresponding prices BS, T, in a Black-Scholes model σ = σ 1. Regime switching model Black Scholes Boyle, Draviam [7] matrix W. H. analytic matrix W. H. LS, T, LS, T, LS, T, LS, T, S = S = S = S = S = Table 4 Lookback strike put option prices LS, T, in a regime switching model comparing the comparing the backward finite elements scheme by Boyle and Draviam [7] left, t =.1, S min =, S max =, N = 1 1 and the matrix Wiener Hopf factorization middle, b min =1e-5, b max = 1 in Remark 7. The parameter sets are taken from Boyle and Draviam [7]: S = K = 1, Z = 1, Q 1, = Q, 1 =.5, r =.5, σ 1 =.15, σ =.5, T = 1. The right column gives the corresponding prices LS, T, in a Black-Scholes model σ = σ 1. 11

12 4 Conclusion nite element scheme see Table 11 in Boyle and Draviam [7]. Again note that the prices by Boyle and Draviam [7] slightly underestimate the analytical expressions an indicator for a small discretization bias. In Tables 1 to 4, one observes that the analytical Black-Scholes prices marked by a D are closely recovered by Fourier integrals and matrix Wiener Hopf factorization. 4 Conclusion This paper demonstrates how one can price exotic options i.e. digital, barrier, and lookback options in a Markovian, regime-switching, Black-Scholes model. In Fourier space, analytical expressions for the option prices are given as functionals of the matrix Wiener Hopf factorization. In the case of or 3 regimes or in the case of a zero interest rate the matrix Wiener Hopf factorization can be solved analytically see, for example, Section 3 or Hieber [1]. This approach turns out to be easy to implement; the resulting option prices are accurate and unbiased. In a numerical case study we confirm that the presence of regime switching in financial data has a significant effect on option prices especially for digital options. One can generalize the presented results in several directions regarding both the considered financial model and the payoff streams of the financial contracts. However, the analytical tractability might be reduced or lost. First, if one adds exponential or phase-type jumps, one can still derive modified matrix Wiener Hopf factorizations see, for example, Jiang and Pistorius [8]. Secondly, the Fourier transform of the two-sided first-passage time can also be represented as a functional of the matrix Wiener Hopf factorization, a result that allows us to price, for example, double barrier options. References M. Abramowitz and I. A. Stegun. Handbook of Mathematical Functions. Dover Publications Inc., New York, S. Asmussen. Stationary distributions for fluid flow models with or without Brownian noise. Communications in Statistics. Stochastic models, Vol. 11, No. 1:pp. 1 49, M. Barlow, L. Rogers, and D. Williams. Wiener-Hopf factorization for matrices. Lecture Notes in Mathematics, Vol. 784:pp , 199. S. Boyarchenko and S. Levendorskiĭ. Exit problems in regime-switching models. Journal of Mathematical Economics, Vol. 44:pp. 18 6, 8. P. Boyle and T. Draviam. Pricing exotic options under regime switching. Insurance: Mathematics and Economics, Vol. 4:pp. 67 8, 7. J. Buffington and R. J. Elliott. American options with regime switching. International Journal of Theoretical and Applied Finance, Vol. 5:pp ,. P. Carr and D. B. Madan. Option valuation using the fast Fourier transform. Journal of Computational Finance, Vol. :pp ,

13 References R. Elliott, T. Siu, and L. Chan. On pricing barrier options with regime switching. Journal of Computational and Applied Mathematics, Vol. 56:pp , 14. R. J. Elliott, L. Chan, and T. K. Siu. Option pricing and Esscher transform under regime switching. Annals of Finance, Vol. 1, No. 4:pp , 5. P. Eloe, R. Liu, and J. Sun. Double barrier option under regime-switching exponential mean-reverting process. International Journal of Computer Mathematics, Vol. 86, No. 6:pp , 9. M. Escobar, P. Hieber, and M. Scherer. Efficiently pricing barrier derivatives in stochastic volatility models. Review of Derivatives Research, in press, 13. M. B. Goldman, H. B. Sosin, and M. Ann Gatto. Path dependent options: buy at the low, sell at the high. Journal of Finance, Vol. 34, No. 5:pp , X. Guo. When the Bull meets the Bear A first passage time problem for a hidden Markov process. Methodology and Computing in Applied Probability, Vol. 3, No. :pp , 1a. X. Guo. An explicit solution to an optimal stopping problem with regime switching. Journal of Applied Probability, Vol. 38:pp , 1b. J. D. Hamilton. A new approach to the economic analysis of nonstationary time series and the business cycle. Econometrica, Vol. 57, No. :pp , P. Hieber. First-passage times of regime switching models. Working paper, 1. P. Hieber. A correction note on: When the Bull meets the Bear A first passage time problem for a hidden Markov process. Methodology and Computing in Applied Probability, DOI 1.17/s , 13. P. Hieber and M. Scherer. Efficiently pricing barrier options in a Markov-switching framework. Journal of Computational and Applied Mathematics, Vol. 35:pp , 1. Z. Jiang and M. Pistorius. On perpetual American put valuation and first-passage in a regimeswitching model with jumps. Finance and Stochastics, Vol. 1, No. 3:pp , 8. J. Kennedy and D. Williams. Probabilistic factorization of a quadratic matrix polynomial. Mathematical Proceedings of the Cambridge Philosophical Society, Vol. 17:pp , 199. M. Kim, B.-G. Jang, and H.-S. Lee. A first-passage-time model under regime-switching market environment. Journal of Banking & Finance, Vol. 3:pp , 8. O. Kudryavtsev and S. Levendorskiĭ. Fast and accurate pricing of barrier options under Lévy processes. Finance and Stochastics, Vol. 13, No. 4:pp , 1. C. Lo, H. Lee, and C. Hui. A simple approach for pricing barrier options with time-dependent parameters. Quantitative Finance, Vol. 3:pp , 3. R. London, H. McKean, L. Rogers, and D. Williams. A martingale approach to some Wiener-Hopf problems. Lecture Notes in Mathematics, Vol. 9:pp. 68 9,

14 References S. Metwally and A. Atiya. Using Brownian bridge for fast simulation of jump-diffusion processes and barrier options. Journal of Derivatives, Vol. 1:pp ,. A. Mijatović and M. Pistorius. Continuously monitored barrier options under Markov process. Mathematical Finance, Vol. 3, No. 1:pp. 1 38, 13. V. Naik. Option valuation and hedging strategies with jumps in the volatility of asset returns. Journal of Finance, Vol. 48, No. 5:pp , S. Raible. Lévy processes in finance: Theory, numerics, and empirical facts. PhD thesis, Freiburg University,. E. Reiner and M. Rubinstein. Breaking down the barriers. Risk 4, Vol. 8:pp. 8 35, L. Rogers. Fluid models in queueing theory and Wiener-Hopf factorization of Markov chains. Annals of Applied Probability, Vol. 4, No. :pp , L. Rogers and Z. Shi. Computing the invariant law of a fluid model. Journal of Applied Probability, Vol. 31, No. 4:pp ,