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Pricing exotic options in a regime switching economy: a Fourier transform method

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Abstract

This article considers the valuation of digital, barrier, and lookback options in a Markovian, regime-switching, Black–Scholes model. In Fourier space, integral representations for the option prices are derived via the theory on matrix Wiener–Hopf factorizations. Our main focus is on the 2-state case where the matrix Wiener–Hopf factorization is available analytically. A comparison to several numerical alternatives (analytical approximations, the Brownian bridge algorithm and a finite element scheme) demonstrates that the pricing formulas are easy to implement and lead to accurate price estimates.

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Notes

  1. (Numerical) solutions to the (matrix) Wiener–Hopf factorization are possible in more general model settings [see, among others, Boyarchenko and Levendorskiĭ (2008), Jiang and Pistorius (2008), Kudryavtsev and Levendorskiĭ (2012), Mijatović and Pistorius (2013)].

  2. An intensity matrix has negative diagonal and non-negative off-diagonal entries. Each row sums up to zero.

  3. Note that in the single barrier case there might be a positive probability that the barrier is never hit. In this case, \(f_{a,\,-\infty }(t,\pi _0)\) is still a density if the probability of never hitting the barrier is attributed to \(t=\infty \).

  4. We denote by \(\mathcal {Q}_M\) the class of irreducible \(M\times M\) generator matrices (non-negative off-diagonal entries and non-positive row sums).

  5. This contains the implicit assumption that the eigenvectors \(v_i\) form a basis, an assumption that turned out to be sufficient in practical applications [see, e.g., Rogers and Shi (1994)]. It is possible to construct artificial examples where such a basis does not exist. The following steps of Algorithm 1 can then be modified using a basis of Jordan vectors.

  6. In the Black–Scholes model with volatility \(\sigma \), option prices for digital and lookback options are given by [see, for e.g., Reiner and Rubinstein (1991)]

    $$\begin{aligned} \overline{\mathcal {D}}(S_0,T,\varvec{\pi }_{\varvec{0}})&= \Phi \Bigg ( \frac{\ln (B/S_0) - \big (r-\sigma ^2/2\big )T}{\sigma \sqrt{T}} \Bigg ) - e^{ \big (\frac{2r}{\sigma ^2}-1\big )\ln (S_0/K)}\, \Phi \Bigg ( \frac{\ln (B/S_0) + \big (r-\sigma ^2/2\big )T}{\sigma \sqrt{T}} \Bigg ),\\ \overline{\mathcal {L}}(S_0,T,\varvec{\pi }_{\varvec{0}})&= S_0\,e^{-rt} \Big ( 1 - \frac{\sigma ^2}{2r} \Big )\, \Phi \Bigg ( \frac{\big (-r+\sigma ^2/2\big )T}{\sigma \sqrt{T}} \Bigg ) + S_0 \Big ( 1 + \frac{\sigma ^2}{2r} \Big )\, \Phi \Bigg ( \frac{\big (-r-\sigma ^2/2\big )T}{\sigma \sqrt{T}} \Bigg ), \end{aligned}$$

    where \(B:=\exp (b)\).

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Acknowledgements

I want to thank the editor and the anonymous referees for valuable comments that helped to improve earlier versions of this article.

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  1. Department of Mathematics and Economics, University of Ulm, Helmholtzstr. 20, 89069, Ulm, Germany

    Peter Hieber

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Correspondence to Peter Hieber.

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This research was carried out while Peter Hieber was a research assistant at TU Munich.

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Hieber, P. Pricing exotic options in a regime switching economy: a Fourier transform method. Rev Deriv Res 21, 231–252 (2018). https://doi.org/10.1007/s11147-017-9139-1

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