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Review of Derivatives Research

Erschienen in:

25.09.2017

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

verfasst von: Peter Hieber

Erschienen in: Review of Derivatives Research | Ausgabe 2/2018

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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.

Vorheriger Artikel The volatility target effect in structured investment products with capital protection

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(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)].

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

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 $.

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).

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.

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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Titel: Pricing exotic options in a regime switching economy: a Fourier transform method
verfasst von: Peter Hieber
Publikationsdatum: 25.09.2017
Verlag: Springer US
Erschienen in: Review of Derivatives Research / Ausgabe 2/2018
Print ISSN: 1380-6645
Elektronische ISSN: 1573-7144
DOI: https://doi.org/10.1007/s11147-017-9139-1