Abstract
Imposing a symmetry condition on returns, Carr and Lee (Math Financ 19(4):523–560, 2009) show that (double) barrier derivatives can be replicated by a portfolio of European options and can thus be priced using fast Fourier techniques (FFT). We show that prices of barrier derivatives in stochastic volatility models can alternatively be represented by rapidly converging series, putting forward an idea by Hieber and Scherer (Stat Probab Lett 82(1):165–172, 2012). This representation turns out to be faster and more accurate than FFT. Numerical examples and a toolbox of a large variety of stochastic volatility models illustrate the practical relevance of the results.
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Notes
A comment on generalizations to stochastic interest rates is given in Remark 2.
Using that \(\sin \big (\frac{n\pi (x-b)}{a-b}\big ) = (-1)^{n}\,\sin \big (\frac{n\pi (x-a)}{a-b}\big )\), one obtains the results in Hieber and Scherer (2012), Theorem 2 (\(\mu = -1/2\), \(\sigma =1\), a generalization to \(\mu \in \mathbb {R}\) and \(\sigma >0\) is straightforward).
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Peter Hieber acknowledges funding by the German Academic Exchange Service (DAAD).
Appendices
Appendix 1: Parameters of the Stein–Stein model
The functions \(L(u)\), \(M(u)\), and \(N(u)\) in the characteristic function are defined as
Appendix 2: Single barrier limit
In our series representation the limit \({a:=\ln (P)\rightarrow \infty }\) cannot be exchanged with the infinite summation over \(n\) as the series representation for \(X_{D,\infty }^{g(S_T)}(S_0)\) is not absolutely convergent. To derive the limiting expression, one has to change the series representation. Then, the limiting option price \(X_{D,\infty }^{g(S_T)}(S_0)\) is given by Theorem 1. For \(a:=\ln (P)\), \(b:=\ln (D)\), and \(x:=\ln (S_0)\), we obtain
If we change the parameterization [see He et al. 1998, Equations (2.3) and (2.4)], we get
This series is absolutely convergent, thus we can change limit and summation. In the limit \(a\rightarrow \infty \) only the “\(n=0\)” term remains, i.e.
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Escobar, M., Hieber, P. & Scherer, M. Efficiently pricing double barrier derivatives in stochastic volatility models. Rev Deriv Res 17, 191–216 (2014). https://doi.org/10.1007/s11147-013-9094-4
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DOI: https://doi.org/10.1007/s11147-013-9094-4