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04-06-2020 | Research Article

Efficient Asian option pricing under regime switching jump diffusions and stochastic volatility models

Published in: Annals of Finance | Issue 3/2020

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Abstract

Utilizing frame duality and a FFT-based implementation of density projection we develop a novel and efficient transform method to price Asian options for very general asset dynamics, including regime switching Lévy processes and other jump diffusions as well as stochastic volatility models with jumps. The method combines continuous-time Markov chain approximation, with Fourier pricing techniques. In particular, our method encompasses Heston, Hull-White, Stein-Stein, 3/2 model as well as recently proposed Jacobi, \(\alpha \)-Hypergeometric, and 4/2 models, for virtually any type of jump amplitude distribution in the return process. This framework thus provides a ‘unified’ approach to pricing Asian options in stochastic jump diffusion models and is readily extended to alternative exotic contracts. We also derive a characteristic function recursion by generalizing the Carverhill-Clewlow factorization which enables the application of transform methods in general. Numerical results are provided to illustrate the effectiveness of the method. Various extensions of this method have since been developed, including the pricing of barrier, American, and realized variance derivatives.

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In between regime switching times, the asset price process follows a regular jump diffusion with constant drift rate and instantaneous volatility rate. Note that this formulation allows for a different jump distribution in each regime.

Which can be seen by applying Cauchy’s inequality to \([a\sqrt{{v}_t}+\frac{b}{\sqrt{{v}_t}}]\ge 2\sqrt{a\sqrt{v_t} \frac{b}{\sqrt{v_t}}}=2\sqrt{ab}>0\) for \(a,b>0\).

For more details on basis theory and its applications in finance, see Kirkby and Deng (2019).

Details on the derivation of \( \widehat{\widetilde{\varphi }}(\xi )\) for the cubic basis can be found in Kirkby (2017b).

For stochastic volatility models, the initial variance state \(v_0\) is not necessarily a member of the variance grid. One approach is to apply linear interpolation with equation (46) for \(j = j_0\) and \(j_0+1\), where \(v_{j_0}\le v_0 <v_{j_0+1}\). However, the grid can be easily adjusted so that \(v_0=v_{j_0}\) is a member.

Accessed on September 20th 2017.

We can increase \(\gamma \) to sufficiently cover the domain of \(v_t\). From numerical experimentation, we find that \(\gamma =4.5\) is sufficient for the models considered in this work.

If moments of the variance process are unknown, the grid can be fixed using \(v_1 = \beta _1 v_0\) and \(v_{m_0} = \beta _2 v_0\). For example, \(\beta _1 = 10^{-3}\) and \(\beta _2 = 4\).

This keeps an “anchor” at the boundary in the case where \(v_0 \approx 0\).

To avoid excessive notation, we continue to suppress the superscript on \(\{\varphi _{a,k}^{m-1}\}=\{\varphi _{a,k}\} \), where the grid shift is understood.

Note that for the purpose of this proof we have defined \(\beta ^j_{Y_{m-1},k}\) so that \( a^{1/2}\Upsilon _{a,N}\beta ^j_{Y_{m-1},k}=\langle f^j_{Y_{m-1}},\widetilde{\varphi }_{a,k} \rangle \).

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Title: Efficient Asian option pricing under regime switching jump diffusions and stochastic volatility models
Publication date: 04-06-2020
Published in: Annals of Finance / Issue 3/2020
Print ISSN: 1614-2446
Electronic ISSN: 1614-2454
DOI: https://doi.org/10.1007/s10436-020-00366-0

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