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Pricing vulnerable options with jump risk and liquidity risk

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Abstract

In this paper, we consider vulnerable options with jump risk and liquidity risk. In the proposed framework, we allow discontinuous changes in the information processes and the liquidity discount factors as well, and default risk is taken into consideration. Specially, we investigate the effect of jumps in the liquidity discount factors and find that the effects of jumps in the liquidity discount factors are stable for different maturities and alternative moneynesses. Further, option prices behave differently with respect to alternative intensities of common jumps, depending on whether there are jumps in the liquidity discount factors or not.

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Notes

  1. Stock prices with liquidity risk (see, e.g., Amihud and Mendelson 1991; Acharya and Pedersen 2005) and corporate bonds with liquidity risk (see, e.g., Fu et al. 2012; He and Xiong 2012; He and Milbradt 2014) have also been studied in the literature.

  2. A partial list of the studies on vulnerable European options includes Johnson and Stulz (1987), Klein and Inglis (1999, 2001), Liao and Huang (2005), Liang and Ren (2007), Yang et al. (2014), Lee et al. (2016), Niu and Wang (2016), Wang et al. (2017a), Wang (2017, 2018), Yang et al. (2019), and Liang and Wang (2020). In addition, there are a few papers focusing on vulnerable Asian options including Tsao and Liu (2012), Jeon et al. (2016) and Wang (2020a, 2020b).

  3. In the proposed framework, the asymmetric effects of good versus bad news are not considered. The authors thank the referee for raising this issue. In order to investigate the asymmetric effects, we can consider the following dynamics,

    $$\begin{aligned} L_1(t)= & {} L_1(0)\exp \Big \{ -\beta _1 \Big (\int _0^t a(s){\mathrm {d}}s + \int _0^t a(s) {\mathrm {d}}W(s)+ \sum _{k=1}^{N(t)}(\theta Z_k +\theta _a Z_k\mathbf {1}(Z_k<0))\Big )\Big \}, \end{aligned}$$

    where \(\theta _a\) is a non-negative constant. Obviously, with a positive value of \(\theta _a\), the effects of the positive and negative news on the liquidity discount factor are asymmetric. In Sect. 3, we will illustrate asymmetric effects on option prices numerically.

  4. Monte Carlo simulation methods are used to obtain the option prices. The simulations are run with daily time steps and 25, 000 paths are generated for one price.

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Acknowledgements

The authors would like to thank the anonymous referee and the editor for their helpful comments and valuable suggestions that led to several important improvements. All errors are our responsibility.

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  1. School of International Trade and Economics, University of International Business and Economics, Beijing, 100029, China

    Xingchun Wang

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The author acknowledges financial support from the National Natural Science Foundation of China (11701084, 11671084).

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Wang, X. Pricing vulnerable options with jump risk and liquidity risk. Rev Deriv Res 24, 243–260 (2021). https://doi.org/10.1007/s11147-021-09177-5

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