A Markov chain approximation scheme for option pricing under skew diffusions

Abstract

In this paper, we propose a general valuation framework for option pricing problems related to skew diffusions based on a continuous-time Markov chain approximation to the underlying stochastic process. We obtain an explicit closed-form approximation of the transition density of a general skew diffusion process, which facilitates the unified valuation of various financial contracts written on assets with natural boundary behavior, e.g. in the foreign exchange market with target zones, and equity markets with psychological barriers. Applications include valuation of European call and put options, barrier and Bermudan options, and zero-coupon bonds. Motivated by the presence of psychological barriers in the market volatility, we also propose a novel ‘skew stochastic volatility’ model, in which the latent stochastic variance follows a skew diffusion process. Numerical results demonstrate that our approach is accurate and efficient, and recovers various benchmark results in the literature in a unified fashion.

Keywords:

• Skew diffusion
• Local time
• Continuous-time Markov chain
• Option pricing
• Target zone
• Psychological barriers

AMS subject classifications:

• 91G80
• 93E11
• 93E20

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 See empirical evidence documented in Balduzzi et al. (1997), Driessen et al. (2011) and references therein.

2 For related study of stochastic volatility models, please refer to Cui (2013), Bernard et al. (2017) and references therein.

3 We thank the anonymous referee for the suggestion to call a a threshold.

4 $\tilde{\alpha }$ is a density parameter and observe that a smaller $\tilde{\alpha }$ leads the grid more massed near the initial value ${\tilde{y}}_0.$

5 The endpoints of grid corresponding to ${\mathbf{\text{Q}}}^{\left(2\right)}$ are given by $y_{n_1+1}=(1-p)(x_{\mathrm{m}\mathrm{i}\mathrm{n}}-a)+a$ and $y_{n_1+n_2}=(1-p)(x_{\mathrm{m}\mathrm{a}\mathrm{x}}-a)+a$.

6 See https://www.macroption.com/vix-all-time-high/ for a record of the highest VIX level, and note that during the 2008 financial crisis, the VIX level breached 80% on a few days.

7 Rossello (2012) and Zhu and He (2018) consider a geometric skew Brownian motion (GSBM) model which is given by a skew Brownian motion instead of a standard Brownian motion W in the classical GBM model.

8 The domain of X will be covered sufficiently as we increase γ to make it large enough. We choose $\gamma =4.5$ and $\overline{x}=0.00001$, which are found to be sufficient for this model from pilot numerical experiments.

9 We thank the anonymous referee for suggesting us randomized the parameters.

10 A larger M has little effect on the results of CTMC.

11 The choice of boundaries is consistent with Cai et al. (2015), in which the nonuniform grid construction is the same as in this paper. It is also sufficient for skew CIR model and the following skew CEV model from our numerical experiments.

12 Thanks to the anonymous referee for suggesting us consider the low interest rate environment.

Additional information

Funding

Supported by the NSFC [grant numbers 71532001 and 11631004], and the LPMC at Nankai University.