Abstract
This paper proposes an analytic solution for pricing options in markets with daily price limits. The Black–Scholes model is a nested case in which the daily price limit approaches infinity. Compared to the Black–Scholes model, our solution may solve the mispricing problem and could yield consistent results with existing numerical methods. Practitioners trading options in price-limit markets may resort to the finite difference method or Monte Carlo simulations. However, applying these numerical methods is often time consuming, thereby further illustrating the importance of an analytic solution.
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Notes
Normally, the Taiwan Stock Exchange sets its daily price limit for each stock at 7% based on the previous day's closing price of each stock. However, daily price limits have been temporarily adjusted to 3.5% to stabilize stock markets when some unusual events occur, such as the 921 earthquake of 1999, the attacks of September 11, 2001, and the global financial crisis of 2008.
Available online at http://www.futuresindustry.org/downloads/FIMag/mayjune08/May-June_Volume.pdf.
Table 1 of Kim and Yang (2004) presents various price limits in the futures markets of the Chicago Board of Trade, while Table 2 includes a list of price limit regulations from various stock markets around the world. However, the web site of each exchange should be consulted for historical or up-to-date regulations because price limit rules may change over time.
The measure that meets the requirement of \( \tilde{E}\left[ {e^{ - rT} S_{T} |S_{0} } \right] = S_{0} \) is called the risk-neutral measure (see Kou and Wang 2004).
Please refer to Bhattacharya and Waymire (1990) for details of the backward equation.
Note that \( p\left( {t,x,y} \right) \) = \( \hat{p}\left( {t,\hat{x},\hat{y}} \right) \) with the specification of \( x = \hat{x} + L \) and \( y = \hat{y} + L \).
Please refer to Bhattacharya and Waymire (1990) for details of the proof.
Individual stock price volatility is often much higher than that of the stock index. However, the equity market can become much more volatile when unusual events happen. For example, the S&P 500 annual volatility reached more than 130% during the 1987 crash and more than 90% during the 2008 credit crunch.
The authors thank the anonymous referee for helpful comments and suggestions.
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Guo, JH., Chang, LF. A generalization of option pricing to price-limit markets. Rev Deriv Res 23, 145–161 (2020). https://doi.org/10.1007/s11147-019-09160-1
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DOI: https://doi.org/10.1007/s11147-019-09160-1
Keywords
- Daily price limit
- Analytic solution
- Local times
- Backward equation
- Characteristic function
- Fast Fourier transform