Because the DJIA index and S&P500 index exhibit the phenomenon of stock market cycles, this section performs sensitivity analysis by the estimated parameters.
Tables 5 and 6 report sensitivity analysis of European call price assuming stock returns follow Markov switching model in DJIA estimated parameters. The base volatility of state 1 (4%2%) and the basic volatility of state 2 (1%+1% and 1%-0.05%) show the effect of the volatility of the Brownian motion with respect to European call option. Hence, the volatility of daily stock return are 2%, 4% and 6%
for state 1；and 0.5%, 1% and 2% for state 2. According to the sensitivity analysis of Table 4, other parameters are fixed, there is a positive relationship between volatility and option value in state 1 and state 2, implying the larger the volatility, the higher the probability of increasing stock price, hence higher call price. In addition, other parameters held constant, there is a positive relationship between
p and call value,
11 because the volatility of the state 1 stay longer time whenp is closed to 1. The
11 higher thep , the lower the probability that the economy will switch from state 1 to
11 state 2.That is, in the long term, the longer the duration of state 1, which has higher volatility, the higher the call value will be. On the contrary, there is a negative relationship betweenp and call value. The higher the
22p , the lower the probability
22 the economy will switch from state 2 to state 1.In the long term, the longer theduration of state 2,which has lower volatility, the smaller the call value.
This paper also discusses the influence of jump volatility on call price. Table 5 illustrates the sensitivity analysis of the impact of jump size and jump frequency on call price. Other things held constant, there is a positive relationship between average jump size and call price. Since call price increases at expiration when stock price increases, the bigger jump size implies larger stock price upside volatility, hence higher call price.
The relationship between the standard deviation of jump size and call price is concave, because the martingale condition will be satisfied. Finally, other parameters held constant, The higher jump frequency indicates more frequent jump volatility.
Therefore, the option value is higher
5. Conclusion
This study proposes a regime-switching model with jump risks to price the European option value. To capture the dynamics of stock returns over expansion-recession cycles and the occurrences of the catastrophe events, we assume the index return would follow the regime-switching model with jump risks.
In this study, we show that compare to Black-Scholes model and regime-switching model, the regime-switching model with jump risks can better
explain the dynamics of DJIA index and S&P 500 stock indices. In addition, both of the regime-switching model and the regime-switching model with jump risks can address the leptokurtic feature of the asset return distribution, volatility smile, and the volatility clustering phenomenon. Next, we examine the influence of parameters on European value under the regime-switching model with jump risks, and find that option prices increase along with the probability of staying in the recession state, but decrease along with the probability of staying in the expansion state. Moreover, the increases of standard deviation in either state, the mean of jump sizes, the standard deviation of jump sizes, and the mean of jump times, would all increase option prices.
The differences among valuations under the Black-Scholes model, the regime-switching model and the regime-switching model with jump risks suggest that it is critical to value a European call option precisely by an appropriate model.
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Table 2: The moments in the regime-switching model with jump risks
Model Regime-switching model with jump risks Mean 1 1 2 2y
Table 3 The descriptive statistics of the DJIA index and S&P500 index Panel A The descriptive statistics of the DJIA index
DJIA 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 Total
Number 251 252 248 252 252 252 252 251 251 253 252 2766
Mean 0.0008 -0.0002 -0.0003 -0.0007 0.0009 0.0001 0.0000 0.0006 0.0002 -0.0016 0.0006 0.0001 Std. 0.0101 0.0130 0.0135 0.0160 0.0104 0.0068 0.0064 0.0062 0.0091 0.0238 0.0152 0.0129
±2% 16 29 24 51 16 0 1 0 14 72 45 268
±2%Mean 0.0054 -0.0066 -0.0020 -0.0002 0.0068 - 0.0203 - -0.0080 -0.0046 0.0023 -0.0001
±3% 0 7 9 14 4 0 0 0 1 39 15 89
±3%Mean - -0.0076 -0.0075 0.0132 0.0162 - - - -0.0334 -0.0045 -0.0020 -0.0013