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Option pricing under stock market cycles with jump risks: evidence from the S&P 500 index

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Abstract

This study incorporates the Markov switching model with return jumps to depict the behavior of stock returns. Based on the daily Standard & Poor’s 500 index (hereafter SPX) and the daily closing price of the call option, we use the particle filtering algorithm to fit the parameter of the model. The joint log-likelihood evaluates the model performance: the weighted average log-likelihood with the rate of return of the SPX and the relative implied volatility root-mean-squared error for the SPX call options. The empirical results identify that the pricing model with jump risks improves the pricing performance to the median-term call options. According to the sensitivity analysis, option prices increase with the probability of remaining in the recession state but decrease with the probability of remaining in the expansion state. Moreover, the call option prices are positively associated with the volatility in each market state and the factors of jump risk.

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Funding

Funding was provided by Ministry of Science and Technology, Taiwan (Grant No. NSC 105-2410-H-004-087-).

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Correspondence to Shih-Kuei Lin.

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Appendices

Appendix A: European call option pricing formula

Assume that the jump risk premiums are the same in the market state 1 and 2, i.e. \(h_{2,1} = h_{2,2} = h_{2}\). Given the filtration \({\mathbb{F}}\left( t \right)\), the days belonging to the market state 1 (denoted by \(m\)), the stock return between time \(t\) and \(T\), \(\ln S\left( T \right) - \ln S\left( t \right) = \mathop \sum \limits_{u = t + 1}^{T} R\left( u \right)\), under the \({\mathbb{Q}}\) measure is

$$\ln \frac{S\left( T \right)}{S\left( t \right)} = r\left( {T - t} \right) - \frac{1}{2}V_{m} - \lambda^{{\mathbb{Q}}} \left[ {g_{Y}^{{\mathbb{Q}}} \left( { - i} \right) - 1} \right]\left( {T - t} \right) + \sqrt {V_{m} } \varepsilon^{{\mathbb{Q}}} \left( t \right) + \mathop \sum \limits_{n = 1}^{{N\left( {T - t} \right)}} Y_{n} ,$$
(20)

where \(V_{m} = \sigma_{1}^{2} k + \sigma_{2}^{2} \left( {T - t - m} \right)\) is the variance with \(m\) days visiting the market state 1 between time \(t\) and \(T\). And, the random variables under the \({\mathbb{Q}}\) measure: \(\varepsilon^{{\mathbb{Q}}} \left( t \right) \sim {\text{Normal}}\left( {0,1} \right)\) is the Gaussian noise process, \(N\left( {T - t} \right) \sim {\text{Poisson}}\left[ {\lambda^{{\mathbb{Q}}} \left( {T - t} \right)} \right]\) is the number of jumps during \(\left( {t,T} \right]\), and \(Y_{n} \mathop \sim\limits^{\text{IID}} f_{Y}^{{\mathbb{Q}}} \left( {y;h_{2} } \right)\) is the jump size. Let the random variable \(X\) be the log-price of stock at time \(T\), i.e. \(X = \ln S\left( T \right) = \ln S\left( t \right) + \mathop \sum \limits_{u = t + 1}^{T} R\left( u \right)\), and \({\mathbb{G}}\left( t \right) = {\mathbb{F}}\left( t \right) \vee m\) be enlarged filtration. Then, we can derive its conditional characteristic function as follow:

$$\begin{aligned} g_{{X|{\mathbb{G}}\left( t \right)}}^{{\mathbb{Q}}} \left( \phi \right) & = {\text{E}}_{{{\mathbb{G}}\left( t \right)}}^{{\mathbb{Q}}} \left( {e^{i\phi X} } \right) \\ & = e^{{i\phi \left( {\ln S\left( t \right) + r\left( {T - t} \right) - \frac{1}{2}V_{m} - \lambda^{{\mathbb{Q}}} \left[ {g_{Y}^{{\mathbb{Q}}} \left( { - i} \right) - 1} \right]\left( {T - t} \right)} \right) - \frac{{\phi^{2} }}{2}V_{m} + \lambda^{{\mathbb{Q}}} \left[ {g_{Y}^{{\mathbb{Q}}} \left( \phi \right) - 1} \right]\left( {T - t} \right)}} . \\ \end{aligned}$$
(21)

Consider a \(T\)-maturity European call option with the strike price \(K\). Given the filtration \({\mathbb{F}}\left( t \right)\), the option pricing formula is

$$\begin{aligned} {\text{Call}}\left( {t;K,T} \right) & = {\text{E}}_{{{\mathbb{F}}\left( t \right)}}^{{\mathbb{Q}}} \left[ {e^{{ - r\left( {T - t} \right)}} \left( {e^{X} - e^{\ln K} } \right)1_{{\left\{ {X > \ln K} \right\}}} } \right] \\ & = e^{{ - r\left( {T - t} \right)}} {\text{E}}_{{{\mathbb{F}}\left( t \right)}}^{{\mathbb{Q}}} \left[ {{\text{E}}_{{{\mathbb{G}}\left( t \right)}}^{{\mathbb{Q}}} \left( {e^{X} 1_{{\left\{ {X > \ln K} \right\}}} } \right)} \right] - Ke^{{ - r\left( {T - t} \right)}} {\text{E}}_{{{\mathbb{F}}\left( t \right)}}^{{\mathbb{Q}}} \left[ {{\text{E}}_{{{\mathbb{G}}\left( t \right)}}^{{\mathbb{Q}}} \left( {1_{{\left\{ {X > \ln K} \right\}}} } \right)} \right], \\ \end{aligned}$$
(22)

where the second equation is derived by the law of iterated expectation. By the inverse Fourier transform of the characteristic function, we can rewrite Eq. (22) as follows:

$$\begin{aligned} {\text{E}}_{{{\mathbb{G}}\left( t \right)}}^{{\mathbb{Q}}} \left( {e^{X} 1_{{\left\{ {X > \ln K} \right\}}} } \right) & = \mathop \smallint \limits_{\ln K}^{\infty } e^{x} f_{{X|{\mathbb{G}}\left( t \right)}}^{{\mathbb{Q}}} \left( x \right)dx \\ & = \frac{1}{2\pi }\mathop \smallint \limits_{ - \infty }^{\infty } \mathop \smallint \limits_{\ln K}^{\infty } e^{{ - i\left( {\phi + i} \right)x}} g_{{X|{\mathbb{G}}\left( t \right)}}^{{\mathbb{Q}}} \left( \phi \right)dxd\phi \\ & = S\left( t \right)e^{{r\left( {T - t} \right)}} \left[ {\frac{1}{2} + \frac{{e^{{ - r\left( {T - t} \right)}} }}{\pi S\left( t \right)} \cdot {\text{Re}}\left( {\mathop \smallint \limits_{0}^{\infty } \frac{{e^{ - i\phi \ln K} g_{{X|{\mathbb{G}}\left( t \right)}}^{{\mathbb{Q}}} \left( {\phi - i} \right)}}{i\phi }d\phi } \right)} \right] \\ & \text{ := }S\left( t \right)e^{{r\left( {T - t} \right)}}\Pi _{1,m} , \\ \end{aligned}$$
(23)
$$\begin{aligned} {\text{E}}_{{{\mathbb{G}}\left( t \right)}}^{{\mathbb{Q}}} \left( {1_{{\left\{ {X > \ln K} \right\}}} } \right) & = \mathop \smallint \limits_{\ln K}^{\infty } f_{{X|{\mathbb{G}}\left( t \right)}}^{{\mathbb{Q}}} \left( x \right)dx \\ & = \frac{1}{2\pi }\mathop \smallint \limits_{ - \infty }^{\infty } \mathop \smallint \limits_{\ln K}^{\infty } e^{ - i\phi x} g_{{X|{\mathbb{G}}\left( t \right)}}^{{\mathbb{Q}}} \left( \phi \right)dxd\phi \\ & = \frac{1}{2} + \frac{1}{\pi } \cdot {\text{Re}}\left( {\mathop \smallint \limits_{0}^{\infty } \frac{{e^{ - i\phi \ln K} g_{{X|{\mathbb{G}}\left( t \right)}}^{{\mathbb{Q}}} \left( \phi \right)}}{i\phi }d\phi } \right){ := \varPi }_{2,m} , \\ \end{aligned}$$
(24)

The second equations in Eqs. (23) and (24) are computed by the Fubini theorem, and the third equations in Eqs. (23) and (24) are derived by the property of complex conjugate. A similar proof can be shown in Bakshi and Madan (2000). Therefore, the pricing formula in Eq. (22) is rewritten by

$$\begin{aligned} {\text{Call}}\left( {t;K,T} \right) & = S\left( t \right){\text{E}}_{{{\mathbb{F}}\left( t \right)}}^{{\mathbb{Q}}} \left( {\Pi _{1,m} } \right) - Ke^{{ - r\left( {T - t} \right)}} {\text{E}}_{{{\mathbb{F}}\left( t \right)}}^{{\mathbb{Q}}} \left( {\Pi _{2,m} } \right) \\ & = \mathop \sum \limits_{m = 0}^{T - t} \frac{{p_{21} \psi_{T,m|q\left( t \right) = 1} + p_{12} \psi_{T,k|q\left( t \right) = 2} }}{{p_{12} + p_{21} }}\left[ {S\left( t \right)\Pi _{1,m} - Ke^{{ - r\left( {T - t} \right)}}\Pi _{2,m} } \right], \\ \end{aligned}$$
(25)

where \(\psi_{T,m|q\left( t \right) = j}\) presents the probability that the stock market belongs to the state 1 for \(k\) days while the time-to-maturity is \(T - t\) days given the initial market state \(q\left( t \right) = j\). It is proved in Duan, Popova, and Ritchken (2002) with a hidden Markov Chain.

Appendix B: Benchmark models: stochastic volatility processes

Our model comprises a Markov switching dynamics for the volatility and it can be categorized in the stochastic volatility models. Under the discrete-time framework, Li (2019), Ornthanalai (2014), Christoffersen et al. (2012), Wu (2006) Heston and Nandi (2000) employ the affine GARCH models to depict the stock return and its variance. That is,

$$R\left( t \right) = \mu - \frac{1}{2}V\left( t \right) - \lambda \left[ {g_{Y}^{{\mathbb{P}}} \left( { - i} \right) - 1} \right] + \sqrt {V\left( t \right)} Z^{{\mathbb{P}}} \left( t \right) + \mathop \sum \limits_{n = 1}^{{\Delta N\left( t \right)}} Y_{n} ,$$
(26)
$$V\left( t \right) = \sigma^{2} + \gamma \left[ {Z^{{\mathbb{P}}} \left( {t - 1} \right) - \eta \sqrt {V\left( {t - 1} \right)} } \right]^{2} + \xi V\left( {t - 1} \right),$$
(27)

where \(\mu\) is the mean of stock return, \(V\left( t \right)\) is the conditional variance of stock return at time \(t\), \(\sigma^{2}\) is the intercept term of the variance equation, \(\gamma\) is the coefficient of leverage effect, \(\eta\) measures the level of asymmetric leverage effect between the good and bad events, \(\xi\) is the parameter of volatility clustering effect. The variance is s stationary process if and only if \(\gamma \eta^{2} + \xi < 1\) is satisfied. Also, note that \(V\left( t \right)/2\) and \(\lambda \left[ {g_{Y}^{{\mathbb{Q}}} \left( { - i} \right) - 1} \right]\) are the convexity adjustment terms, which make the stock return equal to \(\mu\) under the \({\mathbb{P}}\) measure, i.e. \({\text{E}}_{{{\mathbb{F}}\left( {t - 1} \right) \vee q\left( t \right)}}^{{\mathbb{P}}} \left[ {S\left( t \right)} \right] = S\left( {t - 1} \right)e^{{\mu_{q\left( t \right)} }}\). Equations (26) and (27) are called by the GARCH model with return jumps (GARCH-RJ). If there are no jumps, \(\lambda = 0\), Eqs. (26) and (27) reduce to Heston and Nandi (2000)’s GARCH model.

On the other hand, the stock return is also measured under the continuous-time framework like Bates (2012, 2000, 1996), Eraker (2004), Eraker et al. (2003), Bakshi et al. (1997), Eisenberg and Jarrow (1994), Heston (1993). The dynamic of stock log-price under the stochastic volatility model is

$$d\ln S\left( t \right) = \left( {\mu - \frac{1}{2}V\left( t \right) - \lambda \left[ {g_{Y}^{{\mathbb{P}}} \left( { - i} \right) - 1} \right]} \right)dt + \sqrt {V\left( t \right)} dW^{{\mathbb{P}}} \left( t \right) + d\mathop \sum \limits_{n = 1}^{N\left( t \right)} Y_{n} ,$$
(28)
$$dV\left( t \right) = \left( {1 - \xi } \right)\left[ {\sigma^{2} - V\left( t \right)} \right]dt + \gamma \sqrt {V\left( t \right)} d\hat{W}^{{\mathbb{P}}} \left( t \right) + d\mathop \sum \limits_{n = 1}^{{\hat{N}\left( t \right)}} \hat{Y}_{n} ,$$
(29)

where \(\mu\) is the instantaneous mean, \(V\left( t \right)\) is the instantaneous variance at time \(t\), \(1 - \xi\) is the mean-reverting speed, \(\sigma^{2}\) is the long-run mean level, and \(\gamma\) is the volatility of variance. Moreover, \(W^{{\mathbb{P}}} \left( t \right) \sim {\text{Normal}}\left( {0,t} \right)\) and \(\hat{W}^{{\mathbb{P}}} \left( t \right) \sim {\text{Normal}}\left( {0,t} \right)\) are the correlated Brownian motions with the correlation coefficient \(\rho\). \(N\left( t \right) \sim {\text{Poisson}}\left( {\lambda t} \right)\) and \(\hat{N}\left( t \right) \sim {\text{Poisson}}\left( {\hat{\lambda }t} \right)\) are the independent number of jumps for the return and the variance with the time-homogeneous arrival rate. \(Y_{n} \mathop \sim \limits^{\text{IID}} {\text{Normal}}\left( {\theta ,\nu^{2} } \right)\) and \(\hat{Y}_{n} \mathop \sim \limits^{\text{IID}} {\text{Exponential}}\left( {\hat{\theta }} \right)\) are the independent jump sizes for the return- and the variance-jumps, respectively. Equations (28) and (29) are called by the stochastic volatility model with return-jumps and volatility-jumps (SV-RJ-VJ). If there are no jumps for volatility, \(\hat{\lambda } = 0\), the model reduces to the stochastic volatility model with return jumps (SV-RJ) like Bates (2012, 2000) and Bakshi et al. (1997). Next, if the stock market does not exist the jump risks for the return and volatility, \(\lambda = \hat{\lambda } = 0\), the dynamic of stock log-price degenerates to Heston (1993)’s SV model. For comparing under the same assumption, in this paper, we suppose that the jump frequencies of return and volatility are time-homogeneous under the model with jumps.

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Wang, SY., Chuang, MC., Lin, SK. et al. Option pricing under stock market cycles with jump risks: evidence from the S&P 500 index. Rev Quant Finan Acc 56, 25–51 (2021). https://doi.org/10.1007/s11156-020-00885-x

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