Black-Scholes Option Pricing Model and Empirical Analysis on 50ETF Option

In the previous blog, I tried Binomial-Tree Model to price 50ETF option in China, where I found it significantly underestimated the price. In the following analysis, I choose Black-Scholes Model instead, but found that although the difference of the predicted prices are significantly different with a p-value around 0.03, the results of prediction is pretty similar, leading to the same conclusion: Black-Scholes Model also underestimates the price of 50ETF option in China.

The formula of Black-Scholes is as follow:

I have also written the codes of estimating prices under BS Model:

from scipy import stats
def Black_Scholes(r,sig,S0,K,T,isCall):
    # European Option Pricing
    d1 = (np.log(S0/K) + (r+sig**2/2)*T)/(sig*np.sqrt(T))
    d2 = d1 - sig*np.sqrt(T)
    if isCall == True:
        C0 = S0*stats.norm.cdf(d1,0,1)-np.exp(-r*T)*K*stats.norm.cdf(d2,0,1)
        return C0, stats.norm.cdf(d1,0,1)
    else:
        P0 = -S0*stats.norm.cdf(-d1,0,1)+np.exp(-r*T)*K*stats.norm.cdf(-d2,0,1)
        return P0, -stats.norm.cdf(-d1,0,1)

Keep all the other methods the same with the previous blog, only replace the function to price option under binomial-tree model with the function above, and increase the test period from the range of (2015-07-31, 2015-11-30) to (2015-07-31, 2016-03-22), the results of the prices are stored in this file: binomialResults_Binomial&BS.csv

Visualizing the difference between the two models, and we find that these two models nearly predict the same results:

plt.figure(dpi=200)
plt.title('Binomial Against Black-Scholes Price')
plt.xlabel('Binomial Price')
plt.ylabel('Black-Scholes Price')
plt.xlim(0,0.6); plt.ylim(0,0.6)
plt.plot(df_results.BinomialPrice,df_results.BSPrice,ls=' ',marker='.')
plt.savefig('Binomial Against Black-Scholes Price.png')

Although statistical analysis (t-test on the differences) shows that the results are significantly different with a p-value of around 0.03:

diff = df_results.BinomialPrice-df_results.BSPrice
ttest_1sample = stats.ttest_1samp(diff, 0)

In [1]: ttest_1sample
Out[1]: Ttest_1sampResult(statistic=2.1480372929383886, pvalue=0.031870221099299899)

Then we may conclude that although slightly different, Binomial-Tree and Black-Scholes Models predict nearly the same prices on 50ETF option from 2015-07-31 to 2016-03-22. Plot Black-Scholes Price against Real Price, and we can find it also significantly underestimates the price of options.

plt.figure(dpi=200)
plt.title('Black-Scholes Price against Real Price')
plt.xlabel('Black-Scholes Price')
plt.ylabel('Real Price')
plt.xlim(0,0.6); plt.ylim(0,0.6)
plt.plot(df_results.BinomialPrice,df_results.Close,ls=' ',marker='.')
plt.savefig('Black-Scholes Price against Real Price.png')