统计学在期权定价和交易中的应用

期权定价与内在价值
[h1]常见的期权内在价值被定义为0与期权立刻执行所能获得回报的较大值，即max(S-K, 0)(看涨期权的情形)和max(K-S, 0)(看跌期权的情形)。[/h1]

# Option Strike
K = 8000
# Graphical Output
S = np.linspace(7000, 9000, 100) # index level values
h = np.maximum(S - K, 0) # inner values of call option
plt.figure()plt.plot(S, h, lw=2.5) # plot inner values at maturity
plt.xlabel('index level $S_t$ at maturity')plt.ylabel('inner value of European call option')plt.grid(True)

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下面的代码是使用BSM模型来对期权进行定价：

无收益资产欧式看涨期权的定价公式：

无收益资产欧式看跌期权的定价公式:

下面画出了根据BSM模型计算出的期权的现值

# Model and Option Parameters
K = 8000 # strike price
T = 1.0 # time-to-maturity
r = 0.025 # constant, risk-less short rate
vol = 0.2 # constant volatility
# Sample Data Generation
S = np.linspace(4000, 12000, 150) # vector of index level values
h = np.maximum(S - K, 0) # inner value of option
C = [BSM_call_value(S0, K, 0, T, r, vol) for S0 in S] # calculate call option values# Graphical Output
plt.figure()plt.plot(S, h, 'g-.', lw=2.5, label='inner value') # plot inner value at maturity
plt.plot(S, C, 'r', lw=2.5, label='present value')
# plot option present valueplt.grid(True)plt.legend(loc=0)plt.xlabel('index level $S_0$')plt.ylabel('present value $C(t=0)$')

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历史波动率刻画
已经知道，B-S-M期权定价公式中的期权价格取决于下列五个参数：标的资产的市场价格，执行价格，到期期限，无风险利率和标的资产价格波动率。这五个参数中，前三个都是交易获得确定的值，在发达的经济市场，无风险利率也很容易估计，而估计标的资产的波动率要比估计无风险利率要困难的多。估计标的资长价格波动率有两种方法：历史波动率和隐含波动率。

所谓历史波动率，就是从标的资产价格的历史数据中计算出价格对数收益率的标准差，具体方法一般有以下两种：

直接估计（矩估计）
GARCH模型

很遗憾，GARCH模型在中国市场无效。
模拟数据

gbm = simulate_gbm()print_statistics(gbm)

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RETURN SAMPLE STATISTICS
---------------------------------------------
Mean of Daily Log Returns 0.000294
Std of Daily Log Returns 0.012479
Mean of Annua. Log Returns 0.074109
Std of Annua. Log Returns 0.198092
---------------------------------------------
Skew of Sample Log Returns 0.008369
Skew Normal Test p-value 0.861095
---------------------------------------------
Kurt of Sample Log Returns 0.059948
Kurt Normal Test p-value 0.495981
---------------------------------------------
Normal Test p-value 0.781084
---------------------------------------------
Realized Volatility 0.198147
Realized Variance 0.039262

quotes_returns(gbm)

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上证50ETF的实际波动率

etf50 = DataAPI.MktFunddGet(ticker=u"510050",beginDate=u"20150206",endDate=u"20160929",field=[u"closePrice", u"tradeDate"] ,pandas="1")etf50.set_index=("tradeDate")etf50['returns'] = np.log(etf50['closePrice'] /etf50['closePrice'].shift(1))
# Realized Volatility (eg. as defined for variance swaps)
etf50['rea_var'] = 252 * np.cumsum(etf50['returns'] ** 2) / np.arange(len(etf50))etf50['rea_vol'] = np.sqrt(etf50['rea_var'])print_statistics(etf50)

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RETURN SAMPLE STATISTICS
---------------------------------------------
Mean of Daily Log Returns -0.000067
Std of Daily Log Returns 0.021153
Mean of Annua. Log Returns -0.016917
Std of Annua. Log Returns 0.335787
---------------------------------------------
Skew of Sample Log Returns -0.734791
Skew Normal Test p-value  0.000000
---------------------------------------------
Kurt of Sample Log Returns 4.597651
Kurt Normal Test p-value  0.000000
---------------------------------------------
Normal Test p-value  0.000000
---------------------------------------------
Realized Volatility  0.335789
Realized Variance  0.112754

# histogram of annualized daily log returns
def return_histogram(data): ''' Plots a histogram of the returns. ''' plt.figure(figsize=(9, 5)) x = np.linspace(min(data['returns'][1:]), max(data['returns'][1:]), 100) plt.hist(np.array(data['returns'][1:]), bins=50, normed=True) y = dN1(x, np.mean(data['returns'][1:]), np.std(data['returns'][1:])) plt.plot(x, y, linewidth=2) plt.xlabel('log returns') plt.ylabel('frequency/probability') plt.grid(True)return_histogram(etf50)

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etf50.index = etf50['tradeDate']quotes_returns(etf50)

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期权的Greeks
期权的Greeks主要描述了期权价格对其标的的资产价格、到期时间、波动率和无风险利率四个参数值的敏感性指标。

plot_greeks(BSM_delta, 'delta')

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plot_greeks(BSM_gamma, 'gamma')

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plot_greeks(BSM_theta, 'theta')

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plot_greeks(BSM_rho, 'rho')

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plot_greeks(BSM_vega, 'vega')

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二叉树模型和BSW模型

BSM_benchmark = BSM_call_value(S0, K, 0, T, r, sigma)BSM_benchmark

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10.45058357218553

CRR_option_value(S0, K, T, r, sigma, 'call', M=2000)

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10.449583775457942

plot_convergence(10, 1011, 20)

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plot_convergence(10, 1011, 25)

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动态对冲

S, po, vt, errs, t = BSM_hedge_run(p=25)

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APPROXIMATION OF FIRST ORDER ----------------------------- step | S_t | Delta 1 | 97.10 | -0.46 2 | 100.34 | -0.40 3 | 101.03 | -0.39 4 | 104.38 | -0.30 5 | 103.81 | -0.32 6 | 106.85 | -0.27 7 | 109.36 | -0.22 8 | 111.16 | -0.21 9 | 111.09 | -0.20 wrong 10 | 115.83 | -0.15
（部分）

plot_hedge_path(S, po, vt, errs, t)

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pl_list = BSM_dynamic_hedge_mcs(M=200, I=150000)

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Value of American Put Option is  4.461
Delta t=0 is  -0.011
run 1000  p/l  0.044
run 2000  p/l  0.042
run 3000  p/l  0.305
run 4000  p/l  0.969
run 5000  p/l  -0.330
run 6000  p/l  0.093
run 7000  p/l  -0.438
run 8000  p/l  -0.156
run 9000  p/l  -0.126
run 10000  p/l  0.187
SUMMARY STATISTICS FOR P&L
---------------------------------
Dynamic Replications 10000
Time Steps 200
Paths for Valuation 150000
Maximum  2.358
Average 0.008
Median 0.006
Minimum -1.641
---------------------------------

来源：量化投资与机器学习