统计学在期权定价和交易中的应用

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)。

# 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)

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 $C(t=0)$')

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)

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)

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)

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)

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

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')

plot_greeks(BSM_gamma, 'gamma')

plot_greeks(BSM_theta, 'theta')

plot_greeks(BSM_rho, 'rho')

plot_greeks(BSM_vega, 'vega')

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

10.45058357218553

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

10.449583775457942

plot_convergence(10, 1011, 20)

plot_convergence(10, 1011, 25)

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

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)

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

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 

---------------------------------