看跌-看涨平价

殊途同归

构造两个组合。A：买入一份看涨期权，并把行权价的现值 K * exp(-rT) 存入银行（或买无风险债券）。B：买入一份看跌期权，并买入股票。到期时，若股价高于 K，A 是看涨期权的 S - K 加银行的 K，即 S；B 是股票 S 加一份一文不值的看跌期权，即 S。若股价低于 K，看涨期权作废，A 只有银行的 K，B 是看跌期权的 K - S 加股票 S，即 K。两种情况下两个组合到期时价值相同，所以今天的成本也必须相同。

import math

spot, strike, rate, t = 100, 100, 0.05, 1
pv_strike = strike * math.exp(-rate * t)             # what must be put in the bank today to own K at expiry
print(round(pv_strike, 4))                            # 95.1229

call = 10.4506                                        # a call price we observe
put = call - spot + pv_strike                         # the parity relation
print(round(put, 4))                                  # 5.5735
print(round(put + spot - pv_strike, 4))               # 10.4506   and back to the call

平价被打破时：套利

如果市场报价不满足这个关系，就存在无风险利润。定义偏差 (C - P) - (S - K e^(-rT))。偏差为正，说明相对于看跌期权，看涨期权太贵了：卖出看涨、买入看跌、买入股票，并借入行权价的现值。这个组合期初不花钱，无论股价如何到期时都不需支付，并把偏差在今天收入囊中。偏差为负则每一条腿反向操作。真实市场有成本和买卖价差，所以小偏差无法交易，这就是容差的含义。

def parity_gap(call, put, spot, strike, rate, t):
    return (call - put) - (spot - strike * math.exp(-rate * t))

print(round(parity_gap(11.0, put, 100, 100, 0.05, 1), 4))         # 0.5494   the call at 11.00 is too expensive: sell it
print(round(parity_gap(call, put, 100, 100, 0.05, 1), 4))         # 0.0      consistent prices: nothing to do

def arbitrage_action(call, put, spot, strike, rate, t, tolerance=0.01):
    gap = parity_gap(call, put, spot, strike, rate, t)
    if gap > tolerance:
        return "sell call, buy put, buy stock"
    if gap < -tolerance:
        return "buy call, sell put, sell stock"
    return "none"

print(arbitrage_action(11.0, put, 100, 100, 0.05, 1))             # sell call, buy put, buy stock

# The package of the arbitrage, at today's prices: sell the call at 11, buy the put, buy the stock, borrow the strike's present value
cash_today = 11.0 - put - 100 + 100 * math.exp(-0.05 * 1)
print(round(cash_today, 4))                # 0.5494   you are paid the gap now and owe nothing at expiry

# A call minus a put is a synthetic forward: it behaves like the stock bought with borrowed money
print(round(10.4506 - put, 4), round(100 - pv_strike, 4))        # 4.8771 4.8771

• C + K * exp(-rT) = P + S：看涨加债券等于看跌加股票，所以 P = C - S + K * exp(-rT)。
• 偏差 (C - P) - (S - K e^(-rT)) 为正：卖看涨、买看跌和股票；为负：反向操作。
• 平价与模型无关，可用来检验任何定价模型；有股息或提前行权时变成不等式。

下一节：布莱克-斯科尔斯公式，看涨与看跌期权价格的闭式解。

实验：评分内容

• 用 parity_put_price 由看涨价格推出看跌价格25 分
• 用 parity_call_price 由看跌价格推出看涨价格25 分
• 用 parity_gap 衡量价格偏离平价多远25 分
• 用 arbitrage_action 给出无风险交易25 分