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数字期权价格与 call 的关系

European digital option

专题
Finance / 金融
难度
L4

题目详情

Consider the European digital option (or "binary option") that pays a constant HH if the stock price is above strike price XX at expiration and zero otherwise. What is the price of this option, and how is it related to the price of the standard Black- Scholes European call option? Explain carefully.

解析

现金或无数字（cash-or-nothing）call：到期支付 H1{ST>X}H\mathbf{1}_{\{S_T>X\}}。

BS 下价格为折现的风险中性概率：

V=He−r(T−t)N(d2).\boxed{V=He^{-r(T-t)}N(d_2)}.

并且 vanilla call 可拆为 asset-or-nothing 与 cash-or-nothing：

C=StN(d1)−Xe−r(T−t)N(d2).C=S_tN(d_1)-Xe^{-r(T-t)}N(d_2).

因此 N(d2)N(d_2) 就是数字期权的“命中概率”项，数字也可由 call spread 极限复制。

−∂KC 与数字价格等价（差一个贴现因子）.\boxed{-\partial_K C\ \text{与数字价格等价（差一个贴现因子）}}.