二元对冲（跨 K 买卖）错在哪

该策略的关键问题不是交易成本，而是：它在 $S=K$ 处是不连续/不可导的“二元持仓规则”，而连续时间下价格过程会在 $K$ 附近发生无穷多次穿越，导致组合无法保持自融资（需要无穷次、无穷小时间内的调仓）。

直观：一旦接近 $K$，你会被迫在极短时间内反复买入/卖出，策略的现金流不再可控，因此不是一个可行的无套利复制策略。

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英文解析

This paradox is more than a simple puzzle. The question is called the stop- go paradox and was discussed in several publications (Seidenverg (1988) Carr (1989) Ingersoll (1987) El Karoui (1978)). Generally many interview candidates invoke transaction costs, liquidity or the impossibility to hit an exact price. But all these answers are incorrect because Black Scholes assumptions allow you to build this portfolio. The second type of answer is usually about the portfolio not being self financing because the trader would need to start with $K$ in cash. This is correct but could be addressed using forward contracts for example. We could also borrow the needed cash and the paradox would still be unsolved if the rates are zero.

The correct short answer is that this portfolio is not continuously derivable at $K$ , this discontinuity can be crossed an infinity of times by the stochastic process, making it not self- financed.

Let us break the paradox mathematically. We construct the portfolio

$$V(t) = -\mathbb{1}_{\{S_t > KP(t)\}}KP(t) + \mathbb{1}_{\{S_t > KP(t)\}}S_t$$

where $\mathrm{P(t)}$ is the bond's price. This portfolio replicates the terminal payoff and it must satisfy the following equation for all $t$ in order to be self- financed

$$V(t) = V(0) + \int_{0}^{t}m(u)dP(u) + \int_{0}^{t}n(u)dS_u$$

For simplicity we take rates constant equal to zero (the general case can be reduced with a change of numeraire). The portfolio is then

$$V(t) = -\mathbb{1}_{\{S_t > K\}}K + \mathbb{1}_{\{S_t > K\}}S_t$$

and the self- financing condition becomes

$$V(t) = V(0) + \int_{0}^{t}n(u)dS_u$$

where

$$n(u) = \mathbb{1}_{\{S_u > K\}}$$

the portfolio is self- financed only if the following equation holds

$$\begin{array}{r}V(t) - V(0)\overset {?}{=}\int_{0}^{t}\mathbb{1}_{\{S_u > K\}}dS_u\\ g(S_t) = \mathbb{1}_{\{S_t > K\}}(S_t - K) - (S_0 - K)^+\overset {?}{=}\int_{0}^{t}\mathbb{1}_{\{S_t > K\}}dS_u \end{array}$$

The key here is that $g$ is not $C^2$ and we cannot apply the usual Itô's lemma, but we can use the Tanaka's formula (see page 90) because $g$ is $C^2$ outside a finite set of points.

$$g\left(S_{t}\right) = g\left(S_{0}\right) + \int_{0}^{t}g^{\prime}\left(S_{u}\right)d S_{u} + \lim_{\epsilon \to 0}\frac{1}{2\epsilon}\left|\left\{u\in [0,t];S_{u}\in [K - \epsilon ,K + \epsilon ]\right\} \right|$$

where $g^{\prime}$ is the weak derivative of $g$ and $|A|$ is the Lebesgue measure of $A$ . Therefore

$$V(t) - V(0) = \int_{0}^{t} \mathbb{F}\{s_{u} > K\} dS_{u} + \lim_{\epsilon \to 0} \frac{1}{2\epsilon} |\{u \in [0, t]; S_{u} \in [K - \epsilon , K + \epsilon ]\} |$$

The last term does not converge towards zero and the portfolio is not self- financed, breaking the apparent paradox.

In real business conditions this hedging method is not used because of liquidity and the additional risk associated with this book management method. The delta hedging method is preferred, the trader accepts to pay small regular hedging costs in exchange for a much lower risk.