The Skorokhod Embedding Problem is well known and has many solutions. Now let $B=(B_t)_{t\ge 0}$ be a standard Brownian motion and $\tau$ be an embedding to the centered distribution $\mu$, i.e. the stopped process $B=(B_{\tau\wedge t})_{t\ge 0}$ is uniformly integrable and $B_{\tau}$ has the law $\mu$. My question is the following: Given a set of real numbers $(K_i)_{1\le i\le m}$, for any $\varepsilon, \delta>0$, could we find some bounded stopping time $\sigma$ such that

$$P\left[|\tau-\sigma|~\ge~ \delta\right]~~\le~~ \varepsilon$$

and

$$E[(B_{\sigma}-K_i)^+]~~=~~E[(B_{\tau}-K_i)^+] \mbox{ for all } i=1,\ldots, m.$$

Here we assume that the filtration for the Brownian motion $B$ is sufficiently rich. A natural idea for me is to take first $\tau_n=\tau\wedge n := \min(\tau,n)$, then clearly one has

$$\lim_{n\to\infty}\tau_n~~=~~\tau~~\mbox{ and }~~\lim_{n\to\infty}E[(B_{\tau_n}-K_i)^+]~~=~~E[(B_{\tau}-K_i)^+]$$

However, I don't find a good way to modify properly $\tau_n$ to $\sigma$, such that introduce some random variable $G_1, G_2, \ldots$ that are independent of $B$ and $\tau$ and construct a functional $\sigma=\sigma(\tau_n, G_1, G_2, \ldots)$. If someone has a idea, I look forward to that! Many thanks!