Pricing Secured Barrier Call 2

Question

EDIT: OK, I understand the reasoning for the initial answer now; however, I don't understand why we would need the digital call with a strike of 33 in this question. Is it just there to serve as a red herring of sorts?

I'm confused by the logic behind a solution to a problem that was answered here before. I was hoping to comment for an elaboration, but I do not have the reputation to do that yet. I am hoping that I can get an explanation of how this works. The problem was:

A European barrier call with barrier $B=50$, expiration $T=31$, and strike $K=33$ costs $12. The investor is interested in a product that, unlike this barrier call, offers some protection for the case that the stock goes above the barrier 50. The investor wants to buy an investment product called Secured Barrier Call whose payoff structure is

$$ \text{Payoff = } \begin{cases} S(31)-33, \text{if } S(31)\ge33 \text{ and } S(t) < 50, \forall t\le31 \\ 50, \text{ if } S(t)\ge 50 \text{ for some } t\le31\\ 0, \text{ otherwise} \end{cases} $$

An American digital call with strike 33 and expiration 31 costs 0.73, and the American digital call with strike 50 and expiration 31 costs 0.70. I need to compute the price of the Secured Barrier Call.

This was answered as follows:

The goal of this exercise is to replicate the payoff of the Secured Barrier Call by a linear combination of the known products: European up-out call (cost 12), digital strike 33 (cost 0.73) and digital strike 50 (cost 0.7).

Looks to me it is sufficient to buy:

• 1x up-out call

• 50 x digital strike 50

The payout at expiry of this linear combination would be:

• $(S(31)-33)^+$ if $S(t) < 50$ for all $t \le 31$

• 50 if S(t) touched 50 at any time

• 0 otherwise

Can someone please explain the intuition behind how to determine the groupings of the different types of known options that are needed to determine the cost of this new secured barrier option? Thank you!

Also, thanks to @Zizou23 for posting the question and @mbison for providing an answer in the first place. Here's a link to the original problem:

Original Post

Answer 1

Formally, let \begin{align*} \tau = \inf\{t: t \ge 0, S_t \ge 50 \}. \end{align*} Then \begin{align*} \text{Payoff} &= \left(S(31)-33 \right)^+\pmb{1}_{\tau >31} + 50\times \pmb{1}_{\tau \le 31}. \end{align*} That is, an up-out barrier call plus 50 digital up-in barrier options.