Replicating a Secured Barrier Call with Barrier Options
Summary
The document explains a payoff decomposition for a secured barrier call. The product pays the positive difference between the terminal stock price and the strike if the upper barrier is never reached; if the stock reaches that barrier before expiration, it instead pays a fixed amount. Otherwise, it pays nothing. This structure can be expressed as an up-and-out call plus a fixed-payout up-and-in digital barrier option.
The answer formalizes the decomposition using the first time the stock reaches the barrier. The up-and-out call supplies the conditional call payoff when the barrier remains untouched, while the up-and-in digital supplies the fixed payment when the barrier is hit. The source question also lists prices for a barrier call and American digital calls, but the answer does not explain how to use those quoted digital prices to price the required barrier digital. It therefore clarifies the payoff grouping without establishing a complete valuation from the quoted instruments; additional pricing information or assumptions would be needed.
Key ideas
- The secured barrier call pays a call payoff if the upper barrier is never touched.
- A fixed payment is due when the underlying reaches the barrier before expiration.
- The payoff separates into an up-and-out call and a fixed-payout up-and-in digital barrier option.
- The listed American digital prices do not by themselves establish the price of the required barrier digital.
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# Pricing Secured Barrier Call 2
# Pricing Secured Barrier Call 2
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 by Gordon (score 2, accepted)
https://quant.stackexchange.com/a/37585
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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.