Replicating a Secured Barrier Call with a Digital Option
Summary
The document explains how to construct a secured barrier call from an up-and-out European call and a cash-or-nothing digital call. The target pays the ordinary call payoff if the underlying remains below the barrier through expiry, pays the barrier level if the barrier is touched, and otherwise pays nothing. The proposed portfolio holds one up-and-out call and enough digitals at the barrier strike to produce the fixed payment after a touch.
This payoff-matching argument shows why a digital at the lower strike is not needed: the up-and-out call already supplies the ordinary call payoff in the no-touch case. The example gives prices for the barrier call and two American digitals, but the accepted explanation does not use the lower-strike digital or explicitly calculate a final cost. Its replication relies on the stated contracts having the specified payoff conventions and on the digital paying the desired amount whenever the barrier is reached. It does not discuss discounting, exercise details, or market frictions.
Key ideas
- A secured barrier call can be replicated by combining an up-and-out call with a digital option at the barrier level.
- The up-and-out call supplies the vanilla call payoff when the barrier is never touched.
- The digital option supplies a fixed payment when the underlying reaches the barrier.
- Payoff matching identifies which components are needed before prices are combined.
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Full text
# Pricing Secured Barrier Call
# Pricing Secured Barrier Call
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
$$\text{Payoff}= \begin{cases} S(31)-33\quad, & \text{if} \hspace{2mm} S(31)\ge33 \hspace{2mm} \text{and} \hspace{2mm} S(t) < 50\quad,\hspace{5mm} \forall \,t\le 31 \\ 50\hspace{2.2cm}, & \text{if} \hspace{2mm} S(t)\ge50 \hspace{2mm}\text{for some}\,\,t\le31 \\ 0\hspace{2.45cm}, & \text{o.w} \end{cases}$$
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. After computation, I got $46.94$.
That's what I've done: $C_0=12-2\times0.73+52\times0.70=46.94$. But I am not confident about what I've got.
Could you please confirm or help me with any hint if it's wrong? Thank you.
P.S.: I recently started working on quantitative finance, and it's a problem that I found in book for practicing.
## Answer by mbison (score 2, accepted)
https://quant.stackexchange.com/a/31948
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 <= 31
- 50 if S(t) touched 50 at any time
- 0 otherwiseShown 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.