Interpreting a Knock-In Barrier Contract with Conditional Payoffs
Summary
The question describes a path-dependent contract with one payoff if the underlying stays above a barrier and a different payoff if it crosses below it. It asks how to balance the component instruments to give the contract zero initial value and how to solve for the barrier level. The response clarifies that crossing the barrier activates the alternate payoff; it does not make the contract expire. It also interprets the activated payoff as a long call and short put at the same strike, equivalent to a forward payoff.
The answer gives conditions for a zero terminal payoff in certain combinations of barrier path and final price, but it does not derive fair value, discount the payoffs, or provide a method for solving the barrier. It also questions the American-style label: the description specifies a path condition and terminal payoff, but no right to exercise early. Pricing would require contract details and market inputs that the post does not supply.
Key ideas
- A down-in barrier crossing activates the alternate payoff rather than terminating the contract.
- A call minus a put at the same strike produces a forward-like payoff.
- The zero-payoff cases depend on both the barrier path and the terminal underlying price.
- A path-dependent barrier feature does not by itself make a contract American-style.
- The response leaves fair-value calculation and barrier calibration unresolved.
Tags
Full text
# Complicated barrier options
# Complicated barrier options
> We have the following contract consisting of barrier options: If $S_t$ is above the barrier level $B$ during the contract duration, we receive $N\cdot \max (S_T-4.45,0), 4.45>B$ from the bank, where $N$ - nominal value. If $S_t$ droped below level $B$ during the contract duration, then if $S_T\ge4.65$ we will receive $N\cdot (S_T-4.65) $ from the bank, and otherwise we will pay the bank $N\cdot (4.65-S_T)$.
- Derive the condition that the contract parameters must meet so that its value is zero at the time of conclusion, i.e. balance the current values of the instruments that make up this contract.
- Create a script that will determine the value of $B$ given the remaining parameters of this contract.
My understanding of the problem: 1. The contract consists of a down-in-call barrier option that expires if $S_t$ falls below $B$. Then the option is activated, consisting of a put issued to the bank and a call purchased by us for the same exercise price of $4.65$. Therefore, at the time of concluding the contract, the following condition should be met: $$\text{call } K_1 = \text{call } K_2 - \text{put }K_2$$
However, I don't know how to discount barrier options and I haven't found any help in the literature. Could someone tell me what to use or send me some tips?
2. Our contract is an American-type barrier option. Therefore, the final payout depends on the relationship between $\min{S_t}$ and $B$ during the contract period. For this task, I have given market data, I can use the vanna-volga method to calculate various values, but I do not understand at all how I can determine the value of the $B$ barrier. I don't expect anyone to write the code for me, just explain to me what conditions I can use to arrive at the formula for $B$.
## Answer by KaiSqDist (score 2, accepted)
https://quant.stackexchange.com/a/79804
Based on my understanding of what you wrote, I would propose the following solutions:
- The down-in-call barrier doesn't expire when $S_t < B$, it just activates and changes into the long call short put OR a long futures with strike $4.65$ as you have explained. If during the whole duration $S_t \geq B$, the it just stays as the call with strike $4.45$. But to answer the conditions that the down-in-call barrier expires with zero value:
If ($S_t > B$ for the whole duration AND $S_T < 4.45$) OR ($S_t < B$ for $\geq 1$ time in the whole duration AND $S_T = 4.65$).
- I don't think this is an American-barrier? No where in what you wrote does it say that the option can be exercised at anytime, it sounds more like an European (correct me if I am wrong).
Also, "Create a script that will determine the value of $B$ given the remaining parameters of this contract." Are all the remaining parameters given? Can you specify them here?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.