# 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 $$N\cdot (4.65-S_T)$$.
1. 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.
2. 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 $$B$$.

Based on my understanding of what you wrote, I would propose the following solutions:

1. 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 $$S_T = 4.65$$).

1. 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?

• That's right, the option can't be terminated at any time. I thought I could call this an American option in that we control the relationship between $S_t$ and $B$ over the life of the option, not just at time $T$. I have table with ATM, 25RR, 10RR, 25BF, 10BF, FXspot, EUR rates, swap points for tenors ON, 1W, 2W, 1M, 2M, 3M, 6M, 9M, 1Y. Commented Jun 19 at 21:41
• Is the price of the option provided? It seems unlikely to be able to determine $B$ unless you have that. If you have the price of the option, you could back out $B$ via Monte Carlo simulation (at least that is how I would do it). Commented Jun 19 at 22:09
• I don't have price of the option explicite. But if I understand correctly, I have the market data to find the option price using methods such as black-scholes and vanna-volga. So let's assume I manage to price the option. Could you tell me what your idea would be to find barrier $B$? Commented Jun 20 at 8:02
• That seems like a good idea. I would simulate the spot using GBM (you can find the formula online), then specify the parameters accordingly - you should keep everything else constant but vary the barrier level $B$ until you find that the Monte Carlo simulation produces the same price that you have with Black-Scholes/Vanna-Volga. Commented Jun 20 at 9:22