Replicating a Piecewise Exotic Payoff with European Options
Summary
The discussion presents a method for pricing a maturity-only payoff defined as the minimum of two linear functions of the terminal stock price. It advises plotting the payoff against the terminal price, identifying where its slope changes, and using those slopes and breakpoints to construct a replicating portfolio of European calls, puts, and potentially the underlying. Short positions can invert the contribution of an option. If the portfolio reproduces the payoff at every terminal price, no-arbitrage pricing equates the exotic claim’s current value to the portfolio’s cost.
The response is instructional rather than a completed Black–Scholes derivation: it does not provide an explicit formula or calculate a price. It also flags ambiguity in the contract description. The minimum payoff may be signed, and the problem does not clarify whether exercise occurs only when the payoff is positive; any such optionality would change the payoff and must be specified before pricing.
Key ideas
- Plot the terminal payoff and identify its linear segments and slope changes.
- Use European calls and puts at the breakpoints to replicate a piecewise linear payoff.
- A replicating portfolio and the exotic claim must have the same price under no-arbitrage conditions.
- Clarify whether the payoff can be negative or whether exercise is conditional on a positive payoff.
- The discussion gives a replication procedure but does not complete the requested Black–Scholes formula.
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# Special Exotic Option Pricing Approach
# Special Exotic Option Pricing Approach
I am currently stuck with the following problem:
You need to price the following exotic option, where the share price of Stock ABC is the underlying:
• Time to maturity: 2 years
• Right to exercise: Only at maturity
• Payoffs: You receive or pay the minimum of (𝑆𝑇 − 𝑏) and (𝑎 − 𝑆𝑇), where 𝑆𝑇 is the stock price at maturity 𝑇. 𝑎 and 𝑏 are positive constants, set to 𝑎 = 𝐸𝑈𝑅 80 and 𝑏 = 𝐸𝑈𝑅 35 in our contract.
QUESTION:
Derive a pricing formula for the exotic option described above (using BS)
I am not sure what type of exotic option I am encountering, may someone of you can give me a clue?
## Answer by FP0 (score 3)
https://quant.stackexchange.com/a/71658
For homework, I think that people in these forums like when the author explains his current progress/ideas/intuitions.
Try to follow the following steps:
- Create a plot with axes: $x=S_T, y=\text{payoff}$.
- Draw the individual payoffs.
- Use the previous lines in order to determine the global payoff of your product.
- Usually, the idea behind these exercises is to train you to identify a portfolio of options and/or underlying asset which could replicate these payoffs. In this case, since the payoff can only be determined at maturity, can you find a portfolio of European Call/Put options which could replicate the total payoff of your product ? Hint 1: The key here is to use the slopes of the payoff, and the prices $S_T$ where they change. Hint 2: Remember that you can "invert" a payoff simply by shorting (selling) an option.
- To avoid arbitrage opportunities, if you can find a portfolio which can replicate the payoff of your financial product for all possible prices $S_T$ at maturity, then the price of your product now has to be the same as the price of the replicating portfolio now.
Just as a final remark, your question lets me think that your product is an option, but I cannot see any optionality in your payoff. Maybe the product is only exercised when the available payoff is greater than 0 ? I do not know, it depends on your problem, but such an optionality would have to be taken into account in steps 3 and 4.
AFTER you have finished these steps, I suggest you have a look at this page.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.