Approximating a Bounded Cash Payoff with Call Spreads

Summary

The document asks how to create a derivative that pays a fixed cash amount when the underlying asset’s price at maturity lies within a specified interval, and pays nothing outside it. The proposed construction uses two call spreads: buy one around the lower boundary and sell another around the upper boundary. The spread widths should be made as small as practical, and the position size is scaled inversely with width so the combined payoff approaches the desired cash amount inside the interval.

This is a limiting approximation rather than an exact payoff when using finite-width spreads. In real markets, available strikes constrain how closely the structure can match the target, and narrower spreads may require larger notionals. The document offers the construction but provides no pricing, transaction-cost, liquidity, or risk analysis. It therefore explains the payoff replication idea, while leaving practical implementation and valuation to the trader.

Key ideas

• Two call spreads can approximate a payoff that is nonzero only between two price boundaries.
• The lower-strike spread is bought and the upper-strike spread is sold.
• Position notional scales inversely with the width of the call spreads.
• The desired payoff is approached as spread width shrinks.
• Available strikes limit the precision of the replication in practice.

Tags

# How to synthesize this derivative security using plain vanilla call options?


# How to synthesize this derivative security using plain vanilla call options?












A derivative security pays a cash amount c if the spot price of the underlying asset at maturity is between K1 and K2, where 0< K1 < K2 and expires worthless otherwise.

Q: how to construct this derivative security using plain vanilla call options?

Could anyone tell me how to construct this type of derivative security specifically and generally? Thanks!

## Answer by dm63 (score 2, accepted)

https://quant.stackexchange.com/a/31812

Buy a [K1, K1+w] call spread and sell a [K2, K2+w] call spread, where w is as small as possible. The amount of notional of these you will need is c/w. The limit of this as w goes to zero is the payoff you desire. In practice you will be limited to whatever strikes are available to trade.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.