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Replicating a Piecewise Linear Payoff with Put Options

Article Quant Q&A · Author: Wolfy
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Summary

The document shows how to express a capped, piecewise linear payoff using two European puts with the same maturity and different strikes. It first writes the payoff by regions: it is constant below the lower strike, declines linearly between the strikes, and is zero above the upper strike. Indicator functions then identify each region and simplify the expression into a difference of positive-part terms.

The result is a long put at strike 2K combined with a short put at strike K. The explanation is algebraic and applies to the depicted payoff shape under the stated strike relationship; it does not discuss option prices, market assumptions, or whether the position is optimal for any trading objective.

Key ideas

  • Indicator functions can separate a piecewise payoff into price regions.
  • A payoff that declines linearly between two strikes can be built from puts at those strikes.
  • The shown payoff equals a long put at 2K minus a put at K with the same expiry.
  • The decomposition describes terminal payoff, not the strategy’s cost or profitability.

Tags

OptionsDerivatives pricing
Full text
# Payoff of option


# Payoff of option












Consider the payoff $g(S_T)$ shown the figure:

I believe the payoff represented as a linear combination of the payoffs of some options with different strike and same maturity $T$ is $$g(S_T) = (2K - S_T)_{+} - (K - S_T)_{+}$$

I am not exactly sure this is correct, any suggestions is greatly appreciated.

## Answer by Gordon (score 2, accepted)

https://quant.stackexchange.com/a/25628

For this type of question, you basically need only to write the payoff with certain indicator functions. In particular, for the above payoff, we have that \begin{align*} \textrm{Payoff} &= K\, 1_{S_T \le K} + (2K-S_T)\,1_{K < S_T \le 2K}\\ &=K\, 1_{S_T \le K} + (2K-S_T)\big(1_{S_T \le 2K} - 1_{S_T \le K} \big)\\ &=(2K-S_T)\,1_{S_T \le 2K} - (K-S_T)1_{S_T \le K}\\ &=(2K-S_T)^+ - (K-S_T)^+. \end{align*}

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