Analytical Pricing of FX Options in Reflecting Target Zones

Summary

This note develops a pedagogical approach to pricing foreign exchange options when the exchange rate is confined to a target zone with attainable, reflecting boundaries. It explains a no-arbitrage implication of this setting: the difference between foreign and domestic short rates cannot be deterministic. For a narrow band, the exchange-rate process can be chosen for computational convenience, provided the choice supports a tractable solution.

With an appropriate process specification, the pricing partial differential equation can be solved analytically, and European option values can be represented as fast-converging series of elementary functions. The note outlines the general solution approach and gives explicit examples, including mean-reverting models beyond the Ornstein–Uhlenbeck case. Its focus is model-based valuation rather than empirical testing or trading performance. The results depend on the target-zone assumptions and selected process dynamics, so applying the formulas requires checking that those assumptions fit the market and contract being priced.

Key ideas

• The note studies FX options when exchange-rate target-zone boundaries are attainable and reflecting.
• No-arbitrage requires the foreign-minus-domestic short-rate differential to be non-deterministic in this setting.
• For a narrow band, computationally convenient exchange-rate processes can yield analytical pricing solutions.
• European option values are expressed as fast-converging series, with examples using non-Ornstein–Uhlenbeck mean reversion.

Tags

# FX Options in Target Zone


# FX Options in Target Zone









In this note we discuss - in what is intended to be a pedagogical fashion - FX option pricing in target zones with attainable boundaries. The boundaries must be reflecting. The no-arbitrage requirement implies that the differential (foreign minus domestic) short-rate is not deterministic. When the band is narrow, we can pick the functional form of the FX rate process based on computational convenience. With a thoughtful choice, the FX option pricing problem can be solved analytically. The European option prices are expressed via (fast converging) series of elementary functions. We discuss the general approach to solving the pricing PDE and explicit examples, including analytically tractable models with (non-Ornstein-Uhlenbeck) mean-reversion.

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