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Long-term option pricing with a lower reflecting barrier

Thomas, R.G. (2023) Long-term option pricing with a lower reflecting barrier. Annals of Actuarial Science, 17 (2). pp. 358-384. ISSN 1748-4995. E-ISSN 1748-5002. (doi:10.1017/S1748499522000227) (KAR id:102099)


This paper considers the pricing of long-term options on assets such as housing, where either government intervention or the economic nature of the asset limits large falls in prices. The observed asset price is modelled by a geometric Brownian motion (“the notional price”) reflected at a lower barrier. The resulting observed price has standard dynamics but with localised intervention at the barrier, which allows arbitrage with interim losses; this is funded by the government’s unlimited powers of intervention, and its exploitation is subject to credit constraints. Despite the lack of an equivalent martingale measure for the observed price, options on this price can be expressed as compound options on the arbitrage-free notional price, to which standard risk-neutral arguments can be applied. Because option deltas tend to zero when the observed price approaches the barrier, hedging with the observed price gives the same results as hedging with the notional price and so exactly replicates option payoffs.

Hedging schemes are not unique, with the cheapest scheme for any derivative being the one which best exploits the interventions at the barrier. The price of a put is clear: direct replication has a lower initial cost than synthetic replication, and the replication portfolio always has positive value. The price of a call is ambiguous: synthetic replication has a lower initial cost than direct replication, but the replication portfolio may give interim losses. So the preferred replication strategy (and hence price) of a call depends on what margin payments need to be made on these losses.

Item Type: Article
DOI/Identification number: 10.1017/S1748499522000227
Uncontrolled keywords: Long-term option pricing; No-negative-equity guarantee; Put option; Arbitrage; Reflecting barrier
Subjects: H Social Sciences > HG Finance
Divisions: Divisions > Division of Computing, Engineering and Mathematical Sciences > School of Mathematics, Statistics and Actuarial Science
Funders: University of Kent (
Depositing User: Guy Thomas
Date Deposited: 16 Jul 2023 09:40 UTC
Last Modified: 09 Jan 2024 10:21 UTC
Resource URI: (The current URI for this page, for reference purposes)

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