arXiv 2302.058082023-02-11q-fin.PRq-fin.MFq-fin.RM

具有低反射障碍的长期期权定价

Long-term option pricing with a lower reflecting barrier

R. Guy Thomas

备注：: Accepted for publication in Annals of Actuarial Science
DOI：: 10.1017/S1748499522000227

摘要：

本文考虑了住房等资产的长期期权定价问题，假设政府干预或资产的经济性质限制了价格的大幅下跌。观察到的资产价格是由几何布朗运动（“名义价格”）反映在一个较低的障碍。由此产生的观察价格具有标准动态，但在障碍处有局部干预，这允许套利与中期损失;这是由政府的无限干预权力资助的，其利用受到信贷约束。尽管缺乏观测价格的等价鞅测度，这个价格上的期权可以表示为无仲裁名义价格上的复合期权，标准的风险中性参数可以应用于此。因为当观察价格接近障碍时，期权增量趋于零，所以用观察价格进行套期保值与用名义价格进行套期保值的结果是相同的，因此完全复制了期权收益。对冲方案并不是唯一的，任何衍生品的最便宜方案都是最好地利用障碍干预的方案。看跌期权的价格很明显：直接复制的初始成本低于合成复制，复制组合总是有正价值。看涨期权的价格是模糊的：合成复制的初始成本比直接复制低，但复制投资组合可能会产生临时损失，因此看涨期权的首选复制策略（以及价格）可能取决于需要对这些损失支付多少保证金。

英文摘要：

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 is assumed to limit 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, and so the preferred replication strategy (and hence price) of a call may depend on what margin payments need to be made on these losses.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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英文摘要：

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