Maximum Drawdown Insurance Using Options

Chapters 7–9 are dedicated to applications of drawdown and maximum drawdown in hedging, insurance and trading, respectively. As the building blocks for more sophisticated options on drawdown or maximum drawdown, in Chapter 7 we study two types of digital options written on them, namely, the digital call on drawdown preceding a drawup, and the digital call on maximum drawdown. Such options are of interest to an asset manager who knows in advance that her portfolio risk is being evaluated wholly or in part by the portfolio’s maximum drawdown or drawdowns. To determine the fair value of these digital options, we aim to develop self-financing replicating portfolios with the least possible time instances in which trading/rebalance is involved. Using results in Chapter 2, we develop a static model-free replication of the former digital option with one-touch knockouts. By using reflection principle and the expansion of the Canadized option price, we also develop semi-static replication strategies for both digital options with gradually more liquid instruments such as one-touches or path-independent options. We demonstrate that these strategies are valid for models such as Black model, geometric Brownian motion (GBM) model and models obtained from their continuous time-changes.