Optimal Dynamic Hedging of Cliquets
by Andrea Petrelli of Credit-Suisse,
Abstract: Analyzed here is a Cliquet put option (ratchet put option) defined as a resettable strike put with a payout triggered by the reference asset falling below a specified fraction of its value at a prior look-back date. The hedging strategy that minimizes P&L volatility over discrete hedging intervals is assessed. Examples are provided for an asset exhibiting jumpy returns (kurtosis > 3) and temporal correlation between the squared residual returns. The limited liquidity of the asset limits the discrete hedging frequency. Each of the realities of discrete hedging intervals and fat-tailed asset return distributions render the attempted replication imperfect. A residual risk dependent premium is added to the average cost of attempted replication (i.e., average hedging cost) based on a target expected return on risk capital. By comparing the P&L distribution of a derivative seller-hedger with that of a delta-one trader holding a long position in the underlying asset, relative-value based bounds on pricing of vanilla options and Cliquets are presented.
Keywords: Gap-Risk, Cliquet, Crash-Cliquet, Kurtosis, Hedging, Residual-Risk, Option-Traders's-P&L.