Academic journal article Journal of Risk and Insurance

Pricing and Hedging of Discrete Dynamic Guaranteed Funds

Academic journal article Journal of Risk and Insurance

Pricing and Hedging of Discrete Dynamic Guaranteed Funds

Article excerpt

ABSTRACT

We derive a risk-neutral pricing model for discrete dynamic guaranteed funds with geometric Gaussian underlying security price process. We propose a dynamic hedging strategy by adding a gamma factor to the conventional delta. Simulation results demonstrate that, when hedging discretely, the risk-neutral gamma-adjusted-delta strategy outperforms the dynamic delta hedging strategy by reducing the expected hedging error, lowering the hedging error variability, and improving the self-financing possibility. The discrete dynamic delta-only hedging not only causes potential overcharge to clients but also could be costly to the issuers. We show that a naive application of continuous-time hedging formula to a discrete-time hedging setting tends to worsen these possibilities.

INTRODUCTION

We investigate the hedging performance of the discretely monitored dynamic guaranteed fund using our risk-neutral analytic pricing and hedging formulas that are developed by adding a gamma factor to the conventional delta under geometric Gaussian naked fund prices. We demonstrate that a gamma-adjusted-delta dynamic hedging strategy can curtail both the expected total hedging errors and the hedging uncertainty.

Equity-linked insurance products are typically linked to some reference portfolios, for example, a stock market index. Brennan and Schwartz (1976) value a single-period maturity guaranteed equity-linked contract as an insurance contract with an embedded put option. Boyle and Schwartz (1977) determine an optimal investment policy between investing in the reference portfolio and a riskless reserve for a fund issuer to hedge against the investment risk of these guarantees.

In previous investigation of the distribution of the discrete hedging errors of the continuously monitored Black-Scholes European options, Boyle and Emanuel (1980) show that when they use a delta hedging strategy, the mean discrete hedging error is zero. On the other hand, Leland (1985) derives discrete Black-Scholes European option pricing and the terminal-payoff-replication hedging formulas under proportional transaction costs. He shows that the transaction costs increase to infinity when the portfolio rebalancing time interval approaches zero. This result suggests that discrete dynamic hedging is not only practical but also sensible. Various methods are introduced to price and hedge options with transactions costs (e.g., Toft, 1996). Wilmott (1994) shows that the period-to-period variance-minimizing discrete hedging formula adds an adjustment factor to the delta-hedging parameter. The adjustment factor involves a market drift rate to account for the market incompleteness under discrete hedging. In this article, to be consistent with the complete market risk-neutral option valuation theory (Harrison and Kreps, 1979), we introduce the risk-neutral discrete pricing and hedging strategy by adding a gamma factor to the conventional delta.

Although static hedging strategy can be useful for replicating the terminal payoffs for some structured guaranteed products, it is not always easy to find the suitable and tradable options to replicate a long time-to-maturity complicated guarantee. Boyle and Hardy (1997) investigate the costs and benefits for hedging guarantees between using cash reserve strategy based on stochastic simulation of future investment returns and option rebalancing strategy. However, the property of the distribution of the discrete dynamic hedging errors of complicated guarantees such as a dynamic guaranteed fund has yet to be investigated. By applying the discrete probability density function of the random minimum in AitSahlia and Lai (1998), we derive both the analytic pricing and hedging formulas for the discrete dynamic guaranteed funds. These analytic formulas facilitate an accurate study of the hedging error property. We also demonstrate that the conventional delta of our discrete dynamic guaranteed fund valuation model has a call feature, while its gamma has a put feature. …

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