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Static Hedging of Standard Options

Author

Listed:
  • Peter Carr

    (New York University & Bloomberg)

  • Liuren Wu

    (Zicklin School of Business, Baruch College)

Abstract

We consider the hedging of options when the price of the underlying asset is always exposed to the possibility of jumps of random size. Working in a single factor Markovian setting, we derive a new spanning relation between a given option and a continuum of shorter-term options written on the same asset. In this portfolio of shorter-term options, the portfolio weights do not vary with the underlying asset price or calendar time. We then implement this static relation using a finite set of shorter-term options and use Monte Carlo simulation to determine the hedging error thereby introduced. We compare this hedging error to that of a delta hedging strategy based on daily rebalancing in the underlying futures. The simulation results indicate that the two types of hedging strategies exhibit comparable performance in the classic Black-Scholes environment, but that our static hedge strongly outperforms delta hedging when the underlying asset price is governed by Merton (1976)'s jump-diffusion model. The conclusions are unchanged when we switch to ad hoc static and dynamic hedging practices necessitated by a lack of knowledge of the driving process. Further simulations indicate that the inferior performance of the delta hedge in the presence of jumps cannot be improved upon by increasing the rebalancing frequency. In contrast, the superior performance of the static hedging strategy can be further enhanced by using more strikes or by optimizing on the common maturity in the hedge portfolio. We also compare the hedging effectiveness of the two types of strategies using more than six years of data on S&P 500 index options. We find that in all cases considered, a static hedge using just five call options outperforms daily delta hedging with the underlying futures. The consistency of this result with our jump model simulations lends empirical support for the existence of jumps of random size in the movement of the S&P 500 index. We also find that the performance of our static hedge deteriorates moderately as we increase the gap between the maturity of the target call option and the common maturity of the call options in the hedge portfolio. We interpret this result as evidence of additional random factors such as stochastic volatility.

Suggested Citation

  • Peter Carr & Liuren Wu, 2004. "Static Hedging of Standard Options," Finance 0409016, EconWPA.
  • Handle: RePEc:wpa:wuwpfi:0409016
    Note: Type of Document - pdf; pages: 61
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    References listed on IDEAS

    as
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    Cited by:

    1. Pedro Santa-Clara & Shu Yan, 2004. "Jump and Volatility Risk and Risk Premia: A New Model and Lessons from S&P 500 Options," NBER Working Papers 10912, National Bureau of Economic Research, Inc.
    2. repec:wsi:ijtafx:v:13:y:2010:i:08:n:s0219024910006157 is not listed on IDEAS
    3. Nteukam T., Oberlain & Planchet, Frédéric & Thérond, Pierre-E., 2011. "Optimal strategies for hedging portfolios of unit-linked life insurance contracts with minimum death guarantee," Insurance: Mathematics and Economics, Elsevier, vol. 48(2), pages 161-175, March.
    4. Artur Sepp, 2012. "An approximate distribution of delta-hedging errors in a jump-diffusion model with discrete trading and transaction costs," Quantitative Finance, Taylor & Francis Journals, vol. 12(7), pages 1119-1141, May.

    More about this item

    Keywords

    Static hedging; jumps; option pricing; Monte Carlo; S&P 500 index options; stochastic volatility;

    JEL classification:

    • G12 - Financial Economics - - General Financial Markets - - - Asset Pricing; Trading Volume; Bond Interest Rates
    • G13 - Financial Economics - - General Financial Markets - - - Contingent Pricing; Futures Pricing
    • C52 - Mathematical and Quantitative Methods - - Econometric Modeling - - - Model Evaluation, Validation, and Selection

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