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Robust Replication of Volatility and Hybrid Derivatives on Jump Diffusions

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  • Peter Carr
  • Roger Lee
  • Matthew Lorig

Abstract

We price and replicate a variety of claims written on the log price $X$ and quadratic variation $[X]$ of a risky asset, modeled as a positive semimartingale, subject to stochastic volatility and jumps. The pricing and hedging formulas do not depend on the dynamics of volatility process, aside from integrability and independence assumptions; in particular, the volatility process may be non-Markovian and exhibit jumps of unknown distribution. The jump risk may be driven by any finite activity Poisson random measure with bounded jump sizes. As hedging instruments, we use the underlying risky asset, a zero-coupon bond, and European calls and puts with the same maturity as the claim to be hedged. Examples of contracts that we price include variance swaps, volatility swaps, a claim that pays the realized Sharpe ratio, and a call on a leveraged exchange traded fund.

Suggested Citation

  • Peter Carr & Roger Lee & Matthew Lorig, 2021. "Robust Replication of Volatility and Hybrid Derivatives on Jump Diffusions," Papers 2107.00554, arXiv.org.
    • Handle: RePEc:arx:papers:2107.00554
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    References listed on IDEAS

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    1. Jimin Lin & Matthew Lorig, 2019. "On Carr and Lee’s Correlation Immunization Strategy," Applied Mathematical Finance, Taylor & Francis Journals, vol. 26(2), pages 131-152, March.
    2. Tim Leung & Matthew Lorig, 2016. "Optimal static quadratic hedging," Quantitative Finance, Taylor & Francis Journals, vol. 16(9), pages 1341-1355, September.
    3. Breeden, Douglas T & Litzenberger, Robert H, 1978. "Prices of State-contingent Claims Implicit in Option Prices," The Journal of Business, University of Chicago Press, vol. 51(4), pages 621-651, October.
    4. Jimin Lin & Matthew Lorig, 2018. "On Carr and Lee's correlation immunization strategy," Papers 1809.10256, arXiv.org, revised Mar 2019.
    5. Heston, Steven L, 1993. "A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options," The Review of Financial Studies, Society for Financial Studies, vol. 6(2), pages 327-343.
    6. Tim Leung & Ronnie Sircar, 2015. "Implied Volatility of Leveraged ETF Options," Applied Mathematical Finance, Taylor & Francis Journals, vol. 22(2), pages 162-188, April.
    7. Tim Leung & Matthew Lorig & Andrea Pascucci, 2014. "Leveraged {ETF} implied volatilities from {ETF} dynamics," Papers 1404.6792, arXiv.org, revised Apr 2015.
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