IDEAS home Printed from https://ideas.repec.org/p/arx/papers/2308.04130.html
   My bibliography  Save this paper

Options are also options on options: how to smile with Black-Scholes

Author

Listed:
  • Claude Martini
  • Arianna Mingone

Abstract

We observe that a European Call option with strike $L > K$ can be seen as a Call option with strike $L-K$ on a Call option with strike $K$. Under no arbitrage assumptions, this yields immediately that the prices of the two contracts are the same, in full generality. We study in detail the relative pricing function which gives the price of the Call on Call option as a function of its underlying Call option, and provide quasi-closed formula for those new pricing functions in the Carr-Pelts-Tehranchi family [Carr and Pelts, Duality, Deltas, and Derivatives Pricing, 2015] and [Tehranchi, A Black-Scholes inequality: applications and generalisations, Finance Stoch, 2020] that includes the Black-Scholes model as a particular case. We also study the properties of the function that maps the price normalized by the underlier, viewed as a function of the moneyness, to the normalized relative price, which allows us to produce several new closed formulas. In connection to the symmetry transformation of a smile, we build a lift of the relative pricing function in the case of an underlier that does not vanish. We finally provide some properties of the implied volatility smiles of Calls on Calls and lifted Calls on Calls in the Black-Scholes model.

Suggested Citation

  • Claude Martini & Arianna Mingone, 2023. "Options are also options on options: how to smile with Black-Scholes," Papers 2308.04130, arXiv.org.
  • Handle: RePEc:arx:papers:2308.04130
    as

    Download full text from publisher

    File URL: http://arxiv.org/pdf/2308.04130
    File Function: Latest version
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Merton, Robert C, 1974. "On the Pricing of Corporate Debt: The Risk Structure of Interest Rates," Journal of Finance, American Finance Association, vol. 29(2), pages 449-470, May.
    2. Arianna Mingone, 2022. "Smiles in delta," Papers 2209.00406, arXiv.org.
    3. Claude Martini & Arianna Mingone, 2021. "Explicit no arbitrage domain for sub-SVIs via reparametrization," Papers 2106.02418, arXiv.org.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Gady Jacoby & Chuan Liao & Jonathan A. Batten, 2007. "A Pure Test for the Elasticity of Yield Spreads," The Institute for International Integration Studies Discussion Paper Series iiisdp195, IIIS.
    2. ilya, gikhman, 2006. "Fixed-income instrument pricing," MPRA Paper 1449, University Library of Munich, Germany.
    3. Gordian Rättich & Kim Clark & Evi Hartmann, 2011. "Performance measurement and antecedents of early internationalizing firms: A systematic assessment," Working Papers 0031, College of Business, University of Texas at San Antonio.
    4. Gerardo Manzo & Antonio Picca, 2020. "The Impact of Sovereign Shocks," Management Science, INFORMS, vol. 66(7), pages 3113-3132, July.
    5. Christophe Hurlin & Jérémy Leymarie & Antoine Patin, 2018. "Loss functions for LGD model comparison," Working Papers halshs-01516147, HAL.
    6. Neus, Werner, 2014. "Eigenkapitalnormen, Boni und Risikoanreize in Banken," Die Unternehmung - Swiss Journal of Business Research and Practice, Nomos Verlagsgesellschaft mbH & Co. KG, vol. 68(2), pages 92-107.
    7. Giordani, Paolo & Jacobson, Tor & Schedvin, Erik von & Villani, Mattias, 2014. "Taking the Twists into Account: Predicting Firm Bankruptcy Risk with Splines of Financial Ratios," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 49(4), pages 1071-1099, August.
    8. Ulrike Malmendier & Vincenzo Pezone & Hui Zheng, 2023. "Managerial Duties and Managerial Biases," Management Science, INFORMS, vol. 69(6), pages 3174-3201, June.
    9. Wei, Yu & Wang, Yizhi & Vigne, Samuel A. & Ma, Zhenyu, 2023. "Alarming contagion effects: The dangerous ripple effect of extreme price spillovers across crude oil, carbon emission allowance, and agriculture futures markets," Journal of International Financial Markets, Institutions and Money, Elsevier, vol. 88(C).
    10. Lily Y. Liu, 2017. "Estimating Loss Given Default from CDS under Weak Identification," Supervisory Research and Analysis Working Papers RPA 17-1, Federal Reserve Bank of Boston.
    11. Kern, Markus & Rudolph, Bernd, 2001. "Comparative analysis of alternative credit risk models: An application on German middle market loan portfolios," CFS Working Paper Series 2001/03, Center for Financial Studies (CFS).
    12. Jeremy Leake, 2003. "Credit spreads on sterling corporate bonds and the term structure of UK interest rates," Bank of England working papers 202, Bank of England.
    13. Xin Huang & Hao Zhou & Haibin Zhu, 2012. "Systemic Risk Contributions," Journal of Financial Services Research, Springer;Western Finance Association, vol. 42(1), pages 55-83, October.
    14. Milne, Alistair, 2014. "Distance to default and the financial crisis," Journal of Financial Stability, Elsevier, vol. 12(C), pages 26-36.
    15. Boulanouar, Zakaria & Alqahtani, Faisal & Hamdi, Besma, 2021. "Bank ownership, institutional quality and financial stability: evidence from the GCC region," Pacific-Basin Finance Journal, Elsevier, vol. 66(C).
    16. Richardson, Grant & Taylor, Grantley & Lanis, Roman, 2015. "The impact of financial distress on corporate tax avoidance spanning the global financial crisis: Evidence from Australia," Economic Modelling, Elsevier, vol. 44(C), pages 44-53.
    17. Zhijian (James) Huang & Yuchen Luo, 2016. "Revisiting Structural Modeling of Credit Risk—Evidence from the Credit Default Swap (CDS) Market," JRFM, MDPI, vol. 9(2), pages 1-20, May.
    18. Masahiko Egami & Rusudan Kevkhishvili, 2020. "Time reversal and last passage time of diffusions with applications to credit risk management," Finance and Stochastics, Springer, vol. 24(3), pages 795-825, July.
    19. Pesaran, M. Hashem & Schuermann, Til & Treutler, Bjorn-Jakob & Weiner, Scott M., 2006. "Macroeconomic Dynamics and Credit Risk: A Global Perspective," Journal of Money, Credit and Banking, Blackwell Publishing, vol. 38(5), pages 1211-1261, August.
    20. Zhou, Fanyin & Fu, Lijun & Li, Zhiyong & Xu, Jiawei, 2022. "The recurrence of financial distress: A survival analysis," International Journal of Forecasting, Elsevier, vol. 38(3), pages 1100-1115.

    More about this item

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:arx:papers:2308.04130. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: arXiv administrators (email available below). General contact details of provider: http://arxiv.org/ .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.