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A note on options and bubbles under the CEV model: implications for pricing and hedging

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Abstract

The discounted price process under the constant elasticity of variance (CEV) model is not a martingale (wrt the risk-neutral measure) for options markets with upward sloping implied volatility smiles. The loss of the martingale property implies the existence of (at least) two option prices for the call option: the price for which the put-call parity holds and the (risk-neutral) price representing the lowest cost of replicating the call payoff. This article derives closed-form solutions for the Greeks of the risk-neutral call option pricing solution that are valid for any CEV process exhibiting forward skew volatility smile patterns. Using an extensive numerical analysis, we conclude that the differences between the call prices and Greeks of both solutions are substantial, which might yield significant errors of analysis for pricing and hedging purposes.

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Notes

  1. For \(\beta =2\) (the lognormal case), both zero and infinity are natural boundaries.

  2. A similar regularization scheme can be applied at small price levels for the CEV process with \(\beta <2\) to avoid absorption at zero.

  3. We note that Eq. (10) corrects the misprint error of Heston et al. (2007, Page 367) highlighted in Veestraeten (2017, Footnote 1).

  4. There is a small typo in Heston et al. (2007, Page 367): when setting \(K=0\) in Heston et al. (2007, Equation 9), and using their notation, instead of \(G_{1}=S_{t}e^{-q\tau }\) one obtains our Eq. (13).

  5. And the forward stock price is only a local martingale under the risk-neutral measure because \(\frac{\Gamma \left( v,x\right) }{\Gamma \left( v\right) }=Q\left( 2x;2v,0\right) \in \left( 0,1\right) \).

  6. The same problem occurs also in the early exercise premium contained in Kim and Yu (1996, Equation 36).

  7. In the case of the call, however, we note that there is no recovery payment to be paid at maturity upon default, i.e. \(c_{t}^D (S_t, K, T)=0\), because such contingent claim has payout only on the upside region of the underlying asset value, and, therefore, \(c_{t}^0 (S_t, K, T) = e^{-r\tau } \, \mathbb {E}_{\mathbb {Q}}\left[ \left. \left( S_T - K \right) ^{+} 11_{\{\tau _0>T\}} \right| \mathcal {F}_{t}\right] \) is also obtained through Schroder (1989, Equation 3).

  8. Nevertheless, Barone-Adesi and Sala (2019) argue that such loss of monotonicity is attenuated once both (risk-neutral and physical) measures of the pricing kernel are estimated using the same filtration set.

  9. The assumption \(\mu >r-q\) is mild given the paramount evidence of a positive equity risk premium.

  10. For instance, the unrestricted Greeks solutions of Larguinho et al. (2013) have been proved to be crucial in a wide variety of option pricing applications under the CEV model—see, for example, Ruas et al. (2013), Dias et al. (2015), Nunes et al. (2015), Cruz and Dias (2017, 2019).

  11. However, similarly to the case of plain-vanilla calls, the presence of bubbles will imply at least two different solutions for European-style asset-or-nothing calls, range asset options, gap call options and (single and double) barrier call options whenever \(\beta >2\).

  12. For example, Choi and Longstaff (1985, Page 252) reported \(\beta \) values ranging from 4.38 to 5.02 in their application of the CEV model for pricing options on agricultural futures that uses the Emanuel and MacBeth (1982) solution.

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Correspondence to José Carlos Dias.

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Dias, J.C., Nunes, J.P.V. & Cruz, A. A note on options and bubbles under the CEV model: implications for pricing and hedging. Rev Deriv Res 23, 249–272 (2020). https://doi.org/10.1007/s11147-019-09164-x

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