A note on options and bubbles under the CEV model: implications for pricing and hedging
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A note on options and bubbles under the CEV model: implications for pricing and hedging José Carlos Dias1,2
· João Pedro Vidal Nunes1,2 · Aricson Cruz1,2
© Springer Science+Business Media, LLC, part of Springer Nature 2019
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. Keywords Bubbles · CEV model · Greeks · Option pricing · Put-call parity · Local martingales JEL Classification G13
1 Introduction A stock bubble is characterized by the existence of a risky asset whose discounted price process is a strict local martingale under the risk-neutral probability measure but not a martingale, i.e. by the existence of dominating strategies that yield the same payoff as the risky asset, but at a lower price. The presence of bubbles in spot and
Financial support from FCT’s Grant Number UID/GES/00315/2019 is gratefully acknowledged.
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José Carlos Dias [email protected]
1
Instituto Universitário de Lisboa (ISCTE-IUL), Edifício II, Av. Prof. Aníbal Bettencourt, 1600-189 Lisbon, Portugal
2
Business Research Unit (BRU-IUL), Lisbon, Portugal
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J. C. Dias et al.
option prices imply that many standard no-arbitrage results from option pricing theory do not hold. Hence, it is with no surprise that this issue has attracted much attention in the literature—see, for example, Loewenstein and Willard (2000), Cox and Hobson (2005), Heston et al. (2007), Ekström and Tysk (2009), Pal and Protter (2010), Guasoni and Rásonyi (2015) and Veestraeten (2017), just to name a few. In particular, Cox and Hobson (2005), Heston et al. (2007) and Pal and Protter (2010) show that unusual properties in option values arise in the presence of bubbles. For instance, put-call parity fails—consequently, one can choose either put-call parity or risk-neutral option pricing but not both—, the price of an American-style call on a non-dividend paying stock exceeds the price of a similar European-style option, American-style call options on non-dividend paying stocks have no optimal exercise policy, the price of a European-style call is no longer increasing in maturity (for a fixed strike price), call prices do not tend to zero as strike increases to infinity and lookback call options have infinite value. We recall that the equ
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