Shapes of Implied Volatility with Positive Mass at Zero

Marco, Stefano De; Hillairet, Caroline; Jacquier, Antoine

doi:10.2139/ssrn.2335445

Cited by 10 publications

(19 citation statements)

References 26 publications

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“…where the right hand side is worth −∞ if P(S T = 0) (see also [4,Thm 3.6]). As discussed in the Introduction, our Assumption 2.1 (ii) on the coefficient…”

Section: The Limiting Case θϕ(1 + |ρ|) = 4 and Checking Assumption 2mentioning

confidence: 99%

“…Note we have the following nice interpretation of (4): in the Black-Scholes model, where S T = F e σW T − 1 2 σ 2 T is a geometric Brownian motion with constant volatility parameter σ = v √ T , one has E S T F p = e 1 2 p(p−1)v 2 = R e 1 2 p(p−1)v 2 φ(z)dz. Therefore, we can see Equation (4) as an extension of the pricing formula for power payoffs, from the Black-Scholes world to models with non-constant implied volatility.…”

Section: Introductionmentioning

confidence: 99%

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Moment generating functions and normalized implied volatilities: unification and extension via Fukasawa’s pricing formula

Marco

Martini²

2017

Quantitative Finance

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We extend the model-free formula of [Fukasawa 2012] for E[Ψ(XT )], where XT = log ST /F is the logprice of an asset, to functions Ψ of exponential growth. The resulting integral representation is written in terms of normalized implied volatilities. Just as Fukasawa's work provides rigourous ground for Chriss and Morokoff's (1999) model-free formula for the log-contract (related to the Variance swap implied variance), we prove an expression for the moment generating function E[e pX T ] on its analyticity domain, that encompasses (and extends) Matytsin's formula [Matytsin 2000] for the characteristic function E[e iηX T ] and Bergomi's formula [Bergomi 2016] for E[e pX T ], p ∈ [0, 1]. Besides, we (i) show that put-call duality transforms the first normalized implied volatility into the second, and (ii) analyze the invertibility of the extended transformation d(p, ·) = p d1 + (1 − p)d2 when p lies outside [0, 1]. As an application of (i), one can generate representations for the MGF (or other payoffs) by switching between one normalized implied volatility and the other.

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“…where the right hand side is worth −∞ if P(S T = 0) (see also [4,Thm 3.6]). As discussed in the Introduction, our Assumption 2.1 (ii) on the coefficient…”

Section: The Limiting Case θϕ(1 + |ρ|) = 4 and Checking Assumption 2mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Moment generating functions and normalized implied volatilities: unification and extension via Fukasawa’s pricing formula

Marco

Martini²

2017

Quantitative Finance

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“…Moreover, as De Marco, Hillairet, and Jacquier (2013) show P(S T > 0) can be calculated from call options using the identity of Breeden and Litzenberger (1978) and is given by…”

Section: Lower Bounds With Zero Equity Recoverymentioning

confidence: 99%

Equity Option Implied Probability of Default and Equity Recovery Rate

Chang

Orosi

2016

Journal of Futures Markets

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There is a close link between prices of equity options and the default probability of a firm. We show that in the presence of positive expected equity recovery, standard methods that assume zero equity recovery at default misestimate the option‐implied default probability. We introduce a simple method to detect stocks with positive expected equity recovery by examining option prices and propose a method to extract the default probability from option prices that allows for positive equity recovery. We demonstrate possible applications of our methodology with examples that include financial institutions in the United States during the 2007–09 subprime crisis. © 2016 Wiley Periodicals, Inc. Jrl Fut Mark 37:599–613, 2017

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“…Roger Lee [55] was the first to study extreme strike asymptotics, and further works on this have been carried out by Benaim and Friz [6,7] and in [39,40,41,31,23,19]. Large-maturity asymptotics have only been studied in [67,27,46,45,29] using large deviations and saddlepoint methods.…”

Section: Xt T≥0mentioning

confidence: 99%

Large-Maturity Regimes of the Heston Forward Smile

Jacquier

Roome

2014

SSRN Journal

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Abstract. We provide a full characterisation of the large-maturity forward implied volatility smile in the Heston model. Although the leading decay is provided by a fairly classical large deviations behaviour, the algebraic expansion providing the higher-order terms highly depends on the parameters, and different powers of the maturity come into play. As a by-product of the analysis we provide new implied volatility asymptotics, both in the forward case and in the spot case, as well as extended SVI-type formulae. The proofs are based on extensions and refinements of sharp large deviations theory, in particular in cases where standard convexity arguments fail.

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Shapes of Implied Volatility with Positive Mass at Zero

Cited by 10 publications

References 26 publications

Moment generating functions and normalized implied volatilities: unification and extension via Fukasawa’s pricing formula

Moment generating functions and normalized implied volatilities: unification and extension via Fukasawa’s pricing formula

Equity Option Implied Probability of Default and Equity Recovery Rate

Large-Maturity Regimes of the Heston Forward Smile

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