Smile Asymptotics II: Models with Known Moment Generating Functions

Benaim, Shalom; Friz, Peter K.

doi:10.1017/s0021900200003922

Cited by 34 publications

(68 citation statements)

References 7 publications

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“…Although people have known for a long time that, numerically, two‐sided jumps can produce flexible volatility smiles for call and put options on stocks, a complete theoretical justification of this is still needed. For example, only recently Benaim and Friz (2006a, 2006b) gave a rigorous proof of the asymptotic shape of the implied volatility curves, as the strike price K goes to infinity or to zero under various models, including jump models. On the other hand, Renault and Touzi (1996) and Sircar and Papanicolaou (1999) proved that under a stochastic volatility model, if K is within the neighborhood of the initial stock price S 0 , then the implied volatility curves should be convex.…”

Section: Resultsmentioning

confidence: 99%

Credit Spreads, Optimal Capital Structure, and Implied Volatility With Endogenous Default and Jump Risk

Chen

Kou

2009

Mathematical Finance

138

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We propose a two-sided jump model for credit risk by extending the Leland-Toft endogenous default model based on the geometric Brownian motion. The model shows that jump risk and endogenous default can have significant impacts on credit spreads, optimal capital structure, and implied volatility of equity options: (1) Jumps and endogenous default can produce a variety of non-zero credit spreads, including upward, humped, and downward shapes; interesting enough, the model can even produce, consistent with empirical findings, upward credit spreads for speculative grade bonds.(2) The jump risk leads to much lower optimal debt/equity ratio; in fact, with jump risk, highly risky firms tend to have very little debt. (3) The two-sided jumps lead to a variety of shapes for the implied volatility of equity options, even for long maturity options; although in general credit spreads and implied volatility tend to move in the same direction under exogenous default models, this may not be true in presence of endogenous default and jumps. Pricing formulae of credit default swaps and equity default swaps are also given. In terms of mathematical contribution, we give a proof of a version of the "smooth fitting" principle under the jump model, justifying a conjecture first suggested by Leland and Toft under the Brownian model.

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Section: Resultsmentioning

confidence: 99%

Credit Spreads, Optimal Capital Structure, and Implied Volatility With Endogenous Default and Jump Risk

Chen

Kou

2009

Mathematical Finance

138

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“…In order to study the tail behaviour of a M i x e d T S ( μ , β , α , λ + , λ − )−Γ( a , b ) that denotes a MixedTS distribution with Gamma mixing r.v., we need first to recall the structure of its moment generating function where without loss of generality, we require μ =0. If Y ∼ M i x e d T S (0, β , α , λ + , λ − )−Γ( a , b ), the moment generating function is defined as

M_{Y} (u) = E ((), e^{u Y}) = {([], \frac{b}{b - ((), β u + {normal Φ}_{H} (u))})}^{a} .

We recall some useful results on the study of asymptotic tail behaviour given in Benaim and Friz (). Given the moment generating function M of a r.v.…”

Section: Univariate Mixed Tempered Stablementioning

confidence: 99%

“…X with cumulative distribution function F ( x ) defined as

M (u) : = \int e^{u x} d F (x)

we consider r ⋆ and q ⋆ defined respectively as

- q^{⋆} : = \inf ({}, u : M (u) < \infty),

and

r^{⋆} : = \sup ({}, u : M (u) < \infty),

where r ⋆ , q ⋆ ∈(0, ∞ ). Criterion I in Benaim nd Friz () for asymptotic study of tails states…”

Section: Univariate Mixed Tempered Stablementioning

confidence: 99%

“…Considering the right tail of a standardised CTS, we have r ⋆ = λ + and

M_{C T S} ((), r^{⋆} - s) = M_{C T S} ((), λ_{+} - s)

converges to a constant as s →0 + . Standardised CTS case—1: Under the assumption that α ∈(0,1), we apply criterion 1 in Benaim and Friz () checking that the first derivative of M C T S satisfies

M_{C T S}^{(1)} ((), r^{⋆} - s) = s^{- ρ} l_{1} ((), 1 / s)

for some ρ >0, l 1 ∈ R 0 as s →0 + . The first derivative of M C T S in is

M_{C T S}^{(1)} (u) = M_{C T S} (u) [\frac{{(λ_{+} - u)}^{α - 1} - λ_{+}^{α - 1} - {(λ_{-} + u)}^{α - 1} + λ_{-}^{α - 1}}{(1 - α) (λ_{+}^{α - 2} + λ_{-}^{α - 2})}] .

Evaluating

M_{C T S}^{(1)} (u)

at point λ + − s and computing the limit for s →0 + , we obtain

\lim_{s \to 0^{+}} M_{C T S}^{(1)} (λ_{+} - s) \sim M_{C T S} (λ_{<...>})

…”

Section: Univariate Mixed Tempered Stablementioning

confidence: 99%

“…We compute the first‐order derivative of M Y ( u ) and verify if criterion 1 in Benaim and Friz () is satisfied. We consider separately the two cases α ∈(0,1) and α ∈[1,2). r ⋆ = λ + and α ∈(0,1)

\begin{matrix} M_{Y}^{(1)} (u) & = a {([], \frac{b}{b - ((), β u + {normal Φ}_{H} (u))})}^{a - 1} \frac{- b}{{((), b - ((), β u + {normal Φ}_{H} (u)))}^{2}} ((), - β - {normal Φ}_{H}^{'} (u)) \\ = {([], \frac{b}{b - ((), β u + {normal Φ}_{H} (u))})}^{a} \frac{a ((), β + {normal Φ}_{H}^{'} (u))}{((), b - ((), β u + {normal Φ}_{H} (u)))} \\ = M_{Y} (u) \frac{a ((), β + {normal Φ}_{H}^{'} (u))}{((), b - ((), β u + {normal Φ}_{H} (u)))} \end{matrix}

Observing from that

{normalΦ}_{H}^{'} (u) = {(λ_{+} - u)}^{α - 1} - λ_{+}^{α - 1} - (...

…”

Section: Univariate Mixed Tempered Stablementioning

confidence: 99%

See 2 more Smart Citations

On Properties of the MixedTS Distribution and Its Multivariate Extension

Hitaj

Hubalek

Mercuri

et al. 2018

Int Statistical Rev

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A review of the univariate MixedTS is given and some new results on the asymptotic tail behaviour are derived. The multivariate version of the Mixed Tempered Stable, which is a generalisation of the Normal Variance Mean Mixtures, is discussed. Characteristics of this distribution, its capacity in fitting tails and in capturing dependence structure between components are investigated. We discuss a random number generating procedure and introduce an estimation methodology based on the minimisation of a distance between empirical and theoretical characteristic functions. Asymptotic tail behaviour of the univariate Mixed Tempered Stable is exploited in the estimation procedure in order to obtain a better fitting on tails. Advantages of the multivariate Mixed Tempered Stable distribution are discussed and illustrated via numerical analysis.

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