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12
A new look at shortterm implied volatility in asset price models with jumps
 IN MATHEMATICAL FINANCE
, 2013
"... We analyse the behaviour of the implied volatility smile for options close to expiry in the exponential Lévy class of asset price models with jumps. We introduce a new renormalisation of the strike variable with the property that the implied volatility converges to a nonconstant limiting shape, w ..."
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We analyse the behaviour of the implied volatility smile for options close to expiry in the exponential Lévy class of asset price models with jumps. We introduce a new renormalisation of the strike variable with the property that the implied volatility converges to a nonconstant limiting shape, which is a function of both the diffusion component of the process and the jump activity (BlumenthalGetoor) index of the jump component. Our limiting implied volatility formula relates the jump activity of the underlying asset price process to the short end of the implied volatility surface and sheds new light on the difference between finite and infinite variation jumps from the viewpoint of option prices: in the latter, the wings of the limiting smile are determined by the jump activity indices of the positive and negative jumps, whereas in the former, the wings have a constant modelindependent slope. This result gives a theoretical justification for the preference of the infinite variation Lévy models over the finite variation ones in the calibration based on shortmaturity option prices.
The smallmaturity Heston forward smile
 SIAM J. on Financial Mathematics
, 2013
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Efficient solution of backward jumpdiffusion PIDEs with splitting and matrix exponentials
 Journal of Computational
"... We propose a new, unified approach to solving jumpdiffusion partial integrodifferential equations (PIDEs) that often appear in mathematical finance. Our method consists of the following steps. First, a secondorder operator splitting on financial processes (diffusion and jumps) is applied to thes ..."
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Cited by 3 (3 self)
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We propose a new, unified approach to solving jumpdiffusion partial integrodifferential equations (PIDEs) that often appear in mathematical finance. Our method consists of the following steps. First, a secondorder operator splitting on financial processes (diffusion and jumps) is applied to these PIDEs. To solve the diffusion equation, we use standard finitedifference methods, which for multidimensional problems could also include splitting on various dimensions. For the jump part, we transform the jump integral into a pseudodifferential operator. Then for various jump models we show how to construct an appropriate first and second order approximation on a grid which supersets the grid that we used for the diffusion part. These approximations make the scheme to be unconditionally stable in time and preserve positivity of the solution which is computed either via a matrix exponential, or via Páde approximation of the matrix exponent. Various numerical experiments are provided to justify these results. 1
The smallmaturity implied volatility slope for Lévy models. Available at arXiv:1310.3061
, 2014
"... Abstract. We consider the atthemoney strike derivative of implied volatility as the maturity tends to zero. Our main results quantify the growth of the slope for infinite activity exponential Lévy models. As auxiliary results, we obtain the limiting values of short maturity digital call options, ..."
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Abstract. We consider the atthemoney strike derivative of implied volatility as the maturity tends to zero. Our main results quantify the growth of the slope for infinite activity exponential Lévy models. As auxiliary results, we obtain the limiting values of short maturity digital call options, using Mellin transform asymptotics. Finally, we discuss when the atthemoney slope is consistent with the steepness of the smile wings, as given by Lee’s moment formula. 1.
UNIFORM BOUNDS FOR BLACK–SCHOLES IMPLIED VOLATILITY
"... Abstract. The Black–Scholes implied total variance function is defined by VBS(k, c) = v ⇔ Φ ( − k/√v +√v/2) − ekΦ( − k/√v −√v/2) = c. The new formula VBS(k, c) = inf x∈R Φ−1 c + ekΦ(x)) − x]2 is proven. Uniform bounds on the function VBS are deduced and illustrated numerically. As a byproduct of ..."
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Abstract. The Black–Scholes implied total variance function is defined by VBS(k, c) = v ⇔ Φ ( − k/√v +√v/2) − ekΦ( − k/√v −√v/2) = c. The new formula VBS(k, c) = inf x∈R Φ−1 c + ekΦ(x)) − x]2 is proven. Uniform bounds on the function VBS are deduced and illustrated numerically. As a byproduct of this analysis, it is proven that F is the distribution function of a logconcave probability measure if and only if F (F−1(·) + b) is concave for all b ≥ 0. From this, an interesting class of peacocks is constructed. 1.
A note on highorder shorttime expansions for ATM option prices under the CGMY model
, 2013
"... The shorttime asymptotic behavior of option prices for a variety of models with jumps has received much attention in recent years. In the present work, a novel thirdorder approximation for ATM option prices under the CGMY Lévy model is derived, and extended to a model with an additional independen ..."
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The shorttime asymptotic behavior of option prices for a variety of models with jumps has received much attention in recent years. In the present work, a novel thirdorder approximation for ATM option prices under the CGMY Lévy model is derived, and extended to a model with an additional independent Brownian component. Our results shed new light on the connection between both the volatility of the continuous component and the jump parameters and the behavior of ATM option prices near expiration.
ThirdOrder ShortTime Expansions for ClosetotheMoney Option Prices Under the CGMY Model
, 2015
"... The shorttime asymptotic behavior of option prices for a variety of models with jumps has received much attention in recent years. In the present work, novel thirdorder approximations for closetothemoney European option prices under an infinitevariation CGMY Lévy model are derived, and are th ..."
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The shorttime asymptotic behavior of option prices for a variety of models with jumps has received much attention in recent years. In the present work, novel thirdorder approximations for closetothemoney European option prices under an infinitevariation CGMY Lévy model are derived, and are then extended to a model with an additional independent Brownian component. The asymptotic regime considered, in which the strike is made to converge to the spot stock price as the maturity approaches zero, is relevant in applications since the most liquid options have strikes that are close to the spot price. Our results shed new light on the connection between both the volatility of the continuous component and the jump parameters and the behavior of option prices near expiration when the strike is close to the spot price. In particular, a new type of transition phenomenon is uncovered in which the third order term exhibits two distinct asymptotic regimes depending on whether