Leveraged Etf Implied Volatilities From Etf Dynamics

Leung, Tim; Lorig, Matthew; Pascucci, Andrea

doi:10.1111/mafi.12128

Cited by 16 publications

(3 citation statements)

References 44 publications

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“…It also undescores the need to find alternative perturbations using other "small parameters," in particular, our calculations for the λSABR PDE are less tedious. See also [49,43,41] for related calculations for the small time asymptotics. Going further back in time, one needs to mention, among many other contributions, the works of Henry-Labordere [35,37,36] who used Riemannian geometry heat kernel approximations, the works of Gatheral and his collaborators who used heat kernel asymptotics to study the implied volatility, and the works of Lesniewski and his collaborators [31,32,33], who introduced and studied the SABR model, the work of Fouque, Papanicolaou, Sircar, and Solna [24,23,25], who studied stochastic volatility models and singular perturbation techniques in option pricing, see also [22].…”

Section: Introductionmentioning

confidence: 99%

A Volatility-of-Volatility Expansion of the Option Prices in the SABR Stochastic Volatility Model

2014

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We propose a general, very fast method to quickly approximate the solution of a parabolic Partial Differential Equation (PDEs) with explicit formulas. Our method also provides equaly fast approximations of the derivatives of the solution, which is a challenge for many other methods. Our approach is based on a computable series expansion in terms of a "small" parameter. As an example, we treat in detail the important case of the SABR PDE for β = 1, namely ∂τ u = σ 2 1 2 (∂ 2The terms u j are explicitly computable, which is also a challenge for many other, related methods. Truncating this expansion leads to computable approximations of u that are in "closed form," and hence can be evaluated very quickly. Most of the other related methods use the "time" τ as a small parameter. The advantage of our method is that it leads to shorter and hence easier to determine and to generalize formulas. We obtain also an explicit expansion for the implied volatility in the SABR model in terms of ν, similar to Hagan's formula, but including also the mean reverting term. We provide several numerical tests that show the performance of our method. In particular, we compare our formula to the one due to Hagan. Our results also behave well when used for actual market data and show the mean reverting property of the volatility. Contents32 5.2. The SABR model (with general β) 34 References 35

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Section: Introductionmentioning

confidence: 99%

A Volatility-of-Volatility Expansion of the Option Prices in the SABR Stochastic Volatility Model

2014

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“…Options on LETFs of different leverage ratios are also widely traded on the Chicago Board Options Exchange (CBOE). The pricing of these options under stochastic volatility models has been recently studied in Leung et al (2014); Leung and Sircar (2015). In practice, significantly fewer strikes are available for LETF options, some with wider bid-ask spreads, as compared to the nonleveraged counterparts.…”

Section: Hedging An Letf Option With Options On the Referencementioning

confidence: 99%

Optimal Static Quadratic Hedging

Leung

Lorig

2015

SSRN Journal

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We propose a flexible framework for hedging a contingent claim by holding static positions in vanilla European calls, puts, bonds, and forwards. A model-free expression is derived for the optimal static hedging strategy that minimizes the expected squared hedging error subject to a cost constraint. The optimal hedge involves computing a number of expectations that reflect the dependence among the contingent claim and the hedging assets. We provide a general method for approximating these expectations analytically in a general Markov diffusion market.To illustrate the versatility of our approach, we present several numerical examples, including hedging path-dependent options and options written on a correlated asset.

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“…Leung and Sircar [23] study the implied volatility of LETF options assuming that the underlying ETF follows a fast mean-reverting volatility process, and obtain the same scaling as [4]. Leung, Lorig, and Pascucci [21] study the implied volatility of LETF options assuming that the underlying ETF follows a general local-stochastic volatility model. They obtain the same scaling as [4] at the zeroth-order, but caution that the scaling alone is not sufficient to capture the full effect of leverage on the implied volatility, and they derive higher order corrections to the scaling.…”

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confidence: 99%

Short-Time Expansions for Call Options on Leveraged ETFs Under Exponential Lévy Models with Local Volatility

Figueroa-López¹,

Gong²,

Lorig³

2018

SIAM J. Finan. Math.

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In this article, we consider the small-time asymptotics of options on a Leveraged Exchange-Traded Fund (LETF) when the underlying Exchange Traded Fund (ETF) exhibits both local volatility and jumps of either finite or infinite activity. We show that leverage modifies the drift, volatility, jump intensity, and jump distribution of a LETF in addition to inducing the possibility of default, even when the underlying ETF price remains strictly positive. Our main results are closed-form expressions for the leading order terms of off-the-money European call and put LETF option prices near expiration, with explicit error bounds. These results show that the price of an out-of-the-money European call on a LETF with positive (negative) leverage is asymptotically equivalent, in short-time, to the price of an out-of-the-money European call (put) on the underlying ETF, but with modified spot and strike prices. Similar relationships hold for other off-the-money European options. These observations, in turn, suggest a method to hedge off-the-money LETF options near expiration using options on the underlying ETF. Finally, we derive a second order expansion for the implied volatility of an off-the-money LETF option and show both analytically and numerically how this is affected by leverage.

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Leveraged Etf Implied Volatilities From Etf Dynamics

Cited by 16 publications

References 44 publications

A Volatility-of-Volatility Expansion of the Option Prices in the SABR Stochastic Volatility Model

A Volatility-of-Volatility Expansion of the Option Prices in the SABR Stochastic Volatility Model

Optimal Static Quadratic Hedging

Short-Time Expansions for Call Options on Leveraged ETFs Under Exponential Lévy Models with Local Volatility

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