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Root’s barrier: Construction, optimality and applications to variance options
, 2011
"... of understanding the Skorokhod embedding originally proposed by Root for the modelindependent hedging of variance options. Root’s work shows that there exists a barrier from which one may define a stopping time which solves the Skorokhod embedding problem. This construction has the remarkable prope ..."
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Cited by 26 (4 self)
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of understanding the Skorokhod embedding originally proposed by Root for the modelindependent hedging of variance options. Root’s work shows that there exists a barrier from which one may define a stopping time which solves the Skorokhod embedding problem. This construction has the remarkable property, proved by Rost, that it minimizes the variance of the stopping time among all solutions. In this work, we prove a characterization of Root’s barrier in terms of the solution to a variational inequality, and we give an alternative proof of the optimality property which has an important consequence for the construction of subhedging strategies in the financial context.
The smallmaturity smile for exponential Lévy models
, 2012
"... We derive a smalltime expansion for outofthemoney call options under an exponential Lévy model, using the smalltime expansion for the distribution function given in FigueroaLópez&Houdré[FLH09], combined with a change of numéraire via the Esscher transform. In particular, we find that the e ..."
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Cited by 18 (6 self)
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We derive a smalltime expansion for outofthemoney call options under an exponential Lévy model, using the smalltime expansion for the distribution function given in FigueroaLópez&Houdré[FLH09], combined with a change of numéraire via the Esscher transform. In particular, we find that the effect of a nonzero volatility σ of the Gaussian component of the driving Lévy process is to increase the call price by 1 2σ2t 2 e k ν(k)(1+o(1)) as t → 0, where ν is the Lévy density. Using the smalltime expansion for call options, we then derive a smalltime expansion for the implied volatility ˆσ 2 t(k) at logmoneyness k, which sharpens the first order estimate ˆσ 2 t(k) ∼ 1 2 k2 tlog(1/t) given in [Tnkv10]. Our numerical results show that the second order approximation can significantly outperform the first order approximation. Our results are also extended to a class of timechanged Lévy models. We also consider a smalltime, small logmoneyness regime for the CGMY model, and apply this approach to the smalltime pricing of atthemoney call options; we show that for Y ∈ (1,2), limt→0t −1/Y E(St−S0)+ = S0E ∗ (Z+) and the corresponding atthemoney implied volatility ˆσt(0) satisfies limt→0 ˆσt(0)/t 1/Y −1/2 = √ 2πE ∗ (Z+), where Z is a symmetric Ystable random variable under P ∗ and Y is the usual parameter for the CGMY model appearing in the Lévy density ν(x) = Cx −1−Y e −Mx 1{x>0} +Cx  −1−Y e −Gx  1{x<0} of the process.
Martingale Property and Pricing for Timehomogeneous Diffusion Models in Finance
"... I hereby declare that I am the sole author of this thesis. This is a true copy of the thesis, including any required final revisions, as accepted by my examiners. I understand that my thesis may be made electronically available to the public. ..."
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Cited by 3 (2 self)
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I hereby declare that I am the sole author of this thesis. This is a true copy of the thesis, including any required final revisions, as accepted by my examiners. I understand that my thesis may be made electronically available to the public.
Recursive Algorithms for Pricing Discrete Variance options and Volatility Swaps under Timechanged Lévy Processes
, 2015
"... We propose robust numerical algorithms for pricing variance options and volatility swaps on discrete realized variance under general timechanged Lévy processes. Since analytic pricing formulas of these derivatives are not available, some of the earlier pricing methods use the quadratic variation a ..."
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We propose robust numerical algorithms for pricing variance options and volatility swaps on discrete realized variance under general timechanged Lévy processes. Since analytic pricing formulas of these derivatives are not available, some of the earlier pricing methods use the quadratic variation approximation for the discrete realized variance. While this approximation works quite well for longmaturity options on discrete realized variance, numerical accuracy deteriorates for options with low frequency of monitoring or short maturity. To circumvent these shortcomings, we construct numerical algorithms that rely on the computation of the Laplace transform of the discrete realized variance under timechanged Lévy processes. We adopt the randomization of the Laplace transform of the discrete log return with a standard normal random variable and develop a recursive quadrature algorithm to compute the Laplace transform of the discrete realized variance. Our pricing approach is rather computationally efficient when compared with the Monte Carlo simulation and works particularly well for discrete realized variance and volatility derivatives with low frequency of monitoring or short maturity. The pricing behaviors of variance options and volatility swaps under various timechanged Lévy processes are also investigated.
Arbitrage bounds for prices of options on realised variance
, 2011
"... We develop robust pricing and hedging of a weighted variance swap when market prices for a finite number of co–maturing put options are given. We assume the given prices do not admit arbitrage and deduce noarbitrage bounds on the weighted variance swap along with super and sub replicating strateg ..."
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We develop robust pricing and hedging of a weighted variance swap when market prices for a finite number of co–maturing put options are given. We assume the given prices do not admit arbitrage and deduce noarbitrage bounds on the weighted variance swap along with super and sub replicating strategies which enforce them. We find that market quotes for variance swaps are surprisingly close to the modelfree lower bounds we determine. We solve the problem by transforming it into an analogous question for a European option with a convex payoff. The lower bound becomes a problem in semiinfinite linear programming which we solve in detail. The upper bound is explicit. We work in a model–independent and probability–free setup. In particular we use and extend Föllmer’s pathwise stochastic calculus. Appropriate notions of arbitrage and admissibility are introduced. This allows us to establish the usual hedging relation between the variance swap and the ‘log contract ’ and similar connections for weighted variance swaps. Our results take form of a FTAP: we show that the absence of (weak) arbitrage is equivalent to the existence of a classical model which reproduces the observed prices via risk–neutral expectations of discounted payoffs.
unknown title
, 2012
"... Fourier transform algorithms for pricing and hedging discretely sampled exotic variance products and volatility derivatives under additive processes ..."
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Fourier transform algorithms for pricing and hedging discretely sampled exotic variance products and volatility derivatives under additive processes
Root’s and Rost’s Embeddings: Construction, Optimality and Applications to Variance Options
, 2011
"... ..."
unknown title
"... Pricing options on discrete realized variance with partially exact and bounded approximations ..."
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Pricing options on discrete realized variance with partially exact and bounded approximations