Forward equations for option prices in semimartingale models

Bentata, Amel; Cont, Rama

doi:10.1007/s00780-015-0265-z

Cited by 18 publications

(15 citation statements)

References 29 publications

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“…A central ingredient is Dupire's formula, which allows to obtain the diffusion coefficient σ loc directly from the market (or a more complicated reference model), σ 2 loc (K, T ) = 2∂ T C/K 2 ∂ KK C, (1.1) in terms of (call) option prices at various strikes and maturities. 1 Of course, there are ill-posedness issues how to compute derivatives when only given discrete (market) data; this inverse problem is usually solved by fitting market (option or implied vol) data via a smooth parametrization, from which σ loc is then computed. On a more fundamental level, given an arbitrage-free option price surface…”

Section: Failure Of Dupire's Formula In Non-diffusion Settingmentioning

confidence: 99%

“…In fact, if one views (1.1) as a (forward) PDE for call option prices as function of K, T , then the analogous formula in a jump setting is a (forward) PIDE, which is thus the "natural generalization" of Dupire's formula in presence of jumps; cf. [1] and the references therein, in particular towards so-called local Lévy models [2].…”

Section: Failure Of Dupire's Formula In Non-diffusion Settingmentioning

confidence: 99%

“…For instance, in Lévy jump diffusion models, σ loc tends to the volatility σ of the jump diffusion part as T → 0. To see this, recall the forward PIDE for the call price [1,2,4]:…”

Section: Appendix: Local Volatility Blowup In Some Jump Modelsmentioning

confidence: 99%

See 2 more Smart Citations

How to make Dupire’s local volatility work with jumps

Friz¹,

Gerhold

Yor

2014

Quantitative Finance

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There are several (mathematical) reasons why Dupire's formula fails in the non-diffusion setting. And yet, in practice, ad-hoc preconditioning of the option data works reasonably well. In this note we attempt to explain why. In particular, we propose a regularization procedure of the option data so that Dupire's local vol diffusion process recreates the correct option prices, even in manifest presence of jumps.

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Section: Failure Of Dupire's Formula In Non-diffusion Settingmentioning

confidence: 99%

Section: Failure Of Dupire's Formula In Non-diffusion Settingmentioning

confidence: 99%

See 1 more Smart Citation

How to make Dupire’s local volatility work with jumps

Friz¹,

Gerhold

Yor

2014

Quantitative Finance

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“…The purpose of the resulting mimicking processes was to give the opportunity to apply Markovian techniques and tackle both analytical and computational aspects of the initial processes with path-dependent distributional characteristics. Such processes were efficiently used by Dupire [10] and Klebaner [26] for solving option pricing problems in models in which the dynamics of the risky assets are described by general continuous semimartingales and by Bentata and Cont [3] for the discontinuous case. Guyon [20] showed that diffusion-type processes with path-dependent coefficients are conveniently helpful in order to replicate the spot volatility dynamics of the financial market, particularly through the extremum processes such as the running maximum.…”

Section: Introductionmentioning

confidence: 99%

On the drawdowns and drawups in diffusion-type models with running maxima and minima

Gapeev

Rodosthenous

2016

Journal of Mathematical Analysis and Applications

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This document is the author's final accepted version of the journal article. There may be differences between this version and the published version. You are advised to consult the publisher's version if you wish to cite from it.On the drawdowns and drawups in diffusion-type models with running maxima and minima Pavel V. Gapeev * Neofytos Rodosthenous † To appear in Journal of Mathematical Analysis and ApplicationsWe obtain closed-form expressions for the values of joint Laplace transforms of the running maximum and minimum of a diffusion-type process stopped at the first time at which the associated drawdown and drawup processes hit constant levels. It is assumed that the coefficients of the diffusion-type process are regular functions of the running values of the process itself, its maximum and minimum, as well as its maximum drawdown and maximum drawup processes. The proof is based on the solution to the equivalent boundary-value problems and application of the normal-reflection conditions for the value functions at the edges of the state space of the resulting five-dimensional Markov process. We show that the joint Laplace transforms represent linear combinations of solutions to the systems of first-order partial differential equations arising from the application of the normal-reflection conditions.

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“…A drawback with this approach is that a new PDE needs to be derived for each type of payoff. In, e.g., the articles [22,24,25,26,27] Dupire-like equations are derived for some other types of options. In practice, the Dupire equation is used more often for the purpose of calibrating volatility surfaces, than for pricing options.…”

Section: Introductionmentioning

confidence: 99%

Forward deterministic pricing of options using Gaussian radial basis functions

Rad

Höök

Larsson

et al. 2018

Journal of Computational Science

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The price of a fixed-term option is the expected value of the payoff at the time of maturity. When not analytically available, the option price is computed using stochastic or deterministic numerical methods. The most common approach when using deterministic methods is to solve a backward partial differential equation (PDE) such as the Black-Scholes equation for the option value. The problem can alternatively be formulated based on a forward PDE for the probability of the asset value at the time of maturity. This enables simultaneous pricing of several contracts with different payoffs written on the same underlying asset. The main drawback is that the initial condition is a (non-smooth) Dirac function. We show that by using an analytical expansion of the solution for the first part of the time interval, and applying a high-order accurate radial basis function (RBF) approximation in space, we can derive a competitive forward pricing method. We evaluate the proposed method on European call options and barrier options, and show that even for just one payoff it is more efficient than solving the corresponding backward PDE.

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Forward equations for option prices in semimartingale models

Cited by 18 publications

References 29 publications

How to make Dupire’s local volatility work with jumps

How to make Dupire’s local volatility work with jumps

On the drawdowns and drawups in diffusion-type models with running maxima and minima

Forward deterministic pricing of options using Gaussian radial basis functions

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