Closed form asymptotics for local volatility models

We obtain new closed-form pricing formulas for contingent claims when the asset follows a Dupire-type local volatility model. To obtain the formulas we use the Dyson-Taylor commutator method that we have recently developed in [5, 6, 8] for short-time asymptotic expansions of heat kernels, and obtain a family of general closed-form approximate solutions for both the pricing kernel and derivative price. A bootstrap scheme allows us to extend our method to large time. We also perform analytic as well as a numerical error analysis, and compare our results to other known methods.Comment: 30 pages, 10 figure

Option Pricing from Path Integral for Non-Gaussian Fluctuations. Natural Martingale and Application to Truncated L\'evy Distributions

Within a path integral formalism for non-Gaussian price fluctuations we set up a simple stochastic calculus and derive a natural martingale for option pricing from the wealth balance of options, stocks, and bonds. The resulting formula is evaluated for truncated L\'evy distributions.Comment: Author Information under http://www.physik.fu-berlin.de/~kleinert/institution.html. Latest update of paper (including all PS fonts) at http://www.physik.fu-berlin.de/~kleinert/33

Asymptotics for dd-dimensional L\'evy-type processes

We consider a general d-dimensional Levy-type process with killing. Combining the classical Dyson series approach with a novel polynomial expansion of the generator A(t) of the Levy-type process, we derive a family of asymptotic approximations for transition densities and European-style options prices. Examples of stochastic volatility models with jumps are provided in order to illustrate the numerical accuracy of our approach. The methods described in this paper extend the results from Corielli et al. (2010), Pagliarani and Pascucci (2013) and Lorig et al. (2013a) for Markov diffusions to Markov processes with jumps.Comment: 20 Pages, 3 figures, 3 table

Option pricing models without probability: a rough paths approach

We describe the pricing and hedging of financial options without the use of probability using rough paths. By encoding the volatility of assets in an enhancement of the price trajectory, we give a pathwise presentation of the replication of European options. The continuity properties of rough-paths allow us to generalise the so-called fundamental theorem of derivative trading, showing that a small misspecification of the model will yield only a small excess profit or loss of the replication strategy. Our hedging strategy is an enhanced version of classical delta hedging where we use volatility swaps to hedge the second order terms arising in rough-path integrals, resulting in improved robustness

Introducing and solving generalized Black–Scholes PDEs through the use of functional calculus

We introduce some families of generalized Black–Scholes equations which involve the Riemann–Liouville and Weyl space-fractional derivatives. We prove that these generalized Black–Scholes equations are well-posed in (L1−L∞)-interpolation spaces. More precisely, we show that the elliptic-type operators involved in these equations generate holomorphic semigroups. Then, we give explicit integral expressions for the associated solutions. In the way to obtain well-posedness, we prove a new connection between bisectorial-like operators and sectorial operators in an abstract setting. Such a connection extends the scaling property of sectorial operators to a wider family of both operators and the functions involved

Modelling FX smile : from stochastic volatility to skewness

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New stochastic processes to model interest rates : LIBOR additive processes

In this paper, a new kind of additive process is proposed. Our main goal is to define, characterize and prove the existence of the LIBOR additive process as a new stochastic process. This process will be de.ned as a piecewise stationary process with independent increments, continuous in probability but with discontinuous trajectories, and having "càdlàg" sample paths. The proposed process is specifically designed to derive interest-rates modelling because it allows us to introduce a jump-term structure as an increasing sequence of Lévy measures. In this paper we characterize this process as a Markovian process with an infinitely divisible, selfsimilar, stable and self-decomposable distribution. Also, we prove that the Lévy-Khintchine characteristic function and Lévy-Itô decomposition apply to this process. Additionally we develop a basic framework for density transformations. Finally, we show some examples of LIBOR additive processes