Symmetry and duality in Lévy markets

Fajardo, José; Mordecki, Ernesto

doi:10.1080/14697680600680068

Cited by 51 publications

(78 citation statements)

References 32 publications

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“…Since an infinitely divisible random variable ξ is symmetric if and only if γ vanishes and the Lévy measure is symmetric, the above proof is very short in the univariate case and immediately yields the corresponding univariate result stated in [19,12].…”

Section: Theorem 41 Let η Be An Integrable Random Vector Under the mentioning

confidence: 93%

Multivariate Extension of Put-Call Symmetry

Molchanov

Schmutz

2010

SIAM J. Finan. Math.

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Abstract. Multivariate analogues of the put-call symmetry can be expressed as certain symmetry properties of basket options and options on the maximum of several assets with respect to some (or all) permutations of the weights and the strike. The so-called self-dual distributions satisfying these symmetry conditions are completely characterized and their properties explored. It is also shown how to relate some multivariate asymmetric distributions to symmetric ones by a power transformation that is useful to adjust for carrying costs. Particular attention is devoted to the case of asset prices driven by Lévy processes. Based on this, semistatic hedging techniques for multiasset barrier options are suggested.Key words. barrier option, dual market, Lévy process, multiasset option, put-call symmetry, self-dual distribution, semistatic hedging AMS subject classifications. 60E05, 60G51, 91B28, 91B70DOI. 10.1137/0907541941. Introduction. Consider European options on S T = S 0 e rT η being the price of a (say nondividend paying) asset at the maturity time T , where S 0 is the spot price and e rT η is the factor by which the price changes, r is the (constant) risk-free interest rate, and η is an almost surely positive random variable. In arbitrage-free and complete markets, the option price equals the discounted expected payoff, where the expectation can be taken with respect to the unique equivalent martingale measure. In this case Eη = 1 and the discounted price process (S t e −rt ) t∈ [0,T ] becomes a martingale. Unless indicated by a different subscript, all expectations in this paper are understood with respect to the probability measure Q, which is not necessarily a martingale measure. In this paper we do not address the choice of a martingale measure in incomplete markets.One of the most basic relationships between options in arbitrage-free markets is the European call-put parity. This parity can be expressed by

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Section: Theorem 41 Let η Be An Integrable Random Vector Under the mentioning

confidence: 93%

Multivariate Extension of Put-Call Symmetry

Molchanov

Schmutz

2010

SIAM J. Finan. Math.

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“…This characterization seems to be new even for the standard put-call symmetry (4), though the important particular cases of underlying price processes being Lévy processes or processes with price independent compensator are well studied, see [1], [19]. We shall also extend the theory to time non-homogeneous processes, related notion of duality being referred to in [1] as the put -call reversal.…”

Section: Main Objectivesmentioning

confidence: 99%

“…We can refer to papers [11], [30], [17] for detailed reviews of recent developments. Let us mention specifically papers [1], [19], where put -call symmetry was analyzed for markets based on diffusions with price independent jumps and Lévy processes respectively. Paper [10] developed the theory for American options and papers [22], [21] for Asian options.…”

Section: Bibliographical Commentsmentioning

confidence: 99%

Stochastic Monotonicity and Duality ofkth Order with Application to Put-Call Symmetry of Powered Options

Kolokoltsov

2015

Journal of Applied Probability

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We introduce a notion of kth order stochastic monotonicity and duality that allows one to unify the notion used in insurance mathematics (sometimes refereed to as Siegmund's duality) for the study of ruin probability and the duality responsible for the so-called put -call symmetries in option pricing. Our general kth order duality can be financially interpreted as put -call symmetry for powered options. The main objective of the present paper is to develop an effective analytic approach to the analysis of duality leading to the full characterization of kth order duality of Markov processes in terms of their generators, which is new even for the well-studied case of put -call symmetries.

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“…It is based on the put-call duality from [12]. In the COS pricing formula (10), r, q, ν(dx) are essential in the definition of the characteristic function φ , whereas S and K enter the formula for the Fourier cosine coefficients, V k .…”

Section: The Put-call Dualitymentioning

confidence: 99%

“…In these steps, the influence of an exponentially-increasing payoff can be significant as for European call options. Here, we modify the pricing algorithm for Bermudan call options employing put-call parity (12).…”

Section: The Put-call Paritymentioning

confidence: 99%

Fourier Cosine Expansions and Put–Call Relations for Bermudan Options

Zhang

Oosterlee

2012

Springer Proceedings in Mathematics

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In this chapter we describe the pricing of Bermudan options by means of Fourier cosine expansions. We propose a technique to price early-exercise call options with the help of the (European) put-call parity and put-call duality relations. Direct pricing of call options with cosine expansions may give rise to some sensitivity regarding the choice of the size of the domain in which the Fourier expansion is applied. By employing the put-call parity or put-call duality relations, this can be avoided so that call options governed by fat-tailed asset price distributions can be priced as robust and efficiently as put options.

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Symmetry and duality in Lévy markets

Cited by 51 publications

References 32 publications

Multivariate Extension of Put-Call Symmetry

Multivariate Extension of Put-Call Symmetry

Stochastic Monotonicity and Duality ofkth Order with Application to Put-Call Symmetry of Powered Options

Fourier Cosine Expansions and Put–Call Relations for Bermudan Options

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