Multivariate Extension of Put-Call Symmetry

Abstract

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

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

“…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.…”

“…If one has the inequality ≤ rather than the equality in (30), then the dual Y y t exists as a sub-Markov process.…”

“…exists. Summing up the formulas for the dual operators to the drift and diffusive parts leads to (31) under the condition (30).…”

Symmetry and Bates’ rule in Ornstein–Uhlenbeck stochastic volatility models

“…Our symmetry concept has been extended in many directions to obtain other applications, such as the jointly self-dual concept introduced by Molchanov and Schmutz (2010) to obtain a multivariate version of the PCS, while the self-dual concept is analogous to ours. Now, since the OU-SV process is not necessarily a Lévy process, 3 to address its symmetry property, we need to obtain the triplet characteristic of the dual process X , which we denote by = (B X , C X , ν X ), from the characteristic triple = (B X , C X , ν X ) of the original process X .…”

“…Also, it has been proved that under the above symmetry, the implied volatility given by the Black-Scholes formula is symmetric with respect to the log-moneyness, as shown by Bates (1997) (for diffusion with jumps), Fajardo and Mordecki (2006) (for Lévy processes) and Carr and Lee (2009) (for local/stochastic volatility and time-changed Lévy processes). Finally, under symmetry, it is possible to obtain semi-static hedging for barrier options, an application that is very important due to the fact that static hedging is much cheaper than dynamic hedging, as shown by Carr and Lee (2009) (for a single asset) and Schmutz (2011) and Molchanov and Schmutz (2010) (for multiple assets).…”