Pricing Onion Options: A Probabilistic Approach

As argued by Ebenfeld, Mayr and Topper (2002), Onion options may be decomposed into one-touch double barrier binary options (ODBs). Using this idea, these authors provide an arbitrage-free pricing formula for Onion options within the Black-Scholes framework. Their approach rests upon solving the underlying partial differential equation. In this paper, we take an alternative and more direct route: Based on a probabilistic approach, we compute the risk-neutral valuation formula for an ODB. Then, by inverting the decomposition of an Onion option, we are able to derive an alternative pricing formula for this type of an option.


Introduction
Among the vast group of Exotic Options the so called Onion options gained some popularity, (Note 1) and as a consequence, they became regularly traded at, for example, the Deutsche B¨orse in Frankfurt, Germany and at the stock exchange in Stuttgart, Germany. Onion options may be viewed as nested digital double-barrier options. That is, they are composed of a couple of digital knock-out options each of which is characterized by a corridor for the price of the underlying. Ebenfeld, Mayr and Topper (2002) used this composition of Onion options in order to derive an arbitrage-free pricing formula. More precisely, they proposed to decompose such an option into a series of so-called one-touch double barrier binary options (ODBs). Then, after deriving an arbitrage-free price for an ODB, the linearity of the pricing rule (i. e.. the fundamental theorem of finance) may then be applied to obtain an arbitrage-free price for an Onion option. In order to find a valuation formula for an ODB, Ebenfeld, Mayr and Topper (2002) applied the classical Black-Scholes model. In particular, they used the Black-Scholes partial differential equation (PDE) in order to derive an arbitrage-free pricing formula for an ODB -and thus for an Onion option. In fact, it can be shown that this formula is the unique solution of the PDE and thus the unique value for an Onion option in the Black-Scholes model.
In this paper we provide an alternative approach to the derivation of a valuation formula for an Onion option. While Ebenfeld, Mayr and Topper, after suitable transformation of variables, transformed the Black-Scholes PDE into the well known heat equation, we follow a completely different approach here, which we believe to be more direct and thus more intuitive: It is based on a probabilistic approach by computing the risk-neutral valuation formula for currency options. A similar procedure is also pursued by Geman and Yor (1996) who price double barrier options via a probabilistic approach. Similarly, Kunitomo and Ikeda (1992) price double barrier options with curved boundaries via a suitable probability measure.
More recently, the pricing of double barrier options has advanced, and correspondingly, the amount of academic research on this class of options has surged substantially. For example, Luo (2001) derives closed-form solutions for eight types of European-style double-barrier options. Guillaume (2003) examines window double barrier option (options where the monitoring period starts after the beginning of the contract and terminates before its expiry). Labart and Lelong (2009) study double barrier Parisian options, options where the payoff condition depends on the time spent in a row above or below a barrier (or a series of barriers), and not just on the hitting time(s). Buchen (2009) where We have th we just hav

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In this sect used to com order to cal As a first e USD/Euro   In this case, there is still the margin from the beginning -a riskless profit. In the other scenario the long position yields a positive payoff at maturity, the exact amount of which of course depends on the barriers which were hit. Yet, irrespective of which barriers are hit, the options in the short position have reached the same barriers and can, therefore, not yield a higher payoff. Therefore, use the payoff of the short-running option to either settle the short position at maturity, or if it expires worthless before maturity, an additional profit is collected. -Remarkably, this arbitrage opportunity lasted for almost two months until at least the relative price difference was reduced in the market.
This type of mispricing is hard to explain within the Black-Scholes model: There is no reason for any investor to buy the longer-running option, and the emitting bank runs the risk of potential losses if customers become aware of this arbitrage opportunity. In fact, if someone had detected this opportunity, prices should have adjusted quickly. Since this obviously did not occur, we may conclude that the Black-Scholes model is only of limited scope within this framework. In particular, within the Black-Scholes model, it is assumed that the market is frictionless, implying that, among other things, a single deal has no direct effect on the option's price. However, the thinner the trading volume is, the more unrealistic is this assumption -and notably exotic derivatives frequently have a thin trading volume. Also, the Black-Scholes model may simply not be applicable, and correspondingly the issuer may have decided not to price the option according to the arbitrage-free price within the Black-Scholes model.

Conclusions
In this paper we have considered a particular type of an exotic option: Onion options. This type of an option may be viewed as a modified double barrier option: it has several corridors and each time the price of the underlying touches either the upper or the lower barrier of a corridor, the payoff of the option is reduced by some given amount. If up to the time of maturity of the option, the barriers of all corridors are touched, the option ceases to exist and the payoff is nil, otherwise the payoff is determined by those corridors the barriers of which are not hit.
Although the payoff of an Onion option is determined by a simple rule, their valuation is by no means simple. In order to obtain an arbitrage-free formula within the Black-Scholes model, two features turn out to be of significant importance. Firstly, an Onion option may be decomposed into several one-touch double barrier binary options (ODB); and secondly, both the Onion option and the ODB may be regarded as to be path independent as their payoff does not directly depend on the path of the underlying provided that the option is alive at maturity. Given these two features one can derive pricing formulas for the Onion option. Ebenfeld, Mayr, and Topper (2002) provide such a formula based on the well known Black-Scholes PDE.
We recapitulate this result and then provide an alternative pricing rule based upon the risk-neutral valuation formula for currency options. Although both formulas are derived in different ways and look differently -in fact both formulas require the calculation of an infinite sum -, the prices they yield for a given option necessarily coincide. So both the formula of Ebenfeld, Mayr, and Topper as well as ours allow for calculating the arbitrage-free price of an Onion option, and then to compare this price with the actual spot market price.