A new pricing formula is presented to use for European-type single barrier–style contracts when the underlying process is driven by Lévy processes, such as the CGMY (Carr, Gemen, Madan, and Yor 2002) model, generalized hyperbolic model, and Meixner model.
The author presents an alternative way to compute the price of some digital options under Lévy processes using symmetry properties. In the absence of symmetry, he has devised a relationship that allows for the pricing of some single barrier–style contracts.
How Is This Research Useful to Practitioners?
The relationship between the implied volatility symmetry and other symmetry concepts has important applications, such as the construction of semistatic hedges for exotic options and multivariate derivatives.
The author presents an alternative way to price single barrier–style options, such as digital call and put options, asset-or-nothing options, and down-and-in power options, based on symmetry properties. The underlying asset of these options needs to be modeled by an important family of Lévy processes, including the CGMY (Carr, Gemen, Madan, and Yor 2002) model, generalized hyperbolic model, and Meixner model. These models present a very good fit with real data and can capture the excess kurtosis and asymmetry observed in market data for one-dimensional and multidimensional cases.
The author shows how to obtain close- and simple-pricing formulas for the down-and-in power and digital call options by transforming asymmetrical processes into symmetrical ones. He presents two ways of computing prices through approximations. The first approach, based on Taylor approximations, is suitable for only very short maturities and small moneyness. The second approach is based on a quadratic approximation of the implied volatility; it allows for the computation of prices for any maturity and moneyness.
How Did the Author Conduct This Research?
The author creates a Lévy market that is a model of a financial market with two assets: a deterministic savings account and a stock. He investigates whether the risk-neutral distribution of the discounted asset under the new risk-neutral measure remains the same when the numéraire of a Lévy market is changed. A Lévy market is defined as symmetrical when this condition is satisfied. Under symmetry, prices of some digital options are computed. In the absence of symmetry, the price of some digital options is calculated based on a Taylor approximation or quadratic approximation of implied volatility.
The author shows how to obtain close- and simple-pricing formulas for the down-and-in power and digital call options by transforming asymmetrical processes into symmetrical ones. His findings can be useful to better understand the relationship between digital call option prices and implied volatility slopes. And from a regulatory point of view, the findings can be used to compute probabilities of trigger events, such as the probability of a stock price crossing a determined barrier at the end of some given period of time.
This study warrants further research to explore the possible application of the techniques to price contingent convertible bonds (CoCos), for which there is a need to compute the probabilities of trigger events, or the pricing of CoCos with cancelable coupons, for which stock price variations affect the computation of the probability of having coupons.