Option Pricing Where the Underlying Assets Follow a Gram/Charlier Density of Arbitrary Order

Abstract

Abstract. If a probability distribution is sufficiently close to a normal distribution, its density can be approximated by a Gram/Charlier Series A expansion. In option pricing, this has been used fit risk-neutral asset price distributions to the implied volatility smile, ensuring an arbitrage-free interpolation of implied volatilities across exercise prices. However, the existing literature is restricted to truncating the series expansion after the fourth moment. This paper presents an option pricing formula in terms of the full (untruncated) series and discusses a fitting algorithm, which ensures that a series truncated at a moment of arbitrary order represents a valid probability density. While it is well known that valid densities resulting from truncated Gram/Charlier Series A expansions do not always have sufficient flexibility to fit all market-observed option prices perfectly, this paper demonstrates that option pricing in a model based on these densities is as tractable as the (far less flexible) original model of Black and Scholes (1973), allowing non-trivial higher moments such as skewness, excess kurtosis and so on to be incorporated into the pricing of exotic options: Generalising the Gram/Charlier Series A approach to the multiperiod, multivariate case, a model calibrated to standard option prices is developed, in which a large class of exotic payoffs can be priced in closed form. Furthermore, this approach, when applied to a foreign exchange option market involving several currencies, can be used to ensure that the volatility smiles for options on the cross exchange rate are constructed in a consistent, arbitrage-free manner.

Factor Distributions Implied by Quoted CDO Spreads

“…For the ITRX.EUR 2 tranche quotes on 21 March 2005, the fit is not as good. In fact, an unconstrained calibration of skewness and kurtosis to the market quotes in this case would not result in a valid densitythe density shown here is the result of a constrained optimisation as suggested by Schlögl (2008). Difficulties are encountered in particular in fitting the senior tranche spread -this appears to be a problem common to most (possibly all) variations of the Vasicek factor approach.…”

“…Reproduced from Schlögl (2008) for the reader's convenience in the appendix. Note that this Lemma is a special case of scaling and translation results well-known in white noise theory, see e.g.…”

“…Calculations can be performed directly at the level of the infinite series expansion, without the need to truncate the series before deriving the desired results (as has been the practice in the option pricing literature using these expansions). See Schlögl (2008) for an application to traditional option pricing.…”

A Monte Carlo multi-asset option pricing approximation for general stochastic processes

“…3 Schlögl (2013) provides an multi-asset option approximation using a multivariate Gram-Charlier A series expansion; however, there are assumptions over the risk-neutral density, and an additional methodology is needed to extract the moments inside the expansion from the Hermite polynomials. 4 Our results complement the results of Filipović et al (2013), as we provide a thorough study of the higher-order moments Jarrow and Rudd (1982) formula the value of the European option is equal to the Black and Scholes price plus corrections based on the difference of the moments of the lognormal distribution and the real market distribution.…”

Easy Gram–Charlier Valuations of Options

“…More-recent contributions incorporating the GC method within the context of quantitative finance exist as well. On the one hand, Corrado 2 [2007] and Schlögl [2010] applied the GC analysis directly to the BSM formula by adding a correction term that accounts for skewness and kurtosis. On the one hand, Corrado 2 [2007] and Schlögl [2010] applied the GC analysis directly to the BSM formula by adding a correction term that accounts for skewness and kurtosis.…”