On skewed, leptokurtic returns and pentanomial lattice option valuation via minimal entropy martingale measure

Abstract

This article develops, a lattice-based approach for pricing contingent claims when parameters governing the logs of the underlying asset dynamics are modelled by generalized hyperbolic distribution and normal inverse Gaussian distribution. The pentanomial lattice is constructed using a moment matching procedure. Moment generating functions of generalized hyperbolic distribution and normal inverse Gaussian distribution are utilized to compute probabilities and jump parameters under historical measure $\mathbb{P}$. Minimal entropy martingale measure (MEMM) is used to value European call option with a view of comparing the results with some of the existing benchmark model such as Black Scholes model. Empirical data from S&P500 index, RUTSELL2000 index and RUI1000 index are used to demonstrate how the model works. There is a significant difference especially for long term maturity (six months and above) type of contracts, the proposed model outperform the benchmark model, while performing poorly at short term contracts. Pentanomial NIG models seems to outperform the other models especially for long dated maturities.

Keywords:

• binomial
• pentanomial lattice
• generalized hyperbolic distribution
• normal inverse Gaussian
• minimal entropy martingale measure
• European call option

Public Interest Statement

This study develops a simplified approach to value financial contracts such as calls and puts. Features of daily relative changes of price are known to be non-normal in contrast to the common assumption. The proposed model accommodates such several aspects of reality observed in financial markets in recent times. Subjected to observed market data, the proposed valuation model seems to outperform existing benchmark models. The simplified approach may be used to value different types of contracts which may not have closed-form solutions.

Acknowledgements

Financial support from the International Science Programme (ISP) Sweden in collaboration with the Eastern African Universities Mathematics programme, is greatly acknowledged. This article was fine tuned while visiting University of Cape town, South Africa as part of my post doctoral study invited by the African Collaboration for Quantitative Finance and Risk Research (ACQuFRR) directed by Prof. David Taylor. Thanks to the anonymous reviewer for his helpful corrections and suggestions.

Additional information

Funding

This work was supported by International Science program/EAUMP.

Notes on contributors

Ivivi J. Mwaniki

Ivivi J. Mwaniki completed his PhD at the University of Nairobi Kenya in collaboration with Uppsala University Sweden under East African Universities Mathematics Programme/International Science Programme. Currently he works for the University of Nairobi School of Mathematics. This article is part of the ongoing research in relation to option pricing in general. His research interest include option pricing, optimization, time series analysis and mixtures of various distributions, insurance and risk management.