On option pricing models in the presence of heavy tails

Abstract

We propose a modification of the option pricing framework derived by Borland which removes the possibilities for arbitrage within this framework. It turns out that such arbitrage possibilities arise due to an incorrect derivation of the martingale transformation in the non-Gaussian option models which are used in that paper. We show how a similar model can be built for the asset price processes which excludes arbitrage. However, the correction causes the pricing formulas to be less explicit than the ones in the original formulation, since the stock price itself is no longer a Markov process. Practical option pricing algorithms will therefore have to resort to Monte Carlo methods or partial differential equations and we show how these can be implemented. An extra parameter, which needs to be specified before the model can be used, will give market makers some extra freedom when fitting their model to market data.

Keywords:

• Contingent claim pricing
• Heavy-tailed distributions
• Stochastic volatility

Acknowledgements

The authors would like to thank an anonymous referee for his very helpful comments, which have improved this paper considerably. Both authors would like to thank the Derivatives Technology Foundation for partial funding of their research, and for drawing attention to some of the problems addressed in this paper.

Notes

We use a slightly different notation than the one in Borland's paper to emphasize which constants are positive or negative, α = q − 1 in the Borland paper. We also take α smaller than instead of Borland's to make sure that the expectation of quadratic variation is finite for all t∈[0,T ], as shown later. Also note that the constant ξ that we introduce here was taken ξ = 1 in the Borland paper.

The formulation from the book of Karatzas and Shreve (1988) has been used

Note that in Borland's paper, the constant ξ was taken to be one, but then will not satisfy the Fokker–Planck equation. We thank the anonymous referee for pointing this out to us

Local martingales with finite quadratic variation processes are martingales, see Protter (2003) II.6 coll. 3.

See, for example, the book by Protter (2003), II.6 coll. 3.