An Exploration on the Use of q-Derivatives in Asset Pricing

This presentation builds further on the work contained in [1]. We discuss again the meaning of a q derivative [2][3] and interpret the value of q in a financial environment. We are particularly interested in this presentation to discuss how the stochastics are affected when we use such derivatives. In particular, why is the Ito lemma (see [4] for instance) not usable? We also expand on the use of this type of derivative in finance.

References:

[1]  E. Haven (2009). Quantum calculus (q-calculus) and option pricing: a brief introduction. In: P. Bruza, D. Sofge, W. Lawless, K. van Rijsbergen (Eds): Quantum Interaction: Third International Symposium, Saarbrücken (Germany). Lecture Notes in Artificial Intelligence (subseries of Lecture Notes in Computer Science). Springer

[2] G. E. Andrews, R. Askey, R. Roy, Special functions, Cambridge University Press, 1999.

[3] V. Kac, P. Cheung, Quantum calculus, Springer, 2002.

[4] B. Øksendal, Stochastic differential equations, Springer, 1992.