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March  2014, 19(2): 435-446. doi: 10.3934/dcdsb.2014.19.435

## Volatility in options formulae for general stochastic dynamics

 1 School of Mathematical Sciences, Monash University, Victoria 3800, Australia, Australia 2 School of Mathematical Sciences, Monash University Vic 3800

Received  May 2013 Revised  December 2013 Published  February 2014

It is well-known that the Black-Scholes formula has been derived under the assumption of constant volatility in stocks. In spite of evidence that this parameter is not constant, this formula is widely used by financial markets. This paper addresses the question of whether an alternative model for stock price exists for which the Black-Scholes or similar formulae hold. The results obtained in this paper are very general as no assumptions are made on the dynamics of the model, whether it be the underlying price process, the volatility process or how they relate to each other. We show that if the formula holds for a continuum of strikes and three terminal times then the volatility must be constant. However, when it only holds for finitely many strikes, and three or more maturity times, we obtain a universal bound on the variation of the volatility. This bound yields that the implied volatility is constant when the sequence of strikes increases to cover the entire half-line. This recovers the result for a continuum of strikes by a different approach.
Citation: Kais Hamza, Fima C. Klebaner, Olivia Mah. Volatility in options formulae for general stochastic dynamics. Discrete and Continuous Dynamical Systems - B, 2014, 19 (2) : 435-446. doi: 10.3934/dcdsb.2014.19.435
##### References:
 [1] D. Aldous, Stopping times and tightness, Annals of Probability, Volume 6 (1978), 335-340. doi: 10.1214/aop/1176995579. [2] K. Hamza and F. C. Klebaner, On nonexistence of non-constant volatility in the Black-Scholes formula, Discrete and Continuous Dynamical Systems, Volume 6 (2006), 829-834. doi: 10.3934/dcdsb.2006.6.829. [3] K. Hamza and F. C. Klebaner, On one inverse problem in financial mathematics, Journal of Uncertain Systems, Volume 1 (2007), 246-255. [4] K. Hamza and F. C. Klebaner, On the implicit Black-Scholes formula, Stochastics: An International Journal of Probability and Stochastic Processes, Volume 80 (2008), 97-102. doi: 10.1080/17442500701607706. [5] K. Hamza and F. C. Klebaner, Martingales in the Itô-Tanaka formula with applications,, Submitted., (). [6] K. Hamza, S. Jacka and F. C. Klebaner, The EMM conditions in a general model for interest rates, Advances in Applied Probability, Volume 37 (2005), 415-434. doi: 10.1239/aap/1118858632. [7] F. C. Klebaner and R. Liptser, When a stochastic exponential is a true martingale. Extension of a method of Beneŝ, Teoriya Veroyatnostei i ee Primeneniya, Volume 58 (2013), 53-80. [8] I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus, Graduate Texts in Mathematics, Springer-Verlag, New York, second edition, 1991. doi: 10.1007/978-1-4612-0949-2. [9] A. T. Wang, Generalized Ito's formula and additive functionals of Brownian motion,, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 41 (): 153.  doi: 10.1007/BF00538419.

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##### References:
 [1] D. Aldous, Stopping times and tightness, Annals of Probability, Volume 6 (1978), 335-340. doi: 10.1214/aop/1176995579. [2] K. Hamza and F. C. Klebaner, On nonexistence of non-constant volatility in the Black-Scholes formula, Discrete and Continuous Dynamical Systems, Volume 6 (2006), 829-834. doi: 10.3934/dcdsb.2006.6.829. [3] K. Hamza and F. C. Klebaner, On one inverse problem in financial mathematics, Journal of Uncertain Systems, Volume 1 (2007), 246-255. [4] K. Hamza and F. C. Klebaner, On the implicit Black-Scholes formula, Stochastics: An International Journal of Probability and Stochastic Processes, Volume 80 (2008), 97-102. doi: 10.1080/17442500701607706. [5] K. Hamza and F. C. Klebaner, Martingales in the Itô-Tanaka formula with applications,, Submitted., (). [6] K. Hamza, S. Jacka and F. C. Klebaner, The EMM conditions in a general model for interest rates, Advances in Applied Probability, Volume 37 (2005), 415-434. doi: 10.1239/aap/1118858632. [7] F. C. Klebaner and R. Liptser, When a stochastic exponential is a true martingale. Extension of a method of Beneŝ, Teoriya Veroyatnostei i ee Primeneniya, Volume 58 (2013), 53-80. [8] I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus, Graduate Texts in Mathematics, Springer-Verlag, New York, second edition, 1991. doi: 10.1007/978-1-4612-0949-2. [9] A. T. Wang, Generalized Ito's formula and additive functionals of Brownian motion,, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 41 (): 153.  doi: 10.1007/BF00538419.
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