Implied Volatility: General Properties

SCIENTIA

MA

NU

NTE

E T ME

Implied Volatility: General Properties and Asymptotics
October 14, 2009
A thesis presented to

The University of New South Wales in fulfilment of the thesis requirement for the degree of

Doctor of Philosophy by M ICHAEL PAUL V ERAN R OPER

For Gail

PLEASE TYPE
THE UNIVERSITY OF NEW SOUTH WALES
Thesis/Dissertation Sheet
Surname or Family name: Roper
First name: Michael

Other name/s:Paul Veran

Abbreviation for degree as given in the University calendar:
School: Mathematics and Statistics

Faculty: Science

Title: Implied Volatility: General Properties and Asymptotics

Abstract 350 words maximum: (PLEASE TYPE)

This thesis investigates implied volatility in general classes of stock price models.
To begin with, we take a very general view. We find that implied volatility is always, everywhere, and for every expiry well-defined only if the stock price is a non-negative martingale. We also derive sufficient and close to necessary conditions for an implied volatility surface to be free from static arbitrage. In this context, free from static arbitrage means that the call price surface generated by the implied volatility surface is free from static arbitrage. We also investigate the small time to expiry behaviour of implied volatility. We do this in almost complete generality, assuming only that the call price surface is non-decreasing and right continuous in time to expiry and that the call
+
surface satisfies the no-arbitrage bounds (S-K) ≤ C(K, τ)≤ S. We used S to denote the current stock price,
K to be a option strike price, τ denotes time to expiry, and C(K, τ) the price of the K strike option expiring in τ time units. Under these weak assumptions, we obtain exact asymptotic formulae relating the call price surface and the implied volatility surface close to expiry.
We apply our general asymptotic formulae to determining the small time to expiry behaviour of implied volatility

Bibliography: [AP07] L. B. G. Andersen and V. V. Piterbarg, Moment explosions in stochastic volatility models, Finance and Stochastics 11 (2007), no. 1, 29–50. & Sons Inc., New York, 1984. [BBF02] H. Berestycki, J. Busca, and I. Florent, Asymptotics and calibration of local volatility models, Quantitative Finance 2 (2002), no [BBF04] , Computing the implied volatility in stochastic volatility models, Communications on Pure and Applied Mathematics 57 (2004), no. 10, 1352–1373. Black Scholes implied volatility, Advances in Futures and Options Research 8 (1996), 15–29. [BGKW01] A. Brace, B. Goldys, F. Klebaner, and R. Wommersley, Market model of stochastic implied volatility with application to the BGM model, 2001, [BGM97] A. Brace, D. Gatarek, and M. Musiela, The market model of interest rate dynamics, Mathematical Finance 7 (1997), 127–155. [BL78] D. Breeden and R. Litzenberger, Prices of state contingent claims implicit in option prices, Journal of Business 51 (1978), 621–651. [BS88] M. Brenner and M. G. Subrahmanyam, A simple formula to compute the implied standard deviation, Financial Analysts Journal 44 (1988), [Bue06] H. Buehler, Expensive martingales, Quantitative Finance 6 (2006), no [CCIV06] J. Chargoy-Corona and C. Ibarra-Valdez, A note on Black-Scholes implied volatility, Physica A 370 (2006), no (1996), no. 1, 329–359. prices, Finance and Stochastics 9 (2005), no. 4, 477–492. [Cha96] D. M. Chance, A generalized simple formula to compute the implied volatility, The Financial Review 31 (1996), no [CJ90] P. Carr and R. Jarrow, The Stop-Loss Start-Gain Paradox and Option Valuation: A New Decomposition into Intrinsic and Time Value, Review of Financial Studies 3 (1990), no [CM96] C. J. Corrado and T. W. Miller, Jr., A note on a simple, accurate formula to compute implied standard deviations, Journal of Banking and Finance 20 (1996), 595–603. [CM05] P. Carr and D. Madan, A note on sufficient conditions for no arbitrage, Finance Research Letters 2 (2005), no [CN01] D. R. Chambers and S. K. Nawalkha, An improved approach to computing implied volatility, The Financial Review 38 (2001), no. 3, 89– 100. [Com70] L. Comtet, Inversion de yα eγ et y logα y au moyen des nombres de Stirling, C. R. Acad. Sci. Paris S´ r. A-B 270 (1970), A1085–A1088. Chapman & Hall/CRC, Boca Raton, FL, 2004. [CW03] P. Carr and L. Wu, What type of process underlies options? A simple robust test, The Journal of Finance 58 (2003), no Boston, MA, 1992, Translated from the second Portuguese edition by Francis Flaherty. Mathematical Finance 17 (2007), no. 1, 1–14. [DHS07] T. Daglish, J. Hull, and W. Suo, Volatility surfaces: theory, rules of thumb, and empirical evidence, Quantitative Finance 7 (2007), no [DL01] D. Davydov and V. Linetsky, Pricing and hedging path-dependent options under the CEV process, Management Science 47 (2001), no. 7, 949–965. [Dup94] B. Dupire, Pricing with a smile, Risk 7 (1994), no. 1, 18–20. stanford.edu/∼ valdo/papers/FmImplied2SpotVols.pdf, 2005.