Abstract
It is commonly observed across many underlying assets that the implied volatility of the Black Scholes model varies across exercise price and time-to-maturity and has a pattern known as the volatility smile. In this chapter, we first address the volatility smile using the stochastic volatility models which may underestimate the size of the smile. We then develop an approach to calibrate the smile by choosing the volatility function as a deterministic function of the underlying asset price and time so as to fit the model option price to the observed volatility smile.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
Notes
- 1.
In the standard binomial model the up and down probabilities are respectively \(e^{\sigma \sqrt{\varDelta t}}\) and \(e^{-\sigma \sqrt{\varDelta t}}\) where σ is the constant volatility (see Sect. 17.4). Thus \(S_{1} = s_{1}e^{-\sigma \sqrt{\varDelta t}}\) and \(S_{2} = s_{1}e^{\sigma \sqrt{\varDelta t}}\) from which Eq. (18.22) follows.
References
Derman, E., & Kani, I. (1994). Riding on a smile. Risk, 7(2), 32–39.
Dupire, B. (1994). Pricing with a smile. Risk, 7(1), 18–20.
Hull, J., & White, A. (1987). The pricing of options on assets with stochastic volatilities. Journal of Finance, 42, 281–299.
Rubinstein, M. (1994). Implied binomial trees. Journal of Finance, 49, 771–818.
Xu, G., & Taylor, S. (1994). The magnitude of implied volatility smiles: Theory and empirical evidence for exchange rates. Review of Futures Markets, 13, 355–380.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Chiarella, C., He, XZ., Nikitopoulos, C.S. (2015). Volatility Smiles. In: Derivative Security Pricing. Dynamic Modeling and Econometrics in Economics and Finance, vol 21. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-45906-5_18
Download citation
DOI: https://doi.org/10.1007/978-3-662-45906-5_18
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-662-45905-8
Online ISBN: 978-3-662-45906-5
eBook Packages: Business and EconomicsEconomics and Finance (R0)