Book contents
- Frontmatter
- Contents
- Introduction
- 1 The Black–Scholes Theory of Derivative Pricing
- 2 Introduction to Stochastic Volatility Models
- 3 Volatility Time Scales
- 4 First-Order Perturbation Theory
- 5 Implied Volatility Formulas and Calibration
- 6 Application to Exotic Derivatives
- 7 Application to American Derivatives
- 8 Hedging Strategies
- 9 Extensions
- 10 Around the Heston Model
- 11 Other Applications
- 12 Interest Rate Models
- 13 Credit Risk I: Structural Models with Stochastic Volatility
- 14 Credit Risk II: Multiscale Intensity-Based Models
- 15 Epilogue
- References
- Index
9 - Extensions
Published online by Cambridge University Press: 07 October 2011
- Frontmatter
- Contents
- Introduction
- 1 The Black–Scholes Theory of Derivative Pricing
- 2 Introduction to Stochastic Volatility Models
- 3 Volatility Time Scales
- 4 First-Order Perturbation Theory
- 5 Implied Volatility Formulas and Calibration
- 6 Application to Exotic Derivatives
- 7 Application to American Derivatives
- 8 Hedging Strategies
- 9 Extensions
- 10 Around the Heston Model
- 11 Other Applications
- 12 Interest Rate Models
- 13 Credit Risk I: Structural Models with Stochastic Volatility
- 14 Credit Risk II: Multiscale Intensity-Based Models
- 15 Epilogue
- References
- Index
Summary
We present in this chapter several extensions to the asymptotic approach developed in the previous chapters. In Section 9.1, we extend the results of Chapter 4 to the case where the short rate is also varying, driven by the same factors driving the volatility, and the stock is paying dividends. In Sections 9.3 and 9.4, we derive the second-order corrections generated by the fast and the slow factors. These second terms produce the smile of implied volatilities as illustrated with the Heston model in Chapter 10. In Section 9.5, we show that a periodic daily component can easily be incorporated in the model without additional difficulty in the asymptotic analysis. In Section 9.6, we introduce jumps in the fast volatility factor. We indicate how to generalize the perturbation method to multidimensional models in Section 9.7.
Dividends and Varying Interest Rates
In this section, we present generalizations of the first-order approximation to the cases where dividends are paid and/or where the interest rate is varying as a function of the factors (Y, Z). We also present the probabilistic representations of these approximations.
Dividends
In this section we indicate how the perturbation theory explained above can easily be modified to incorporate dividend modeling into the stock price model. For simplicity we consider a continuous dividend yield D, which is the fraction of the stock price received by a stockholder per unit of time.
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- Publisher: Cambridge University PressPrint publication year: 2011