Jump processes and the implied volatility curve

It is a commonly held belief that markets tend to both over-react and under-react to the arrival of good or bad news. Market confidence can drive up values of stocks, and market fear can lead to large negative dips in stock value. In early September 2008, Bloomberg mistakenly republished a 2002 article where it set out that United Airlines had sought bankruptcy protection from its creditors[1]. Investors believed it was current, and within three minutes of the republication, United Airlines stock had plummeted 75%.Markets are volatile; they tend to fluctuate rapidly and regularly. Investor option is a major reason for these fluctuations and should be considered when price setting options.Many of todays option pricing models are derived from the robust Black-Scholes model[3].The Black-Scholes model uses mainly observable variables and disregards many outside variables, such as investor option…


1. Introduction
2. Model
2.1. Background
2.2. Assumptions
2.3. Model Formulation
2.4. Derivation of the PDE
3. Solving the PDE
3.1. Background
3.2. Finite Differencing of PDE
3.3. Stability and convergence of finite differencing
3.4. Boundaries
4. Numerical Implementation and Analysis
4.1. Matrix Form of the PDE
4.2. Jump Matrix
4.3. Calculation of Option Prices
5. Implied Volatility
5.1. Background
5.2. Calculations
5.3. Numerical Analysis
6. Conclusion
Appendix A.
A.1. PDE Derivation
A.2. Matrix Algebra
A.3. Matlab Code

Author: Skoog, Daniel

Source: Uppsala University Library

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