1. Road Map

2. Why Need Option Spanning?

3. Step 1: Put-Call Parities and Boundaries

4. Step 2: Spanning of Options

5. Step 3: Valuation of Variance

6. Step 4: VIX

[Appendix]  

• Put-Call Parity [PDF]

• Calculus Derivatives of Option Values in Strike Price [PDF]

• Option Spanning I: Breeden and Litzenberger(1978) [PDF]

• Option Spanning II: Carr and Madan(1998) [PDF]

• Valuation of Variance and VIX [PDF]

1. Concepts of Volatilities: Historical vs. Implied

2. Shape of Volatility Smile

• Symmetry of the Smile: Risk Reversal

• Curvature of the Smile: Butterfly

3. Considering Smile Effects in Pricing

4. Possible Reasons for the Smile

• Non-Geometric Price Process

• Possibility of Crash(or Fear)

• Two Risk Factors

• Option Sellers under Capital Requirement Reg.

[Appendix]  

• Assete Pricing with Two Factors [PDF]

1. Counting Process(=Jump Process)

2. Merton(1976)’s Jump Model

3. Pricing Jump Models

[Appendix]  

• Counting Process(Jump Process) [PDF]

• Merton(1976)'s Jump Model [PDF]

• Pricing Jump Models [PDF]