Stochastic Volatility and the Pricing of Financial Derivatives
byAntoine Petrus Cornelius van der Ploeg

357 pages

Contents

- I - Stochastic Volatility and the Pricing of Financial Derivatives: An Introduction
1. Motivation 1
1.1 Summary of main goals ..................................................................6
2. Outline..................................................................................................6
3. Overviews of research ............................................................................9

-IIThe Multifactor Affine Stochastic Volatility Derivative Pricing Model
1. Introduction ..........................................................................................11
2. Stock market setting of the model............................................................12
3. Derivative pricing ...................................................................................17
3.1 Valuation of a general contingent claim.............................................17
3.2 Call and put option valuation ...........................................................22
4. Hedging derivatives ................................................................................24
4.1 Delta, vega and other Greeks ..........................................................25
4.2 Hedging in incomplete markets ........................................................26
4.3 Example: hedging in 1-factor SV models ...........................................26
4.3.1 Hedging path-independent derivatives....................................27
4.3.2 Hedging path-dependent derivatives ......................................29
4.3.3 Hedging in practice ..............................................................30
5. Derivative betas and asset returns ...........................................................31
5.1 CAPM and stock and volatility betas..................................................31
5.2 Asset returns and the volatility risk premium.....................................33
5.3 Asset returns and the relation with Arbitrage Pricing Theory ................34
5.4 Derivative returns and the hedging strategy ......................................37
5.5 Asset returns and the stochastic discount factor .................................38

6. Volatility risk and compensation for volatility risk .......................................39
6.1 Market-price-of-volatility-risk assumptions in the literature .................40
6.2 Investment strategies dominated by volatility risk ..............................40
6.2.1 Straddles: bets on volatility ..................................................41
6.2.2 Variance swaps: trades on realized variance ...........................42
6.2.3 Delta hedging......................................................................44
6.3 Empirical evidence on volatility-risk compensation..............................45
6.3.1 Discussion of the evidence ....................................................46

7. A theory on the negative volatility risk premium ........................................50
7.1 The investor problem......................................................................51
7.2 Relating derivative returns to optimal consumption.............................54
7.3 Consumption smoothing and the volatility risk premium......................56

8. Foreign exchange markets.......................................................................58
8.1 Garman-Kohlhagen (1983) model.....................................................58
8.2 The Bates (1996) model and empirical evidence.................................58
8.3 Our model in a FX-markets context ..................................................59
9. From theory to practice...........................................................................60

Appendix
IIa. The bond-stock market is arbitrage-free: A proof ...............................62
IIb. Derivative betas .............................................................................63
IIc. Variance-swap analysis ...................................................................64

-III A State Space Approach to the Estimation of the
Multifactor Affine Stochastic Volatility Derivative Pricing Model
1. Introduction ..........................................................................................67
2. Current estimation methods of SV models and the state space framework.....68
2.1 Current methods and motivating the state space approach..................68
2.1.1 Combining stock and option data for estimation.......................69
2.1.2 Our approach and advantages of our approach........................71
2.2 A review of the linear state space representation, the Kalman
filter and smoother, and QML estimation ...........................................72
3. Deriving a state space representation .......................................................75
3.1 Tackling the SDE of the volatility-driving factors.................................76
3.2 Extracting information from stock returns..........................................77
3.2.1 ARSV(1) model and relation with the earlier literature ..............79
3.2.2 Relation with the GARCH(1,1) model for stock returns..............79

3.3 Extracting information from realized volatilities ..................................82
3.3.1 Realized volatility.................................................................82
3.3.2 Overview of research on RV .................................................83
3.3.3 An equation for the average variance .....................................86
3.3.4 A measurement equation for realized volatilities ......................88
3.3.5 A closer look at the measurement equation for RV ...................90
3.4 Extracting information from option prices ..........................................93
3.4.1 Linearizing the call price formula ...........................................94
3.4.2 Using the theoretical relationship in practice ...........................95
3.4.3 Preliminary insight in the quality of the linearization .................98
3.4.4 Linearizing around the integrated variance instead...................100
3.4.5 Higher-order approximations.................................................102
3.5 Possible state space models.............................................................105
3.5.1 Refresher ............................................................................105
3.5.2 System matrices for return - option data ................................106
3.5.3 System matrices for RV - option data .....................................107
4. Concluding remarks................................................................................108

Appendix
IIIa. Proof of { } tω being white noise and uncorrelated with {ut +Δt} ............109
IIIb. Investigating the correlation between , { } RV t ϖ and { } t +Δt u .................111
IIIc. The expected value of the realized variance 2RV,t t σ + Δ ..........................113

- IV - Monte Carlo and Empirical Results for Ornstein-Uhlenbeck SV:
Examining UK Financial Markets
1. Introduction ..........................................................................................117
2. FTSE100-index data: An explorative analysis .............................................119
2.1 Data collection and construction.......................................................119
2.2 Analyzing the volatility term structure...............................................124
3. A Monte Carlo study to the 1-factor OU SV model ......................................127
3.1 The 1-factor OU SV option pricing model ...........................................127
3.2 Simulating from the 1-factor OU SV option pricing model ....................128
3.2.1 Assumptions .......................................................................128
3.2.2 Set up of the Monte Carlo experiment ....................................128
3.3 State space model..........................................................................131
3.4 Simulation results for 1-factor OU SV................................................133
3.4.1 Quality of the call price approximations ..................................133
3.4.2 Disturbance terms of the state space model ............................135
3.4.3 Performance of the state space estimation method ..................137

4. FTSE100-index data: Results for 1-factor OU SV ........................................141
4.1 Parameter estimates.......................................................................141
4.2 Volatilities and in-sample fit.............................................................142
4.3 Compensation for volatility risk and straddle returns...........................147
4.4 Diagnostic checking ........................................................................149
4.5 1-factor OU SV: Estimation results using return, SM, MM
and LM option data jointly ...............................................................152
5. FTSE100-index data: Results for multifactor OU SV ....................................156
5.1 The model in case of multifactor OU SV.............................................156
5.2 Estimation results for 2-factor OU SV................................................157
5.3 Estimation results for 3-factor OU SV 160
5.3.1 Decomposing stock volatility: long-memory, mediumterm
and short-term volatility trends 162
5.3.2 Another interpretation: Level, slope and curvature factors ........163
5.3.3 Risk premia, straddle returns and consumption smoothing ........166
5.3.4 Fit of the volatility term structure. 169
5.3.5 Diagnostic checking . 171
6. Summary 173
Appendix