Pricing Models of Volatility Products and Exotic Variance Derivatives

Description

Pricing Models of Volatility Products and Exotic Variance Derivatives summarizes most of the recent research results in pricing models of derivatives on discrete realized variance and VIX. The book begins with the presentation of volatility trading and uses of variance derivatives. It then moves on to discuss the robust replication strategy of variance swaps using portfolio of options, which is one of the major milestones in pricing theory of variance derivatives. The replication procedure provides the theoretical foundation of the construction of VIX. This book provides sound arguments for formulating the pricing models of variance derivatives and establishes formal proofs of various technical results. Illustrative numerical examples are included to show accuracy and effectiveness of analytic and approximation methods. Features Useful for practitioners and quants in the financial industry who need to make choices between various pricing models of variance derivatives Fabulous resource for researchers interested in pricing and hedging issues of variance derivatives and VIX products Can be used as a university textbook in a topic course on pricing variance derivatives Cover Half Title Series Page Title Page Copyright Page Dedication Contents Preface 1. Volatility Trading and Variance Derivatives 1.1. Implied volatility and local volatility 1.2. Volatility trading using options 1.2.1. Taking volatility position using straddles and strangles 1.2.2. Volatility exposure generated by delta hedging options 1.3. Derivatives on discrete realized variance 1.3.1. Swaps and options on realized variance and volatility 1.3.2. Generalized variance swaps 1.3.3. Timer options 1.3.4. Target volatility options 1.4. Replication of variance swaps 1.4.1. Replication of continuous variance swaps 1.5. Practical implementation of replication: Finite strikes and discrete monitoring 1.5.1. Continuously sampled realized variance replicated by options of finite strikes 1.5.2. VIX: Extracting model-free volatility from S&P 500 index options 1.5.3. Replication of swaps on discrete realized variance Appendix 2. Lévy Processes and Stochastic Volatility Models 2.1. Compound Poisson process 2.1.1. Poisson process 2.1.2. Random jump sizes 2.1.3. Stochastic integration 2.1.4. Jump measure and Lévy measure 2.2. Jump-diffusion models 2.2.1. Itô’s formula 2.2.2. Asset price process: Geometric Brownian motion with compound Poisson jumps 2.2.3. Merton’s model with Gaussian jumps 2.2.4. Kou’s model with exponential jumps 2.3. Lévy processes 2.3.1. Definition 2.3.2. Infinite divisibility 2.3.3. Characteristic exponent and Lévy-Khintchine representation 2.3.4. Lévy-Itô decomposition theorem 2.3.5. CGMY model: Dampened power law as Lévy measure 2.3.6. Generalized Hyperbolic model 2.3.7. Martingale condition on drift under risk neutral measure 2.4. Time-changed Lévy processes 2.4.1. Time-change techniques: Subordinators and activity rates 2.4.2. Variance Gamma model 2.4.3. Normal Inverse Gaussian model 2.4.4. Barndorff-Nielsen and Shephard model 2.5. Stochastic volatility models with jumps 2.5.1. Distribution formulas of instantaneous variance of CIR type 2.5.2. Pricing of swap on continuous realized variance 2.6. Affine jump-diffusion stochastic volatility models 2.6.1. Joint moment generating function of the affine model 2.6.2. Numerical valuation of complex algorithms and Heston trap 2.6.3. Schobel-Zhu model 2.7. 3/2 stochastic volatility model 2.7.1. Model formulation 2.7.2. Partial Fourier transform of the triple joint density 2.7.3. Partial Fourier transform of the joint density function of (X,V) 2.7.4. Joint characteristic function of (X,I) Appendix 3. VIX Derivatives under Consistent Models and Direct Models 3.1. VIX, variance swap rate and VIX derivatives 3.1.1. Relation between variance swap rate and VIX2 under jumps 3.1.2. VIX derivatives 3.2. Pricing VIX derivatives under consistent models 3.2.1. Affine stochastic volatility models 3.2.2. 3/2-model with jumps in index value 3.2.3. Barndorff-Nielsen and Shephard model 3.2.4. GARCH type models 3.3. Direct modeling of VIX 3.3.1. Multifactor affine jump-diffusion models 3.3.2. 3/2 plus models Appendix 4. Swap Products on Discrete Variance and Volatility 4.1. Direct expectation of square of log return 4.2. Nested expectation via partial integro-differential equation 4.2.1. Vanilla variance swaps under the Heston stochastic volatility model 4.2.2. Variance swaps under the 3/2-model 4.3. Moment generating function methods 4.3.1. Variance swap and gamma swap 4.3.2. Corridor type swaps 4.3.3. Numerical tests of the convergence for discretely monitored variance swaps 4.3.4. Volatility swaps 4.4. Variance swaps under time-changed Lévy processes 4.4.1. Multiple of log contract for pricing swaps on continuous realized variance 4.4.2. Swaps on discrete realized variance 4.4.3. Generalized variance swaps 4.4.4. Convergence of fair strikes 4.4.5. Conditions on convergence in expectation Appendix 5. Options on Discrete Realized Variance 5.1. Adjustment for discretization effect via lognormal approximation 5.1.1. Discrete realized variance under the lognormal model 5.1.2. Approximation formulas for moment generating function 5.2. Normal approximation to conditional distribution of discrete realized variance 5.2.1. Conditional normal approximation pricing scheme 5.2.2. Simplified conditional pricing schemes 5.2.3. Non-simulation asymptotic approximation pricing scheme 5.3. Partially exact and bounded approximation for options on discrete realized variance 5.3.1. Lower bound with known characteristic function 5.3.2. Partially exact and bounded approximation 5.3.3. Numerical calculations of partially exact and bounded approximation 5.4. Small time asymptotic approximation 5.4.1. Small time asymptotics under Lévy models 5.4.2. Small time asymptotics under the semimartingale models 5.4.3. Option pricing using small time asymptotic approximation 6. Timer Options 6.1. Model formulation 6.1.1. Governing partial differential equation 6.2. Pricing perpetual timer options 6.2.1. Conditional expectation based on Black-Scholes type formula 6.2.2. Integral price formulas under the Heston model 6.2.3. Perturbation approximation 6.3. Finite maturity discrete timer options 6.3.1. Fourier inversion integral price formula 6.3.2. Fourier space time stepping numerical algorithm Appendix Bibliography Index