List of symbols
Acknowledgements
Preface
1 Introduction to Derivatives
1.1 Hedging with Forward Contracts
1.2 Speculation with Forward Contracts
1.3 Arbitrage
1.4 Vanilla Options
1.5 Interest Rates
1.6 Valuing a Forward Contract
1.7 Key Points
1.8 Further Reading
2 Stochastic Calculus
2.1 Brownian Motion
2.2 Stochastic Model for Stock Price Evolution
2.3 Ito's Lemma
2.4 The Product Rule
2.5 Log Normal Stock Price Evolution
2.6 The Markov Property
2.7 Term Structure
2.8 Ito's Lemma in More than One Dimension
2.9 Key Points
2.10 Further Reading
3 Martingale Pricing
3.1 Setting the Scene
3.2 Tradeable Assets
3.3 Zero Coupon Bond
3.4 Rolling Money Market Account
3.5 Choosing a Numeraire
3.6 Changing the Measure
3.7 Girsanov's Theorem
3.8 Martingales
3.9 Continuous Martingales
3.10 Black-Scholes Formula for a Call Option
3.11 At-The-Money Options
3.12 The Black-Scholes Equation
3.13 An Elegant Derivation of the Black-Scholes Formula
3.14 Key Points
3.15 Further Reading
4 Dynamic Hedging and Replication
4.1 Dynamic Hedging in the Absence of Interest Rates
4.2 Dynamic Hedging With Interest Rates
4.3 Delta Hedging
4.4 The Greeks
4.5 Gamma, Vega and Time Decay
4.6 Vega and Volatility Trading
4.7 Key Points
4.8 Further Reading
5 Exotic Options in Black-Scholes
5.1 European Options
5.2 Asian Options
5.3 Continuous Barrier Options
5.3.1 The Reflection Principle
5.3.2 The Reflection Principle with Log-Normal Dynamic
5.3.3 Valuing Barrier Options in Black-Scholes
5.3.4 Discretely Monitored Barrier Options
5.4 Key Points
5.5 Further Reading
6 Smile Models
6.1 The Volatility Smile
6.2 Smile Implied Probability Distribution
6.3 The Forward Kolmogorov Equation
6.4 Local Volatility
6.5 Key Points
6.6 Further Reading
7 Stochastic Volatility
7.1 Properties of Stochastic Volatility Models
7.2 The Heston Model
7.2.1 What Makes the Heston Model Special
7.2.2 Solving for Vanilla Prices
7.2.3 The Feller Boundary Condition
7.3 The SABR Model
7.4 The Ornstein-Uhlenbeck Process
7.5 Mixture Models
7.6 Regime Switching Model
7.7 Calibrating Stochastic Volatility Models
7.8 Key Points
7.9 Further Reading
8 Numerical Techniques
8.1 Monte Carlo
8.1.1 Monte Carlo in One Dimension
8.1.2 Monte Carlo in More than One Dimension
8.1.3 Variance Reduction in Monte Carlo
8.1.4 Limitations of Monte Carlo
8.2 The PDE Approach
8.2.1 Stable and Unstable Schemes
8.2.2 Choice of Scheme
8.2.3 Other Ways of Improving Accuracy
8.2.4 More Complex Contracts in PDE
8.2.5 Solving Higher Dimension PDEs
8.3 Key Points
8.4 Further Reading
9 Local Stochastic Volatility
9.1 The Fundamental Theorem of On-smile Pricing
9.2 Arbitrage in Implied Volatility Surfaces
9.3 Two Extremes of Smile Dynamic
9.3.1 Sticky Strike Dynamics
9.3.2 Sticky Delta Dynamics
9.4 Local Stochastic Volatility
9.5 Simplifying Models
9.5.1 Spot-Volatility Correlation
9.5.2 Term Structure Vega for a Barrier Option
9.5.3 Simplifying Stochastic Volatility Parameters
9.5.4 Risk Managing with Local Stochastic Volatility Models
9.6 Practical Calibration
9.7 Impact of mixing on contract values
9.8 Key Points
9.9 Further Reading
10 Volatility Products
10.1 Overview
10.2 Variance Swaps
10.2.1 The Variance Swap Contract
10.2.2 Idealised Variance Swap Trade
10.2.3 Valuing the Idealised Trade
10.2.4 Beauty in Variance Swaps
10.2.5 Delta and Gamma of a Variance Swap
10.2.6 Practical Considerations
10.3 Volatility Swaps
10.3.1 Volatility Swap in Stochastic Volatility Models and LSV
10.3.2 Volatility Swap Versus Variance Swap
10.3.3 Valuing a Volatility Swap
10.3.4 Stochastic Versus Local Volatility
10.4 Forward Volatility Agreements
10.4.1 Practicalities
10.5 Key Points
10.6 Further Reading
11 Multi-Asset
11.1 Overview
11.2 Local Volatility with Constant Correlation
11.3 Copulas
11.4 Correlation Smile
11.5 Marking Correlation Smile
11.5.1 Common Correlation Products
11.5.2 The Triangle Rule
11.6 Modelling
11.6.1 Local Correlation
11.6.2 Practicalities
11.6.3 Local stochastic correlation
11.7 Valuing European Contracts
11.7.1 Special Properties of Best-of Options
11.7.2 Valuing a Best-of Option in Black-Scholes
11.7.3 Construction of a Joint PDF
11.7.4 Using the Density Function for Pricing
11.8 Numeraire Symmetry
11.9 Baskets as Correlation Instruments
11.10 Summary
11.11 Key Points
11.12 Further Reading
Afterword
Appendix: Measure Theory and Girsanov's Theorem
References
Further Reading
Index