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thesis by Hans Buhler
I Consistent Modelling 16
2 Consistent Variance Curve Models 17
2.1 Problem Statements and Overview . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.1.1 Review of the Stochastic Volatility Case . . . . . . . . . . . . . . . . . . . 19
2.2 General Variance Curve Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.2.1 The Martingale Property and Explosion of Variance . . . . . . . . . . . . 24
2.2.2 Fixed Time-to-Maturity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.2.3 Fitting the Market with Exponential Variance Curve Models . . . . . . . 27
2.3 Consistent Variance Curve Functionals . . . . . . . . . . . . . . . . . . . . . . . . 29
2.3.1 Markov Variance Curve Market Models . . . . . . . . . . . . . . . . . . . 30
2.3.2 HJM-Conditions for Consistent Parameter Processes . . . . . . . . . . . . 31
2.3.3 Extensions to Manifolds: When does Z stay in Z ? . . . . . . . . . . . . . 32
2.4 Variance Curve Models in Hilbert Spaces . . . . . . . . . . . . . . . . . . . . . . 35
3 Examples 37
3.1 Exponential-Polynomial Variance Curve Models . . . . . . . . . . . . . . . . . . . 37
3.2 Exponential Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.3 Variance Swap Volatility Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.4 Fitting Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
II Hedging 47
4 Theory of Replication 48
4.1 Problem Statements and Overview . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.2 Hedging in Complete Markets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.2.1 General Complete Markovian Markets . . . . . . . . . . . . . . . . . . . . 51
4.2.2 Pricing with Local Martingales . . . . . . . . . . . . . . . . . . . . . . . . 58
4.2.3 Hedging with Variance Swaps . . . . . . . . . . . . . . . . . . . . . . . . . 59
2
CONTENTS 3
4.2.4 Hedging in classic Stochastic Volatility Models . . . . . . . . . . . . . . . 65
5 Hedging in Practice 66
5.1 Model and Market . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
5.1.1 Additional Market Instruments . . . . . . . . . . . . . . . . . . . . . . . . 69
5.2 Parameter Hedging in Practise . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
5.2.1 Constrained Parameter-Hedging in Practise . . . . . . . . . . . . . . . . . 74
5.3 Dynamic Arbitrage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
5.3.1 Entropy Swaps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
III Practical Implementation 82
6 A variance curve model 83
6.1 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
6.1.1 Existence, Uniqueness and the Martingale Property . . . . . . . . . . . . 87
6.2 Pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
6.2.1 Pricing General Payo