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Non-Gaussian Merton-Black-Scholes Theory

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Table of Contents

Preface
Ch. 1 Introduction 1
1.1 The Gaussian Merton-Black-Scholes theory 1
1.2 Regular Levy Processes of Exponential type 8
1.3 Pricing of contingent claims 13
1.4 The Generalized Black-Scholes equation 23
1.5 Analytical methods used in the book 28
1.6 An overview of the results covered in the book 31
Ch. 2 Levy processes 39
2.1 Basic notation and definitions 39
2.2 Levy processes: general definitions 45
2.3 Levy processes as Markov processes 52
2.4 Boundary value problems for the Black-Scholes-type equation 62
Ch. 3 Regular Levy Processes of Exponential type in 1D 67
3.1 Model Classes 67
3.2 Two definitions of Regular Levy Processes of Exponential type 82
3.3 Properties of the characteristic exponents and probability densities of RLPE 85
3.4 Properties of the infinitesimal generators 87
3.5 A "naive approach" to the construction of RLPE or why they are natural from the point of view of the theory of PDO 87
3.6 The Wiener-Hopf factorization 89
Ch. 4 Pricing and hedging of contingent claims of European type 97
4.1 Equivalent Martingale Measures in a Levy market 97
4.2 Pricing of European options and the generalized Black-Scholes formula 104
4.3 Generalized Black-Scholes equation and its properties for different RLPE and different choices of EMM, and implications for parameter fitting 111
4.4 Other European options 113
4.5 Hedging 115
Ch. 5 Perpetual American Options 121
5.1 The reduction to a free boundary problem for the stationary generalized Black-Scholes equation 121
5.2 Perpetual American put: the optimal exercise price and the rational put price 124
5.3 Perpetual American call 139
5.4 Put-like and call-like options: the case of more general payoffs 143
Ch. 6 American options: finite time horizon 151
6.1 General discussion 151
6.2 Approximations of the American put price 153
6.3 American put near expiry 159
Ch. 7 First-touch digitals 165
7.1 An overview 165
7.2 Exact pricing formulas for first-touch digitals 166
7.3 The Wiener-Hopf factorization with a parameter 169
7.4 Price near the barrier 177
7.5 Asymptotics as [tau] [approaches] + [infinity] 183
Ch. 8 Barrier options 185
8.1 Types of barrier options 185
8.2 Down-and-out call option without a rebate 187
8.3 Asymptotics of the option price near the barrier 197
Ch. 9 Multi-asset contracts 199
9.1 Multi-dimensional Regular Levy Processes of Exponential type 199
9.2 European-style contracts 203
9.3 Locally risk-minimizing hedging with a portfolio of several assets 209
9.4 Weighted discretely sampled geometric average 216
Ch. 10 Investment under uncertainty and capital accumulation 221
10.1 Irreversible investment and uncertainty 221
10.2 The investment threshold 223
10.3 Capital accumulation under RLPE 225
10.4 Computational results 227
10.5 Approximate formulas and the comparative statics 230
Ch. 11 Endogenous default and pricing of the corporate debt 231
11.1 An overview 231
11.2 Endogenous default 233
11.3 Equity of a firm near bankruptcy level and the yield spread for junk bonds 239
11.4 The case of a solvent firm 242
11.5 Endogenous debt level and endogenous leverage 247
11.6 Conclusion 248
11.7 Auxiliary results 249
Ch. 12 Fast pricing of European options 255
12.1 Introduction 255
12.2 Transformation of the pricing formula for the European put 258
12.3 FFT and IAC 260
12.4 Comparison of FFT and IAC 264
Ch. 13 Discrete time models 267
13.1 Bermudan options and discrete time models 267
13.2 A perpetual American put in a discrete time model 269
13.3 The Wiener-Hopf factorization 272
13.4 Optimal exercise boundary and rational price of the option 278
Ch. 14 Feller processes of normal inverse Gaussian type 281
14.1 Introduction 281
14.2 Constructions of NIG-like Feller process via pseudodifferential operators 284
14.3 Applications for financial mathematics 289
14.4 Discussion and conclusions 294
Ch. 15 Pseudo-differential operators with constant symbols 295
15.1 Introduction 295
15.2 Classes of functions 297
15.3 Space S'(R[superscript n]) of generalized functions on R[superscript n] 300
15.4 Pseudo-differential operators with constant symbols on R[superscript n] 305
15.5 The action of PDO in the Sobolev spaces on R[superscript n] [subscript plus or minus] 312
15.6 Parabolic equations 316
15.7 The Wiener-Hopf equation on a half-line I 324
15.8 Parabolic equations on [0,T] x R[subscript +] 338
15.9 PDO in the Sobolev spaces with exponential weights, in 1D 344
15.10 The Sobolev spaces with exponential weights and PDO on a half-line 354
15.11 Parabolic equations in spaces with exponential weights 358
15.12 The Wiener-Hopf equation on a half-line II 358
15.13 Parabolic equations of R x R[subscript +] with exponentially growing data 363
Ch. 16 Elements of calculus of pseudodifferential operators 365
16.1 Basics of the theory of PDO with symbols of the class S[actual symbol not reproducible](R[superscript n] x R[superscript n]) 366
16.2 Operators depending on parameters 373
16.3 Operators with symbols holomorphic in a tube domain 377
16.4 Proofs of auxiliary technical results 379
16.5 Change of variables and pricing of multi-asset contracts 381
16.6 Pricing of barrier options under Levy-like Feller processes 382
Bibliography 385
Index 393
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