Non-Gaussian Merton-Black-Scholes Theory Book

	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