Bibliography. Principles of Infinitesimal Stochastic and Financial Analysis Downloaded from

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1 Bibliography 1.Anderson, R.M. (1976) " A Nonstandard Representation for Brownian Motion and Ito Integration ", Israel Math. J., 25, Berg I.P. van den ( 1987) Nonstandard Asymptotic Analysis, Springer Verlag, Berlin. 3.Berg, I.P. van den and Koudjeti, F. (1997) " From binomial expectations to the Black-Scholes formula : the main ideas ", Ann. Math. Blaise Pascal, 4, Black, F. and Scholes, M. (1973) "The pricing of Options and Corporate Liabilities"Journal of Political Economy, 81, Cox, J.C., Ross, S.A. and Rubinstein, M. (1979) "Option pricing, a simplified approach' 'Financial Economics, 7, N.Cutland, E.Kopp, and W. Willinger ( 1991) " A nonstandard approach to option pricing ", Mathematical Finance, 9 (16), 1. 7.Deledicq, A. and Diener, M. (1989) Lesons de calcul infinitesimal, Armand Colin, Paris. 8.Diener, F. and Diener, M., editors ( 1995 ) Nonstandard Analysis in Practice, Springer Verlag, Berlin. 9.Diener, F. and Reeb, G. (1989 ) Analyse Non Standard, Hermann, Paris. 10.Duffie, D. (1988 ) Security Markets, Stochastic Models, Academic Press, New York. 11.Feller, W. (1968 ) An Introduction to Probability Theory and Its Applications, John Wiley & sons, Inc., New York. 12.Keisler, H. J. (1976) Elementary Calculus, Prindle, Weber and Schmidt. 13.Koudjeti, F. (1995 ) Elements of external calculus, with an application to mathematical finance, Labyrint Publication, Capelle a/d IJssel, The Netherlands. 14.Lamberton, D. and Lapeyre, B. (1993) Introduction au calcul stochastique applique d la finance, Ellipses, Paris. 15.Malkiel, R.G. (1990) A random walk down Wall Street, Norton, Ne N York. 131

2 132 Bibliography 16.Nelson, E. (1987) Radically elementary probability theory, Princeton University Press, Princeton. 17.Petry, A. (1996) Analyse infinitesimale, Imprisil, Liege, Belgium. 18.Robert, A. (1988) Nonstandard Analysis, John Wiley and Sons Inc. New York. a., a li.-4 -" _ {..._e,,, 11

3 Index Drit, 67 r, 111 W, 36 W (a), 38 [a..b], 11 r,, IOO At,x, 44 A t + X, 46 A,: 46 a t, 100 Pu 100 «W t, 36 %, 32 7j t, 32 77(t,x)+, 46 7j(t,x)", 46 E r, 104 E (o), 24 E.,48 E.,.,51 Var t,x, 51, 10 b 0) 19 pr t,48 prt,., 51 T, 1 0, 10 tf,46 A t, A.,32 r. in,,83 Pa, 4 pr r, HI C, 1 c(tt,\/&), 1 ft, 1 C[O..T], 1 M, 14 Af, 14 Q, 14 ft (6t, y/6tj, 10 adapted -process, 61 adapted to time, 59 - geometrically, 62 random variable -, 62 American option, 99 arbitrage, 119 artificial probability, 103 Asean option, 99 backward process, 74, 104 barrier option, 128 binomial, 1 - function, 3

4 . "*,. a..1,11 4 't-inn.., + -FJt" 41, 11, 1,I, Index: - process, 31, 53, 55 - triangle, 37, 45 - coefficients, 1, 16 - cone, 1, 84 - distribution, 1 - process, triangle, 1, 122 bivalent process, 42 Black-Scholes - formula, 30, 117, model, v, 118 butterfly option, 126 call option European -, 98 Cauchy - law, 14 - principle, 13 change of scale, 4 Chebyshev inequality, 9, 15 compound interest, 101 conditional, 46 - expectation, 46, 48, 51, 70 - probability, 46, 48, 51 - standard deviation, 51, 87 reduced -, 85 - variance, 51 reduced -, 84 time- - probability, 46 conditionally independent increments, 55, 76 cone binomial -, 1 Cox, Ross, Rubinstein - formula, 121, model, v, 101, 102, 118 De Moivre-Laplace theorem, 9, 19, 28, 116 delivery date of an option, 98 derivative, 61 -process, 67 Dirac distribution, 126 discrete geometric Brownian motion 83 discrete interval, 11 discrete path-integral, 114, 116 discrete surface, 31, 44, 88, 96 distribution Dirac -, 126 Standard Normal -, 14, 123 binomial -, 1 normal -, 14, 39 Poisson -, 28 Euler approximation, 116 European option, 97, 113 call, 98 put, 98 price of a -, 114 Feynman-Kac formula, 116 financial strategy, 100, 119 forward contract, 118 function of S-exponential order, 23 Gauss function, 5 geometric Brownian motion, 83, 101, 104, 113 Girsanov -theorem, 57 heat equation, 96 hedge of an option, 100 hedging strategy, 100 Hypotheses (H), 114 increment of a process, 32 independent random variables, 34 conditionally -, 55, 76 indirect volatility, 127 Ito calculus, 91 it,., 2 jump-process, 128 jumps, 87, 91

5 Index 135 Leibnitz rules, 10 lookback option, 99 lower movement, 42 lower prolongation, 42 Markov process, 53, 76, 86 martingale, 67, 71, 78, 91 mass concentration lemma, 15 mass of a random variable, 14, 15, 91 model for a stock price, 85 movement, 42 lower -, 42 upper -, 42 near-interval, 11 normal distribution, 22, 39 option, 97, 104, 113 American -, 99 Asean -, 99 barrier -, 128 butterfly -, 126 call -, 98 delivery date of -, 97 European -, 97 hedge of -, 100 lookback -, 99 payoff of -, 97 put -, 98 striking price of an -, 98 option price, 105 order of magnitude, 9 of the - of, 9, 15 of the same - as, 9 symbols for -, 9 payoff at time T, 97 permanence principle, 13 Poisson - distribution, 28, random walk, 120 Poisson random walk, 74 portfolio, 88, 100 value of -, 100 predictable -process, 61, 68 price of an option, 105, 113, 114 process, 31 - replicating an option, 104 backward -, 104 increment of a stochastic -, 32 jump-, 128 underlying -, 98 volatility of -, 85 backward -, 74 binomial -, 53 bivalent -, 42 Markov -, 53 Poisson -, 74 recombining -, 42, 63 stochastic -, 31 Wiener -, 36 prolongation, 42 put option European -, 98 put-call parity, 119 random variable - geometrically adapted to the time t, 62 truncated -, 121 mass and tail of a -, 14 rate of return, 84 - of a stock, 86 recombining process, 42, 63, 83 reduced conditional standard deviation, 85 reduced conditional variance, 84 replicating (process - an option), 104 Riemann sum, 11, 22, 116 risk limited -, 127 risk-neutral probability, 104, 125 riskless investment, 100, 101, 128 Robinson's lemma, 13 S-continuous, 91

6 136 Index S-exponential order (function of -), 23 self-financing strategy, 100, 101, 104, 105, 108 shadow - of a Riemann-sum, 11 Standard Normal distribution, 123 state of a trajectory, 45 Stirling's formula, 29 stochastic -difference equation, 74 stochastic process, 31 stock price, 83 striking price of an option, 98 stroboscopy, 28 surface discrete -, 88, 96 tail of a random variable, 14 t,,, 2 trajectory, 31 trajectory state of a -, 45 trajectory of a stochastic process, 31 trend, 61, 67, 71, 84, 87 triangle binomial -, 1 underlying process, 98 upper movement, 42 upper prolongation, 42 value of a portfolio, 100 vibrations, 87, 91 volatility, 86, 93, 127 indirect -, 127 volatility of a process, 85 weak form of market efficiency, 85 Wiener walk, 36, 39, 68, 72 ^v,jt ^