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Introduction to the Mathematics of Financial Derivatives

Salih N. Neftci

posted on 21 December 2005

reviewed by Joseph L. McCauley

Attractive at first sight, but not systematic enough on closer perusal. Too often tries to motivate continuum derivations by discrete arguments, and then the continuum argument is presented at best incompletely. I would prefer to see a systematic development with full explanation via Ito calculus, one can then discretize when needed. But the nontrivial stochastic differential equations required to describe real finance market data (the empirically correct equations) dx=R(x,t)dt+?D(x,t)dB(t) where especially D(x,t) depends nontrivially on both x and t, with x=lnp(t)/po, are not discussed in Neftci or any other financial engineering text.




Some bad mistakes in Ito calculus and also conceptually can be found in ch. 21: eqn. (75) is wrong, a partial time derivative is missing on the rhs (there are no stationary processes in finance, including the Osborne-Black-Scholes Gaussian returns model). This mistake propagates into another serious mistake on pg. 285: eqns. (80-82) are certainly neither a backward nor forward Kolmogorov equation, are simply wrong. See either Stratonovich ("Topics in the theory of Random noise", vol. 1) or McCauley ("Dynamics of Markets: Econophysics and Finance") for forward and backward time Kolmogorov pdes. See also the basically unreadable but highly stimulating books by Steele ("Stochastic Calculus") and Durrett ("Brownian Motion and Martingales")for extensive discussions of the 'Kac-Feynman' formulae and several different pdes (one can follow their discussions of the latter topics).