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November 2004
Elementary Stochastic Calculus
Book Review by Alex Castaldo
Nowadays all PhD programs in Finance and most Master's programs with a quant
orientation require a class called Continuous Time Finance or Stochastic
Calculus in Finance. The idea is to teach enough Stochastic Calculus that
students will be able to follow the derivation of the Black-Scholes PDE from
scratch.
The problem is that Stochastic Calculus is a very advanced area of modern
mathematics and the textbooks used in Math departments (like Karatzas and
Shreve) are not suitable for a Finance audience. They are far too abstract and
rigorous, and assume a very mathematically sophisticated reader.
In recent years new Stochastic calculus textbooks have appeared that are more
applied and "dumbed down" for a Finance audience. The most well known is
probably Oksendal "Stochastic Differential Equations". These books are still
Math (not Finance) books, with a Definition, Theorem, Proof style of exposition
and a complicated notation. The Mikosch book, Elementary Stochastic Calculus is
in this category. It requires less background than Oksendal and is easier to
understand; it seems to be the easiest of these books that I know of (to a
mathematician, the sloppiest). It still requires some previous exposure to
modern probability theory, which is based on measure theory, and some exposure
to convergence theorems and limits.
To someone trading or analyzing options, the value of knowing Stochastic
Calculus is probably limited. One can understand and use the Black-Scholes
theory without having seen a rigorous mathematical proof of its correctness.
And many difficult option problems have to be solved on a computer in discrete
time in any case, bypassing stochastic calculus altogether. But for people
wanting to learn the mathematics behind Black-Scholes, Mikosch's book may be of
interest.
Thomas Mikosch: Elementary stochastic calculus, with finance in view
World Scientific 1998 (reprinted 2004)
ISBN 9810235437
Price 24 GBP