Algorithms for Worst-Case Design and Applications to Risk Management

By Berç Rustem; Melendres Howe | Go to book overview

COMMENTS AND NOTES

CN 1: Brownian Motion

The term Brownian Motion has been used to describe the motion of a particle that is subject to a large number of small molecular shocks. In financial calculus, Brownian Motion is used interchangeably with the term Wiener Process which is a particular type of Markov stochastic process. Markov stochastic processes are processes where only the present value of a variable is relevant for predicting the future. Models of stock price behavior are usually expressed as a Wiener Process or Brownian Motion (see Hull, 1997).


CN 2: Ito's Lemma

The price of a stock option is a function of the underlying stock's price and time. More generally, we can say that the price of any derivative is a function of the stochastic variables underlying the derivative and time. Ito's lemma states that if a variable x follows an Ito Process, that is, that

where dz is a Wiener Process and a and b are functions of x and t, then another function G of x and t follows the process where dz is the same Wiener Process. Thus, G also follows a Wiener Process.


CN 3: Cumulative Normal Distribution Function

To solve the Black and Scholes formula, one needs to calculate the cumulative normal distribution function, Θ. The function can be evaluated directly using numerical procedures. Alternatively, a polynomial approximation can be used that provides values for Θ(d) with a six-decimal-place accuracy. The following have been extracted from Hull (1997):

where

-244-

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