Sufficient condition for monotonicity in constructing the distribution function with Bernoulli scheme

A. D. Vedenyapin ; S. A. Mitasov

Matematičeskaâ fizika i kompʹûternoe modelirovanie, no. 3 (2015), pp. 45-54.

Résumé

This paper is the construction of the distribution function using the Bernoulli scheme, and is also designed to correct some of the mistakes that were made in the article [2]. Namely, a function built in [2] need not be monotonous, and some formulas need to be adjusted. The idea of building as well as in [2], is based on the model of Cox–Ross–Rubinstein “binary market”. The essence of the model was to divide time into $N$ steps, and assuming that the price of an asset at each step can move either up to a certain value with probability $p$, or down also by some certain value with probability $q = 1 - p$. Prices in step $N$ can take only a finite number of values. “Success” or “failure” was the changing price for some fixed value in the model of Cox–Ross–Rubinstein. Here as a “success” or “failure” at every step we consider the affiliation of changing the index value to the section $[r, S]$ either to the interval $[I, r)$. Further a function $P(r)$ was introduced, which at any step gives us the probability of “success”. The maximum index value increase for the all period of time $[T, 2T]$ will be equal $nS$, and the maximum possible reduction will be equal $nI$. Then let $x \in [nI, nS]$. This segment will reflect every possible total variation that we can get at the end of a period of time $[T, 2T]$. The further introduced inequality $k \ge \frac{x - nI}{S - I}$ gives us the minimum number of successes that needed for total changing could be in the section $[x, nS]$ if was $n - k$ reductions with the index value to $I$. Then was introduced the function $r(x, k_{min})$ which is defined on the interval $(nI, nS]$ and provided us some assurance that the total index changing could be in the section $[x, nS]$ if successful interval is $[r (x, k_{min}), S]$ and the amount of success is satisfying to our inequality. The probability of $k$ “successes” and $n - k$ “failures” is calculated according to the formula of Bernoulli, where the probability of “success” is determined by the function $P(r)$, and $r$ is determined by the function $r(x, k_{min})$: $$ P_n^k = C_n^k{[P(r(x, k_{min}))]}^k {[1 - P(r(x, k_{min}))]}^{n-k}. $$ We defined the probability that the total index change will be in the section $[x, nS]$ as the sum of the probabilities of incompatible events, for which the number of successes satisfies to entered inequality. Then obviously we defined the probability that the total index changing will be in the interval $[nI, x)$. Then the function $F(x)$ was introduced, defined on the whole line, which is identical to the amount (probability that the total index changing will be in the interval) in the interval $(nI, nS]$, and is identical to zero and one in the additions. Some properties of the distribution function $F(x)$ are satisfied automatically. A sufficient condition for the monotonicity is presented in the form of Theorem 2.

Détail

Keywords: binary market, distribution function, probability, test-step, mathematical model.
Mots-clés : Bernoulli scheme

Bibliographie

[1] ZAO «Investitsionnaya kompaniya “IT Invest”», Binomial Model of the Cox–Ross–Rubinstein, , Title from screen http://www.itinvest.ru/editorfiles/File/options/Lection7.pdf

[2] A.\;D. Vedenyapin, S.\;A. Mitasov, “Applying Bernoulli Scheme for Constructing the Distribution Function of the Absolute Change in Price of an Asset”, Analysis, Modeling and Forecasting of Economic Processes, Data of the V International Scientific and Practical Internet Conference (December 15, 2013–February 15, 2014), eds. L.\;Yu. Bogachkova (Volgograd State University), V.\;V. Davnis (Voronezh State University), Konsalt Publ., Volgograd, 2014, 55–59

[3] John C. Cox, Stephen A. Ross, M. Rubinstein, “Option Pricing: a Simplified Approach”, Journal of Financial Economics, 7:3 (1979), 229–263 | DOI | Zbl

[4] A. Penati, G. Penacchi, The Cox–Ross–Rubinstein Option Pricing Model, , Title from screen http://home.cerge-ei.cz/petrz/FM/f400n10.pdf