![]() ![]() Although Brownian motions are continuous everywhere they are differentiable nowhere. This means that if the sign of all negative gradients were switched to positive, then $B$ would hit infinity in an arbitrarily short time period. Brownian motions have unbounded variation.The sequence of discrete random variables representing the coin toss is $Z_i \in \_i$ was chosen carefully in order that in the limit of large $N$, $B$ was both finite and non-zero. Concurrently the payoff returned from each coin toss will be modified. Hence the coin tosses will be spaced equally in time. In this interval $N$ coin tosses will be carried out, which each take a time $T/N$. However, the manner in which they are increased must occur in a specific fashion, so as to avoid a nonsensical (infinite) result.Ĭonsider a continuous real-valued time interval $$, with $T > 0$. In order to achieve this, the number of time steps will need to be increased. The current goal is to work towards a continuous-time random walk, which will provide a more sophisticated model for the time-varying price of assets. In the previous discussion on the Markov and Martingale properties a discrete coin toss experiment was carried out with an arbitrary number of time steps. It will be shown that a standard Brownian motion is insufficient for modelling asset price movements and that a geometric Brownian motion is more appropriate. In this article Brownian motion will be formally defined and its mathematical analogue, the Wiener process, will be explained. In both of these articles it was stated that Brownian motion would provide a model for path of an asset price over time. ![]() The Markov and Martingale properties have also been defined in order to prepare us for the necessary mathematical tools used to model asset price paths. In a previous article on the site we have introduced stochastic calculus in the context of its role in quantitative finance. ![]()
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