Encoding Market View via a Randomized Brownian Bridge

Trading decisions often depend on the trader’s subjective belief of the distribution of the asset price on a given future date. For example, if a trader anticipates a big price movement for a company stock after its earnings announcement, then perhaps a long straddle position makes sense. There are many instances like this.

And as time progresses, the trader will learn more about the price distribution by observing price fluctuations, which in turn will inform trading decisions.

Mathematically, the trader’s market view can be described by a prior distribution of the asset price on some future date. The price process reveals itself over time, but there is also uninformative noise embedded in the asset price dynamics. This leads to the question: how to build a stochastic model that appropriately reflects the trader’s market view dynamically over time?

A randomized Brownian bridge is a perfect candidate for this. In our recent paper, we present a randomized Brownian bridge (rBb) model, whereby the log-price of the asset follows a Brownian bridge with a randomized endpoint representing the random terminal log-price.

The trader’s belief is described by a real-valued random variable D to be realized at some future time T so that the terminal log-price of the asset. The standard Brownian bridge always converges to 0 at time T. The log-price process X is a randomized Brownian bridge since it has a random end-point. Exponentiating the log-price recovers the underlying asset price process.

To utilize our model, the trader would specify a distribution for the stock price at some future time (e.g. end of the month, option expiration date, earnings announcement date). In contrast to almost all existing models, the trader here has total freedom in choosing any future marginal distribution as desired. It can be as simple as a two-point discrete distribution, the normal distribution, or other more sophisticated.