Expected Median Of A Shifted Brownian Motion: Theory and Calculations
58 Pages Posted: 12 Jan 2021 Last revised: 14 Feb 2021
Date Written: December 28, 2020
Financial derivatives linked to the median, which is the 50%-th percentile of a distribution, have not been extensively studied in realistic models of financial markets, as such derivatives simply did not exist until recently. The Libor reform that brought a seismic change to the interest rate markets - the largest market in the world - linked derivatives worth hundreds of trillions of notional to a median of interest rate spreads, making the median, arguably, the most important "number" to study and understand in all of the financial markets at the moment.
Numerically-efficient algorithms for calculating the fair value of the median that incorporate both the historical observations and the future dynamics of the Libor vs. the risk-free rate spreads in a realistic model have already been developed in our previous work on the subject. In this paper, we go significantly deeper than this important special case. Here, we focus on calculating the expected value of the median in a model of a history-less Brownian motion with a time-dependent shift, a model that provides rich mathematical structure to investigate, while also being very relevant to the current financial markets. We combine a newly-established linearization property under the large-volatility limit of the median, a universal white-noise approximation, and novel Machine Learning techniques to derive a general, numerically efficient algorithm for calculating the expected median. Theoretical advances and practical solutions to currently-topical problems are presented.
Keywords: Libor, Libor reform, Median, Quantile, Percentile, RFR, OIS, Risk-Free Rates, Sonia, Fallback, Arc-Sine, Occupation Time, Brownian motion with drift, Fallback Spread, Libor Adjustment Spread
JEL Classification: C61, G13, G15, G18, G21, C51
Suggested Citation: Suggested Citation