Cap Calculation

Cap and Floor Calculation

The objective of this work is to delineate a methodology for determining probabilistic bounds within which a specified index is unlikely to deviate significantly. This methodology is designed to minimize counterparty risk and optimize capital efficiency by establishing statistically robust upper and lower thresholds (cap and floor) for the index. The premise is to ensure the probability of the index's realized ratio R, exceeding these bounds is statistically minimal, historically less than 5%.

Methodology overview

We employ a non-parametric approach to calculate the cap and floor based on historical performance data. This method is advantageous for its replicability and its departure from traditional parametric techniques, which may not accurately reflect complex market dynamics. The derivation of the cap involves the following three steps:

1. Historical percentile-based bounds Define the cap initially to be at least the xth percentile of the realized ratio, R, over the preceding n days. This premise is grounded in the notion that the index's recent performance is reflective of current market conditions. We set x = 0.99 (99th percentile) and n = 180 days.

2. Incorporating delta change Historical analysis reveals a negative correlation between the magnitude of R and its delta change. To accommodate this, the cap includes an additional buffer, based on the discrete delta change of R over the preceding n days. To avoid overfitting, we segment the historical data into s equally sized groups over the same n-day window. We then calculate the xth percentile of R's delta changes, conditional on the segmented sample.

3. Constraint by longitudinal historical performances The cap is further restrained by the xth percentile of R over a longer historical span, generally five years.

Definitions and formulation

Given an index with a realized ratio R(t, l, q), the upper bound (UB(t, l, k)) and lower bound (LB(t, l, k)) for a contract of length l, starting k days ahead from the current time t, are determined by the following expressions:

where:

is the function to calculate the moving average of index at time t with q days of averaging

q is the moving average lookback window, e.g. q = 1 means spot rate and q = 7 means weekly average

m = min(R) over 5 years.

s = 4 (quartile);

is a function to find y’s percentile of realized ratio days delta change conditional on sampling with the group that is evenly divided into s that level of , l falls into from the last n days.