Overview
Pnl Attribution seeks to understand what caused the price of a derivative to change. That is, the value of a derivative is typically a function of a number of external factors including :
- Interest Rates
- Underlying Price
- Underlying Volatility
- Passage of Time
As the price changes from one period to the next, it is helpful to understand which of the underlying factors affected the price and by how much.
The Derivative Attribution Problem
Pnl Attribution of derivatives is an example of the attribution problem. In this case, we assume that the value of a derivative at time {% t %} is a function of a set of factors.
{% value_t = f(x_{1,t}, x_{2,t} , ... ,x_{n,t}) %}
then the value at time t+1 is
{% value_{t+1} = f(x_{1,t+1}, x_{2,t+1} , ... ,x_{n,t+1}) %}
That means that the PnL between times {% t %} and {% t+1 %} is given by
{% PnL = value_{t+1}-value_t %}
The attribution problem is then to compute how much of the change in value, or PnL, was attributable
to the change that ocurred in each of the underlying pricing factors.
Attribution Methodologies
There exist various methodologies for running an attribution analysis. Each methodology has its own pros and cons.
- Scalloping : is an approach that explains the entire difference (or PnL) between two times, however, the factors are evaluated in a preset order, and answer will be different depending on the order chosen.
- Gradient Based : the Gradient based approach has the benefit that the answer does not depend on an arbitrary choice (as in the scalloping order), however, it does not typically explain the whole difference. That is, there will be an unexplained portion.