# Building Correlated Distributions

# Simulations are key #

Whenever you're dealing with economical data, the stuff that is within the

finance and business realm, you will always be looking at how much money you

will make or lose. This I will loosely define as the payoff of a product, say

you buy $100 worth of gummy bears and the next day all the supply of gummy

bears run out, and suddenly your gummy bears are worth $200. So the payoff

would be $100, since you've made $100 essentially. When you're dealing with

prices that will fluctuate over time, you will want to be able to model what

you think the price will be at a certain time.

In the field of electricity wholesale markets, often you will be trading

multiple products in order to hedge the risk of your portfolio. In other to

determine which products you want to buy/sell, you will need to form an

educated opinion on what is going to happen in the future. One of the ways to

do this is to have a payoff distribution model.

## Payoff Distribution Model #

I should have began this post by stating that I am by no means an expert on

this matter, and all this technical jargon is pretty much whatever I think is

correct. I've been working within the electricity industry for not a very long

time, and there are still so many intricacies that I don't know or understand.

So to me, a payoff distribution model is a tool that allows you to build

bespoke distributions that you use to evaluate performance and risk of your

portfolio. These distributions are simulated prices for the underlying

product, which can be made to contain all the correlations you desire.

Within the New Zealand Electricity market, the FTR products are offered during

2 auctions per month, which can allow for 2 opportunities to take a bite of

the apple. In order to form an opinion on what price a product might sell for

in an auction, you can create a distribution of possible prices. This is where

a payoff distribution model comes in. We begin with the historical

distribution of payoffs for each product, which gives us a starting point for

the distributions.

## Keeping things correlated #

The input into the model will be the historical payoffs for each product in

the auction, and the desired output will be a distribution of prices for each

product. This new payoff distribution will be possible gains or losses from

the result of the auction. This will give a performance measure of the total

portfolio, as a combination of all the payoff distributions. Because of the

nature of electricity, electrons traveling through the grid, there will always

be correlation between all the FTR products.

One method to keep the correlations in tact is to sample from the historical

distributions by grouping the products into clusters first. The clustering of

historical payoffs can then be given labels, depending on if there is an

historical event that can be attributed to those product times in history.

Then for each cluster, we can adjust the probability of the event happening to

each cluster. Using these adjustment values as probability weights, we can

sample from the historical data to get a simulated distribution.

```
distribution = sample(historical_payoffs,
size = 10000,
replace = TRUE,
prob = Weights)
```

This methodology keeps the correlation of the products since within the

historical payoffs, the correlation is there. The flaws in this procedure lie

in the fact that clustering doesn't allow for custom tailored distributions

for each product, but only for each cluster. This results in product payoff

distributions that are not statistically detailed enough. So ideally we want

to able to select each product, view the historical payoff distributions,

build a target distribution, then create the resulting payoff by sampling

using the target and historical distribution.

But then how do we deal with the correlations?