README.md

Introduction

This repo is for the JPMorgan Quant Research program

Task One

After asking around for the source of the existing data, you learn that the current process is to take a monthly snapshot of prices from a market data provider, which represents the market price of natural gas delivered at the end of each calendar month. This data is available for roughly the next 18 months and is combined with historical prices in a time series database. After gaining access, you are able to download the data in a CSV file. "Nat_Gas.csv" You should use this monthly snapshot to produce a varying picture of the existing price data, as well as an extrapolation for an extra year, in case the client needs an indicative price for a longer-term storage contract.

Download the monthly natural gas price data. "Nat_Gas.csv"

• Each point in the data set corresponds to the purchase price of natural gas at the end of a month, from 31st October 2020 to 30th September 2024.

• Analyze the data to estimate the purchase price of gas at any date in the past and extrapolate it for one year into the future.

• Your code should take a date as input and return a price estimate.

Try to visualize the data to find patterns and consider what factors might cause the price of natural gas to vary. This can include looking at months of the year for seasonal trends that affect the prices, but market holidays, weekends, and bank holidays need not be accounted for.

Task Two

Great work! The desk now has the price data they need. The final ingredient before they can begin trading with the client is the pricing model. Alex tells you the client wants to start trading as soon as possible. They believe the winter will be colder than expected, so they want to buy gas now to store and sell in winter in order to take advantage of the resulting increase in gas prices. They ask you to write a script that they can use to price the contract. Once the desk are happy, you will work with engineering, risk, and model validation to incorporate this model into production code.

The concept is simple: any trade agreement is as valuable as the price you can sell minus the price at which you are able to buy. Any cost incurred as part of executing this agreement is also deducted from the overall value. So, for example, if I can purchase a million MMBtu of natural gas in summer at $2/MMBtu, store this for four months, and ensure that I can sell the same quantity at $3/MMBtu without incurring any additional costs, the value of this contract would be ($3-$2) *1e6 = $1million. If there are costs involved, such as having to pay the storage facility owner a fixed fee of $100K a month, then the 'value' of the contract, from my perspective, would drop by the overall rental amount to $600K. Another cost could be the injection/withdrawal cost, like having to pay the storage facility owner $10K per 1 million MMBtu for injection/withdrawal, then the price will further go down by $10K to $590K. Additionally, if I am supposed to foot a bill of $50K each time for transporting the gas to and from the facility, the cost of this contract would fall by another $100K. Think of the valuation as a fair estimate at which both the trading desk and the client would be happy to enter into the contract.

You need to create a prototype pricing model that can go through further validation and testing before being put into production. Eventually, this model may be the basis for fully automated quoting to clients, but for now, the desk will use it with manual oversight to explore options with the client.

You should write a function that is able to use the data you created previously to price the contract. The client may want to choose multiple dates to inject and withdraw a set amount of gas, so your approach should generalize the explanation from before. Consider all the cash flows involved in the product.

The input parameters that should be taken into account for pricing are:

• Injection dates.
• Withdrawal dates.
• The prices at which the commodity can be purchased/sold on those dates.
• The rate at which the gas can be injected/withdrawn.
• The maximum volume that can be stored.
• Storage costs.

Write a function that takes these inputs and gives back the value of the contract. You can assume there is no transport delay and that interest rates are zero. Market holidays, weekends, and bank holidays need not be accounted for. Test your code by selecting a few sample inputs.

Task Three

You have now moved to a new team assisting the retail banking arm, which has been experiencing higher-than-expected default rates on personal loans. Loans are an important source of revenue for banks, but they are also associated with the risk that borrowers may default on their loans. A default occurs when a borrower stops making the required payments on a debt.

The risk team has begun to look at the existing book of loans to see if more defaults should be expected in the future and, if so, what the expected loss will be. They have collected data on customers and now want to build a predictive model that can estimate the probability of default based on customer characteristics. A better estimate of the number of customers defaulting on their loan obligations will allow us to set aside sufficient capital to absorb that loss. They have decided to work with you in the QR team to help predict the possible losses due to the loans that would potentially default in the next year.

Charlie, an associate in the risk team, who has been introducing you to the business area, sends you a small sample of their loan book and asks if you can try building a prototype predictive model, which she can then test and incorporate into their loss allowances.

The risk manager has collected data on the loan borrowers. The data is in tabular format, with each row providing details of the borrower, including their income, total loans outstanding, and a few other metrics. There is also a column indicating if the borrower has previously defaulted on a loan. You must use this data to build a model that, given details for any loan described above, will predict the probability that the borrower will default (also known as PD: the probability of default). Use the provided data to train a function that will estimate the probability of default for a borrower. Assuming a recovery rate of 10%, this can be used to give the expected loss on a loan.

You should produce a function that can take in the properties of a loan and output the expected loss. You can explore any technique ranging from a simple regression or a decision tree to something more advanced. You can also use multiple methods and provide a comparative analysis.

Task Four

Now that you are familiar with the portfolio and personal loans and risk are using your model as a guide to loss provisions for the upcoming year, the team now asks you to look at their mortgage book. They suspect that FICO scores will provide a good indication of how likely a customer is to default on their mortgage. Charlie wants to build a machine learning model that will predict the probability of default, but while you are discussing the methodology, she mentions that the architecture she is using requires categorical data. As FICO ratings can take integer values in a large range, they will need to be mapped into buckets. She asks if you can find the best way of doing this to allow her to analyze the data.

A FICO score is a standardized credit score created by the Fair Isaac Corporation (FICO) that quantifies the creditworthiness of a borrower to a value between 300 to 850, based on various factors. FICO scores are used in 90% of mortgage application decisions in the United States. The risk manager provides you with FICO scores for the borrowers in the bank’s portfolio and wants you to construct a technique for predicting the PD (probability of default) for the borrowers using these scores.

Charlie wants to make her model work for future data sets, so she needs a general approach to generating the buckets. Given a set number of buckets corresponding to the number of input labels for the model, she would like to find out the boundaries that best summarize the data. You need to create a rating map that maps the FICO score of the borrowers to a rating where a lower rating signifies a better credit score.

The process of doing this is known as quantization. You could consider many ways of solving the problem by optimizing different properties of the resulting buckets, such as the mean squared error or log-likelihood (see below for definitions). For background on quantization, see here.

Mean squared error

You can view this question as an approximation problem and try to map all the entries in a bucket to one value, minimizing the associated squared error. We are now looking to minimize the following:

$$ \text{MSE} = \frac{1}{n} \sum_{i=1}^{n} \left( Y_i - \hat{Y_i} \right)^2 $$

Log-likelihood

A more sophisticated possibility is to maximize the following log-likelihood function:

$$ LL(b_1, \ldots, b_{r-1}) = \sum_{i=1}^r \left[k_i \ln p_i + (n_i - k_i) \ln(1 - p_i)\right] $$

Where:

• $b_i$ is the bucket boundaries,
• $n_i$ is the number of records in each bucket,
• $k_i$ is the number of defaults in each bucket, and
• $p_i = \frac{k_i}{n_i}$ is the probability of default in the bucket.

This function considers how rough the discretization is and the density of defaults in each bucket. This problem could be addressed by splitting it into subproblems, which can be solved incrementally (i.e., through a dynamic programming approach). For example, you can break the problem into two subproblems, creating five buckets for FICO scores ranging from ​600 to 850. Refer to this page for more context behind a likelihood function. This page may also be helpful for background on dynamic programming.