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The pricing models and neural network representations used in part one of the paper "Empirical analysis of rough and classical stochastic volatility models to the SPX and VIX markets".
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This project implements the pricing models used in part one of the analysis of [1] as well as fast neural network approximations of these.
We start by outlining the models: Let S(t) denote the time t price of an asset and let r(t) and q(t) denote the risk-free interest rate and the continuously compounded dividend yield respectively; r(t) and q(t) are assumed deterministic. Under some standard assumptions, we then have
under the risk-neutral measure. Here V(t) is some process (the instantaneous variance) and we write W for Brownian motions.
The models for V(t) that we consider are defined below.
Remark: The notation differs slightly between the paper and the code. Keep this in mind.
In the Heston model of [2], we have
We consider also a rough Heston model akin to [3] defined by
The rough Bergomi model of [4] assumes
Under the extended rough Bergomi model we assume that
In the code, we use the reparameterisation:
The code first and foremost implements pricing algorithms for puts and calls on S(t). You should consult the code or the paper for a description of what methods and schemes are used. This part of the project is only implemented in Matlab.
Secondly, there are Matlab scripts for generating large datasets of option prices for different model parameters. Neural networks are then trained (in Python and using Keras) to represent the datasets for each model.
Thirdly, the code implements interfaces to evaluate the neural networks. The interfaces are available in Matlab, Python and R.
The network weights are located in the folder .../code/neural_networks/data/neural_network_weights if one wants to implement the neural networks in other languages. The training and test datasets are rather large (almost 30 GB in total) and are thus not included in the repository. They can instead be provided upon request.
With neural networks we can compute an entire volatility surface in around 1 millisecond on a standard laptop. A typical calibration to SPX options can then be performed in less than a second. Read the paper for the details, or try it yourself.
The networks are also highly accurate as exemplified with the rough Bergomi model below:
The parameters are:
A more detailed analysis of the approximation error can be found in the paper.
There are a number of scripts to help get you started. They are explained below.
| Language | Folder | Description |
|---|---|---|
| Matlab | .../get_started/neural_networks_in_matlab | Examples of using the neural networks in Matlab. |
| Python | .../get_started/neural_networks_in_python | Examples of using the neural networks in Python. |
| R | .../get_started/neural_networks_in_R | Examples of using the neural networks in R. |
| Matlab | .../get_started/models_in_matlab | Examples of using the underlying pricing models in Matlab. |
Remarks:
The following external packages and libraries are included in the project:
Selected from shared topics, language and repository description—not editorial ratings.
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The overall goal of this project is to help the CMs focus on areas that will foster economic development for their respective states. Since the most common measure of economic development is the GDP, we will analyse the GDP of the various states of India and suggest ways to improve it. Understanding GDP Gross domestic product (GDP) at current prices is the GDP at the market value of goods and services produced in a country during a year. In other words, GDP measures the 'monetary value of final goods and services produced by a country/state in a given period of time'. GDP can be broadly divided into goods and services produced by three sectors: the primary sector (agriculture), the secondary sector (industry), and the tertiary sector (services). It is also known as nominal GDP. More technically, (real) GDP takes into account the price change that may have occurred due to inflation. This means that the real GDP is nominal GDP adjusted for inflation. 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The data for GDP analysis of the Indian states is divided into two parts: Data I-A: This dataset consists of the GSDP (Gross State Domestic Product) data for the states and union territories. Data I-B: This dataset contains the distribution of GSDP among three sectors: the primary sector (agriculture), the secondary sector (industry) and the tertiary sector (services) along with taxes and subsidies. There is separate dataset for each of the states. We are expected to read the dataset for the available states and join these (in Python) if needed. There are two parts to this project. In the first part, we will analyse and compare the GDPs of various Indian states (both total and per capita). The GDP of a state is referred to as the GSDP (Gross State Domestic Product). Then, we will divide the states into four categories based on the GDP per capita, and for each of these four categories, we will analyse the sectors that contribute the most to the GDP (such as agriculture, real estate, manufacturing, etc.). In the second part, we will analyse whether GDP per capita is related to dropout rates in schools and colleges. Part-I: GDP Analysis of the Indian States For each of the following steps of analysis, choose an appropriate type of plot for comparing the data. Also, ensure that the plots are in increasing or decreasing order for better comparison. For example, if we make a bar plot to compare the GDPs of the states, ensure that the bars are in either increasing or decreasing order of GDP. Part I-A: For the analysis below, use the Data I-A. First, we need to load the data in Python properly and then clean it. This also involves the treatment of missing values, we can choose to drop the row or column as well. Remember this will affect our next analysis and results drastically. Plot a graph for rows " % Growth over previous year" for all the states (not union territories) whose data is available, use as much data as possible for this exercise. Use the best fit line to represent the growth for each state. Draw a similar line graph for the nation as well. How will we compare the growth rates of any two states? Which states have been growing consistently fast, and which ones have been struggling? Rank top 3 fastest and 3 slowest-growing states. What is the Nation's growth rate? What has been the growth rate of my home state, and how does it compare to the national growth rate? Plot the total GDP of the states for the year 2015-16: Which Plot will we use for this? Why? (Remeber to plot the graph in a way such as it is easier to read and compare) Identify the top 5 and the bottom 5 states based on total GDP. What insights can we draw from this graph? What states are performing poorly? 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