Fundamentals of Numerical Methods for Quant Finance
Introduction
Some of us may have come across some form of numerical techniques at a certain point of time in our lives — be it in school, undergraduate/postgraduate level or on the job. Numerical techniques are useful not just in finance but across industries. We may imagine them like a key concept that is industry agnostic and can be applied for solving a wide variety of problems / tasks that we come across in the industry. Numerical methods are one of the most widely researched areas in quant finance. Further, consistent efforts are on, to enhance the existing numerical techniques Or design new numerical techniques which can help increase the speed and accuracy with which we can estimate solutions to problems.
While there are a plethora of numerical methods out there, in this article we will focus on a few of the numerical methods for pricing options. Further, in my pursuit to make this article useful to a wider audience, we will not be discussing the actual mathematical formulas that go into the models. I will try to explain the numerical techniques in simple words for everyone’s benefit.
So why do we require options pricing?
Options are a key class of derivatives products that market participants use mainly for trading and hedging purposes. Options trade on both — regulated Exchanges (eg: National Stock Exchange of India) Or on the Over the Counter (OTC) market. The volume of options traded on the OTC market is far bigger than the volume that trades on Exchanges.
Options are a part of the trading book for a bank or any institution. It is important to know what is the daily valuation (or mark to market, or fair valuation) of each position that is sitting on the trading book.
A few reasons as to why market participants do options pricing are:
o Trading
o Daily portfolio valuation
o Risk Management
o Regulatory capital charge calculation
o Counterparty exposure monitoring
o Stress testing
o Collateral management
o Reporting — Internal MIS as well as Regulatory reporting etc.
The above points explain the reason i.e. “the Why” for options pricing. Now that we have an idea as to why it is important to price options, it gives us motivation to study “the How” part of options pricing i.e. understanding various techniques that one may apply to do options pricing
Why do we need to work on numerical methods? Can’t we have a ready formula all the time?
While it would be wonderful to have a ready formula for pricing options, that may not be possible all the time. Many problems that we come across in the industry may not have a ready formula i.e. a closed form solution. Thus, when a closed form solution does not exist, we need to resort to a numerical method which helps us make an estimation of the option price on that particular day. In the section below we will discuss a few popular techniques for pricing options.
Numerical Methods for Options pricing
We will touch upon three popular numerical methods that are used for pricing options. While these techniques are quantitatively intensive, we will keep this article light on the applied mathematics part, so that it becomes more appealing to a wider audience. In practice, understanding and implementing these techniques requires one to understand: finance + applied maths + programming
1. Trees / Lattice
The binomial model for options pricing is a very popular form of the trees method for options pricing. Binomial model is easy to understand and represent in the form of a simple diagram as shown below:
The reason why it’s called as a binomial tree is due to the structure it follows. From each node there are two branches that come out and connect to the next layer of nodes. The tree extends from today (i.e. t) uptil option maturity date (i.e. T). From the above diagram, we can imagine that S0 is the underlying asset price today i.e. time t, whereas, the last set of nodes farthest on the right represent option maturity date i.e. time T. So one can imagine that there is a time axis along the X-axis of the above chart and we are mapping the nodes at respective time steps. Binomial model is a discrete time options pricing model.
While in this article, we will not be discussing the mathematical formulas that go into building the tree, following are a few important points to note:
o The number of time steps i.e. from t to T can be decided by the model developer
o The approach to construct the tree is to have stock prices estimated at each nodes from left to right i.e. from the root node till the expiration nodes. Whereas, options prices are calculated in the backward direction i.e. from T to t i.e. through backward induction
o The aim is to calculate the option price at time t i.e. today
o Binomial model can be used for pricing European options Further, one can conveniently incorporate an early exercise feature that is present in a few options including American options and ESOPs.
In words, we say that the option price is the expected value of the future payoffs discounted at the risk free rate of interest
2. Finite Difference
Finite difference (“FD”) technique follows a grid approach. We will represent option price V as a functional form i.e. V(S,t) where option price is a function of asset price and time.
Below is a diagrammatic representation of the FD scheme:
The above is a simple 2-D grid with:
· Time along the X-axis
· Asset Price along the Y-axis
The main idea behind the finite difference approach is to convert a partial differential equation into a difference equation. The Difference equation is our effort to convert a continuous time system represented by a partial differential equation into a discrete time system as represented by the finite difference grid.
Ignoring the mathematical formulas that go into the FD scheme, following are a few important points to note:
o Finite difference is a very powerful technique for pricing options which can be applied to a wide variety of options types
o Under the FD scheme, we estimate the option prices at each of the cross section nodes. We begin with the terminal nodes and step back along the grid until we reach the set of root nodes i.e. representing time t as today.
o Implementing the FD scheme requires one to know some programming language be it VBA, Python, etc.
o It can easily price options right from European options to exotic structures
o It’s a remarkably stable model
I personally love the finite difference scheme. This is something that I extensively use on my client projects on model validations. There are different types of FD method including:
a. Explicit FD
b. Implicit FD
c. Crank-Nicholson FD scheme etc.
3. Monte Carlo simulation
Another popular technique that we use in the industry is the Monte Carlo simulation. The idea of monte carlo simulation is to generate as many possible paths that the underlying asset may follow from today until option maturity. This involves simulating thousands of such paths.
Below is a diagrammatic representation of the monte carlo simulation:
We attempt to estimate a distribution of asset prices at maturity. Simulation paths can be generated for a variety of assets. We require a stochastic differential equation (SDE) for simulation of the monte carlo paths. A popular model is the Geometric Brownian Motion (GBM) that we use for stock price simulation.
dS = r.S dt + σ.S dX
Once simulation is completed we can use the individual paths in our calculation to estimate the option price.
A few points to note regarding Monte carlo simulation:
o It is easy to implement
o Most of the complex maths gets handled conveniently and therefore the onus is not on the modeler to work on the complex maths
o It can handle exotic options
o It is computationally intensive, therefore, if the number of simulated paths are very large, one may consider using parallel computing or using a cluster of processors
o Monte carlo shares a very close connection with Expectations algebra making it very appealing to use while studying Martingales approach for options pricing.
Learning Quant Finance and Analytics:
We, at FinQuest Institute LLP offer a Quantitative Risk and Finance Program (QRFP) certificate. The QRFP certificate is a training program that focuses on quant finance, risk analytics and quant modelling in Python. The QRFP program is designed for the industry practitioner. The modules of this program are fully aligned with the industry requirements. This enables candidates of the QRFP certificate to gear up for exciting career roles including:
o Model development/ model validation,
o Derivatives pricing,
o Risk management,
o Rates modelling quants;
o Treasury Analytics;
o Front Office desk quant,
o Business Analyst for risk projects etc
Numerical methods is one of the modules of the QRFP certificate as well !
Conclusion
Numerical techniques are highly quantitative in nature. It requires one to understand the application maths, finance and programming so that one can build a model that is conceptually robust and is very stable. With systematic effort, one can master these numerical methods which are extensively used in the industry
Numerical methods are very exciting to learn and implement. There is a lot of research that has happened in numerical methods till date. However, there is still room for improvement. Numerical methods can also be considered as a potential topic for a doctoral thesis as well.
From job market perspective, there is a good demand for candidates who understand and are can implement numerical methods. Banks, Investment banks, consulting firms, asset managers etc. require employees with these kind of quantitative skills. Therefore, it would be wise for anyone to master numerical methods and get ready to tackle on a variety of interesting roles that the industry has to offer in the domain of quant finance.
