Quantitative finance is a real-world field that draws on tools from many different mathematical disciplines. Additionally, there are numerous modelling approaches to financial modelling. It’s odd how passionate supporters of various mathematical specialities become when debating the advantages and disadvantages of their own and their “opponents'” methodologies. What is all the hullabaloo about differential equations and martingales, is this a territorial issue, and will there be tears before bedtime?
Here is a summary of the several modelling approaches along with a few helpful tools. It will become evident how a “tool” and a “modelling approach” differ.
Modelling Approaches
- Probabilistic
- Deterministic
- Discrete: difference equations
- Continuous: differential equations
Useful Tools
- Simulations
- Approximations
- Asymptotic analysis
- Series solutions
- Discretization methods
- Green’s functions
These are by no means arbitrary lists, but there is room for improvement and addition. First, let’s examine the modelling methodologies.
Probabilistic
At least in the context of quantitative finance, one of the fundamental presumptions about the financial markets is that asset values are arbitrary. Financial variables are commonly thought of as following a random path, with parameters that describe the asset’s growth and level of unpredictability. Using an average growth rate and its divergence from this average, we may accurately estimate the asset path. Over the past 30 years, this modelling approach has had the biggest influence, contributing to the derivatives markets’ phenomenal rise.
Deterministic
The premise of this strategy is that everything we need to know about the future will be revealed by our model. If we have access to enough data and a large enough brain, we can use equations or algorithms to make future predictions. It’s interesting to note that this includes the topics of chaos and dynamical systems. Furthermore, as you are aware, chaotic systems are practically hard to anticipate due to their extreme sensitivity to beginning conditions. According to the “butterfly effect,” rainfall will “cause” itself over Manchester if a butterfly flaps its wings in Brazil. (And what doesn’t either!) In the realm of finance, this modelling approach, which gained popularity in the early 1990s, has fallen short of expectations.
Discrete/Continous
Regardless of its probability or determinism, the final model you record may be discrete or continuous. Discrete refers to the fact that time and/or asset prices can only be increased in discrete units, such as a day, a year, or a penny. Continuous indicates the absence of such a lower increment. It is frequently easier to understand continuous processes mathematically than discrete ones. However, in the realm of numerical computation, it is necessary to convert a continuous model into a discrete one.
We are left with difference equations in discrete models.
The binomial model used in option pricing is one illustration of this. The time step is the limited amount of time that passes. Differential equations are the result of using continuous models. With continuous asset price and continuous time, the Black-Scholes model is the discrete space equivalent of the binomial model. Both Black-Scholes and binomial models are derived from probabilistic assumptions about the financial world.
Let’s now examine a few of the accessible tools.
Simulations
By executing simulations, we may test the future if the financial world is random.
For instance, the average growth and risk of an asset can be used to indicate its price. Now, let’s see what might happen to this random asset in the future. Should we adopt this strategy, we would want to conduct an enormous number of simulations. Running just one would be pointless; instead, we’d prefer to see a variety of potential outcomes.
In addition, simulations can be applied to non-probabilistic problems. An interpretation of a model generated inside a deterministic framework may be probabilistic simply due to the commonalities among mathematical equations.
Discetization Methods
These can take many different forms and serve as a complement to simulation methods. The most well-known of them are the discretizations of continuous models like Black-Scholes, or finite-difference approaches.
Unless the task is really basic, you will most likely use simulation or finite difference approaches for your number-crunching.
Approximations
In modelling, our goal is to arrive at a solution that represents a significant and practical entity, like an option price. We might not be able to solve the model quickly unless it is quite basic. This is the role of approximations. Approximate solutions may exist for complex models. Furthermore, these approximations may be sufficient for our needs.
Asymptotic Analysis
This is a very helpful technique that is practically unknown in finance until lately, but it is utilised in most disciplines of relevant mathematics. The principle is straightforward: use parameters or variables that are either huge or little, or unique in some other way, to find approximations of solutions to complex problems. For vanilla options, for instance, there are straightforward approximations available near expiration.
Green’s Solutions
This is an extremely unique method that is only effective in specific circumstances. The concept is that solutions to specific solutions of a related problem can be used to build answers to more complex problems.
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