Mathematical Finance | Gentle Reading For Sleep

Drift off with calm, bedtime reading that gently explores mathematical finance while supporting sleep and easing insomnia. This calm, bedtime reading blends clear explanations with a steady rhythm, helping sleep come more easily for listeners experiencing insomnia or restless nights.
As Benjamin reads, you’ll learn about models, probability, and how mathematics shapes modern finance, all presented in a soothing, unhurried way. His calm cadence is designed for bedtime reading and sleep, offering peaceful, fact-filled education without dramatics or performance. This episode supports insomnia relief, stress reduction, and anxious minds while letting curiosity quietly fade into rest. Press play, relax, and allow learning to guide you toward sleep. Happy sleeping!

Transcript

Welcome to the I Can't Sleep Podcast,

Where I help you drift off one fact at a time.

I'm your host,

Benjamin Boster,

And today's sponsored episode from Terry Yee is about mathematical finance.

Mathematical finance,

Also known as quantitative finance and financial mathematics,

Is a field of applied mathematics concerned with mathematical modeling in the financial field.

In general,

There exist two separate branches of finance that require advanced quantitative techniques,

Derivatives pricing on the one hand,

And risk and portfolio management on the other.

Mathematical finance overlaps heavily with the fields of computational finance and financial engineering.

The latter focuses on applications and modeling,

Often with the help of stochastic asset models,

While the former focuses,

In addition to analysis,

On building tools of implementation for the models.

Also related is quantitative investing,

Which relies on statistical and numerical models,

And lately machine learning,

As opposed to traditional fundamental analysis when managing portfolios.

French mathematician Louis Bachelier's doctoral thesis,

Defended in 1900,

Is considered the first scholarly work on mathematical finance.

But mathematical finance emerged as a discipline in the 1970s,

Following the work of Fisher Black,

Myron Scholes,

And Robert Merton on option pricing theory.

Mathematical investing originated from the research of mathematician Edward Thorpe,

Who used statistical methods to first invent card counting in blackjack,

And then applied its principles to modern systematic investing.

The subject has a close relationship with the discipline of financial economics,

Which is concerned with much of the underlying theory that is involved in financial mathematics.

While trained economists use complex economic models that are built on observed empirical relationships,

In contrast,

Mathematical finance analysis will derive and extend the mathematical or numerical models without necessarily establishing a link to financial theory,

Taking observed market prices as input.

The fundamental theorem of arbitrage-free pricing is one of the key theorems in mathematical finance,

While the Black-Scholes equation and formula are amongst the key results.

Today,

Many universities offer degree and research programs in mathematical finance.

There are two separate branches of finance that require advanced quantitative techniques,

Derivative pricing and risk and portfolio management.

One of the main differences is that they use different probabilities,

Such as the risk-neutral probability,

Or arbitrage pricing probability,

Denoted by Q,

And the actual or actuarial probability,

Denoted by P.

The goal of derivatives pricing is to determine the fair price of a given security in terms of more liquid securities,

Whose price is determined by the law of supply and demand.

The meaning of fair depends,

Of course,

On whether one considers buying or selling the security.

Examples of securities being priced are plain vanilla and exotic options,

Convertible bonds,

Etc.

Once a fair price has been determined,

The sell-side trader can make a market on the security.

Therefore,

Derivatives pricing is a complex extrapolation exercise to define the current market value of a security,

Which is then used by the sell-side community.

Quantitative derivatives pricing was initiated by Louis Bauchelet in the theory of speculation,

With the introduction of the most basic and most influential of processes,

Brownian motion,

And its applications to the pricing of options.

Brownian motion is derived using the Langevin equation and the discrete random walk.

Bauchelet modeled the time series of changes in the logarithm of stock prices as a random walk in which the short-term changes at a finite variance.

This causes longer-term changes to follow a Gaussian distribution.

The theory remained dormant until Fischer Black and Myron Scholes,

Along with fundamental contributions by Robert C.

Merton,

Applied the second most influential process,

The geometric Brownian motion,

To option pricing.

For this,

M.

Scholes and R.

Merton were awarded the 1997 Nobel Memorial Prize in Economic Sciences.

Black was ineligible for the prize because he died in 1995.

The next important step was the Fundamental Theorem of Asset Pricing by Harrison and Pliska,

1981,

According to which the suitably normalized current price,

P-naught,

Of security is arbitrage-free,

And thus truly fair only if there exists a stochastic process,

P-sub-T,

With constant expectations.

A process satisfying this condition is called a martingale.

A martingale does not reward risk.

Thus,

The probability of the normalized security price process is called risk-neutral,

And is typically denoted by the blackboard font letter Q.

The relationship in that equation must hold for all times t.

Therefore,

The processes used for derivatives pricing are naturally set in continuous time.

The quants who operate in the Q-world of derivatives pricing are specialists with deep knowledge of the specific products they model.

Securities are priced individually,

And thus the problems in the Q-world are low-dimensional in nature.

Calibration is one of the main challenges of the Q-world.

Once a continuous-time parametric process has been calibrated to a set of traded securities,

Through a relationship such as that equation,

A similar relationship is used to define the price of new derivatives.

The main quantitative tools necessary to handle continuous-time Q-processes are Ito's stochastic calculus,

Simulation,

And partial differential equations,

PDEs.

Risk and portfolio management aims to model the statistically-derived probability distribution of the market prices of all the securities at a given future investment horizon.

This real probability distribution of the market prices is typically denoted by the blackboard font letter P,

As opposed to the risk-neutral probability Q used in derivatives pricing.

Based on the P distribution,

The buy-side community takes decisions on which securities to purchase in order to improve the prospective profit and loss profile of their positions considered as a portfolio.

Increasingly,

Elements of this process are automated.

For their pioneering work,

Markovits and Sharpe,

Along with Merton Miller,

Shared the 1990 Nobel Memorial Prize in Economic Sciences for the first time ever awarded for a work in finance.

The portfolio selection work of Markovits and Sharpe introduced mathematics to investment management.

With time,

The mathematics has become more sophisticated.

Thanks to Robert Merton and Paul Samuelson,

One-period models were replaced by continuous-time,

Brownian motion models,

And the quadratic utility function implicit in mean-variance optimization was replaced by more general,

Increasing,

Concave utility functions.

Furthermore,

In recent years,

The focus shifted toward estimation risk,

I.

E.

,

The dangers of incorrectly assuming that advanced time-series analysis alone can provide completely accurate estimates of the market parameters.

Much effort has gone into the study of financial markets and how prices vary with time.

Charles Dow,

One of the founders of Dow Jones & Company and the Wall Street Journal,

Enunciated a set of ideas on the subject which are now called Dow Theory.

This is the basis of the so-called technical analysis method of attempting to predict future changes.

One of the tenets of technical analysis is that market trends give an indication of the future,

At least in the short term.

The claims of the technical analysts are disputed by many academics.

While numerous empirical studies have examined the effectiveness of technical analysis,

There remains no definitive consensus on its usefulness in forecasting financial markets.

Over the years,

Increasingly sophisticated mathematical models and derivative pricing strategies have been deployed,

But their credibility was damaged by the 2008 financial crisis.

Contemporary practice of mathematical finance has been subjected to criticism from figures within the field,

Notably by Paul Wilmut and by Nassim Nicholas Taleb in his book The Black Swan.

Taleb claims that the prices of financial assets cannot be characterized by the simple models currently in use,

Rendering much of current practice at best irrelevant and at worst dangerously misleading.

Wilmut and Emmanuel Derman published the Financial Modellers Manifesto in January 2009,

Which addresses some of the most serious concerns.

Bodies such as the Institute for New Economic Thinking are now attempting to develop new theories and methods.

In general,

Modelling the changes by distributions with finite variance is increasingly said to be inappropriate.

In the 1960s,

It was discovered by Benoit Mandebrot that changes in prices do not follow a Gaussian distribution,

But are rather modelled better by Levy-Alpha stable distributions.

The scale of change,

Or volatility,

Depends on the length of the time interval to a power a bit more than one half.

Large changes,

Up or down,

Are more likely than what one would calculate using a Gaussian distribution with an estimated standard deviation.

But the problem is that it does not solve the problem as it makes parameterization much harder and risk control less reliable.

Perhaps more fundamental,

Though mathematical finance models may generate a profit in the short run,

This type of modelling is often in conflict with a central tenet of modern macroeconomics,

The Lucas Critique,

Or rational expectations,

Which states that observed relationships may not be structural in nature and thus may not be possible to exploit for public policy,

Or for profit,

Unless we have identified relationships using causal analysis and econometrics.

Mathematical finance models do not therefore incorporate complex elements of human psychology that are critical to modelling modern macroeconomic movements,

Such as the self-fulfilling panic that motivates bank runs.

Quantitative analysis in finance refers to the application of mathematical and statistical methods to problems in financial markets and investment management.

Professionals in this field are known as quantitative analysts,

Or quants.

Quants typically specialize in areas such as derivative structuring and pricing,

Risk management,

Portfolio management,

And other finance-related activities.

The role is analogous to that of a specialist in industrial mathematics working in non-financial industries.

Quantitative analysis often involves examining large data sets to identify patterns such as correlations among liquid assets or price dynamics,

Including strategies based on trend following or mean reversion.

Although the original quantitative analysts were sell-side quants from market-maker firms concerned with derivatives,

Pricing,

And risk management,

The meaning of the term has expanded over time to include those individuals involved in almost any application of mathematical finance,

Including the buy-side.

Applied quantitative analysis is commonly associated with quantitative investment management,

Which includes a variety of methods such as statistical arbitrage,

Algorithmic trading,

And electronic trading.

Some of the larger investment managers using quantitative analysis include Renaissance Technologies,

D.

E.

Shaw & Co.

,

And AQR Capital Management.

Quantitative finance started in 1900 with Louis Bachelier's doctoral thesis,

Theory of Speculation,

Which provided a model to price options under a normal distribution.

Jules Rinald had posited already in 1863 that stock prices can be modeled as a random walk,

Suggesting in a more literary form the conceptual setting for the application of probability to stock market operations.

It was,

However,

Only in the years 1960-1970 that the merit of these was recognized as options pricing theory was developed.

Harry Markowitz's 1952 doctoral thesis,

Portfolio Selection,

And its published version was one of the first efforts in economic journals to formally adapt mathematical concepts to finance.

Mathematics was,

Until then,

Confined to specialized economics journals.

Markowitz formalized a notion of mean return and covariances for common stocks,

Which allowed him to quantify the concept of diversification in a market.

He showed how to compute the mean return and variance for a given portfolio,

And argued that investors should hold only those portfolios whose variance is minimal among all portfolios with a given mean return.

Thus,

Although the language of finance now involves Ito's calculus,

Management of risk in a quantifiable manner underlies much of the modern theory.

In sales and trading,

Quantitative analysts work to determine prices,

Manage risk,

And identify profitable opportunities.

Historically,

This was a distinct activity from trading,

But the boundary between a desk quantitative analyst and a quantitative trader is increasingly blurred,

And it is now difficult to enter trading as a profession without at least some quantitative analysis education.

Front office work favors a higher speed-to-quality ratio,

With a greater emphasis on solutions to specific problems than detailed modeling.

FOQs typically are significantly better paid than those in back office,

Risk,

And model validation.

Although highly skilled analysts,

FOQs frequently lack software engineering experience or formal training,

And bound by time constraints and business pressures,

Tactical solutions are often adopted.

Increasingly,

Quants are attached to specific desks.

Two cases are XVA specialists,

Responsible for managing counterparty risk,

As well as minimizing the capital requirements under Basel III,

And structurers,

Tasked with the design and manufacture of client-specific solutions.

Quantitative analysis is used extensively by asset managers.

Some,

Such as FQ,

AQR,

Or Barclays,

Rely almost exclusively on quantitative strategies,

While others,

Such as PIMCO,

BlackRock,

Or Citadel,

Use a mix of quantitative and fundamental methods.

One of the first quantitative investment funds to launch was based in Santa Fe,

New Mexico,

And began trading in 1991 under the name Prediction Company.

By the late 1990s,

Prediction Company began using statistical arbitrage to secure investment returns,

Along with three other funds at the time,

Renaissance Technologies and D.

E.

Shaw & Co.

,

Both based in New York.

Prediction hired scientists and computer programmers from the neighboring Los Alamos National Laboratory to create sophisticated statistical models using industrial-strength computers in order to build the supercollider of finance.

Machine learning models are now capable of identifying complex patterns in financial market data.

With the aid of artificial intelligence,

Investors are increasingly turning to deep learning techniques to forecast and analyze trends in stock and foreign exchange markets.

Major firms invest large sums in an attempt to produce standard methods of evaluating prices and risk.

These differ from front-office tools in that Excel is very rare,

With most development being in C++,

Though Java,

C Sharp,

And Python are sometimes used in non-performance-critical tasks.

LQs spend more time modeling,

Ensuring the analytics are both sufficient and correct,

Though there is tension between LQs and FOQs on the validity of their results.

LQs are required to understand techniques such as Monte Carlo methods and finite difference methods,

As well as the nature of the products being modeled.

Often,

The highest-paid form of quant,

ATQs,

Make use of methods taken from signal processing,

Game theory,

Gambling Kelly criterion,

Market microstructure,

Econometrics,

And time series analysis.

Risk management has grown in importance in recent years,

As the credit crisis exposed holes in the mechanisms used to ensure that positions were correctly hedged.

A core technique continues to be value-at-risk,

Applying both the parametric and historical approaches,

As well as conditional value-at-risk and extreme value theory.

This is supplemented with various forms of stress test,

Expected shortfall methodologies,

Economic capital analysis,

Direct analysis of the positions at the desk level,

And,

As below,

Assessment of the models used by the banks' various divisions.

After the 2008 financial crisis,

There surfaced the recognition that quantitative valuation methods were generally too narrow in their approach.

An agreed-upon fix,

Adopted by numerous financial institutions,

Has been to improve collaboration.

Model Validation,

MV,

Takes the models and methods developed by front office,

Library,

And modeling quantitative analysts,

And determines their validity and correctness.

The MV group might well be seen as a superset of the quantitative operations in a financial institution,

Since it must deal with new and advanced models and trading techniques from across the firm.

Post-crisis,

Regulators now typically talk directly to the quants in the middle office,

Such as the model validators,

And since profits highly depend on the regulatory infrastructure,

Model validation is gained in weight and importance with respect to the quants in the front office.

Before the crisis,

However,

The pay structure in all firms was such that MV groups struggled to attract and retain adequate staff,

Often with talented quantitative analysts leaving at the first opportunity.

This gravely impacted corporate ability to manage model risk,

Or to ensure that the positions being held were correctly valued.

An MV quantitative analyst would typically earn a fraction of quantitative analysts in other groups with similar length of experience.

In the years following the crisis,

As mentioned,

This has changed.

Quantitative developers,

Sometimes called quantitative software engineers or quantitative engineers,

Are computer specialists that assist,

Implement,

And maintain the quantitative models.

They tend to be highly specialized language technicians that bridge the gap between software engineers and quantitative analysts.

The term is also sometimes used outside the finance industry to refer to those working at the intersection of software engineering and quantitative research.

The non-ergodicity of financial markets and the time dependence of returns are central issues in modern approaches to quantitative trading.

Financial markets are complex systems in which traditional assumptions,

Such as independence and normal distribution of returns,

Are frequently challenged by empirical evidence.

Thus,

Under the non-ergodicity hypothesis,

The future returns about an investment strategy,

Which operates on a non-stationary system,

Depend on the ability of the algorithm itself to predict the future evolutions to which the system is subject.

As discussed by Ola Peters in 2011,

Ergodicity is a crucial element in understanding economic dynamics,

Especially in non-stationary contexts.

Identifying and developing methodologies to estimate this ability represents one of the main challenges of modern quantitative trading.

In this perspective,

It becomes fundamental to shift the focus from the result of individual financial operations to the individual evolutions of the system.

Operationally,

This implies that clusters of trades oriented in the same direction offer little value in evaluating the strategy.

On the contrary,

Sequences of trades with alternating buy and sell are much more significant.

Since they indicate that the strategy is actually predicting a statistically significant number of evolutions of the system.

Because of their backgrounds,

Quantitative analysts draw from various forms of mathematics.

Statistics and probability,

Calculus centered around partial differential equations,

Linear algebra,

Discrete mathematics,

And economics.

Some on the buy side may use machine learning.

The majority of quantitative analysts have received little formal education in mainstream economics.

And often apply a mindset drawn from the physical sciences.

Quants use mathematical skills learned from diverse fields,

Such as computer science,

Physics,

And engineering.

These skills include but are not limited to advanced statistics,

Linear algebra,

And partial differential equations.

As well as solutions to these based upon numerical analysis.

Commonly used numerical methods are finite difference method used to solve partial differential equations.

Monte Carlo method also used to solve partial differential equations.

But Monte Carlo simulation is also common in risk management.

Ordinary least squares used to estimate parameters in statistical regression analysis.

Spline interpolation used to interpolate values from spot and forward interest rates curves and volatility smiles.

Bisection,

Newton,

And secant methods used to find the roots,

Maxima,

And minima of functions,

E.

G.

Internal rate of return,

Interest rate curve building.

Reviews

This was kind of interesting which surprised me! Thankfully not THAT interesting though, or maybe it was your voice that put me to sleep? 😜
Thank you, Benjamin! 😻

So incredibly boring—it went right off my smooth brain and put me straight to sleep!

Perfect - I'm going to bookmark this for those extreme cases of insomnia! 🎧😴