Emergence
noun
social science, science — specialized

The development of particular patterns, properties, or behaviors in parts of complex systems that happen only when the parts of the system interact, and that the parts do not have on their own.

Recently, some juniors and interns at work reached out to me to find out more about quantitative finance. Through the conversation, I reflected upon my career journey and how my view of the industry and quantitative finance developed over the years.

When I first encountered quantitative finance, I saw it mainly as a discipline of models — a way to price derivatives, measure risk, and bring mathematical elegance into financial markets. But the more time I spent in the industry, the more I realized that quantitative finance is not just about building better models. As trading becomes more digital, quantitative finance increasingly becomes one of the key languages of the market. It is essential to understanding how uncertainty forms, how markets react to it, and how that uncertainty eventually becomes tradable.

That is what makes the field so fascinating today.

Things have evolved at lightning speed. Agentic AI is changing how we solve problems. New tradable markets are emerging. Tokenization is challenging the boundaries of market hours. Geopolitics is becoming harder to ignore. Data is no longer scarce. The ability to process it meaningfully is.

All these developments point to a larger shift: quantitative finance is becoming more than a pricing discipline. It is becoming a way to understand the complex financial world that we live and breathe.

Perhaps this makes a good topic to restart this series with — the development of quantitative finance, and the emergence of new forms of tradable uncertainty in financial markets.

Quantitative Finance as a Financial Service

When people think about the history of quantitative finance, the most iconic discovery many would point to is the Black-Scholes-Merton model, also known as the “trillion-dollar equation”. Published in 1973, the model revolutionised the industry by giving market participants a means to price optionality in a risk-neutral manner (Black & Scholes, 1973; Merton, 1973).

In the same year, the first modern option exchange, the Chicago Board Options Exchange (CBOE), was founded. This was a significant moment. Optionality no longer was a bespoke feature negotiated between parties over-the-counter. It became a tradable instrument on a lit market, made accessible to more people (Cboe, 2024).

This marked an era of financial engineering. Early quants moved deeper into financial mathematics, and more advanced models were subsequently developed. Pricing became a service. Investors could buy structured payoffs, while the sell-side manufactured these exposures, hedged them with more vanilla instruments, and earned a spread for intermediating complexity.

Over time, financial engineering expanded beyond options. Quantitative models were no longer used only for derivatives pricing. They were also used for risk quantification, structuring, and repackaging uncertainty into products that could be sold to different investors. Value-at-Risk became a common framework to quantify downside risk (RiskMetrics Group, 1996), while credit derivatives allowed institutions to transfer and repackage credit exposures.

At the height of financial engineering, the Gaussian copula provided the industry with a way to model default correlation and price complex credit products (Li, 2000). Products such as Collateralized Debt Obligations, or CDOs, became popular. The idea was simple in appearance: pool together many credit exposures, slice the pool into tranches with different levels of risk, and sell each tranche to investors with different risk appetites. Some of these tranches were even sold with triple-A ratings.

In principle, this was good product innovation. Risk could be decomposed, repackaged, and transferred to investors who were willing to hold it.

Until the assumptions failed.

The Hard Lesson from the Crisis

The issue with the Gaussian copula was not simply that it was mathematical. The issue was that it relied on assumptions that looked clean in normal times but became fragile under stress.

The model assumed that default correlation could be estimated and applied in a relatively stable way. Under this assumption, a diversified pool of mortgages appeared safer than it truly was. The tranches looked less risky, and therefore could be priced more cheaply than the risks they actually entailed.

As we learned from the Global Financial Crisis (GFC), the default correlation is not static. Defaults across mortgages became highly correlated when the housing bubble burst. Default correlation spiked, contagion spread, and the financial system came under severe stress (Brigo, Pallavicini, & Torresetti, 2009).

The GFC revealed a very important lesson for quants: that quantitative methods are not deterministic hacks of the financial markets.

Quantitative finance is a social science rather than a natural science. Unlike physics, markets do not operate according to static laws. The agents within the system observe, react, adapt, arbitrage, panic, and change their behavior. Once a model becomes widely used, it can even change the very market it is trying to describe.

This means that all models are wrong, even when some are useful.

In such a discipline, understanding model assumptions becomes essential. A model is not truth. It simplifies reality so that we can frame problems in manageable ways and make decisions. It is the quant’s responsibility to know what assumptions are made in a model, when those assumptions apply and that the model can be trusted, when it should be questioned, and when it should be switched off.

Regulatory Reform, Futurization, and Setting the Stage for Quant Trading

The Global Financial Crisis did not only expose model risk. It also exposed structural weaknesses in the financial system.

One of the most important was the opacity of the bilateral over-the-counter derivatives market. Counterparty exposures were difficult to observe. Risk was spread across a web of bilateral relationships. Regulators had limited visibility into where the largest vulnerabilities were building up.

Post-crisis, rules such as the Uncleared Margin Rules were introduced to reduce excessive bilateral exposures and encourage greater use of central clearing. The idea was to make the system more transparent, better margined, and easier to manage in the event of default (Basel Committee on Banking Supervision & International Organization of Securities Commissions, 2020).

Central clearing works by introducing a central counterparty (CCP) between buyers and sellers. The CCP becomes the buyer to every seller and the seller to every buyer. This does not remove risk from the system, but it changes how risk is managed. It allows exposures to be margined, monitored, netted, and default-managed through a centralized structure.

While some OTC instruments moved towards central clearing, others moved towards what is often described as futurization.

Futurization refers to the creation of exchange-traded alternatives to instruments that were traditionally traded over the counter. Instead of negotiating bespoke contracts bilaterally, market participants can trade standardized futures or options on exchange, with transparent pricing, central clearing, and more accessible liquidity.

Futurization increased the universe of tradable assets on an exchange and unveiled a new era of trading. Exchange-traded derivatives, with its transparent pricing and electronic execution, set a good environment for the application of quantitative finance. Attention shifts from derivatives pricing to quantitative trading.

Technology as a Super-cycle

Apart from regulatory reforms and new exchange-traded derivatives post the GFC, the world also evolved technologically. Technology created new possibilities for markets, and to me, three developments stood out that allowed quantitative finance to thrive.

Technology created a new asset class

The GFC not only revealed the weaknesses of intermediaries, it also inspired the creation of Bitcoin — the first decentralized digital money that allows peer-to-peer transactions without the need of a trusted middleman (Nakamoto, 2008).

Bitcoin was a multidisciplinary creation. It drew inspiration from cryptography, computer science, economics, game theory, and distributed systems. More importantly, it showed that anyone could design a new monetary system with code, incentives, and a network of participants.

This set a precedent. Blockchain technology became popularized as people started to create new cryptocurrencies, each attempting to improve on or extend Bitcoin’s design.

The trading of cryptocurrencies was empowered by both centralized crypto exchanges and decentralized platforms. These markets operate 24/7, are fully digital, and are highly accessible through APIs. For quantitative traders, crypto became a natural test bed. Data was relatively accessible, trading infrastructure was open, and strategies could be deployed quickly.

Cryptocurrency therefore did not only create a new asset class. It created a new market structure for experimentation.

Neo-brokerages democratized trading

Trading also became much easier in the mid-2010s with the rise of digital and mobile-first brokerages. Platforms such as Robinhood, Tiger Brokers, Moomoo, and others made the trading experience far more accessible to retail investors. Trades could be entered with a tap of the screen.

This changed how people interacted with markets. Investing and trading became less intimidating. Market access became more digital, interactive and immediate. With that accessibility came greater curiosity. More people started to ask how markets work, how prices move, how strategies are built, and how data can be used to make better decisions. More began learning about quantitative finance, and experimenting with trading, bringing the field closer to mainstream conversations.

Digitization created the new fuel — data

Ultimately, what drives quantitative finance is digitization.

With digitization comes an abundance of digital footprint in the form of data. Prices, transactions, order books, news, satellite images, shipping data, weather patterns, social media, corporate filings, and even alternative datasets can all be transformed into signals.

As more data became available, more sophisticated models could be deployed. These models could process larger amounts of information and form more nuanced trading views. Further advancement in machine learning and natural language processing also enabled unstructured data to be transformed into usable inputs.

This created a reinforcing spiral. Data gave rise to better models, and better models made more data economically useful.

Quantitative finance thrived in such an environment. For once, data was no longer a bottleneck.

Financialization of Everything

With technological advancement and digitization, quantitative finance has evolved from being mainly a tool for financial services into a broader way of forming a market view.

In the earlier phase, quantitative finance was heavily associated with the Q-probability measure — the risk-neutral world of pricing, replication, and hedging. Increasingly, the emphasis has shifted towards the P-probability measure — the real world, of forecasting, signal extraction, positioning, and decision-making.

This is an important shift, where the question is no longer only: what is the fair value of this derivative?

The question is increasingly: what does the data say about the future, and how can that view be expressed in the market?

This is where the financialization of everything begins.

Financialization of Commodities

Commodity trading was historically grounded in the physical world. Producers, consumers, merchants, and physical traders dominated the market. A barrel of oil, a cargo of LNG, a tonne of copper, or a megawatt-hour of electricity was tied to logistics, storage, transportation, and physical delivery.

But with the futurization of commodities, financial players could participate without necessarily touching the physical commodity. They could express views on supply, demand, inventory, weather, geopolitics, and macroeconomic conditions through exchange-traded derivatives. The growing role of financial investors in commodity futures markets has also shaped debates around risk sharing, information discovery, and price formation (Cheng & Xiong, 2014).

Information also became more powerful. Trade data, satellite imagery, vessel tracking, weather forecasts, and news flows allowed financial participants to understand physical markets from a distance. This brought more liquidity, more diversity of participants, and more systematic trading into commodity markets.

As an example, the ICE Dutch TTF natural gas futures price is no longer influenced only by the fundamental demand and supply dynamics of European natural gas. From my experience as an LNG quant, financial flows became one of the factors we started watching more closely in the formation of futures prices.

Through that lens, LNG prices increasingly face two orders of price determination. The first is the demand and supply of physical molecules — storage, weather, infrastructure, shipping, and consumption. The second is the demand and supply of derivatives, which influences the price benchmarks upon which physical commodity trades are often based.

In September 2024, S&P Global reported that hedge funds had been selling out of European natural gas futures, while investment funds accounted for nearly 22% of Dutch TTF futures positions. Traders also pointed to “funds closing their positions” as a key driver behind the fall in European gas and LNG prices, with fund flows amplifying the bearish pressure on both prompt and forward prices (Seth & Blakeway, 2024).

This is a good example of financialization in practice. A physical market remains grounded in molecules, storage, weather, and infrastructure. But once the derivative market becomes sufficiently liquid and widely traded, financial positioning itself becomes part of the price-formation process.

“Financialization of Weekends”

Crypto also challenged a long-standing assumption in traditional finance: that markets sleep.

Unlike traditional financial markets, digital assets trade continuously. They do not pause over the weekend. News, policy changes, geopolitical shocks, or market rumors can be reflected in crypto prices while traditional markets are closed.

Cryptocurrency has introduced a generation into the world of always-on price discovery. And with tokenization, 24/7 trading is no longer limited to cryptocurrencies. The tokenization of real-world assets (RWAs) and introduction of perpetuals on crypto exchanges blurs the boundary between trading hours and non-trading hours of the underlying assets.

For instance, at the onset of the US-Iran conflict on 28 February 2026, traders rushed to Hyperliquid - a native crypto exchange that offers RWA perps - to trade on tokenized oil futures when the traditional exchanges were closed on the weekends (Financial Times, 2026). Even when the main market is closed, adjacent markets may still be reacting to new information.

This introduces competition into traditional markets. Tradfi exchanges are compelled to extend their trading hours in order to retain control over liquidity of their products. 24/7 trading gradually becomes less of a good-to-have, and more a requisite to remain competitive.

For once, the market never truly sleeps.

Financialization of Information

Data fuels every decision making. With the abundance of data we have today, and the processing prowess available on our laptops, we are equipped to assign value to things that were once intangible.

We can model probabilities of future events. We can estimate the likelihood of election outcomes, policy decisions, macroeconomic releases, weather events, corporate actions, or geopolitical developments. Prediction markets and event contracts sit within this broader trend.

These markets transform uncertainty into price. They allow participants to express views not on a traditional asset, but on the probability of an outcome. In doing so, they turn information itself into something tradable.

This is powerful. Markets are no longer limited to pricing assets with cash flows, collateral, or physical value. They can also price expectations, beliefs, probabilities, and information.

At the same time, this development requires caution. Not every uncertainty should become a tradable product. Market design, regulation, and incentives matter. A market can aggregate information, but it can also amplify noise, manipulation, and speculation.

The financialization of information may be one of the most fascinating developments in modern markets, but it also reminds us of the same lesson from earlier phases of quantitative finance: innovation is useful only when we understand its assumptions, incentives, and limits.

Emergence

Technology and data have led to a new form of emergence in financial markets. They have not only democratized trading, but also made new products, new markets, and new ways of expressing views possible. Consequently, Quantitative finance has become richer in scope. It is no longer just about pricing another derivative, but about understanding the complex financial world that we live and breathe, with the data that we can process and analyze.

Perhaps this also calls for a paradigm change. In the past, when data was less abundant and computing power was more limited, parsimony was not just elegant — it was necessary. Models had to simplify aggressively because both information and computation were scarce.

Today, the constraints are changing. With generative AI and agentic AI, new approaches to problem-solving may emerge. We may no longer be limited only by our natural cognition or by the traditional ways we decompose problems. AI can help us explore more possibilities, process more context, and do some of the heavy lifting. But the role of the quant does not disappear. It shifts towards asking better questions, monitoring assumptions, understanding incentives, and knowing when the machine may be confidently wrong.

Perhaps AI is not meant to simply automate how we already work. Perhaps its greater potential is to help us solve problems in ways that humans would not naturally arrive at on our own.

And that, to me, is the next stage of emergence in quantitative finance.

References

Basel Committee on Banking Supervision & International Organization of Securities Commissions. (2020). Margin requirements for non-centrally cleared derivatives. Bank for International Settlements.

Black, F., & Scholes, M. (1973). The pricing of options and corporate liabilities. Journal of Political Economy, 81(3), 637–654.

Brigo, D., Pallavicini, A., & Torresetti, R. (2009). Credit models and the crisis, or: How I learned to stop worrying and love the CDOs.

Cboe. (2024). The creation of listed options at Cboe. Cboe.

Cheng, I.-H., & Xiong, W. (2014). Financialization of commodity markets. Annual Review of Financial Economics, 6, 419–441.

Financial Times. (2026, March 15). Retail traders rush into oil bets as Iran war drives wild price swings. Financial Times.

Li, D. X. (2000). On default correlation: A copula function approach. The Journal of Fixed Income, 9(4), 43–54.

Merton, R. C. (1973). Theory of rational option pricing. The Bell Journal of Economics and Management Science, 4(1), 141–183.

Nakamoto, S. (2008). Bitcoin: A peer-to-peer electronic cash system.

RiskMetrics Group. (1996). RiskMetrics technical document.

Seth, T., & Blakeway, A. (2024, September 11). European gas, LNG prices struggle to find a floor as funds shift position. S&P Global Commodity Insights.

Disclaimer: All views expressed in this article are strictly my own and do not reflect the views, opinions, policies, or positions of my employer, past employers, or any organization with which I am affiliated. Nothing in this article should be interpreted as professional, investment, legal, or financial advice.

This is perhaps an outlier to all my other posts in the QuantSimplified series — one that does not have anything to do with quantitative methods or theory, but a personal reflection (in fact I was contemplating whether to include this or not). A lot had happened in my life at the point of writing: not only have I officially graduated from my Master’s degree in quantitative finance, I have also hit my 1st year milestone as a quantitative analyst. I felt motivated to write about some reflections I had over the past one year whilst on the job. This will be a relatively short article (I promised!), and the following is some key lessons.

The dumbest person in the room

The title of this article might have raised some eyebrows, especially since it kinda went against conventional impressions. Financial engineers were recognised as the “rocket scientists” of Wall Street, and many may regard quants as a group of people who are so well-educated and have attained too high an academic qualification to be called the dumbest person in the room.

This title was inspired by a fellow quant who constantly reminded me that humility is the essence of being a quant — a quant/ scientist is someone who consciously recognises that there are always more brilliant things/ people out there, and knowledge is a constant pursuit. No one is ever too smart, and a real quant always recognises that one can always be better, but never the best. This doctrine is perhaps one of my biggest takeaways in my first year being a quantitative analyst, one that keeps me grounded and constantly stay curious and work hard.

Equally important, in practice there are indeed situations that a quantitative analyst may be the dumbest person in the room. When solving certain problems, a quantitative analyst often starts from zero ground — one may be totally new to the domain of the problem and may not have the required contextual knowledge. In such cases, a quantitative analyst does begin as the dumbest person in the room filled with experts that have the required subject matter expertise.

In such situations, it is essential for quantitative analysts to ask the right questions, and here comes the second learning.

Dumb, but not dumb

When one thinks of a quant, one may imagine a person sitting in front of a computer the whole day facing lines of codes in fancy colors. Well, that is probably accurate, but that is not all to the work of quants. Quants cannot remain quiet and be isolated in a corner. Like in all other roles, communication is key.

Quants are problem solvers; and with all problem solving it requires active listening and prompting right questions. There are many situations in which the problem may be unprecedented, and each stakeholder may hold some key information that could shed light on the potential solutions. Quants being the owners of tools that could qualify as solutions would need to communicate and prompt the right questions in pursuit of that solution. This requires superior problem-framing skills that give one clarity to the essence of the problem, coupled with strong communication in order to prompt the key information leading to the solution.

Apart from communicating to facilitate problem solving, quants need to be the sales person for their solutions. Many solutions are made of advanced mathematical models that not everyone is trained to understand intuitively. In order to keep things moving, quants need to be able to simplify the explanation such that key stakeholders are able to follow the essence of the solutions — techniques like analogies are effective. It is important also to know what is essentially critical and what can, or must not, be reduced. As a quote famously attributed to Albert Einstein reads, “Everything should be made as simple as possible, but no simpler”.

Limitations are more important than solutions

As a matter of fact, in most situations a quant faces there is no perfect solutions. Mathematics, being a man-made tool, ultimately attempts to describe a much more complex and unpredictable world that we live in. Many models and solutions are but an approximation given the best information one has at any point in time. It is important for one to acknowledge that one is wrong; the key is to not be too wrong.

Henceforth, understanding the limitation to a solution is more important than the solution itself. In the same spirit, a more complicated or over-engineered solution or structure may not necessarily be the best, as the more complicated a solution is the more limitations one can expect. Important concepts like principles of parsimony and Pareto’s principles that one learns from econometrics and economics really come into play.

Being owners of the solutions, it is a quant’s responsibilities to always bear in mind and communicate the limitations of a solution, and to stop when progress results in more limitations than the benefits the solution may yield.

I hope that in this simple reflection post one may find some points that resonate well and provide some affirmations and suggestions on the challenges facing quantitative analyst regardless of background or industry. The key message of this post is to highlight the following:

1. Despite being a numerical work, never forget to work on communication skills

2. For aspiring quants — do not only focus on the fanciness of models/ solutions, but be aware and have one’s eyes peeled on the limitations of each model

3. Most importantly, be humble and stay curious

For more content on how to be a successful quant, please check out Igor Tulchinsky’s Finding Alpha, Chapter 32 on “The Seven Habits of Highly Successful Quants”.

Thanks for reading, and looking forward to feedback and opinions! :)

I gotta admit, that writing this series has gotten tougher by the day for various reasons — including

1. An ever increasingly hectic routine as life goes on, having to juggle work and study concurrently (side note: as of the time of writing I’ve completed my Master’s program, finally!),

2. Procrastination, and

3. Most importantly, the fear of being wrong

The last point is perhaps the most striking of all, a phenomenon more famously known as the Dunning-Kruger effect; in the journey of learning, one becomes less confident of one’s competence as one accumulates knowledge in the subject matter.

However, by remembering the initial intention of this article series, I managed to overcome this writing paralysis. The idea of the ‘QuantSimplified’ series was never to be academically rigorous — it was meant to be the opposite. It aims to provide readers with intuitions of difficult but important topics in quantitative finance, without getting caught up with too much technical jargon or complex concepts. The goal is to make things simple and factual, without making them simpler than necessary.

As I work to make this series better, I appreciate all feedback and questions as always. Please enjoy!

An oracle?

Perhaps one of the (if not the) biggest misnomers in financial jargon is the term ‘futures price’. Every day, one can easily find references to a certain future price whenever a news article reports the daily development of a market — statements such as “stock futures dip after latest inflation…”, or “crude oil futures climbed…” were common in daily reports.

But what exactly is a “futures price”? Briefly, a futures contract is a financial instrument (derivative) that binds the parties entering the contract into a future transaction. It is almost identical to a forward contract, which has a rigorous definition as follows:

“A forward contract is an agreement between two counterparties, a buyer and seller, to transact a specific underlying asset (can be physical or financial) at a future date (expiration date) at a pre-agreed upon price (strike price). This contract is an obligation to both parties.”

A future is simply an exchange-traded version of a forward contract, that theoretically comes with lower counterparty risks due to central clearing.

Because of the exchange-traded nature of futures contracts, it is a convenient tool to report up-to-date market sentiment. To elaborate,

• Most major futures (say S&P500, EURO STOXX 50, Brent Crude, Gold) are available for trading almost ‘round-the-clock, 5 days a week; the trading activities of futures — especially during the night hours of the markets they represent — would offer insights to how global participants react to a given news

• Whilst the main indicators of equity markets are the respective equity indices, real-time update of index prices are arguably less accessible than futures prices; futures prices are oftentimes published (at a 15-minute delay) on various platforms and exchange websites, and indices are often hidden behind paywalls. Thus, a more convenient alternative would be preferred.

• Some asset classes, specifically commodities, do not have observable spot prices; futures prices become the beacon that participants use to track the prices of these commodities (in fact, unlike other asset classes, the spot price of a commodity can be said to be a derivative of the futures price).

• Lastly, the trading volumes behind these major futures are substantial; looking at the table below, the dollar (notional) value of the average daily trading activities of some key contracts are in billions of US dollars; these futures markets are so liquid that they are arguably more “forward-looking” than the index prices

Given these reasons, most news outlets report futures prices as a convenient indicator of the most up-to-date market sentiment.

Now that we have established the definition of “futures price” and its common application in news reports, it’s time to address the misnomer — do futures prices have predictability of the prices in future?

The anatomy of futures

In the earlier discussion, we have conveniently left out a detail on futures prices; that there exists not only one but many futures prices for each underlying asset at any given time!

Recall from the definition that the parties entering a futures contract are obligated to transact at a given future date. Theoretically, there can be an infinite number of contract expiries! Practically, the futures expiration dates are determined by the listing exchanges using standardized rules (e.g. the third Friday of the third month in a quarter for most equity index futures) and there is a set number of contracts that will be listed (e.g. up to 72 consecutive months for energy contracts). The objective of standardizing the expiries and limiting the number of available contracts is to prevent the dilution of liquidity. It is a balance between meeting market participants’ hedging requirements and ensuring liquidity for price transparency.

By plotting all the available futures prices, one can derive the forward curve of the S&P 500, as shown in the figure below. The contract whose expiry is closest to the present is known as the front/ prompt month contract and the one whose prices are reported in the news.

If the front-month contract is representative of the current price of the underlying asset, are the further-dated futures representative of the future price of the asset? Are the far-dated futures prices a forecast of the asset price?

Too good to be true.

As some may already expect, when it sounds too good to be true, it is likely untrue.

It is human nature to seek certainty, and it is tempting to associate futures prices with price forecasts. And it doesn’t help with the misnomer.

A futures price works very badly as a price forecast because (in my opinion) its fundamental design is not of a forecast but of a trade-off.

When entering into a futures contract, the key problem one (the hedger) is attempting to solve is to trade a future uncertainty for a certainty today. The hedger may not want to take a chance in the future (due to personal outlook or other business practicalities) and would prefer to lock in a price today. Taking the other side of the hedger would be the futures market maker, who then faces an opposite trade-off — the market maker is forgoing certainty today to take on uncertainty in the future. Because there is no free lunch in the world, the market maker would demand a risk premium — an extra for providing the service to the hedgers. As we describe the problem, it has always been about pricing the trade-off, and not about forecasting the price trend. Even if the need for hedging came from an outlook, the forecast is not priced into the futures contract. Well, it is true that if the overall market is negative, there will be more short interests thus bringing down the futures price. But the magnitude of the drop was due to the increase in the risk premium, and not the forecasted magnitude of the negative price trend. These two may not be the same thing, due to market frictions, liquidity (or the lack thereof), information asymmetry, etc.

But more importantly, a futures price offers a point, not a set of probable prices. The futures price constantly updates itself given new market information of the trade-off. Eventually, as the time of expiry goes to zero, the futures price would converge to the spot price. This convergence is natural, but this does not mean that the futures price is a good forecast. It simply moves with the spot price and constantly reacts to new information. In fact in history, a three-month futures contract at any given point in time was never successful (except for some instances by chance) in predicting the price three months later.

Future comes in many shapes and sizes

Now that we understand that the futures prices are essentially pricing of trade-offs, the next question would be if all the forward curves should look the same.

The short answer is — not quite. It is tempting to think that because one locks in a price today for a future transaction, the time value of money should come into play, making future prices higher than spot prices. This is the case for many equity index futures due to replication. A futures contract can be replicated using a combination of stocks and savings accounts.

Consider a stylized example, in which a market maker goes into a long futures contract of a stock, expiring 3 months later. As a long holder, one is obligated to purchase the underlying stock in exactly 3 months from the counterparty. To do that, the market maker can first borrow the stock, short it, get the proceeds, and save it in the bank account. 3 months later, the market maker can then use the money one saved in the bank account to purchase the stock through the contract and return the stock to whomever one borrowed from. The fair value of this futures contract is thus the current price of the stock multiplied by the accumulation one can expect from saving the amount in a risk-free bank account. The same argument also holds for a market maker going short. In this example, we can expect that the futures price is always higher than the spot price as long as the interest rate offered by the savings account is positive. This is illustrated in the first plot in the figure below, where the forward curve is upward-sloping (contango).

From the same figure, we can see that it is not always the case that the forward curve will be upward-sloping. It could also be downward sloping (backwardat-ed), or exhibiting some kind of seasonality. To put it simply, the shape of the forward curve is driven by the economics of the underlying assets.

Take Brent crude futures as a case in point: a few reasons why the crude futures exhibit a backwardation are:

1. Cost of storage: for a short futures holder to meet one’s obligation in the future delivery, the short holder needs to secure to supply today and keep it for the delivery date. Unlike financial assets that can be easily bought and sold, physical barrels are harder to procure, and forward planning is required. Because storing supplies incurs costs, these costs will be priced into the futures contract and be passed to the buyer.

2. Hedging pressure: In addition to the cost of storage, in some situations, there are more short hedgers than long hedgers. This is usually the case in oil contracts, as producers are more vulnerable to price uncertainty than the ultimate consumers. Consumers can easily change their consumption to react to price changes, but producers have less flexibility in managing their production. Hence, there are likely to be more shorts than longs. Due to this imbalance, market makers would demand more premium to take the other side of the trade, thus bringing the futures prices lower.

Lastly, the long-short imbalances may not be monotonic across expiries, as seen in a natural gas contract. Sometimes, the imbalance itself may exhibit seasonality. In the case of natural gas, there may be more end-customer demand during winter due to a greater need for heating than there is during summer. There is a greater need for consumers to hedge the winter contracts (as prices are expected to be higher and consumers would want to protect against undesirable spikes in prices), resulting in a local contango. Conversely, as demand in summer is generally weaker, there would be relatively more hedging demand from suppliers to lock in summer prices ahead of time.

It is important to note that sometimes “seasonality” can manifest even without imbalances. For instance, some equity index futures may exhibit discontinuities due to corporate actions, such as expected dividend events. The chart below which shows the forward curve of the EURO STOXX 50 is a good example.

Making the best use of a tool — some forewarnings

Even though futures prices are not a good forecast of prices in the future, from the above we can conclude that futures prices are extremely helpful in the following manner

• The front-month futures price is a good proxy to “nowcast” the market, particularly when the underlying spot market is not available — e.g. Eurex Asian trading hours provide good information on market reaction to overnight news when the European cash markets are not open

• The forward curve structure reveals the economics of the underlying asset it seeks to track; comparing forward curves across time provides insights into market sentiments and any changes in the overall outlook of the short-term future relative to today (this is not the same as forecasting)

That being said, as with all types of data, one needs to be cautious of the potential downsides. With regards to the forward curve, apart from paying attention to the price, one needs to question the reliability of the prints. To determine how reliable a print is, two key measures are 1) trading volume, and 2) open interest. The more trading activity there is in a contract, the more reflective the price is of market sentiment. From the chart below, one can see that the trading volume and open interest decay exponentially the further away the expiry is (also to note, the trading volume and open interest are expressed in logarithmic scale).

All in all, despite this being a foundational concept which oftentimes overlooked by many, futures prices and the forward curve structure can offer plenty of information if one manages the information cautiously. It does pay dividends to have a good appreciation of forward and futures pricing, before diving into more complicated derivatives concepts.

I hope this article achieved its goal of providing a good primer into futures pricing and addressing some misconceptions one may have. Thanks for reading!

References:

• CME E-Mini S&P 500 futures: https://www.cmegroup.com/markets/equities/sp/e-mini-sandp500.html

• CME Gold futures: https://www.cmegroup.com/markets/metals/precious/gold.html

• CME Natural gas futures: https://www.cmegroup.com/markets/energy/natural-gas/natural-gas.html

• Eurex EURO STOXX 50 futures: https://www.eurex.com/ex-en/markets/idx/stx/euro-stoxx-50-derivatives/blc/EURO-STOXX-50-Index-Futures-160088

• Intercontinental Exchange Brent Crude futures: https://www.ice.com/products/219/Brent-Crude-Futures

• Korea Exchange KOSPI 200 futures: https://global.krx.co.kr/contents/GLB/02/0201/0201040201/GLB0201040201.jsp

Unlike previous articles in the Quant Simplified series, which largely dived into the mathematics and theories of Quantitative Finance, this article focuses on a practical aspect of the equity market and attempts to analyze index rebalancing with quantitative methods.

The motivations for this article are

1. To introduce the concept of an equity index to beginners

2. To evaluate if an index rebalancing arbitrage exists

3. To explain some observations using ex-post (after-the-facts) analysis

This article is inspired by a project done in one of the courses under Singapore Management University’s Masters in Quantitative Finance program.

Equity Indices — The Multi-Trillion Dollar Fruit Baskets

An index acts as an indicator of the overall performance of the investable universe it represents. Most notably, broad-based equity indices such as the S&P 500, FTSE 100, and the STOXX Europe 600 serve as a measure of market growth in the US, United Kingdom, and Continental Europe, respectively. While indices are not limited to equities as an asset class (there are also indices based on commodities and credit), our analysis focuses on broad-based equity indices only.

Apart from serving as an indicator of equity market growth, equity indices also play a crucial role as benchmarks for index funds. There are two primary types of index funds: a passive fund that aims to closely replicate the index and an actively managed fund where the fund manager endeavors to beat the benchmark. According to an article from VettaFi, the combined Asset Under Management (AUM) of the top 10 equity index ETFs total up to USD 2.3 trillion (VettaFi, 2023). This substantial AUM pertains only to passive ETFs, with more existing in the active fund sphere. Consequently, any alterations made to the benchmark index could result in significant movements of capital, potentially creating tradable opportunities.

Due to its crucial role in the world of investing, index providers must ensure that index construction is clear, concise, and replicable. To create an index, one must first define its objective — what exposure investors would achieve by replicating the index — and its methodology. The methodology should address the following:

1. Eligibility — determining the universe of stocks considered in the index based on its objective.

2. Construction — establishing a systematic process that dictates which stocks within the eligible universe enter or exit the index to fulfill its objective.

3. Calculation — employing a methodical approach to derive the index’s value, accounting for corporate actions and adjustments.

4. Maintenance — implementing a standard operational procedure to safeguard the index’s integrity, ensuring it consistently meets its objective over time.

Rebalancing — A Reshuffle of Hands

To maintain the index, index providers need to incorporate within the methodology a process to assess whether the current set of stocks in the basket still aligns with the index objectives. If not, adjustments to the constituents are necessary. This process is known as index rebalancing.

Index rebalancing occurs at predetermined intervals, often annually, semi-annually, or quarterly. Rebalancing frequencies higher than quarterly are rare due to the increased costs associated with frequent changes, making index maintenance expensive. During rebalancing, significant amounts of money can be expected to shift as passive fund managers sell off stocks excluded from the index and purchase newly included stocks to mirror the index composition. The act of rebalancing generates buying and selling pressures on newly included and excluded stocks, respectively.

Since rebalancing is essentially a compulsory action for passive fund managers to ensure accurate tracking of the index, it can create ‘arbitrage’ opportunities. Stocks anticipated to enter the index may experience price appreciation during the buying process, causing upward price momentum. Conversely, stocks expected to exit the index may undergo depreciation when the selling occurs.

Capturing the elusive — A study on FTSE UK Index Universe

In our attempt to evaluate the feasibility of capturing arbitrage opportunities during rebalancing, we analyzed the FTSE UK Index universe, i.e. the FTSE 100 Index and the FTSE 250 Index.

Our selection of these indices is guided by specific criteria:

1. Simple and objective index methodology — We focused solely on indices with a fixed number of constituents and straightforward quantitative inclusion criteria. Indices incorporating subjective factors like ESG considerations were excluded due to their subjective nature.

2. Transparency — Preference was given to indices where data regarding the index and its constituents are publicly accessible, and obtainable at relatively low costs, ensuring transparency in the analysis process.

3. ETF AUM — e chosen indices needed to have a sufficiently substantial AUM in ETFs to guarantee sufficient liquidity and active trading.

Being the sixth biggest country by GDP, the United Kingdom makes a good target for the index rebalancing study. With a developed equity market, the FTSE 100 ETFs have collectively about 30 yards (20 bn EUR) in AUM, while FTSE 250 (mid-cap) ETFs have approx 3 yards (JustETF.com, 2023).

The data required can also be obtained easily. All data are accessed via Refinitiv Datastream Python API through the WRDS subscription.

Our Hypothesis

In this study, our hypothesis posits the presence of statistically significant non-zero returns on the target indices during rebalancing. As rebalancing may result in a shuffling between the FTSE 100 and FTSE 250 indexes, an exclusion from the FTSE 100 index may mean an inclusion into the FTSE 250 index. Consequently, selling pressure to reduce exposure in FTSE 100 funds is countered by buying pressure to increase exposure in FTSE 250 funds. To explore this phenomenon, we conduct a two-tailed test, assuming a null hypothesis of a zero average return, without specifying the direction of the alternative hypothesis.

This study analyzes a total of 42 rebalancing periods spanning a decade from 2013 to Q2 2023, encompassing 488 unique instances. Additionally, to ensure that observed returns are attributed solely to rebalancing activities rather than other events (such as dividends, trading suspensions, or delistings), we implement the following exclusions.

1. Stocks that were impacted by delisting or extraordinary events (e.g. trading suspensions)

2. Stocks that have corporate actions around the rebalancing period

3. Stocks that the team has not been able to get reliable price data on (delisted stocks or stocks that have ambiguous/ multi-venue listing)

4. Due to the rarity of the event, we do not consider stocks that enter FTSE 100 out of nowhere (only 2 occurrences in the entire history)

The FTSE 100 and FTSE 250 historic additions and deletions are based on published files that can be found on the LSEG website.

Specific to the FTSE UK index series (FTSE Russell, 2023), the inclusion or exclusion of stocks is determined by their free-float market capitalization at the rank date. This date is defined as the Tuesday falling within the week of the 1st Friday of the rebalancing month, approximately T-14 trading days. The FTSE UK series follows a quarterly rebalancing schedule, occurring in March, June, September, and December. The actual rebalancing occurs immediately after the third Friday of the rebalancing month, coinciding with the expiry of the index futures. This date is referred to as the ‘rebal date’.

Our analysis revolves around a set of Ex-ante (pre-event) and Ex-post (post-event) dates. We aim to determine the optimal timing for entering a position to yield significant non-zero returns.

Our results — the expected and unexpected

We categorized the rebalancing into 4 distinct cases

1. Dropping out from FTSE 100 into FTSE 250 (FTSE100_FTSE250)

2. Entering FTSE 100 from FTSE 250 (FTSE250_FTSE100)

3. Dropping out from FTSE 250 into small caps (drop_FTSE250)

4. Entering FTSE 250 (new_FTSE 250)

Our statistical analysis revealed statistically significant non-zero returns in each case, albeit occurring at different timings. The signs of the average returns mostly align with expectations. Typically, average returns tend to be positive in response to perceived good news and negative in reaction to perceived bad news. However, there’s an exception observed in the case of new entrants into FTSE 250. This deviation from the expected trend might be explained by investor risk aversion.

Dropping out from FTSE 100 into FTSE 250

In this scenario, we observed statistically significant negative average returns from 20 trading days before the rebal date to after rebalancing. This outcome aligns with our expectations, considering that stocks dropping from a large-cap to a mid-cap index might be perceived as unfavorable news. One possible explanation for the observed negative returns is linked to risk-averse investor behavior. When a stock is potentially facing exclusion from the index due to its market cap before the rank date, investors are prone to initiating sell-offs, anticipating the impending negative news.

Entering FTSE 100 fromFTSE 250

In contrast to the previous case, we observed positive average returns after the rebalancing in this scenario. However, these positive returns were evident only when compared to prices from 1 day before the rebal date. One potential explanation could be linked to risk-averse investor behavior, where investors might display less sensitivity or hesitancy in reacting to perceived good news. They might be inclined to act cautiously, preferring certainty rather than taking chances on an increase in stock prices prematurely.

Dropping out from FTSE 250 into small caps

Similar to the first case, we observed sharp negative returns on stocks from 20 trading days before the rebal date. The transition from mid-cap (FTSE 250) to small-cap is particularly negatively perceived, given that small-cap companies are often seen as less established and inherently riskier assets.

Due to this perception, risk-averse investors tend to act decisively and at an earlier stage, offloading stocks expected to exit the FTSE 250 universe. Moreover, the AUM of FTSE small-cap funds is notably lower compared to FTSE 250 funds. Unlike FTSE 100 dropouts, where both buying and selling pressures exist, dropouts from the FTSE 250 face substantial selling pressures with limited demand for buying.

An intriguing observation is that despite the initial sharp decline, there are some, albeit less significant, positive returns when trading 1 day before the rebal date up until 20 days post-rebalancing. This could be attributed to an exaggerated reaction to the news of the dropout, causing these stocks to potentially be undervalued. Consequently, some investors might seize the opportunity to buy, resulting in marginal buying pressure, although not as substantial as the sell-off.

New entrants into FTSE 250

Arguably the most counterintuitive among the cases, new entrants into the FTSE 250 exhibited, on average, negative returns when traded from 1 day before the rebal date until 10 days post-rebalancing, despite the ostensibly positive news.

We theorize that even though the inclusion into the FTSE 250 index from the small-cap category might be perceived as favorable news, speculators were likely capitalizing on these smaller-cap stocks to realize their profits. Risk-averse investors typically exhibit a cautious response to positive news, leading to a delayed onset of buying activity compared to the sell-off. Consequently, this led to an initial decrease in value, followed by a return to initial levels much later.

Arbitrage — gotta catch ’em all?

The feasibility of the opportunities is largely dependent on the cost of and ability to capture the arbitrage. One key assumption is that traders can freely buy and short-sell stocks and that the costs incurred are negligible compared to the returns. However, these assumptions do not always hold.

In some instances, short-selling may not be feasible. Short-selling a stock requires locating a willing lender for the stock. While this might be feasible for large-cap stocks, it might prove challenging for small-cap ones. Even when possible, the cost of short-borrowing tends to be inversely correlated with stock availability. The scarcer the stock for short-borrowing, the higher the associated borrowing cost. Consequently, this limitation might impede the feasibility of capitalizing on certain opportunities, especially those reliant on short sales.

All in all, the substantial influence of equity indices on investments and the vast capital invested in index funds can generate trading prospects through changes in index constituents, notably during index rebalancing.

While the price momentum due to index rebalancing is largely in line with expectations, the timing is pretty much impacted by investors’ risk aversion. Our study on the FTSE UK Series emphasizes the critical role of behavioral finance in investment dynamics.

Moreover, even when trading prospects arise, it’s essential to consider the ability and cost of seizing these opportunities to determine their feasibility. Varied indices offer distinct opportunities, making the search for such prospects an exhilarating and dynamically evolving pursuit.

Disclaimer: This article is purely didactical and should not be construed as investment advice. The content of this study is academic in nature and has been written by the author as a graduate student enrolled in Singapore Management University’s Masters in Quantitative Finance program. The views expressed in this article do not represent those of the author’s employing company. All analyses are based on publicly available information.

For a tutorial on Refinitiv Datastream API, go to: https://github.com/Finsinyur/QF603_QAFM_Project/blob/main/notebooks/00_WRDS_tutorial_refinitiv_quant_analytics.ipynb

References

[1] FTSE 100 ETFs, JustETF.com

[2] FTSE 100 Historic Additions and Deletions, FTSE Russell

[3] FTSE 250 ETFs, JustETF.com

[4] FTSE 250 Historic Additions and Deletions, FTSE Russell

[5] Guide to Calculation Methods for the FTSE UK Index Series, FTSE Russell

[6] Largest ETFs: Top 100 ETFs By Assets, VettaFi

One thing I found out in my pursuit of quantitative finance is that being an extremely multi-dimensional discipline, there seem to be endless things to learn. Not only are there countless topics to catch up on, but those topics are highly varied in difficulties. Nothing seems to be too trivial to ignore.

But in such a highly sought-after discipline where current discussions surround the most advanced techniques (unsupervised learning and causal inference are nowadays the darlings in the field), there’s so much franticness to chase after the hype. The fever has inadvertently caused new learners to become anxious and overwhelmed in their learning.

My motivation for creating content for the Quant Simplified series is to provide beginners with some avenues to read and think about the more basic topics, in a manner as relatable as possible, to help bridge the gaps between beginners and the more advanced topics. It is also to add some intuitions to those topics and demystify those scary math formulae for everyone.

Going back to one of the more basic concepts, inspired by a post by Florian Campuzan on LinkedIn which I engaged in one of my sleepless nights, this post shall explore some basic root searching algorithms — the Bisection Method and the Newton-Raphson (aka Newton’s) Method.

Consider a general function where the dependent variable y is a function of an independent variable x, which we denote as y = f(x). The root solution is the value of x in which y = 0.

Root searching algorithms are a set of systematic rules, written in codes, that seek to solve for the root with a given function y.

A Number Guessing Game — Bisection Method

When I was much younger, I used to play a simple game with a group of friends. It’s not exactly fancy, but because it has some elements of luck and gambling (not sure if there’s a high correlation between the love of options and the love of gambling :-) ), it was rather entertaining.

The game works like this: one player will play the “banker” and write down a number on a paper, with no one else knowing of that number. This number is typically a whole number (i.e. no decimal) and is within a range of numbers, say between 1–100. The other players (could be one or more) would need to guess what that number is, and one will win if one made a spot-on guess. Otherwise, if the number is wrong, that number will become the new limit, replacing the upper or lower bound so long as the actual number is within the bounds, narrowing the range the next player needs to pick from and increasing the winning odds. In this game, the guessers are given a limited number of turns, and if no correct guess is made by the end of the round, the banker wins.

This is essentially how the Bisection Method works, with us the users being the “banker” and the algorithm being the “player”. The number that the player is seeking to find is known as the “root”, the solution to an equation.

The only differences here are:

1. As bankers, we don’t know what this number is, we only know the function that we need to solve for

2. The upper and lower bounds will need to be carefully thought through and provided, and the algorithm will always guess the center value within the boundaries until it finds the solution

3. The banker is not guessing a whole number, the solution can have decimals

On point 1: one may think that if we already have the function, we could solve for the root analytically and not go through such a rigorous game. In some situations this is true; we would not want to use a root searching algorithm on a trivial problem like a quadratic equation. However, there are situations (which we will touch upon briefly later) that there is no analytical solution, i.e. formula, that could help us compute for the roots. Thus numerical approaches such as the Bisection Method are needed.

On point 2: In order to find a root solution, we need to provide a set of boundaries (X_lower and X_upper) that, with 100% certainty, encapsulate at least one root solution. The only way to ensure this is to choose the boundaries such that

Meaning that the outputs from the function given the upper limit x_upper and the lower limit x_lower are opposite in sign. This makes sense. If the output is given the upper limit X_upper is more than 0 and the output given the lower limit X_lower is less than 0, there would exist an X that gives an output that is zero.

One needs to note that oftentimes functions have multiple root solutions. This means that the initial boundaries defined matter. Different boundary conditions may give rise to different root solutions if there is more than one root to begin with. So, if we only use the Bisection method, multiple iterations would be required with different bounds to get the desired solutions.

On point 3: This is a crucial point to note and makes all the difference. Because a root solution may not necessarily be a whole number (and more likely than not the solution has decimals), there are infinite numbers to guess from within the boundaries. How does the algorithm deal with this?

Numerical solutions often do not provide an exact solution but are more of an approximated solution. We as users need to provide a tolerance in accuracy — that we are willing to forfeit to trade off for speed. If we look for pinpoint precision, we may be spending disproportionally more time for a marginal improvement in accuracy. Of course, with modern computers, we could afford to set a fairly strict tolerance, say one of a million (10 to the power of -6), without burning up the PC.

Unfortunately, the Bisection Method has its flaws. The most obvious is that it is algorithmically inefficient, and its time performance is not the best. It is important to have more algorithms in the toolkit.

Tuning the Radio — Newton’s Method

One thing that comes to my mind when I think of Newton’s method is the tuning for a radio channel. Well, it’s 2023, and I’m sure the younger crowd here does not know what a radio is, so here’s an image of one for reference. The below analogy also applies to tuning a guitar or zooming a camera lens.

Imagine you have a radio with a knot (the knot beside the word ‘TUNE’ on the above image) that allows you to tune to your favorite channel. In the beginning, you will start at a random position (let’s call this position X0), and you will get feedback from the radio. Usually, it will start with a fuzzy static noise. One would try to turn the knot in the direction that reduces the static noises. Sometimes one may overtune and the static noise may increase again. Then one would turn the knot in the other direction and repeat the process until the feedback from the radio is clear (or to be specific, sufficiently clear based on our audible range).

That’s essentially what the Newton-Raphson formula does mathematically. And you may ask, how so?

The Newton-Raphson Method can be described using the below equation

Essentially, the position of the knot is our dependent variable X. The sound we hear from the radio (i.e. the feedback) is f(X). Our desired channel is f(X_solution) which is equal to 0.

We start with an initial position of X0, and we get the initial feedback f(X0). depending on how loud the static noise is, i.e. how far f(X0) is away from 0, we will move in the direction where the static noise becomes softer, i.e. f(X1) is closer to 0 than f(X0). How much should we tune? It depends on 2 things: 1) If f(X0) is so far away from 0, i.e. the static noise is so loud, we will make a bigger turn on the knot. 2) If a small turn doesn’t reduce too much of the static sound, then we turn the knot much faster. Conversely, if a small turn reduces the static noise tremendously, we will turn the knot slower. This is described by f’(X) in the denominator, which parallels the change in static noise with respect to the turning of the knot.

The negative sign ensures that the algorithm always progresses in the direction of the root solution, i.e. against the direction of increasing errors. Similar to the Bisection method, we would need to apply some tolerances. This is equivalent to our audible range. Our end result is that we do not get a 100% static-free sound, but it is virtually static-free because it’s so marginal that we don’t get affected by it.

One drawback of Newton’s method is that it relies on the derivative of function f(X), i.e. f’(X). Sometimes, this cannot be easily gotten. So here comes a modification of the Newton’s method, namely the Secant Method:

This makes use of the first-order finite difference approximation for the first derivative of the function f(X), as shown below.

The reason why the Secant Method is brought up here is that, together with the Bisection Method (and another ingredient — the inverse quadratic interpolation), the three algorithms make up Brent’s method. Brent’s method is very commonly used due to its rapid convergence, amongst many other benefits, thanks to the synergies among the three building blocks.

Understanding the root of the problem

As addressed briefly earlier, the reason why we implement root search algorithms is that many problems do not have an analytical solution. Such problems are extremely common in finance.

For starters, the pricing formula for a coupon-paying bond is a prime example. The bond pricing formula expresses the price of the bond as a function of the yield-to-maturity, the coupon payout, and the time of payouts and maturity in a closed-form equation. However, for any bond with more than one payout in its lifetime, we could not solve for its yield-to-maturity because we cannot express it as a function of the other variables, i.e. inverting the bond pricing formula.

Another popular formula that cannot be inverted is the Black-Scholes-Merton equation. In option pricing, finding the implied volatility is extremely important as it is the only non-observable variable in the equation (therefore implied — you may refer to my first article of the series for more info on implied volatility). The root search algorithms come in handy for such situations.

Bigger than it seems

Apart from finding yield-to-maturity from a bond pricing formula and implied volatility in the Black-Scholes-Merton model, there are many other situations in finance that we will need numerical methods for. These algorithms are not difficult, but they can be a part of the building blocks of more efficient and advanced algorithms. It is important to build some intuitions for these algorithms as a first step to challenging the more sophisticated models.

I hope you find the analogies in this article helpful in simplifying the root search algorithms. While analogies help to build relatable intuitions, they are still far from perfect. There are some complexities that cannot be accurately explained through simplification. I highly recommend the book “A Primer for the Mathematics of Financial Engineering” by Dan Stefanica, which compiles the most important math for quantitative finance and provides sufficient depth for serious learners. Please do like and share this article if you find it helpful. Thank you!

I always believed that to really understand a subject, one needs to be able to explain said topic in the simplest manner possible. This is the key motivation for me to come up with the QuantSimplified series. This series seeks to achieve 2 objectives — one is to challenge me to really master some fundamental concepts in quantitative finance, and the second is to help aspiring quants pick up some intuitions on these concepts. But truth be told, this is by no means an easy undertaking, and hence it took me about 4 months to think of a second topic to write about after my first.

The topic of this post was inspired by a conversation that I had a couple of months back. The conversation was about quants and the role of quants in communicating complicated concepts to the general layperson. I was asked to explain ‘Ito’s Lemma’ simply, and how it was used in trading and option pricing.

I think the answer I gave back then was decent, but it could do better. So I spent some spare time in the next couple of months building a story that could draw links between the theory and the practice. And before we touch the lemma, we need to first discuss ‘Stochastic processes’.

“Stochastic” is a difficult word

I gotta admit, when I first started my quant finance journey, stochastic calculus has always been one of the more intimidating topics. To begin with, it sounds difficult — and is difficult to pronounce.

But its meaning is surprisingly simple; “stochasticity” means randomness. When it comes to randomness, it leads to a core topic — probability.

Consider the following random game: you are given a fair coin, and you are asked to toss it. At every toss, if you get a ‘head’, you assign a score of ‘1’ to your score sheet. Otherwise, if you get a ‘tail’, you assign a score of ‘-1’ instead. You are asked to play this for many rounds and you are to accumulate the points as you progress into the game.

To introduce some terms,

• Sample space: it is a set of possible outcomes of the random game. In our case, it’s ‘Head’ (‘H’) or ‘Tail’ (‘T’)

• Random variable: it is a numerical value that is assigned to an outcome. Here, the random variable takes a discrete value of either -1 or 1. The score that we earn is our random variable

If we only played one turn, our sample space will be limited to 2 outcomes, and our random variable will be restricted to 2 possible values. The more rounds we play, the sample space grows as the outcome will need to consider a sequence of outcomes (e.g. {‘H’, ‘T’, ‘H’, ‘H’}) and the random variable would also expand to take in more possible values.

If we record our cumulative score at each turn on a sheet, that series of score records can be said to follow a stochastic process. In essence, a stochastic process is a time series of random variables. To illustrate a stochastic process, let us simulate the game and plot our cumulative score after 10 rounds.

Here it is, and oof I do have really bad luck here! Let’s see how this looks after playing for a thousand turns.

That is better, I managed to get a positive score! But wait… ignoring the scale of the score itself, doesn’t this plot look like how a stock price would move?

Asset price dynamics follow a stochastic process

Just like our scores in the coin-tossing game, the price of an asset is a random variable, and its future price is uncertain. In quantitative finance, we model the asset prices in order to determine the value of its derivative instruments. But when it comes to modeling, there are a few problems to solve:

1. What assumption(s) do we make on the distribution of the random variables?

2. What are the value of the parameters of the distribution, after we solved problem 1?

The first problem is likely to be the most difficult one to solve. Unlike our coin-tossing game, in which we know the probability of each outcome, the true distribution of the asset price (or return) is unknown. We often need to make some sort of assumptions based on empirical observations. The most commonly used distribution is a normal distribution. When plotting historical asset returns in a histogram, we observed that the return distribution follows quite closely — but not exactly — a Gaussian distribution. There are other tweaks and variations to the Gaussian distribution, such as adding jumps, but those are for another day.

So how do we relate asset price to the asset return, if we were to model the asset return as a normal distribution? Here, we made use of a small math trick. Assuming that the asset return usually takes on a relatively small value, i.e. (<< 1), we can approximate the asset return as the logarithmic difference of asset price at time t and at time t-1, i.e.

With this approximation, we can now model the asset price as a “lognormal” distribution. In fact, a stochastic process based on a lognormal distribution is known as a “Geometric Brownian Motion”, and it takes a stochastic differential equation (SDE) as follows:

Risk-neutrality

Now that we have made the assumptions necessary to overcome problem 1, we need to deal with the second problem, to determine the mean and the variance of the lognormal distribution. We shall ignore the variance for now, and focus on the mean.

From the above equation, the stochastic process is made up of a deterministic drift and a stochastic component which is the product of the variance and the asset price and a Wiener process. A Wiener process W(T) follows a normal distribution with a mean of 0 and a variance of T. The deterministic drift describes the general direction of the price evolution.

The estimation of the drift is highly model dependent. Different models with highly differentiated assumptions would arrive at different values of the drift. It is highly unlikely that two fundamentally different models could give rise to the same drift. But more importantly, it is almost impossible to know if any model is spot on in its guess of the actual drift of the asset price, given that the true distribution of the asset price is not determinable.

For instance, investor A ran her model and is highly bullish on the asset and thinks that the drift would be at 5%. On the contrary, investor B’s model determined that the asset is going to depreciate, and thus the drift should be at -2%. There is no way one can know beforehand which model is correct. These models are working on ‘real-life’ probabilities, or what the industry calls the ‘P’-measure. If we cannot reconcile in the ‘P’-world, what could be an alternative?

For many assets, one can rely on risk-neutral pricing to determine the drift. Without going too much into the details, the key idea is that if we can hedge a portfolio perfectly, the expected value of the portfolio measured with respect to a reference asset would remain the same as the initial value of the portfolio w.r.t. that reference asset.

This reference asset has a unique name in quantitative finance, and it is known as the numeraire. The numeraire has to meet certain conditions; for one it must be a tradable asset, and it cannot have negative values.

As it turns out, to replicate and hedge a forward contract or an option contract, the numeraire is the money market account. Under this numeraire, the drift is simply the risk-free rate, denoted as ‘r’. Therefore this measure is known as the ‘risk-neutral measure’, and the industry calls it the ‘Q’-measure. Below shows the SDE of the asset price under the Q-measure.

Very quickly on the Ito’s Lemma: SDEs based on Geometric Brownian Motion cannot be easily solved as a Wiener process is nowhere differentiable and we can’t use the usual techniques that we apply on non-stochastic equations. Ito’s Lemma introduces a simple technique (based on Taylor’s series expansion) that allows us to solve such SDEs. This is crucial because we need the asset price dynamics for option pricing.

Option pricing = pricing of likelihood

The bulk of this article has been explaining what is a stochastic process and how we determine the parameters of the assumed distribution. We have not explained why we need to model the asset price as a stochastic process. The answer is rather simple — it is because of the non-linearity of the option contract payout.

Consider a forward contract. If one were to sell a forward contract on AAPL today (assume the share price is at $179 per lot), one could remain risk neutral by borrowing $179 from the money market to buy the share today to lock in today’s price. As the seller holds a short position on the money market, he would need to pay the cost of borrowing, let’s say 5.3% annualized. This additional cost needs to be added to the forward contract pricing. Likewise, if during the lifetime of the contract, there is an expected dividend payment of $0.24 per lot which would be paid to the seller who would hold the actual share, this dividend payment (or yield, q) would need to be deducted from the contract price. Because the payout of the forward contract is linear, i.e. for every dollar appreciation in the underlying stock the forward appreciates by a dollar, and likewise if the value of the underlying stock decrease by a dollar, we need not care about the variance of the asset price distribution, but only the mean. In fact, we could potentially derive the analytical solution of the forward price through intuition alone.

However, an option contract has a payout structure that favors the buyer over the seller. Take the call option as a case in point: a buyer of the call would gain a dollar for every dollar the underlying asset exceeds the pre-agreed price level (‘strike price’). On the other hand, if the underlying asset price falls below the strike, the buyer is not obliged to exercise the option and thus can simply let the option expires. This means that for underlying asset prices below the strike, the payout is zero.

Since the option payout structure disadvantages the seller, the seller would demand a premium on the option. To determine this premium amount, one needs to evaluate the probability of being exercised against — meaning the probability that the underlying price would exceed the strike price level. This is where the stochastic process comes in, to allow market makers to determine based on an assumed distribution the pricing of the option.

And this is how stochastic process and calculus are relevant in quantitative finance — it allows us to price derivatives that are non-linear. If one further expands on the equation below, one would arrive at the Black-Scholes option pricing equation. This also explains why there are cumulative distribution functions within the BS equation. We are truly working on determining the likelihood of being exercised against to price the premium.

The key is volatility

Now that we have covered the relevance of stochasticity in quantitative finance, we are now back with one last important point — volatility. We have not mentioned how one determines the variance (which is volatility-squared) to be used for the distribution. Unfortunately, there is no easy answer to this one.

Volatility is important, because the larger the volatility (i.e. the standard deviation), the greater the likelihood the underlying price exceeds the strike price. One would price the option premium higher under a more uncertain environment.

One common method used is to look at the market-implied volatility. If there is a liquid options market, demand and supply would help determine a price level that incorporates the market participants' expectations of volatility. By using the market option prices, one could work in reverse with the BS equation to determine the volatility used — therefore this is called the “implied” volatility. This implied volatility could then be used, and one can adjust based on one’s own expectation of the volatility.

The real world is not that straightforward

I think it is important that we appreciate the basic theory of option pricing, and how stochastic processes help make this possible. But there are still many issues in the real world that cannot be simplified.

Firstly, we made a bold assumption that the underlying asset price follows a lognormal distribution. We know for a fact that the asset return is actually not normal — it tends to be negatively skewed with fat tails. Our assumptions introduced some biases in our pricing models.

Secondly, although we could derive implied volatilities, what if there are no liquid option markets? How do we then determine volatility?

Lastly, what if the option cannot be hedged? For instance, volatility and weather are underlying assets that are intangible and cannot be bought or sold. How do we derive risk-neutral pricing for volatility and weather derivatives?

All these are crucial questions to ask because they are real and prominent in the industry. I hope this article has provided you with a sense of option pricing and the complicated math behind the theory so that you can ponder these difficult questions further.

Thank you for reading the second article of the series, and that you have gained some insights on the basic of option pricing in quantitative finance! For more primer on implied volatility, you may refer to my previous article “The many facets of Risk”. Please do comment to provide your feedback on how to make this series better. Thank you!

The many facets of Risk — Quant Simplified

To help you under the concept discussed here better, here is a brain teaser:

Consider our coin-tossing game again. If the rule of the game is that after tossing the coins for 50 turns if your cumulative score is greater than 6 (i.e. you have tossed 6 more heads than you did tails), I will pay you $5 + a dollar for every point exceeding 6. Ignoring the efforts needed to toss the coin (50 times can be very tiring I know), what is the minimum amount that you should pay to play the game?

I vaguely recalled early in my finance education that among the first topics covered was “risk”. And typically, volatility is often brought up as a key risk metric. When I first learned about risk and volatility, I took it for granted that the higher the volatility, the riskier an investment would be. It’s easy, it makes sense at first thought, and it’s convenient. And many thought the same.

As I gradually got more exposed to the world of derivatives and quantitative finance, more questions came. Risk is no longer that straightforward, and often contextual — the concept and measure of risk vary in different situations. As part of the Quant Simplified series, I hope to share more about risk and volatility and discuss the various context that gives risk a different meaning.

A quick recap — risk and volatility

Volatility, by definition, is a statistical measure of the dispersion of returns for a given asset. Mathematically, it’s referred to as the standard deviation σ and some may refer to the variance of the returns. Consider the daily return on an asset to be r, the daily return volatility follows the same equation as getting the standard deviation of the daily returns, as follows.

One thing to note is that more often than not volatility is quoted as annualized. The annualized daily return volatility would simply be the daily return volatility multiplied by the square root of the number of trading days in a year. Usually, one would assume 252 trading days per year for simplicity.

Volatility is an intuitive measure of risk; the higher the volatility, the higher the dispersion of the asset prices. As Figure 1 shows, the greater the volatility, the wider the range of the asset's daily returns. The more uncertain the price is, the greater the risk. So far so good.

How does one interpret volatility? What does a 1.2% (non-annualized)12-month daily return on the DAX index mean intuitively? If one were to make an assumption that the daily return follows a normal distribution, this would mean that over the past 12 months, 68% of the time the daily return falls within +/- 1.2%. This is a simple application of the 68–95–99.7 rule.

However, it is important for one to be mindful that the 68–95–99.7 rule at best is a rough (and naive) estimate of the return profile. This is because asset return does not necessarily follow a normal distribution.

As observed empirically, drastic price movements do happen especially when there are risk events happening at the backdrop. This creates a “fat-tailedness” in the distribution, in which extreme ends of the distribution have a higher probability density than that of a normal distribution. In addition, there are more often negative price movements than there are positive price movements. This led to a skew in the distribution. Figure 2 illustrates the non-normality of returns perfectly. The DAX index has fat tails (excess kurtosis of 0.48) and a negative skew (-0.17).

Nevertheless, the normal assumption is convenient and often not too wrong. Using the rule would help laymen grasp the concept easily as well.

Portfolio performance — risk and return

One application of volatility is to quantify a portfolio's performance. A portfolio is a collection of tradable assets — across different asset classes in the form of various instruments — held for the purpose of investing. One can either manage one’s own portfolio by trading the assets directly or by taking a share of other’s portfolios and seconding the risk management work to the portfolio managers. The latter comes in the form of fund products, such as exchange-traded funds, mutual funds, or, for accredited investors, hedge funds.

When investing in passive funds, the expectation of the fund's performance is that it tracks the benchmark index that it supposedly follows. Thus there is no expectation for it to outperform the benchmark, and the cost is lower. On the other hand, for actively managed funds, a high premium is paid for management fees. Therefore, there is a need to compare two different funds to determine which performs better and at a reasonable premium.

Comparing average returns is not sufficient most of the time. For two funds with the same average returns, would one prefer the fund with less volatility over the assessment period or the one with more volatility? How much more volatility would one be willing to accept for a fund with a higher mean return? Thankfully, one widely accepted and used portfolio metric is the Sharpe ratio. It takes into account volatility and average return in the assessment of portfolio performance. Loosely interpreted, it is the measure of the excess return of the portfolio (over the benchmark) per unit of risk taken. Intuitively, the higher the Sharpe ratio, the higher the additional return made above the benchmark, rebased on the volatility of the portfolio's excess return.

A commonly used benchmark is the risk-free rate of return. If another benchmark is used, some may refer to this as the information ratio instead.

Risk and Psychology

Well, all is good thus far with measuring risk. However, some might immediately see a small problem with volatility as a measure of risk. Mathematically, ‘risk’ simply refers to the uncertainty of asset return, but does not put different weights on positive deviation and negative deviation. ‘Risk’, in reality, tends to have a negative connotation among people. People love positive returns and hate going into negative. There ought to be some differentiation, ain’t it?

Indeed, the Sharpe ratio does not take into account the psychological interpretation of ‘risk’. Some may use skewness as a rudimentary measure to account for this, but skewness — along with other higher moments of distribution — is mathematically rigorous to compute and not as intuitive to the masses. We ought to introduce something simpler.

One candidate that is widely used in portfolio management is the Maximum Drawdown (MDD). Unlike volatility, MDD is rather simple mathematically. It measures the largest price drop from a peak to a trough over a given period of time. Measured in percentage terms, it gives one a good idea of how much one would lose if one were to enter the position at a peak. Figure 3 illustrates how MDD is calculated on the DAX Index.

Another candidate to measure downside risk is the Sortino ratio. The Sortino ratio looks very similar to the Sharpe ratio, with a slight tweak in the denominator. Instead of using the volatility, the semi-deviation of the return is used.

Semi-deviation differs from standard deviation in that only negative deviations are taken into account, and all positive deviations were given a zero weight. As such, all positive price deviations were ignored, and only the negative price deviations were included in the risk measure.

Risk and Regulation

Apart from measuring portfolio performance, ‘risk’ is a huge regulatory topic. Following various market events from 1992 that led to huge losses in the international markets, the Basel II Accord was introduced. The Accord mandates financial institutions to use standard measurements for credit, operational, and market risks. For market risks specifically, it was stated that the Value-at-Risk (VaR) approach is much preferred.

The VaR is a statistical measure of the riskiness of a portfolio of assets, which gives the maximum dollar amount expected to be lost over a given time horizon. This can be used on a portfolio level or a firm level to assess potential losses. It requires 2 key parameters, namely the significance level (α), and the risk horizon.

The Basel II Accord requires a 1% significance level in VaR calculation, i.e. to calculate a 99% VaR. Consider a firm with an average Profit-and-Loss (PnL) of $66,000 and volatility of $100,000. Figure 4 shows the cut-off point of the 1% significance level on the Profit-and-Loss distribution. The 99% VaR of this firm is $166,635. Loosely interpreted, we say that the firm has less than a 1% chance of losing more than $166,635.

The calculation of VaR is often not an easy topic and may require the use of simulation tools such as the Monte-Carlo simulation. More details of VaR calculation may be covered in future chapters. As a beginning, knowing VaR exists and what it means is important at the beginning of a quant’s journey.

Risk — looking back or looking forward?

In the large part of this chapter, we have looked in depth volatility as a measure of risk. However, the volatility used is based on historical data, and the purpose of risk measurement here is for reporting. Would historical volatility be useful as a measure of the current (and future) risk of the asset? If not, what would be an alternative?

Ideally, we’d want to know the current risk of investing in an asset. However, historical volatility does not equal actual volatility. While there are models that allow one to predict future volatility, predictions are simply educated bets and there is no guarantee that the prediction will always converge to the actual volatility.

Thankfully, though we may not be able to tell what is the actual volatility, we can get a feel of what the market thinks the volatility would be. This relates to the options market, where one can derive the implied volatility from the option prices.

Options, one of the commonly known derivatives, give the buyer the right, but not the obligation, to fulfill the contract. A call (put) option gives buyers the optionality to purchase (sell) an underlying asset at a predetermined (aka strike) price at a future date. Due to their properties, options are an effective tool for hedging and speculating.

Options are generally priced using the Black-Scholes equation, which takes in the strike price, volatility, interest rate, time-to-expiry, and current underlying asset price (aka spot price) to generate the value of the option today.

Generally, out-of-the-money (OTM) options (where the potential payoff is zero) are cheaper. The deeper in-the-money (ITM) an option is, the more expensive it gets. Thus, most trading occurs on at-the-money (ATM) or OTM strikes. And the market price of the options would increase when there is more demand to get into an option position.

Like how one can derive an option price with a given volatility, the reverse is the same. And most of the time, the volatility is unknown, but market prices — driven by the forces of demand and supply — are known. By working to solve the Black-Scholes equation backward, one can then get the “implied volatility” of the option for a given market price. This gives us an indication of how much is the market “pricing the volatility”. The higher the implied volatility, the higher the perceived risk.

Implied volatility is not static across all strike prices. Figure 5 shows a typical implied volatility profile of OTM/ ATM index options. Index options tend to exhibit a “skew/smile”, where implied volatilities are higher at one end compared to the other, with a minimum somewhere in near at-the-money strike. The higher implied volatilities at extreme ends are due to higher trading activities on the OTM strikes (due to the options being cheaper). Significantly higher implied volatilities are observed at lower strikes due to the need for downside protection. Investors enter OTM put options (lower strike range) to hedge a potentially large downside price movement. Thus, the higher the OTM put option implied volatility is, the more fearful investors are.

Now that we have linked implied volatilities to a fear gauge, this brings us to a new asset class — the volatility asset. Because implied volatilities are derived from tradable options, we are able to construct an index that gives one an indication of what the market thinks of the current volatility. This is the essence of volatility indices such as the VSTOXX index (based on EURO STOXX 50 options) and the VIX index (based on S&P 500 options). Both indices offer a glimpse of the implied variance/ volatilities priced into the respective option markets, thus showing the extent of investors’ confidence.

With an index, one is able to trade the derivatives on implied volatilities. The VIX futures and VSTOXX futures allows one to hedge or speculate the forward implied volatilities of the market. Details of the volatility index and derivatives shall be covered in future chapters.

All in all, we have covered risk and volatility to a good extent in this chapter — from the basic understanding of volatility, the various risk metrics that rely upon historical volatility calculation, the limitations of volatility as a risk measurement in portfolio management, and implied volatility as an indication of market perceived level of risk. This chapter also offers one a good glimpse of various topics in quantitative finance and a simple explanation of key terms.

I hope you enjoyed this article, and find this a useful primer (for the unacquainted) or summary (for the well-informed). Do comment with your feedback to help improve the series, and let me know what you’d like to see for future chapters. Thanks for reading!