Pricing Weather Risk: How to Value Agricultural Derivatives with Climate-Based Volatility Inputs

Mathias DUMONT

In this article, Mathias DUMONT (ESSEC Business School, Global Bachelor in Business Administration (GBBA), 2022-2026) explains how weather risk impacts the pricing of agricultural derivatives like futures and options, and how climate-based data can be integrated into stochastic pricing models. Combining academic insights and practical examples, including a mini-case from the SimTrade Blé de France simulation, the article illustrates adjustments to models such as the Black-Scholes-Merton model for temperature and rainfall variables in valuing agricultural contracts.

Introduction

Extreme weather has always been a critical factor in agriculture, but climate change is amplifying the frequency and severity of these events. From prolonged droughts to unseasonal floods, weather shocks can send crop yields and commodity prices on wild rides. This rising uncertainty has given birth to weather derivatives – financial instruments designed to hedge weather-related risks – and has made volatility forecasting a key challenge in pricing agricultural contracts. In fact, as businesses grapple with climate volatility, trading volume in weather derivatives has surged. CME Group saw a 260% increase last year (CME Group, 2023). The question for traders and risk managers is: how do we quantitatively factor weather risk into the pricing of futures and options on crops like wheat and corn?

Weather Risk and Agricultural Markets

Weather directly affects crop supply. A bumper harvest following ideal weather can flood the market and depress prices, whereas a drought or frost can decimate yields and trigger price spikes. These supply swings translate into volatility for agricultural commodity markets. For example, during the U.S. drought of 2012, corn prices skyrocketed, and the implied volatility of corn futures jumped by over 14 percentage points within a month, reaching ~49% in mid-July. Such surges reflect the market rapidly repricing risk as participants absorb new climate information (in this case, worsening crop prospects). Seasonal patterns are also evident: harvest seasons tend to coincide with higher price volatility because that’s when weather uncertainty is at its peak. Studies show that harvesting cycles create predictable seasonal volatility patterns in crop markets – when a critical growth period is underway, any shift in rainfall or temperature forecasts can send prices swinging.

Beyond affecting supply quantity, weather can influence crop quality (e.g., excessive rain can spoil grain quality) and even logistic costs (flooded transport routes, etc.), further feeding into prices. The interconnected global nature of agriculture means a drought in one region can reverberate worldwide. As noted in the SimTrade Blé de France case, weather conditions in France influence the quantity and quality of wheat the company harvests, while weather conditions around the world influence the international wheat price. In the Blé de France simulation (which models a French wheat producer’s stock), participants see how news of floods or droughts translate into stock price moves. For instance, the company might project a 7-million-ton wheat harvest, but analysts’ forecasts range from 6.5 to 7.2 Mt – with the realized level highly weather-dependent in the final weeks of the season. A poor weather turn not only shrinks the crop but boosts global wheat prices, creating a complex revenue impact on the firm. This mini-case underlines that weather risk entails both volume uncertainty and price uncertainty, a double-whammy for agricultural firms and their investors.

Case Study: Weather Shocks in Wheat Markets

To illustrate the impact of weather risk on commodity pricing, consider three simulated scenarios for an upcoming wheat growing season: (1) **Favorable weather**, (2) **Moderate conditions**, and (3) **Severe weather** such as drought. Each scenario generates a distinct price trajectory in the wheat market. Under favorable weather, prices tend to remain stable or decline slightly, particularly at harvest, due to strong yields and potential oversupply. In moderate conditions, prices may rise modestly as the market adjusts to balanced supply and demand. In contrast, severe weather triggers early price rallies as concerns about yield shortfalls emerge, followed by sharp spikes once crop damage becomes evident. For producers and traders, anticipating these divergent price paths is essential for pricing contracts, managing risk exposure, and structuring hedging strategies effectively.

Figure 1. Simulated commodity price paths under three weather scenarios.
Simulated Price Paths
Source: Author’s simulation.

Figure 1. shows the simulation of commodity price paths under three weather scenarios: severe weather (red), moderate weather (orange), and favorable weather (green). A mid-season weather forecast alert (Day 15) triggers a shift in market expectations, causing price divergence. This simulation illustrates how weather shocks and forecasts impact commodity pricing through volatility and revised yield expectations.

From a risk management perspective, tools exist to handle these contingencies. Farmers or firms concerned about catastrophic weather can turn to weather derivatives for protection. Weather derivatives are financial contracts (often based on indexes like temperature or rainfall levels) that pay out based on specific weather outcomes, allowing businesses to offset losses caused by adverse conditions. They have been used by a wide range of players – from utilities hedging warm winters, to breweries hedging late frosts. These instruments can be customized over-the-counter or traded on exchanges. Notably, CME Group lists standardized weather futures and options tied to indices such as heating degree days (HDD) and cooling degree days (CDD) for various cities. The existence of such contracts means that even when commodity producers cannot fully insure their crop yield, they might hedge certain aspects of weather risk (like an unusually hot summer) via financial markets. In our context, a wheat farmer worried about drought could, say, buy a weather option that pays off if rainfall falls below a threshold, providing funds when their crop output (and thus futures position) suffers.

Climate-Based Volatility in Derivatives Pricing

How can weather uncertainty be incorporated into derivative pricing models? Classic option pricing, such as the Black-Scholes-Merton model, assumes a fixed volatility for the underlying asset’s returns. For agricultural commodities, that volatility is anything but constant – it ebbs and flows with the weather and seasonal progress. Practitioners thus often use stochastic volatility models or at least adjust the volatility input over time. For example, one might use higher volatility estimates during the crop’s growing season and lower volatility post-harvest when output is known. This practice parallels how equity traders anticipate higher volatility in stock prices ahead of major earnings or profit announcements, and lower volatility after the announcement of profits by the firm.

Like companies facing performance surprises, weather shocks inject information asymmetry into the market, which must be priced into the option premiums. This aligns with the observed Samuelson effect, where futures contracts on commodities tend to have higher volatility when they are near maturity (coinciding with harvest uncertainty).

Market prices of options themselves reflect these expectations. When a looming weather event is expected to cause turmoil, options premiums will rise. The metric capturing this is implied volatility – the volatility level implied by current option prices. Implied vol is essentially forward-looking and will jump if traders foresee choppy waters ahead. Empirical evidence shows that extreme weather forecasts translate into higher implied vols for crop options. In 2012, as drought fears intensified, corn option implied volatility spiked (alongside futures prices). Conversely, once a forecasted drought started being relieved by rains, implied volatility eased off, signaling that some uncertainty had been resolved. A recent study also found that integrating meteorological data (like rainfall and temperature anomalies) into volatility modeling significantly improves the ability to hedge risk in agricultural markets. In other words, the more information we feed into our models about the climate, the more accurately we can price and hedge these derivatives.

Figure 2. Implied Volatility of Crop Options Over Time with Weather Events
Line chart showing implied volatility of crop options over 12 months with spikes linked to weather events
Source: Author’s simulation.

This simulation illustrates the evolution of implied volatility over a 12-month crop cycle. Forecasted climate events—drought (Month 3), frost (Month 6), heatwave (Month 8), and rainfall shortage (Month 11)—lead to moderate but distinct volatility spikes. As uncertainty resolves, volatility returns to baseline.

One practical approach to pricing under climate uncertainty is to use scenario-based or simulation-based models. Instead of assuming a single volatility number, an analyst can simulate thousands of possible weather outcomes (perhaps using historical climate data or meteorological forecast models) and the corresponding price paths for the commodity. Each simulated price path yields a payoff for the derivative (e.g. an option’s payoff at expiration), and by averaging those payoffs (and discounting appropriately), one can derive a weather-adjusted theoretical price. This Monte Carlo style approach effectively treats weather as an external random factor influencing the commodity’s drift and volatility. It’s particularly useful for complex derivatives or when the payoff depends explicitly on weather indices (such as a derivative that pays out if rainfall is below X mm).

When the derivative’s underlying is the commodity itself (e.g. a corn futures option), traditional risk-neutral pricing arguments still apply, but the challenge is forecasting volatility. Traders often adjust the volatility smile/skew on agricultural options to account for asymmetric weather risks – for instance, if a drought can cause a much bigger upside move than a rainy season can cause a downside move, call options might embed a higher implied volatility (reflecting that upside risk of price spikes). This is observed in practice as well; extreme weather events can distort the implied volatility “skew” of crop options, as out-of-the-money calls become more sought after as disaster insurance.

In contrast, if the derivative’s underlying is a pure weather index (say an option on cumulative rainfall), then pricing becomes more complex because the underlying (rainfall) is not a tradable asset. In such cases, the Black-Scholes-Merton formula is not directly applicable. Instead, pricing relies on actuarial or risk-neutral methodologies that incorporate a market price of risk for weather. For example, one method is to estimate the probability distribution of the weather index from historical data, then add a risk premium to account for investors’ risk aversion to weather variability, and discount expected payoffs accordingly. Another method uses “burn analysis” – taking historical weather outcomes and the associated financial losses/gains had the derivative been in place, to gauge a fair premium. Academic research has proposed models ranging from modified Black-Scholes-Merton-type formulas for rainfall (with adjustments for the non-tradability) to advanced statistical models (e.g. Ornstein-Uhlenbeck processes with seasonality for temperature indices. The key takeaway is that whether it’s directly in commodity options or in dedicated weather derivatives, climate factors force us to go beyond textbook models and embrace more dynamic, data-driven pricing techniques.

Why should I be interested in this post?

For an ESSEC student or a young finance professional, this topic sits at the intersection of finance and real-world impact. Understanding weather risk in markets is not just about farming – it’s about how big data and climate science are increasingly intertwined with financial strategy. Agricultural commodities remain a cornerstone of the global economy, and volatility in these markets can affect food prices, inflation, and even economic stability in various countries. By grasping how to value derivatives with climate-based volatility inputs, you are gaining insight into a growing niche of finance that deals with sustainability and risk management. Moreover, the skills involved – scenario analysis, simulation modeling, blending of economic and scientific data – are highly transferable to other domains (think energy markets or any sector where uncertainty reigns). In a world facing climate change, expertise in weather-related financial products could open career opportunities in commodity trading desks, insurance/reinsurance firms, or specialized hedge funds. Ultimately, this post encourages you to think creatively and interdisciplinarily: the best hedging or valuation solutions may come from combining financial theory with environmental intelligence.

Related posts on the SimTrade blog

   ▶ Camille KELLER Coffee Futures: The Economic and Environmental Drivers Behind Rising Prices

   ▶ Jayati WALIA Implied Volatility

   ▶ Akshit GUPTA Futures Contract

   ▶ Anant JAIN Understanding Price Elasticity of Demand

Useful resources

Chicago Mercantile Exchange (CME) Weather futures and options product information. (Exchange-traded weather derivative contracts on temperature and other indices)

U.S. Energy Information Administration Drought increases price of corn, reduces profits to ethanol producers (2012). (Article discussing the 2012 drought’s impact on corn prices and volatility)

Nature Communications (2024) Financial markets value skillful forecasts of seasonal climate. (Research showing that seasonal climate outlooks have measurable effects on implied volatility and market uncertainty)

Das, S. et al. (2025) Predicting and Mitigating Agricultural Price Volatility Using Climate Scenarios and Risk Models. (Academic study demonstrating the integration of climate data into volatility models and using Black-Scholes to value a government price support as a put option)

Pai, J. & Zheng, Z. (2013) Pricing Temperature Derivatives with a Filtered Historical Simulation Approach. (Discussion of why Black-Scholes is not directly applicable to weather derivatives and alternative pricing approaches)

About the author

The article was written in May 2025 by Mathias DUMONT (ESSEC Business School, Global Bachelor in Business Administration (GBBA), 2022-2026).

Implied Volatility

Jayati WALIA

In this article, Jayati WALIA (ESSEC Business School, Grande Ecole Program – Master in Management, 2019-2022) explains how implied volatility is computed from option market prices and a option pricing model.

Introduction

Volatility is a measure of fluctuations observed in an asset’s returns over a period of time. The standard deviation of historical asset returns is one of the measures of volatility. In option pricing models like the Black-Scholes-Merton model, volatility corresponds to the volatility of the underlying asset’s return. It is a key component of the model because it is not directly observed in the market and cannot be directly computed. Moreover, volatility has a strong impact on the option value.

Mathematically, in a reverse way, implied volatility is the volatility of the underlying asset which gives the theoretical value of an option (as computed by Black-Scholes-Merton model) equal to the market price of that option.

Implied volatility is a forward-looking measure because it is a representation of expected price movements in an underlying asset in the future.

Computation methods for implied volatility

The Black-Scholes-Merton (BSM) model provides an analytical formula for the price of both a call option and a put option.

The value for a call option at time t is given by:

 Call option value

The value for a put option at time t is given by:

Put option value

where the parameters d1 and d2 are given by:,

call option d1 d2

with the following notations:

St : Price of the underlying asset at time t
t: Current date
T: Expiry date of the option
K: Strike price of the option
r: Risk-free interest rate
σ: Volatility of the underlying asset
N(.): Cumulative distribution function for a normal (Gaussian) distribution. It is the probability that a random variable is less or equal to its input (i.e. d₁ and d₂) for a normal distribution. Thus, 0 ≤ N(.) ≤ 1

From the BSM model, both for a call option and a put option, the option price is an increasing function of the volatility of the underlying asset: an increase in volatility will cause an increase in the option price.

Figures 1 and 2 below illustrate the relationship between the value of a call option and a put option and the level of volatility of the underlying asset according to the BSM model.

Figure 1. Call option value as a function of volatility.
Call option value as a function of volatility
Source: computation by the author (BSM model)

Figure 2. Put option value as a function of volatility.
Put option value as a function of volatility
Source: computation by the author (BSM model)

You can download below the Excel file for the computation of the value of a call option and a put option for different levels of volatility of the underlying asset according to the BSM model.

Excel file to compute the option value as a function of volatility

We can observe that the call and put option values are a monotonically increasing function of the volatility of the underlying asset. Then, for a given level of volatility, there is a unique value for the call option and a unique value for the put option. This implies that this function can be reversed; for a given value for the call option, there is a unique level of volatility, and similarly, for a given value for the put option, there is a unique level of volatility.

The BSM formula can be reverse-engineered to compute the implied volatility i.e., if we have the market price of the option, the market price of the underlying asset, the market risk-free rate, and the characteristics of the option (the expiration date and strike price), we can obtain the implied volatility of the underlying asset by inverting the BSM formula.

Example

Consider a call option with a strike price of 50 € and a time to maturity of 0.25 years. The market risk-free interest rate is 2% and the current price of the underlying asset is 50 €. Thus, the call option is ‘at-the-money’. If the market price of the call option is equal to 2 €, then the associated level of volatility (implied volatility) is equal to 18.83%.

You can download below the Excel file below to compute the implied volatility given the market price of a call option. The computation uses the Excel solver.

Excel file to compute implied volatility of an option

Volatility smile

Volatility smile is the name given to the plot of implied volatility against different strikes for options with the same time to maturity. According to the BSM model, it is a horizontal straight line as the model assumes that the volatility is constant (it does not depend on the option strike). However, in practice, we do not observe a horizontal straight line. The curve may be in the shape of the alphabet ‘U’ or a ‘smile’ which is the usual term used to refer to the observed function of implied volatility.

Figure 3 below depicts the volatility smile for call options on the Apple stock on May 13, 2022.

Figure 3. Volatility smile for call options on Apple stock.
Apple volatility smile
Source: Computation by author.

Excel file for implied volatility from Apple stock option

We can also observe that the for a specific time to maturity, the implied volatility is minimum when the option is at-the-money.

Volatility surface

An essential assumption of the BSM model is that the returns of the underlying asset follow geometric Brownian motion (corresponding to log-normal distribution for the price at a given point in time) and the volatility of the underlying asset price remains constant over time until the expiration date. Thus theoretically, for a constant time to maturity, the plot of implied volatility and strike price would be a horizontal straight line corresponding to a constant value for volatility.

Volatility surface is obtained when values for implied volatilities are calculated for options with different strike prices and times to maturity.

CBOE Volatility Index

The Chicago Board Options Exchange publishes the renowned Volatility Index (also known as VIX) which is an index based on the implied volatility of 30-day option contracts on the S&P 500 index. It is also called the ‘fear gauge’ and it is a representation of the market outlook for volatility for the next 30 days.

Related posts on the SimTrade blog

   ▶ All posts about Options

   ▶ Akshit GUPTA Options

   ▶ Jayati WALIA Brownian Motion in Finance

   ▶ Jayati WALIA Brownian Motion in Finance

   ▶ Youssef LOURAOUI Minimum Volatility Factor

   ▶ Youssef LOURAOUI VIX index

Useful resources

Academic articles

Black F. and M. Scholes (1973) “The Pricing of Options and Corporate Liabilities” The Journal of Political Economy, 81, 637-654.

Dupire B. (1994). “Pricing with a Smile” Risk Magazine 7, 18-20.

Merton R.C. (1973) “Theory of Rational Option Pricing” Bell Journal of Economics, 4, 141–183.

Business

CBOE Volatility Index (VIX)

CBOE VIX tradable products

About the author

The article was written in May 2022 by Jayati WALIA (ESSEC Business School, Grande Ecole Program – Master in Management, 2019-2022).

Standard deviation

Standard deviation

Jayati WALIA

In this article, Jayati WALIA (ESSEC Business School, Grande Ecole Program – Master in Management, 2019-2022) presents an overview of standard deviation and its use in financial markets.

Definition

The standard deviation is a measure that indicates how much data scatter around the mean. The idea is to measure how an observation deviates from the mean on average./p>

Mathematical formulae

The first step to compute the standard deviation is to compute the mean. Considering a variable X, the arithmetic mean of a data set with N observations, X1, X2 … XN, is computed as:

img_arithmetic_mean

In the data set analysis, we also consider the dispersion or variability of data values around the central tendency or the mean. The variance of a data set is a measure of dispersion of data set values from the (estimated) mean and can be expressed as:

variance

Note that in the above formula we divide by N-1 because the mean is not known but estimated (usual case in finance). If the mean is known with certainty (when dealing the whole population not a sample), then we divide by N.

A problem with variance, however, is the difficulty of interpreting it due to its squared unit of measurement. This issue is resolved by using the standard deviation, which has the same measurement unit as the observations of the data set (such as percentage, dollar, etc.). The standard deviation is computed as the square root of variance:

standard deviation

A low value standard deviation indicates that the data set values tend to be closer to the mean of the set and thus lower dispersion, while a high standard deviation indicates that the values are spread out over a wider range indication higher dispersion.

Measure of volatility

For financial investments, the X variable in the above formulas would correspond to the return on the investment computed on a given period of time. We usually consider the trade-off between risk and reward. In this context, the reward corresponds to the expected return measured by the mean, and the risk corresponds to the standard deviation of returns.

In financial markets, the standard deviation of asset returns is used as a statistical measure of the risk associated with price fluctuations of any particular security or asset (such as stocks, bonds, etc.) or the risk of a portfolio of assets (such as mutual funds, index mutual funds or ETFs, etc.).

Investors always consider a mathematical basis to make investment decisions known as mean-variance optimization which enables them to make a meaningful comparison between the expected return and risk associated with any security. In other words, investors expect higher future returns on an investment on average if that investment holds a relatively higher level of risk or uncertainty. Standard deviation thus provides a quantified estimate of the risk or volatility of future returns.

In the context of financial securities, the higher the standard deviation, the greater is the dispersion between each return and the mean, which indicates a wider price range and hence greater volatility. Similarly, the lower the standard deviation, the lesser is the dispersion between each return and the mean, which indicates a narrower price range and hence lower volatility for the security.

Example: Apple Stock

To illustrate the concept of volatility in financial markets, we use a data set of Apple stock prices. At each date, we compute the volatility as the standard deviation of daily stock returns over a rolling window corresponding to the past calendar month (about 22 trading days). This daily volatility is then annualized and expressed as a percentage.

Figure 1. Stock price and volatility of Apple stock.

price and volatility for Apple stock
Source: computation by the author (data source: Bloomberg).

You can download below the Excel file for the calculation of the volatility of stock returns. The data used are for Apple for the period 2020-2021.

ownload the Excel file to compute the volatility of stock returns

Related posts on the SimTrade blog

▶ Jayati WALIA Quantitative Risk Management

▶ Jayati WALIA Value at Risk

▶ Jayati WALIA Brownian Motion in Finance

Useful resources

Wikipedia Standard Deviation

About the author

The article was written in November 2021 by Jayati WALIA (ESSEC Business School, Grande Ecole Program – Master in Management, 2019-2022).

VIX index

VIX index

Youssef_Louraoui

In this article, Youssef LOURAOUI (ESSEC Business School, Global Bachelor of Business Administration, 2017-2021) presents the VIX index, which is a financial index that measures the uncertainty in the US equity market.

This article is structured as follows: we begin by defining the grounding notions of the VIX index. We then explain the behavior of this index and its statistical characteristics. We finish by presenting its practical usage in financial markets.

Definition

The CBOE Volatility Index, abbreviated “VIX”, is a measure of the expected S&P 500 index movement calculated by the Chicago Board Options Exchange (CBOE) from the current trading prices of options written on the S&P 500 index.

Known as Wall Street’s “fear index”, the VIX is closely monitored by a broad range of market players, and its level and pattern have become ingrained in market discussion.

Figure 1 illustrates the evolution of the VIX index for the period from 2003 to 2021.
Figure 1 Historical levels of the VIX index from 2003-2021.
VIX_levels_analysis
Source: computation by the author (Data source: Thomson Reuters).

VIX values greater than 20 are regarded to be high by market participants. If the VIX is between 12 and 20, it is considered normal; if it is less than 12, it is considered low. As it is the case with other indices, the VIX is computed using the price of a basket of tradable components (in this case, options expiring within the next month or so). The profit or loss that option buyers and sellers realize during the option’s life will depend, among other things, on how significantly the S&P 500’s actual volatility will differ from the implied volatility given by the VIX at the start of the period (S&P Global Research, 2017).

Behavior of the VIX index

Statistical distribution of the S&P500 index returns and VIX level

Figure 2 displays the statistical distribution of the price variations in the S&P500 index for different levels of the VIX index The higher the VIX index (by convention, greater than 20), the more severe the distribution tends to be, with negative skewness and high kurtosis indicating heightened volatility in the US market, therefore exacerbating both positive and negative swings. An opposite finding may be made for the VIX level at lower levels (often less than 12), when market swings are less evident due to less skewness and lower kurtosis (S&P Global Research, 2017).

Figure 2. The distribution of 30-day return in the S&P500 index for different VIX index levels.
Statistical distribution of the S&P500 index returns
Source: S&P Global Research (2017).

If the VIX is low, market players may benefit by purchasing options; conversely, if the VIX is high, market participants may profit from selling options. The specific utility of anticipated VIX is that it gives us with a more accurate assessment of whether VIX is high, low, or normal at any point in time (S&P Global Research, 2017). Thus, VIX may be regarded of as a crowd-sourced estimate of the S&P 500’s expected volatility. As with interest rates and dividends, one cannot invest directly in them, even though one can guess on their future worth, one cannot invest directly in VIX, and the significance of a specific VIX level is commonly misinterpreted (S&P Global Research, 2017).

Recent volatility in the S&P500 index and VIX level

Figure 3 demonstrates that the VIX index is strongly correlated with recent market volatility. However, there is considerable variance; for example, a recent volatility level of about 20% has been associated with a VIX level of 34 (point B, when VIX was very “high”) and with a VIX level of 12 (point C, when VIX was relatively “low”). Volatility (realized or implied) has a strong propensity to return to its mean. This insight is not especially original, despite its illustrious past. There is an enormous body of data demonstrating that volatility tends to mean revert across markets, and the pioneers of this field were given the Nobel Prize in part for incorporating their results into volatility forecasts and simulations (S&P Global Research, 2017).

Figure 3. Relation between VIX and recent volatility.
VIX_regression_analysis
Source: S&P Global Research (2017).

Realized volatility in the S&P500 index and VIX level

Figure 4 represents the relationship between Realized volatility in the S&P500 index over a period and the VIX level at the begining of the period.

Figure 4. VIX versus next realized volatility.
VIX_realized_graph
Source: S&P Global Research (2017).

Mean reversion

Figure 5 shows how VIX index converge to a certain llong-term level as time passes. This finding is not due to 15% being exceptional in any manner; this figure for M was calculated using historical volatility levels for the S&P 500 and their evolution. It is not implausible that M (else referred to as long-term average volatility in the US equities market) may change over time; changes in the S&P 500’s sector weightings, trade All of these factors have the ability to influence both the pace and the volume and the point at which mean reversion occurs.

Figure 5. Mean-reversion dynamic in recent volatility.
VIX mean reversion
Source: S&P Global Research (2017).

Use of the VIX index in financial markets

There are two methods for determining an asset’s volatility. Either through a statistical calculation of an asset’s realized volatility, also known as historical volatility, which serves as a pointer to the asset’s volatility behavior. This is a limited method that is based on the premise that past volatility tends to replicate itself in the future, without including a forward-looking study of volatility. The second technique is to extract an asset’s volatility from option prices referred to as “implied volatility”.

Why should I be interested in this post?

When investors make investment decisions, they utilize the VIX to gauge the degree of risk, worry, or stress in the market. Additionally, traders can trade the VIX using a range of options and exchange-traded products, or price derivatives using VIX values.

Related posts on the SimTrade blog

   ▶ All posts about Options

   ▶ Akshit GUPTA Options

   ▶ Akshit GUPTA History of Option Markets

   ▶ Jayati WALIA Implied Volatility

   ▶ Youssef LOURAOUI Minimum Volatility Factor

Useful resources

Business analysis

CBOE , 2021. VIX

Nasdaq, 2021. Realized Volatility

Nasdaq, 2021. Vix Index Volatility

S&P Global Research, 2017. Reading VIX: Does VIX Predict Future Volatility?

S&P Global Research, 2017. A Practitioner’s Guide to Reading VIX

About the author

The article was written in September 2021 by Youssef LOURAOUI (ESSEC Business School, Global Bachelor of Business Administration, 2017-2021).

Brownian Motion in Finance

Brownian Motion in Finance

Jayati WALIA

In this article, Jayati WALIA (ESSEC Business School, Grande Ecole Program – Master in Management, 2019-2022) explains the Brownian motion and its applications in finance to model asset prices like stocks traded in financial markets.

Introduction

Stock price movements form a random pattern. The prices fluctuate everyday resulting from market forces like supply and demand, company valuation and earnings, and economic factors like inflation, liquidity, demographics of country and investors, political developments, etc. Market participants try to anticipate stock prices using all these factors and contribute to make price movements random by their trading activities as the financial and economics worlds are constantly changing.

What is a Brownian Motion?

The Brownian motion was first introduced by botanist Robert Brown who observed the random movement of pollen particles due to water molecules under a microscope. It was in the 1900s that the French mathematician Louis Bachelier applied the concept of Brownian motion to asset price behavior for the first time, and this led to Brownian motion becoming one of the most important fundamental of modern quantitative finance. In Bachelier’s theory, price fluctuations observed over a small time period are independent of the current price along with historical behavior of price movements. Combining his assumptions with the Central Limit Theorem, he also deduces that the random behavior of prices can be said to be represented by a normal distribution (Gaussian distribution).

This led to the development of the Random Walk Hypothesis or Random Walk Theory, as it is known today in modern finance. A random walk is a statistical phenomenon wherein stock prices move randomly.

When the time step of a random walk is made infinitesimally small, the random walk becomes a Brownian motion.

Standard Brownian Motion

In context of financial stochastic processes, the Brownian motion is also described as the Wiener Process that is a continuous stochastic process with normally distributed increments. Using the Wiener process notation, an asset price model in continuous time can be expressed as:

brownian motion equation

with dS being the change in asset price in continuous time dt. dX is the random variable from the normal distribution (N(0, 1) or Wiener process). σ is assumed to be constant and represents the price volatility considering the unexpected changes that can result from external effects. μdt together represents the deterministic return within the time interval with μ representing the growth rate of asset price or the ‘drift’.

When the market is modeled with a standard Brownian Motion, the probability distribution function of the future price is a normal distribution.

Geometric Brownian Motion

weiner notation

with dS being the change in asset price in continuous time dt. dX is the random variable from the normal distribution (N(0, 1) or Wiener process). σ is assumed to be constant and represents the price volatility considering the unexpected changes that can result from external effects. μdt together represents the deterministic return within the time interval with μ representing the growth rate of asset price or the ‘drift’.

When the market is modeled with a geometric Brownian Motion, the probability distribution function of the future price is a log-normal distribution.

Properties of a Brownian Motion

  • Continuity: Brownian motion is the continuous time-limit of the discrete time random walk. It thus, has no discontinuities and is non-differential everywhere.
  • Finite: The time increments are scaled with the square root of the times steps such that the Brownian motion is finite and non-zero always.
  • Normality: Brownian motion is normally distributed with zero mean and non-zero standard deviation.
  • Martingale and Markov Property: Martingale property states that the conditional expectation of the future value of a stochastic process depends on the current value, given information about previous events. The Markov property instead focusses on the ‘no memory’ theory that the expected future value of a stochastic process does not depend on any past values except the current value. Brownian motion follows both these properties.

Simulating Random Walks for Stock Prices

In quantitative finance, a random walk can be simulated programmatically through coding languages. This is essential because these simulations can be used to represent potential future prices of assets and securities and work out problems like derivatives pricing and portfolio risk evaluation.

A very popular mathematical technique of doing this is through the Monte Carlo simulations. In option pricing, the Monte Carlo simulation method is used to generate multiple random walks depicting the price movements of the underlying, each with an associated simulated payoff for the option. These payoffs are discounted back to the present value and the average of these discounted values is set as the option price. Similarly, it can be used for pricing other derivatives, but the Monte Carlo simulation method is more commonly used in portfolio and risk management.

For instance, consider Microsoft stock that has a current price of $258.65 with a growth trend of 55.2% and a volatility of 35.92%.

A plot of daily returns represented as a random normal distribution is:

Normal Distribution

The above figure represents the simulated price path according to the Geometric Brownian motion for the Microsoft stock price. Similarly, a plot of 10 such simulations would be like this:

Microsoft GBM Simulations

Thus, we can see that with just 10 simulations, the prices range from $100 to over $600. We can increase the number of simulations to expand the data set for analysis and use the results for derivatives pricing and many other financial applications.

Brownian motion and the efficient market hypothesis

If the market is efficient in the weak sense (as introduced by Fama (1970)), the current price incorporates all information contained in past prices and the best forecast of the future price is the current price. This is the case when the market price is modelled by a Brownian motion.

Related Posts

   ▶ Jayati WALIA Black-Scholes-Merton option pricing model

   ▶ Jayati WALIA Plain Vanilla Options

   ▶ Jayati WALIA Derivatives Market

Useful Resources

Academic articles

Fama E. (1970) Efficient Capital Markets: A Review of Theory and Empirical Work, Journal of Finance, 25, 383-417.

Fama E. (1991) Efficient Capital Markets: II Journal of Finance, 46, 1575-617.

Books

Malkiel B.G. (2020) A Random Walk Down Wall Street: The Time-tested Strategy for Successful Investing, WW Norton & Co.

Code

Python code for graphs and simulations

Brownian Motion

What is the random walk theory?

About the author

The article was written in August 2021 by Jayati WALIA (ESSEC Business School, Grande Ecole Program – Master in Management, 2019-2022).