Bachelier Model

The Bachelier model is a mathematical framework employed in modern quantitative finance to determine the price of European-style options. It is influenced by factors such as the underlying asset's current price, the option's strike price, time to expiration and the underlying asset's volatility.

Louis Bachelier came up with the Brownian motion concept in the 1900s, which the Bachelier model employed to explore price dynamics. Later Fischer Black and Myron Scholes expanded the entire model in the 1970s. One utilizes this method for pricing options in markets where the underlying asset follows a normal or Gaussian distribution. This assumption implies that price changes follow a normal distribution and possess constant volatility.

• The Bachelier model is a mathematical framework that finds application in the pricing of European options on assets that do not pay dividends, assuming that the underlying asset follows a normal distribution.
• It mostly finds use in modern quantitative finance for gauging the prices of less but constantly volatile short-term options, for instance - interest rate options or options on futures contracts.
• The Bachelier model assumes constant price volatility throughout the option's lifespan based upon normal distribution, distinguishing it from the Black-Scholes model that assumes the options' price changes based on the log-normal distribution.

The Bachelier model has been a revolutionary development in stochastic processes in modern quantitative finance. It aids in ascertaining the option price depending upon several factors, including the current price of the underlying asset, the option's strike price, the time to expiration, and the underlying asset's volatility. The model assumes Brownian motion that functions on the phenomenon that the underlying asset price is constantly variable, although such variation may be random and small.

The Brownian motion, which forms the basis of this mathematical model, was discovered by the great French mathematician Louis Bachelier in 1900 during his Ph.D. Thesis 'Théorie de la Spéculation.' He later came to be known as the Father of financial mathematics for his contribution to the world of speculation.

Hence, such a financial model is particularly valuable for pricing options on assets with relatively low volatility or short-term options. However, it may not be suitable for assets or options exhibiting significant skewness or kurtosis, as it assumes a symmetric normal distribution of price changes. It differs from the Black-Scholes model, which assumes that price changes adhere to a log-normal distribution.

While the Bachelier model is a well-established pricing framework, the Black-Scholes model is more widely adopted due to its ability to handle assets with log-normal distributed returns. However, the Black-Scholes was specifically designed for European-style options and may not accurately price other options or complex derivative securities.

In the Bachelier model, one computes the option price through the cumulative distribution function (CDF) of the standard normal distribution. The Bachelier option pricing formula for the European call or put option is as follows:

For Call Option Price: C=e-rT

For Put Option Price: P=e-rT

• Where, d1=(F-K)T;
• F represents the forward price;
• K denotes the strike level or cap rate;
• N(d1) signifies the CDF of the standard normal distribution evaluated at d1;
• N(d1) signifies the CDF of the standard normal distribution evaluated at d1;
• N(-d1) indicates the probability density function (PDF) of the standard normal distribution ascertained at d1;
• σ represents the constant volatility of the underlying asset;
• r denotes a risk-free discount rate;
• T signifies the time to expiration of the option in years.

Check out these examples for a better idea:

Examples #1

Apart from the pricing of European call-and-put options, the Bachelier model is applicable to value interest rate options such as caps and floors. These options are derivatives that offer protection against fluctuations in interest rates. The Bachelier model assumes that the underlying interest rates conform to a normal distribution, facilitating the valuation of these interest rate options.

Example #2

Suppose the current price of an underlying asset for a European call option is $100 while its strike price is $110. Also, the time to expiration is one year, the risk-free interest rate is 0.05 (5%), and the underlying asset volatility is 0.2 (20%).

We can use the given Bachelier model formula for evaluating the call option price:

C=e-rT

Here, the e is the exponential function (approximately 2.71828).

To calculate the option price using a Bachelier model, we need to find out the value of d1 first.

d1=(F-K)T

d₁ = ($100 - $110)/0.2√1 = -$50

Also, calculate N(d1) and n(d1) using statistical tables or software, and find the values of N(d1) and n(d1) corresponding to the calculated d1 value, I.e., -$50.

Finally, evaluate the expression using the obtained values of N(d1) and n(d1) to calculate the price of the European call option using the above-mentioned Bachelier option pricing formula.

Please note that the Bachelier model assumes a normal distribution for the underlying asset's price and constant volatility. If the underlying asset follows a different distribution or the volatility is not constant, the Bachelier model may not be suitable, and alternative models, such as the Black-Scholes model, may be more appropriate.

The Bachelier model and the Black-Scholes model are the financial frameworks that specifically find use in the pricing of European-style options. However, both these models differ from each other in the following ways:

This article has been a guide to what is Bachelier Model. Here, we explain its formula, compare it with Black-Scholes Model, and examples. You may also find some useful articles here -

• Option Pricing
• Options Market
• American Options