# Risk-Neutral Densities

Assuming a complete and arbitrage-free market, a fundamental result of asset pricing theory is the existence of a unique probability function (measure) under which the price of any asset can be represented as the discounted expectation of the asset’s future payoffs. For instance, the price of a call option can be expressed as:

$\displaystyle C(K, \tau, r) = e^{-r \tau} \int_K^{\infty} (S_T - K) f(S_T) dS_T$

where, $K$ is the option’s strike price, $\tau$ the remaining time until expiration, $r$ the risk-free rate, and $S_T$ is a random variable representing the underlying stock price at expiration. By definition, the payoff of a call option is $\max\{S_T - K,0\}$, and hence the integral is taken over the interval in which the payoff is positive (i.e. $S_T > K$).

The function $f(S_T)$ is called the risk-neutral density of $S_T$ and can be intuitively thought of as a standard probability density function that combines investors’ own risk preferences with their beliefs about the true distribution of $S_T$.

This risk-neutral density is not directly observable, however, a simple and yet remarkable result in option pricing theory, known as the Breeden-Litzenberger formula, allows one to estimate $f(S_T)$. This formula is the result of taking the second derivative of $C(K, \tau, r)$ with respect to $K$. That is:

$\displaystyle \frac{\partial^2 C(K, \tau, r)}{\partial K^2}= e^{-r\tau} f(K)$

Or rearranging terms:

$\displaystyle f(K) = e^{r \tau} \frac{\partial^2 C(K, \tau, r)}{\partial K^2}$

Hence, for any given stock, provided that there are a “reasonable” number of call (or put) options with varying strike prices, it is possible to approximate the underlying risk-neutral density $f(S_T)$ as:

$\displaystyle f(K) = e^{r \tau} \frac{C(K + \Delta K, \tau, r) - 2C(K, \tau, r) + C(K - \Delta K, \tau, r)}{(\Delta K)^2}$

A perfect candidate for using this risk-neutral density estimation method is the S&P 500 index. Options on this index are by far the most traded options in the US. These options come in a wide range of strike prices which can be used to implement the risk-neutral density estimation procedure described above.

The next figure is the result of implementing this risk-neutral density estimation on the S&P 500 for the month of October 2008, a period of great turmoil for equity markets around the world (implementation code here). In addition to applying the Breeden-Litzenberger formula, the estimation also employs spline interpolation to generate a smooth risk-neutral density, and the Generalised Extreme Value (GEV) distribution to complete the density tails. Further details about this estimation procedure can be found here.

The result (in blue) is a left-skewed density which encapsulates investors beliefs about the true distribution of the S&P 500 index. As with any other probability density, the area under the curve and between two points on the x-axis can be interpreted as the probability $S_T$ is between those two points at expiration. In addition, the figure shows (in red) the probability density function of a lognormal distribution with the same mean and variance as the risk-neutral density. Lognormal distributions are extensively used in finance to describe how equity prices (such as the S&P 500 index) behave, and thus it serves as a benchmark for comparison. For instance, the probability the S&P 500 index experienced a 40% drop in October 2008 was roughly 8 times higher under the risk-neutral density compared to the traditionally assumed lognormal distribution.

In summary, the Breeden-Litzenberger formula, complemented with a highly liquid options market, allow us to summarise investors beliefs about the probability of different future outcomes in a simple density plot. In a sense, this density plot is a visual representation of all the information there is available on a particular equity asset and its price uncertainty.

# Brownian Motion

As a mathematical model, the Brownian Motion is well known for being explicitly delineated by no other than Albert Einstein in one of his “Annus Mirabilis” papers of 1905. In it, the Brownian Motion played a critical role in ending the then heated debate over the existence of atoms. However, the French mathematician Louis Bachelier was, in fact, the first to model the now called Brownian Motion as part of his PhD dissertation. Unlike Einstein, Bachelier created the Brownian to value stock options and also unlike Einstein, the importance of Bachelier’s contribution was not well appreciated at the time[1].

Nowadays, the Brownian Motion (also called Wiener process) is a cornerstone of current mathematical finance. It has been widely used to model the time trajectory of the price of financial instruments such as stocks, bonds, and options. It is also a key component of the Black-Scholes-Merton’s option pricing model who won Myron Scholes and Robert C. Merton the Nobel Memorial Prize in Economic Sciences in 1997[2].

Until not too long ago, the Brownian Motion was an enigma to me. I appreciated its importance and even recognised how to apply it to some finance related problems but its characteristics, and more importantly its construction, remained puzzling and elusive. This is no doubt why I was surprised to learn that its construction, however complex, can become very intuitive one we start looking closely. One approach to building a Brownian Motion hinges on a special family of wave-like functions called wavelets. Adding a certain kind of randomness to a particular wavelet approximation produces the famous Brownian Motion. For more details have a look at Chapter 3 of Michael Steele’s “Stochastic Calculus and Financial Applications”.

In this animation, I’ve used this wavelet-based construction to simulate and visualise three different Brownian Motions on the time interval 0 to 1. (implementation code here). The way to think about these three instantiations of the Brownian is to picture nature choosing different states of the world (three in this case). For each different state of the world, there is an entire realisation of the Brownian represented by the three different lines in the graph.

However beautiful, the simplicity of this construction runs the risk of understating the importance of a mathematical model that constitutes the bedrock of an entire branch of mathematics called stochastic calculus.