# 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.